THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Gravity Illustrated Spacetime Edition
RICKARD JONSSON
Department of Theoretical Physics Astronomy and Astrophysics group Chalmers University of Technology and Goteborg ¨ University Goteborg, ¨ Sweden 2004
Gravity Illustrated Spacetime Edition Rickard Jonsson ISBN 91-7291-519-6 c Rickard Jonsson, 2004. ! Doktorsavhandlingar vid Chalmers Tekniska Hogskola, ¨ Ny serie nr 2201 ISSN 0346-718x
Department of Theoretical Physics Astronomy and Astrophysics group Chalmers University of Technology and Goteborg ¨ University SE-412 96 Goteborg ¨ Sweden Telephone +46(0)31-772 10 00
Cover: An illustration of the curved spacetime for a line through Earth. The small toy car with the pen is used to draw straight lines on the spacetime – corresponding to the motion of three apples thrown out from the center of the earth. The yellow line corresponds to the motion of the yellow apple visible outside Earth. The figure was created using Matlab and Povray.
Back cover: An illustration of the spacetime for a line outside Earth, and the spacetime trajectory of an apple that has been thrown upwards. The figure was created using Matlab.
Gravity Illustrated Rickard M. Jonsson Department of Theoretical Physics Astronomy and Astrophysics group Chalmers University of Technology and Goteborg ¨ University Abstract This thesis deals with essentially four different topics within general relativity: pedagogical techniques for illustrating curved spacetime, inertial forces, gyroscope precession and optical geometry. Concerning the pedagogical techniques, I investigate two distinctly different methods, the dual and the absolute method. In the dual scheme, I start from the geodesic equation in a 1+1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element. I then find another metric, with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent dual metric can be embedded in ordinary Euclidean space. Freely falling particles correspond to straight lines on the embedded surface. In the absolute scheme, I start from an arbitrary Lorentzian spacetime with a given field of timelike four-velocities uµ . I then perform a coordinate transformation to the local Minkowski system comoving with the given four-velocity at every point. In the local system the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. For the particular case of two dimensions we may embed the absolute geometry as a curved surface. The method is well suited for visualizing gravitational time dilation, cosmological expansion and black holes. Concerning inertial forces, gyroscope precession and optical geometry, the general framework is based on the introduction of a congruence of reference worldlines in an arbitrary spacetime. This allows us to describe the local motion and acceleration of particles in terms of the speed relative to the congruence, the time derivative of the speed and the spatial curvature (project down along the reference congruence) of the corresponding worldline. I present two papers concerning inertial forces in this framework, one formal and one intuitive. I also present two papers concerning gyroscope precession, again one formal and one intuitive. In particular I illustrate how one can explain gyroscope precession in an arbitrary stationary spacetime as a double Thomas precession effect. Introducing a novel type of spatial curvature measure for the worldline of a test particle, we present a natural way of generalizing the theory of optical geometry to include arbitrary spacetimes. The generalized optical geometry allows us to do optical geometry across the horizon of a black hole. Keywords: curved spacetime, embeddings, pedagogical techniques, inertial forces, gyroscope precession, optical geometry
Acknowledgements First of all I would like to express my gratitude towards my supervisor, Marek Abramowicz for allowing me to work on what I find interesting – and to develop as an independent researcher. I would also like to thank Ulf Torkelsson and Sebastiano Sonego for all sorts of comments and advice. I must of course also acknowledge the sole Bohmian-Mechanics-Lover of the house – who is also my friend, colleague and fellow rebel in physics – Hans Westman. We have shared more discussions and laughs (many at the expense of various physicists) than I can remember. ’Wha Wha Wha ...’:) While Achim Tassemark didn’t include me in his acknowledgements, I am a forgiving man and would not in any way hold that against him:) He has been my companion in angling, longbow archery, smithery, horseback riding and much else over the years and likely will be in the future as well. It can be debated whether they deserve it, but I would like to acknowledge also ’the most unlikely creatures of all’ – the Lord of the String PhD-students below us (in every respect). May the spirit of Gert Jonnys guide you for ever:) I would also like to mention the very best longbow shooting nephews of mine that the world has ever seen – David and Alexander:) More important than anyone I must acknowledge Agneta – whose name is still magic, and who taught me that physics is just crap – compared to what matters in life. Lastly I would like to acknowledge my parents, among many other things for making sure that I had something to eat while finishing this thesis. They are, as Tina Turner puts it ’Simply the Best’ (although I very much doubt that Tina was referring to her parents:).
September, 2004 Rickard Jonsson
Grow stronger From the 13’th warrior Hope is for free Fabrizio
C ONTENTS 1 Introduction 1.1 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 An introduction to Einstein’s gravity 2.1 The spacetime diagram . . . . . . 2.2 Flat spacetime . . . . . . . . . . . 2.3 Mass curves spacetime . . . . . . 2.4 Curved spacetime for a staff outside Earth . . . . . . . . . 2.5 The spacetime for a line through the Earth . . . . . . . . . 2.6 Forces and gravity . . . . . . . . 2.7 About mass . . . . . . . . . . . . 2.8 More about mass . . . . . . . . . 2.9 Was Newton wrong and was Einstein right? . . . . . . . .
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3 Making illustrations 3.1 About matlab . . . . . . . . . . . . . 3.1.1 Some shortcomings of matlab 3.2 About povray . . . . . . . . . . . . . 3.3 Matlab to povray . . . . . . . . . . . 3.4 Comments . . . . . . . . . . . . . . .
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4 Making models 4.1 The funnel shaped spacetime . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The bulgy cylinder spacetime . . . . . . . . . . . . . . . . . . . . . . . .
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5 Spacetime visualization 5.1 Paper I . . . . . . . . . 5.1.1 A technical note 5.2 Paper II . . . . . . . . . 5.2.1 A technical note
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6 Inertial forces 6.1 A technical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Gyroscope precession 7.1 A technical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8 Optical geometry 8.1 Generalized optical geometry in technical terms . . . . . . . . . . . . .
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9 Conclusion and outlook
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10 A spherical interior dual metric 10.1 Conditions for spheres . . . . . 10.2 The dual interior metric . . . . 10.3 Approximative internal sphere 10.4 Spheres in the Newtonian limit
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13 Lie transport and Lie-differentiation 13.1 Contravariant Lie differentiation . . . . . . . . . . . . . . . . . . . . . . 13.2 Covariant Lie differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11 The absolute visualization 11.1 The perch skin intuition . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Vacuum field equations for flat absolute metric . . . . . . . . . . . . 11.3 On geodesics and flat metrics . . . . . . . . . . . . . . . . . . . . . . 11.4 Closed dimensions and timelike loops . . . . . . . . . . . . . . . . . 11.5 Warp drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Finding generators to make a single trajectory an absolute geodesic 11.6.1 Considering generators . . . . . . . . . . . . . . . . . . . . . 11.6.2 Considering photons . . . . . . . . . . . . . . . . . . . . . . . 12 Kinematical invariants 12.1 The definitions of the kinematical invariants . . . . . . . . . 12.2 The average rotation . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 A matrix formulation of rotation . . . . . . . . . . . . 12.3 About deformation . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Back to four-dimensional formalism . . . . . . . . . . . . . . 12.4.1 The four dimensional analogue to the rotation vector
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APPENDED PAPERS Paper I Embedding spacetime via a geodesically equivalent metric of Euclidean signature R. Jonsson – Gen. Rel. Grav. (2001) 33 no 7, pp. 1207-1235. Paper II Visualizing curved spacetime R. Jonsson – Accepted for publication in Am.J.Phys. Paper III Inertial forces and the foundations of optical geometry R. Jonsson – To be submitted to Class. Quant. Grav. Paper IV An intuitive approach to inertial forces and the centrifugal force paradox R. Jonsson – To be submitted to Am.J.Phys. Paper V A covariant formalism of spin precession with respect to a reference congruence R. Jonsson – To be submitted to Class. Quant. Grav. Paper VI An intuitive derivation of spin precession in stationary spacetimes R. Jonsson – To be submitted to Am.J.Phys. Paper VII Generalizing optical geometry R. Jonsson, H. Westman – To be submitted to Class. Quant. Grav. Paper VIII Optical geometry across the horizon R. Jonsson – To be submitted to Class. Quant. Grav.
1 Introduction Ever since it was presented in 1916, Einstein’s General theory of Relativity concerning space, time and gravitation, has been extremely successful in explaining all sorts of gravitational phenomena. Two examples from the beginning of the century are the gravitational deflection of light from distant stars passing close to our sun and the precession of the perihelion of Mercury. More recent experiments involve gravitational redshifts, relativistic slowing down of atomic clocks and indirect measurements of gravitational waves. Since the theory was presented it has also puzzled the minds of physicists and people in general. Indeed, for most people the theory is still clouded in mystery. In particular, people find the legendary black holes fascinating yet incomprehensible. To understand Einstein’s theory one must understand its heart – the curved spacetime. Unfortunately it is easier said than done to explain this concept without using mathematics. There are however ways of working around the difficulties and two of my scientific papers are directly related to this. In fact the whole process of explaining relativity has become something of a passion for me. I have spent countless hours doing computer-generated illustrations and more time than I care to remember down in the workshop making real models of curved spacetime. Using these models and illustrations I have given over thirty lectures on General Relativity in university courses as well as at high schools, science festivals and conferences. I believe there is plenty of room for improvement in general of how we teach physics. The author of a book, whether popular or scientific, may have a crystal clear understanding of the field he writes about. As a reader translates the written words into his own mental language, a lot may be lost however. We can minimize this loss by using examples, working on the exact formulations and rewriting the text according to how well it was understood. Still, much of the intuition that we have concerning physics is encoded as mental images and movies. I believe the whole 1
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learning process can become much more efficient if we actually create these images and movies. Of course it takes a lot of effort to produce interesting and pedagogical illustrations. With the aid of better computers and software – our opportunities to do so have however improved. For fields like General Relativity, for which there is even a large interest from the general public, it might be worth the effort. I hope to one day write a book on General Relativity directed towards a general audience. The idea is to use large color images spanning over double pages, to use no mathematics (or maybe just a little, well hidden in an appendix :) and to use analogies. I believe that one can give a very good understanding of General Relativity in this manner for anyone who is interested in the mysteries of Einstein’s theories. The second chapter of this thesis gives an idea of what I want to achieve, although the layout and form of this thesis is limiting to say the least. While all my work is related to General Relativity, it is not all related to popularizing the field. I have also worked on inertial forces and gyroscope precession. An example of an inertial force is the apparent force pushing us outwards as we take a steep curve at high speed with a car. A gyroscope is essentially a rapidly spinning body, that tends to keep its direction of spin. In general relativity we can however make the gyroscope turn or precess just by moving it around. For each of these two fields I have written both a formal paper and a paper more directed towards intuitive understanding. I have also worked on the field of optical geometry. In brief, the idea behind this theory is to consider a rescaled (stretched) version of the standard spacetime. Relative to the rescaled spacetime some effects, like gyroscope precession, can be explained in a more straight-forward manner. In one paper I show, together with my colleague Hans Westman, how one can generalize the standard theory of optical geometry so it applies to a wider class of spacetimes than the standard theory of optical geometry does. In yet another paper I show how one may apply this generalization to consider optical geometry also for the inside of a black hole.
1.1 This thesis This thesis serves many purposes apart from a being a means of collecting the appended papers. One purpose is to present my work, and the general field of work, for a more general audience. Another purpose is to present results that are not included in the papers. Lastly, the thesis provides an opportunity to review some topics that are underlying the appended papers. The thesis is organized in the following way: • Chapter 2 gives an introduction to General Relativity, aimed at a general audience. The main object of this chapter is to explain how spacetime geometry can explain why things are falling towards the Earth when we drop them.
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• Chapter 3 gives a brief introduction to the practical side of making illustrations using computer software like Matlab and PovRay. • Chapter 4 gives an account of how I made the real models that I use when lecturing on General Relativity. • Chapter 5 is a brief introduction to the topic of pedagogical techniques within general relativity, and my work therein. • Chapter 6 introduces the reader to inertial forces and my work within this field. • Chapter 7 is an introduction to gyroscope precession and my work related to this. • Chapter 8 gives an introduction to optical geometry and my two papers concerning this. • Chapter 9 provides some brief conclusions and an outlook. Then follows a part of the thesis that contains technical comments on the papers. Two chapters are introducing new insights and results concerning Papers I & II, and two chapters are reviews. • Chapter 10 is a comment to Paper I, regarding the shape of the spacetime inside a star or a planet. I show that in the Newtonian limit it corresponds to an exact sphere, as depicted on the front page of this thesis. • Chapter 11 is a comment to Paper II where I present some additional results and insights that are not included in Paper II. • Chapter 12 reviews the kinematical invariants defined for a reference congruence of worldlines. This chapter is related to Papers III-VIII. • Chapter 13 reviews Lie transport and Lie differentiation. This chapter is mainly related to Paper III, although it also has some relevance for Papers IV-VIII. Then follow the appended papers.
2 An introduction to Einstein’s gravity Gravity is something that we are all more or the staff a number of events happen: an alarm less accustomed to. If we throw an apple up- clock rings, a fire cracker explodes and a walkwards, or maybe of bowling ball, we know that ing man puts down his feet. it will soon fall down again. One might wonder why everything that we Ring! Ring! throw upwards insists on falling down again. Ring! Poff! The two most renown theories for this were Tap! put forward by Newton (late 17’th century) and by Einstein (early 20’th century). Figure 2.1: Life on a staff. We can mark the events happening on the staff in a spacetime diagram (Fig. 2.2). The later the event happens – the higher up in the diagram it is marked. If the event is far to the right along the staff, it is marked to the right in the diagram. Newton’s theory explains the fact that the apple returns by a gravitational force pulling the apple back towards Earth. In Einstein’s theory there is no gravitational force. Instead it explains the motion of the apple by a law saying that the motion corresponds to a straight line in a curved spacetime. To explain what this really means is the purpose of this chapter. Let us however warm up by explaining what a spacetime diagram is.
Time 7s 6s 5s
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Figure 2.2: The spacetime diagram. Here we can mark everything that happens on the staff. Can we deduce from the diagram in what direction the man is moving?
2.1 The spacetime diagram Imagine that you are living on a straight staff with blue position markings on it (Fig. 2.1). On 5
2. An introduction to Einstein’s gravity
6 In the diagram we can also illustrate motion of objects by so called worldlines (Fig. 2.3). The higher the velocity – the more tilted the worldline (if straight up is considered as not tilted).
Time
Time 7s 6s 5s 4s 3s 2s
Position
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Position 0m 1m 2m 3m 4m 5m 6m 7m
Figure 2.5: A flat spacetime for a staff in space.
Figure 2.3: Worldlines describing motion. The The upright blue lines correspond to fix position
leftmost line corresponds to an object moving to- along the staff. What type of motion does the two wards the left along the staff, the middle to an ob- worldlines correspond to? ject at rest and the rightmost to an object that is first at rest but is then accelerated to high velocity toIf we throw an apple along the staff in space, wards the right.
and let the apple move freely, it will continue with constant velocity along the staff. In EinSoon we will discuss the concept of curved stein’s theory the motion of the apple is explaispacetime illustrated as a more or less curved ned by a law saying that the motion corressurface. The flat spacetime diagram can then ponds to a straight line in the spacetime. be regarded as a map of the curved spacetime, similar to how a map can illustrate the curved The motion of a thrown surface of Earth. free object corres-
ponds to a straight line in the spacetime
2.2 Flat spacetime We now place the staff somewhere out in empty space, where we do not notice any effects of gravity (Fig. 2.4). Poff!
Poff! Poff!
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Figure 2.4: A staff in space. Along the staff various events happen and objects are moving.
In Einstein’s theory we have here a so called flat spacetime which we can illustrate by a flat plane (Fig. 2.5). On the plane we describe events and motion along the staff in space, just like we did earlier in the spacetime diagram.
By free we here essentially mean that nothing is touching the object. In section 2.4 we will give it a more precise meaning. To create a straight line on our flat spacetime surface, we could use a ruler. We will however instead use a little toy car equipped with a downwards directed pen (see the cover of this thesis). The toy car, from now on denoted the drawing car, we can then by hand roll forward. The point of using the drawing car is that it also works on curved surfaces, as will be used later. If we know the initial velocity of a thrown apple, we can use the drawing car to predict how the apple is going to move. We put the drawing car at a point on the spacetime surface corresponding to the starting position of
2.3. Mass curves spacetime
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the apple and the time that we threw the apple. We give the drawing car a direction relative to the surface, corresponding to the starting velocity of the apple, and roll the car forward. The drawn line corresponds to the motion of the apple after the throw. As an example we may predict the motion of two apples, thrown with different velocities from the zero-meter position at zero time (Fig. 2.6). Can we see were the apples are predicted to be after 0.4 seconds?
2.3 Mass curves spacetime We can create a curved spacetime from the flat spacetime we just showed by stretching and bending it. In Einstein’s theory spacetime is curved by mass. The greater the mass the greater the curvature (Fig. 2.8).
Initial directions corresponding to the velocities 1½ m/s och 3 m/s.
Figure 2.8: Mass curves spacetime. The Earth illustrates in what direction the curving mass lies. Strictly speaking the mass should lie within the spacetime – not outside as depicted here.
The straight lines we get, corresponding to the motion of the apples along the staff.
Sometimes one illustrates how mass curves something by putting a metal ball on a rubber lines can be created with a three-wheeled toy car sheet and look at how the sheet bends. Unforequipped with a pen. tunately this analogy tells nothing about how the spacetime is curved which is what we want Fig. 2.7 illustrates how the scenario with to illustrate here. Before we talk more about how mass curves the spacetime we will explain the apples would look in reality. what it means that it is curved. Figure 2.6: Straight lines in a flat spacetime. The
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Figure 2.7: Apples thrown along a staff in space. The images show the position of the apples at consecutive times. Compare with Fig. 2.6.
We see how one, at least in outer space, can predict the motion of thrown apples using a geometric model.
2.4 Curved spacetime for a staff outside Earth We now place the staff upright outside of Earth. Here the mass of the Earth has curved the spacetime as illustrated by the funnel-shaped surface of Fig. 2.9. Time is directed clockwise around the funnel (as seen from above) and one circumference corresponds to one second. An object moving up along the staff at a fixed speed corresponds to a worldline spiraling around the funnel. The faster the object moves, the steeper the spiral. An object at rest corresponds to a horizontal circle.
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Position 1.2m 1.0m 0.8m 0.6m 0.4m 0.2m
Poff! Poff!
Initial directions corresponding to the velocities 1½ m/s och 3 m/s
Poff!
0.0m
The straight lines that we get, corresponding to the motion of the upwards thrown apples along the staff.
Time Figure 2.9: A curved spacetime for a line outside Earth. Strictly speaking the surface should not close in on itself in the time direction. Rather one should come to a new layer after one circumference – as on a paper roll, illustrated on the back cover of this thesis. The Earth is shown in miniature.
Figure 2.11: Straight lines in a curved spacetime.
In Einstein’s theory, here as well as in outer space, the motion of thrown free objects corresponds to straight lines in the spacetime. By free we mean that gravity alone determines the motion of the objects (no air resistance for instance). We can thus predict the motion of the thrown objects by the same method that we used earlier (with the drawing car), although the spacetime surface is now curved.
We thus place the drawing car on the base of the funnel, corresponding to the starting position on the staff. We give it initial directions relative to the funnel, corresponding to the initial velocities of the apples, and roll it forward along the surface of the funnel (Fig. 2.11). We see that both worldlines correspond to apples that initially are moving up along the staff, reach a maximum height, and then fall back again. The light-colored (green) line reaches a higher level on the funnel and takes a longer time before returning to the base of the funnel. This is reasonable since the corresponding apple is thrown upwards with a higher velocity. You may note precisely how high the apples are expected to come, and what time it takes them to return to their initial position. In Fig. 2.12 we illustrate the motion of the apples in reality. We thus have a geometrical model for predicting the motion of apples. Unlike in Newton’s theory there is no gravitational force in this model. It is the shape of spacetime that determines that the worldline returns to the base of the funnel, just like an upwards thrown apple returns to the surface of the Earth. If the spacetime outside of the Earth would
As an example, we study two apples thrown upwards from the zero-meter marking on the staff with different velocities (Fig. 2.6). 0.4 m 3 m/s 0.2 m
1,5 m/s
0.0 m
Figure 2.10: Apple-throwing. As soon as the apples have left the throwing hand, the motion corresponds to straight lines in the spacetime.
Note that the light-colored (green) initial direction corresponds to a precisely twice as high velocity as the dark-colored (red) initial direction. Time is directed clockwise as seen from above.
2.5. The spacetime for a line 0.0 s
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through the Earth 0.3 s
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2.5 The spacetime for a line through the Earth To give another example of a curved spacetime we imagine a hole straight through the Earth, with a staff that follows the hole (Fig. 2.14).
0.0m
Figure 2.12: Upwards thrown apples along a staff outside of Earth. The pictures show the position of the apples at consecutive moments in time – compare with the predictions from Fig. 2.11.
Poff! Poff!
Poff!
have a shape as illustrated to the left in Fig. 2.13 – apples thrown upwards would not return according to the theory. Note also that if we let the drawing car turn – it will not preFigure 2.14: A staff through Earth. Along the staff dict the motion of upwards thrown free objects events happen and objects move. (illustrated to the right in Fig. 2.13. The curved spacetime for the staff is illustrated in Fig. 2.15. Notice that the outer parts of this spacetime correspond to the funnel that we earlier displayed.
Figure 2.13: Left: An alternative shape of spacetime. A drawing car rolled straight forward would follow the dashed lines that never return to the base of the spacetime. Right: An alternative worldline. A drawing car that turns created the solid line which is not returning to the base of the spacetime.
Time
Position
Figure 2.15: Curved spacetime for a staff through
Thus the shape of the spacetime and the rule about straight lines determine that the upwards thrown apples should return to Earth – in Einstein’s theory.
Earth. In this illustration it is about 8 minutes between the time lines (running along the surface) and about 1500 km between the position lines (going around the surface).
In the introduction we mentioned that we would attempt to explain the meaning of a straight line in a curved spacetime, and how this can explain why an upwards thrown apple falls down again. Hopefully this feels rather natural now.
We now move to the center of the Earth where we simultaneously throw three apples, with different velocities, along the staff as illustrated in Fig. 2.16. This time it is perhaps not as evident what is going to happen. But we know that as soon
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Figure 2.16: Apples in the center of the Earth. The dark-colored apple is just released and the others are thrown with the velocities 6 and 18 km/s.
as the apples have left the throwing hand their motion will correspond to straight lines in the spacetime. If we have an actual model of this type of spacetime we can repeat the procedure with the drawing car. We put the drawing car on the spacetime model at a point corresponding to the center of the Earth and a certain time. We give the drawing car three different initial directions corresponding to the three different velocities. The result is shown in Fig. 2.17. Initial directions corresponding to the initial velocities
The straight lines that we get, corresponding to the motion of the upwards thrown apples
These scenarios correspond precisely to what we would expect from reality (although an actual experiment would be difficult to carry out in practice). Once more we see how straight lines in a curved spacetime can explain the motion of thrown apples. As we move further away from the Earth the funnels at the ends of the spacetime will assume a cylindric shape. But cylinders are in fact flat in the sense that we can roll out a cylindric surface to a flat surface. When we are far from the curving effects of the Earth we thus have a flat spacetime, just like in the preceeding section. In this chapter we have illustrated different parts of the spacetime in a certain order. All of these parts are however connected and there is thus only one spacetime (per universe:).
2.6 Forces and gravity In Einstein’s theory there is no gravitational force, but there are forces also in this theory. An example is the force by which we affect a car as we push it forward. A force acting on an object causes the worldline of the object to curve. The greater the force the greater the curvature of the worldline.
The motion of an object affected by a force corresponds to a curved line in the spacetime
Figure 2.17: Straight lines in a curved spacetime. Note that the straight lines correspond to motion along the staff through the Earth.
The dark (red) line corresponds to an apple at rest at the center of the Earth, the semi-light (green) line corresponds to an apple that is oscillating back and forth along the staff, around the center of the Earth. The light (yellow) line corresponds to an apple that passes the surface of the Earth and continues onward into outer space without ever returning.
As an example we study an apple that at first is just hovering near a staff in outer space. The apple then receives a push towards the right and goes off along the staff at a constant speed. How this scenario looks relative to the flat spacetime is illustrated in Fig. 2.18. The law about forces and the curvature of worldlines applies also when the spacetime is curved. As an example we study an apple that we hold by its shaft here at Earth. In Einstein’s theory there is only an upwards directed force from the hand acting on the apple. How the apple can remain at rest even though
2.7. About mass
11 An upwards directed force is acting on the apple...
I
... and makes the worldline curve upwards. Compare with the straight dashed lines.
II I III
Figure 2.18: Apple-pushing in outer space. The worldline deviates from the straight dashed lines Figure 2.20: The resolution of the paradox. An as the force of the push acts on the apple. apple at rest at a height of 0.4 meters. The force makes the worldline curve upwards all the time but Einstein’s theory Newton’s theory the shape of the spacetime insures that it will not get any higher anyway.
apple, such that no forces are acting on it, the worldline would straighten up (Fig. 2.21).
Figure 2.19: An apple held at rest at Earth. In Newton’s theory there is a gravitational force acting on the apple. In Einstein’s theory there is no gravitational force.
Figure 2.21: An apple being released. As soon as the apple is released the worldline follows a
it is affected by a net force upwards may apstraight line. pear pardoxical – but the solution is given by the curved spacetime (Fig. 2.20). Note that there is no gravitational force puIf we were to direct the drawing car along lling the apple down after we have released it. a horizontal position line and roll it forward, it would have to turn upwards to remain at the The apple falls because we cease to curve its same height (the front wheel should be turned worldline and let it follow a straight line. to the right). The apple is thus at all times affected by an upwards force, just like the drawing car is all the time turning in the upwards 2.7 About mass direction – but it does not get upwards because the spacetime is curved! The effect of forces in Einstein’s theory is to To make the apple go upwards we must act curve worldlines. How much a worldline is to on it by a greater force than that required to curve for a certain force depends on the mass keep it at rest. The drawing car must thus turn of the object. The greater the mass the lesser more upwards than what is required to keep it the curvature of the worldline (Fig. 2.22). One at a fix height. The result would be a worldline might say that a large mass makes the drawspiraling up along the funnel. ing car hard to steer – so it tends to roll straight If we on the other hand were to drop the ahead unless a great force acts on the object.
2. An introduction to Einstein’s gravity
12 8 kg
1 kg
shape would be more complicated than those we have shown in this article. It would not be rotation symmetric. On the irregularly shaped surface we would however still be able to use the drawing car to predict the motion of an apple thrown from the Earth towards the moon.
2.9 Was Newton wrong and was Einstein right? Figure 2.22: Apple pushing in space – again. The
We have seen, although in a simplified form
worldline of the heavy apple is curved less by an (that nevertheless gives precisely the right preequal force.
That it is difficult to affect the motion of an object with a large mass is common to both Einstein’s and Newton’s theory. The difference lies in that large mass does not mean large gravitational force in Einstein’s theory – there is no force of gravity there.
2.8 More about mass Apart from the effect that mass has on the curvature of worldlines, it has also the property that it curves spacetime itself. The point is that mass contains energy, and all sorts of energy curve the spacetime. Even a radio signal emitted by a cellular phone contains energy and will curve the spacetime a little. So it is not quite as simple as that an upwards thrown apple corresponds to a straight line in a spacetime whose shape is independent on the apples motion, but rather its a straight line in a spacetime that the apple itself partially curves (or has curved). In the case of the apple, the mass is however so small that it it does not curve the spacetime significantly. If we on the other hand would let for instance the moon fall towards the Earth (and the Earth towards the moon), we would have to take into consideration that the Earth and the moon consist of particles that all curve the spacetime a little. The spacetime would also for this case correspond to an unmoving surface (it would make no sense to have a moving spacetime), but the
dictions), how Einstein’s theory explains the motion of apples by a curved spacetime. Newton on the other hand explained it with a gravitational force. One might then wonder who was right? Both theories describe with good accuracy all sorts of every day gravitation, like how a dropped apple falls towards the ground. There are however situations where the predictions from Newton’s theory differ from those of Einstein’s. A famous example regards the orbit of Mercury around the sun (Fig. 2.23).
Figure 2.23: The orbit of Mercury. The dashed line is the motion as prescribed by Newton’s theory, the solid by Einstein’s. The effect that the ellipse is rotating (in the plane of the paper) is however exaggerated half a million times.
Measurements of the position of Mercury from the 18’th century and onwards show that the almost elliptical orbit of Mercury is in fact rotating in accordance to Einstein’s theory. So Newton’s theory does not describe reality correctly. It is however simple to use for
2.9. Was Newton wrong and calculations compared to Einstein’s theory, and works very well in most cases. But does this mean that Einstein’s theory about the curved spacetime is right? When it comes to physics at a fundamental level, one can never know if a theory is right. Tomorrow the apples may fall upwards and we see no way to explain this in Einstein’s theory. Then we must search for a new theory that explains both why the apples fell down yesterday and why they are falling up today. How nature really works – we do not know. What we can say for sure is that the curved spacetime exists in Einstein’s theory – which seems to be a good description of nature.
Figure 2.24:
Curved spacetime in practice. Demonstration using a spacetime funnel and drawing car.
was Einstein right?
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3 Making illustrations The images of this thesis have mainly been created using Matlab, under the Linux operating system. Post processing has been made in Corel Photo Paint. Images built from several different images or vector graphics have been made in Corel Draw, gimp or xfig. There are considerably more advanced programs for making three-dimensional graphics than matlab. The ideal would likely be ’Maya’ – the program that was used to make the animated movie ’Shrek’ among others. Learning some basic tricks and building your own specialized graphic functions in matlab and utilizing a freeware called povray, one can still come a long way.
3.1 About matlab To create the images of this article I have written over 170 matlab functions and programs of a total of more than 350 kb. To list them all would fill this entire thesis. Just to give a feel for how the various codes are working I give a little example of a matlab code utilizing some of my functions on the next page. The non-standard matlab functions that I call here are rjxsurf, putonsurf and rjsurfset. The function rjxsurf plots the surface much like the standard matlab function surf does. The main difference lies in that all the lines of the vertexes of the surface need not be plotted but for instance one may plot every other line by including ’linesep1’,2 in the arguments. To use this feature one must also send the viewing angles az and el. What rjxsurf does is that it puts the lines out as ordinary lines (using plot3), but then it lifts them a bit towards the eye – so they are visible on top of the surface. This function allows for the creation of smooth surfaces that are not completely riddled with too dense-lying lines. Also setting lighting phong smooths the surface nicely. 15
3. Making illustrations
16 clf N=50; [X,Y,Z]=sphere(N); C=ones(size(X)); az=160;el=20;
%%%% Plot the sphere with coordinate lines %%%%%%%%%% rjxsurf(X,Y,Z,C,’setlines’,’on’,’az’,az,’el’,el,’linesep1’,2,... ’linesep2’,2,’col’,[0 0 1],’linecol’,[1 1 0]); %%%% Set out a ’surface line’ on the sphere %%%%%%%%%% phibase=linspace(0,2 *pi,N+1); % A coordinate basevector thetabase=linspace(0,pi,N+1); % A coordinate basevector phi=linspace(2.8,5.2,200); % Trajectory phi theta=pi/2+sin(phi *10).*(phi-min(phi)) *0.2; % Trajectory theta rjcell{1}={’surfline’,theta,phi,’col’,[1 0 1],’epsilon’,0.1,... ’arrow’,’curved’}; handles=putonsurf(X,Y,Z,thetabase,phibase,{az,el,[]},rjcell) rjsurfset(handles,{0.7,0.3,0.4}) %%%% Fixing some lighting etc %%%%%%%%%% view(az,el); axis off;axis image shading flat light;lighting phong
Figure 3.1: The output from the matlab program above
3.1. About matlab
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Printing to a file at high resolution and resampling (compressing) the image while applying anti-aliasing can give high quality images. The function putonsurf is especially well suited for plotting worldlines on parameterized surfaces. The function rjsurfset is just a quick way to set some reflection properties of surface elements.
3.1.1 Some shortcomings of matlab Matlab does most of what one wants concerning graphics, but it lacks some features that would be useful: • Doing real raytracing, i.e allowing objects to cast shadows and reflections on other objects. In Matlab there is only a cruder shading of surfaces depending on the angle of the surface relative to the light source - but independent on the reflections and shadows of other surfaces. • Doing anti-aliasing while rendering. ’Rendering’ is the process whereby a twodimensional image is formed from a three-dimensional scenario. Anti-aliasing is a certain averaging technique of the color of nearby pixels, taking away the ’staircase’ appearance (pixelization) of the edges of lines and surfaces and making them smooth. • Doing high resolution animation. This is perhaps the most surprising deficit of matlab. It has a simple structure for making animations (movies). There is however no real way to control the resolution of the movie – matlab appears to take a so called screen-shot of every image. In general the screen resolution (and size) are not what one might desire for a reasonably nice (anti-aliased) movie that covers the screen of a standard computer or TV. • Doing subtractions of volumes. For instance one may want to subtract a cylinder from a sphere to make a hole through the sphere. This type of operation one must do ’by hand’ in matlab – and it can be quite time consuming and complicated. Concerning the second point there is a solution. Matlab can print to a file at a very high resolution, for instance issuing the command: print -dtiff -r900 filename The dimensions of this image is 8×6 inches. At 900 dots per inch this is a very large image (typically several tens of Mb). Then one opens the resulting file filename.tif in some program for manipulating two-dimensional images (Corel Photo paint in my case). Then one resamples the image to 600 dpi and to the size that one desires, making sure that the anti-alias check-box is checked. The required resolution of the
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3. Making illustrations
original file to get a good final result depends on the dimensions of the final file. As a rule of thumb one should have at least twice as many pixels (per dimension) in the original image as one does in the final image, to get a nice anti-aliasing.
3.2 About povray There are graphics programs that have the abovementioned features that matlab lacks. In particular there is a program called PovRay, a freeware downloadable from http://www.povray.org. I advice the interested reader to go there and have a look at the ’Hall of Fame’ images. Some are really spectacular in appearance, and still comparably easy to make. PovRay works similar to Matlab, but it is specifically designed to make images rather than to do calculations. Here is an example of a piece of povray code in a file called example.pov: #include "colors.inc" background {color Cyan} camera { location <0, 2, -3> look at <0, 1, 2>} sphere {<0, 1, 3>, 1 texture {pigment{color Red} finish {reflection {0.5} phong 0.7 phong size 10 ambient 0.0 diffuse 0.8}}} plane { <0, 1, 0>, -1 pigment {checker color Green, color Blue}} light source { <2, 4, -0.001> color White parallel point at <0,0,0>}
To process (render) this file one may write (on the Unix or Linux system): povray +P +A +W1200 +H900 example.pov The image is then rendered and the resulting file is output as example.png. If one desires another format, like the postscript file displayed below in Fig. 3.2, one opens example.png in some program (like gimp or Corel Photo Paint) and converts it to the desired format. There are however also downsides to povray compared to Matlab. For instance one cannot interactively rotate and zoom the three-dimensional scenario using the
3.3. Matlab to povray
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Figure 3.2: An example of the output from a short povray program. mouse. Also the matlab standard representation of surfaces as matrixes (which is not standard in povray) is quite practical if one is dealing with parameterized surfaces, as I have been to a large degree.
3.3 Matlab to povray I have written a converter (a matlab script) that creates a povray file from a matlab generated three dimensional scene . The converter is called rjmat2pov.m and is very simple to use. After running your matlab script to create a three dimensional image, type rjmat2pov(’filename’) and you will get a povray file called filename.pov that you may render as any other povray file. What rjmat2pov does is that it finds all the visible surface elements in the matlab scene, and divides them into the natural triangles that they are made of and makes a povray mesh2 object out of them. The colors and reflection properties are also translated although the translation of the reflection properties is rather approximate (different types of reflection properties are used). Any light sources are also translated from the matlab scene to the povray scene. There is plenty of room for improvement of this code. Unless there are very many surface objects and large colormaps it should work just fine however. I used this converter for the cover illustration.
3.4 Comments I plan to put the matlab files required for the examples above on my homepage. There are quite a few help-files required, but so long as you put the whole pack in your personal matlab directory (assuming Linux or Unix) so that matlab can find them – this need not concern you. My current homepage is at http://fy.chalmers.se/˜rico and the files can be found by clicking the matlab symbol.
4 Making models A picture may be worth more than a thousand words, but a model is worth more than a thousand images (or at least more than a couple of images:). If for no other reason than that it was a whole lot of work to make them, and that I would like to be the first to write a thesis on relativity containing phrases like ’turning-lathe’, ’welding’ and ’molding form’, I will now briefly explain how I made the models that I use while lecturing about gravity.
4.1 The funnel shaped spacetime From aluminum sheet metal (5 mm thick) I cut out 10 equal stripes whose shape I had calculated and printed (using a computer). I then waltzed these stripes to give them the right curvature in the direction along the funnel. Next I made several pressforms of different radii and pressed the the stripes in the press-forms to give them the right curvature in the direction around the funnel. Then I TIG-welded together the pieces, see Fig. 4.1. Next I built a rack consisting of an axle and several wooden discs that fitted snuggly to the inside of the funnel. Using the rack I turned the spacetime round and smooth as best as possible with the lathe cutting steel. (I am skipping the parts where I cut through the surface and had to weld it together again:) Then, while spinning it fast in the lathe, I used an angle grinder with a disc of segmented grinding papers to further smoothen the surface. After that followed finer grinding and polishing in the lathe. In the end it shone like a mirror. I then left it to a car lacquerer, Mr Istvan Papp, who lacquered it with a grey metallic and clear varnish (two component). Next I put the funnel into the lathe, spun it slowly, and used a water based felt tip pen in stead of a cutting steel to make the blue circles around the spacetime. I cut the tip of the pen (and spent quite a few pens) to make the lines of varying width 21
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4. Making models
Figure 4.1: A spacetime in the making. To the left the spacetime has just been spotwelded together. To the right the spacetime after turn lathing and angle grinding (narrower towards the top of the funnel). For the black time lines along the funnel, I used masking tape and paintbrush. For the second and meter markers I used sticker letters, the part that remains after you have taken out the actual letter, and used a paint brush to make the letters and numbers. Between applying lines and text one had to put a layer of clear varnish to protect what was already made. I am still indebted to Istvan for helping me out with this. At last I lacquered the inside myself with a blue metallic and clear varnished it.
4.2 The bulgy cylinder spacetime For the second model I used a different technique. First I glued together boards of wood to form a big lump, with an metal axle running through the middle of it. Then I turned and ground this big chunk of wood to a shape corresponding to half the spacetime, see Fig. 4.2. To make it as smooth as possible I clear varnished, waxed and polished the wooden plug. Then I applied gelcoat (a plastic), layers of glass fibre and liquid plastic to make a glass fibre molding form. In this form I molded the two ends of the spacetime, again using glass fibre and liquid plastic. If you want to try this – do not neglect the useful-
4.2. The bulgy cylinder spacetime
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Figure 4.2: From a lump of wood to a glass fibre spacetime. ness of a really good gas mask. After cutting and grinding the ends of the spacetime flat, I molded a means of attachment with bolts between the two ends. I also did some molding at the ends of spacetime allowing me to tightly fit a couple of turned aluminum discs there. Through central holes in these discs I put a steel axle enabling me to put the hole construction in the lathe and to grind away the small differences in radius at the intersection. Then came the Time of Lacquering - that I did myself this time. Below is just the briefest outline of a few things that went wrong in the primary tries. • Metallic paint too old.
• Air holes in glass fibre becoming visible during lacquering. • Partially clogged up jets on the paint sprayer.
• Wrong distance from sprayer to target. • Dust particles on the surface getting trapped under the paint. • Dust particles in the air sticking to the drying surface. • Paint running from a jet getting thrown out on the surface. • Water drops falling on un-dried paint.
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4. Making models • Trying to remove paint from a failed lacquering attempt - the thinner eats through the lower layers of paint creating a canyon in the surface. • The fire department arriving, clearing the entire house, because the lacquering fog set of the alarm. • The color appearing patchy upon rotating the surface - although it looked fine when not rotating it.
Then I built a scaffolding for the entire spacetime allowing me to rotate it. Seeing that there was an un-roundness of something like a millimeter, I decided to put the whole thing in the lathe again. After a lot of spray puttying and grinding it was back to the lacquering box. Then of course all the air holes in the glass fibre surfaced and I had some nice sessions of applying spray putty and grinding it in the lathe again. Then it was back to the lacquering box. To avoid patches becoming visible when rotating the spacetime, I hooked it up to an electric drill rotating the spacetime as I just moved the paint sprayer from one end to the other. A few more things went wrong. • Putting a finger in the un-dried paint to see if the paint is not dry yet -- a classic! • The spacetime spinning to fast - paint drying to fast.
• Un-smoothness of the surface from the last lathing becoming visible as the metallic is applied.
In the end I made more than 15 lacquering attempts – and every time you have to wait for the paint to dry, grind the spacetime, flush the lacquer box and yourself (wearing a rain coat) etc. I even left it to a professional car lacquerer at one point, but I was not happy with what he had done either, so I ground it again and kept going. Suffice it to say that in the end I made it. Oh – just in case you want to try this – some inspectors found out about my technique with the electric drill and they were not overly happy to have a sparkinducing machine inside a flammable fog of thinner:) Putting the coordinate lines on the surface was analogue to what I did on the funnel. Parallel to making the bulgy spacetime, I also made a flat spacetime model, using a plane of PVC-plastic. My models are displayed in Fig. 4.3.
4.2. The bulgy cylinder spacetime
Figure 4.3: Some curved shapes on my bed:)
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5 Spacetime visualization I have written two research papers where I develop two different techniques directed towards popularizing the field of general relativity. Both these techniques allows us to visualize curved spacetime by a curved surface. The ideas underlying the illustrations are however quite different, and they illustrate different aspects of the full theory of General Relativity. Similar techniques have been developed before. There is a popular scientific book called ’Visualizing Relativity’, by L. C. Epstein [1]. Some of the illustrations are quite similar to the ones that I have created (related to Paper I) although the underlying idea and how one interprets the illustrations is completely different1 . Yet another technique is presented by D. Marolf in [2] though interpreting the embedded surface that he considers requires some knowledge of special relativity 2 . There is also a paper by W. Rindler [3] where the basic ideas have some similarity to those of Paper I though he considers a more mathematical approach intended for undergraduate students of physics. 3 . 1
The theory underlying [1] is based on the assumption of an original time-independent, diagonal, Lorentzian line element. Rearranging terms in this line element one can get something that looks like a new line element, but where the proper time is now a coordinate. The ’space-propertime’ can be embedded as a curved surface. 2 In particular he considers the radial line element of a maximally extended black hole. The proper distances can be illustrated by embedding the surface in 2+1 dimensional Minkowski space (visualized as a Euclidean 3-space) 3 Essentially he considers the Newtonian equations of motion in Hamilton’s formalism, and matches them (approximately) with the geodesic equations of motion we get considering a fourdimensional Riemannian (positive definite) line element. The components of the metric are found using qualitative guessing and trial and error. Thus he presents a way of bypassing special relativity to anyway give an understanding of the concept of curved spacetime – given some knowledge of Newtonian mechanics and Riemannian geometry.
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5. Spacetime visualization
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5.1 Paper I In this paper I present a method that allows us to visualize curved spacetime by a curved surface. The method is tailor-made to explain what it means to have straight lines in a curved spacetime and how this can explain why an apple thrown upwards returns to Earth. This method is underlying the introduction to general relativity presented in chapter 2, as well as the cover of this thesis. The idea underlying this method is hard to explain in brief without getting technical, but I will try to give a feeling for what is done in the paper. Consider two events, like snapping your fingers, first with the right hand and then with the left. In Einstein’s theory, one assigns an interval (a kind of distance) between this pair of events. If there is time for a light signal to travel from one event to the other, the interval will be positive otherwise it will be negative 4 . Next consider the curved surface of a sphere on which we have sprinkled grains of sand. The geometry of the sphere can be defined as the set of distances separating all pairs of (nearby) grains of sand on the surface. Similarly, the geometry of spacetime can be defined as the set of intervals separating all pairs of (nearby) events. Relative to this abstract geometry, a canon ball shot from a canon will follow a line that is straight in some (abstract) sense. The fact that we have negative intervals (distances) between events however makes it impossible to illustrate directly the spacetime geometry by a sphere or some other curved surface. What I do in Paper I is that I take the set of spacetime intervals between events, and transform them such that all the intervals become positive, but without changing what is a straight line. We can illustrate the new geometry, that has only positive distances, with a curved surface.
5.1.1 A technical note In technical terms, I consider a Lorentzian two-dimensional, time-independent line element. I then find another line element that is positive definite but has the same geodesic structure. The new geometry can be embedded in Euclidean space. Due to a one-parameter freedom in going from the original to the new dual line element, together with freedoms of the embedding, one can illustrate even the weakly curved spacetime of Earth with a significantly curved surface.
5.2 Paper II In this paper I present another method of visualizing curved spacetime. This method is well suited to explain why clocks slow down near massive bodies, and what that 4
Strictly speaking it is the square of the interval that can be positive or negative – but that is of little importance here.
5.2. Paper II
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really means. The method can also be applied to explain black holes, expanding universes (big bang) and much more. This is a technical paper exploring the possibilities of the method rather than applying the method. However, in section 2 of the paper the method is applied to give a very brief introduction to general relativity. The section is directed to teachers of physics, and uses only a little bit of mathematics, though in principle the method can be used without any reference to mathematics. Even at the current level, a general reader with an interest to learn about curved spacetime may benefit from reading it. Fig. 5.1 gives an example of an illustration from Paper II.
Time
Space Figure 5.1: Cosmological models. From these models one may understand how space itself can expand. For further details, see section 2 of Paper II. As was the case in the previous paper, the idea is to make a transformation turning all of the intervals (distances) between pairs of events into positive distances. This time the idea is however not to preserve straight lines, but to preserve the intervals themselves (as much as possible). The resulting geometry can be illustrated with a curved surface. If one knows how to interpret such an illustration, one will be able to find out everything there is to know about the true spacetime geometry (including the negative distances).
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5. Spacetime visualization
5.2.1 A technical note Technically speaking, I first introduce an arbitrary field of timelike four-velocities uµ . Then, at every point, I perform a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature which for the special case of two dimensions be embedded as a curved surface. On the surface lives small Minkowski systems relative to which special relativity holds.
6 Inertial forces An example of an inertial force is the (apparent) force that pushes us outwards if we are on a rotating platform, or if we drive our car through a roundabout at high speed. This particular inertial force is known as a centrifugal force. In the general theory of relativity one can introduce inertial forces in a similar manner to how one introduces them in Newtonian mechanics. Some effects that occur in relativity are however quite counter-intuitive from the point of view of Newtonian mechanics. In fact if we consider a rocket in orbit near a black hole it will require a greater rocket thrust outwards to keep it from falling into the black hole the faster it rotates around the black hole1 . We might say that the centrifugal force is pointing inwards rather than outwards here.
Earth
Figure 6.1: Left: a rocket orbiting the Earth. The faster the orbital velocity the less outward thrust from the rocket engine is required to keep the rocket on a fix radius. If the orbital velocity is high enough – no outward thrust is required. Right: a rocket in orbit around a black hole. The faster the rocket moves the greater the required rocket thrust needed to keep it from falling into the black hole. 1
This effect occurs between the event horizon (the radius of the black hole) and 1.5 times the radius of the black hole (the so called photon radius where particles of light can move on circles).
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6. Inertial forces
I consider inertial forces in general spacetimes using the mathematical formalism of general relativity in Paper III. I also derive a general formalism of inertial forces using only basic principles of relativity as presented in paper IV. The interested reader who knows a bit about special relativity and vectors may benefit from reading this paper. Among other things, I show how one can explain the abovementioned scenario with the black hole as a natural consequence of relativity. The mathematical results of the formal paper are a bit more general than those of the intuitive paper, but the results agree where comparable.
6.1 A technical note In Paper III we consider a general timelike congruence of reference worldlines in an arbitrary spacetime. The local motion of a test particle can be described in terms of the velocity relative to the congruence, the time derivative of this velocity and the spatial curvature of the test particle worldline projected onto the local slice. In this formalism inertial forces appear naturally in form of the kinematical invariants of the congruence (see chapter 12). While relativistically correct, the resulting equations of motion are effectively three-dimensional. I show that when the congruence is shearing, the projected curvature is not necessarily the most natural measure of spatial curvature, and I present an alternative definition. I also study the effect of conformal rescalings on the inertial force formalism. In Paper IV, via the equivalence principle and basic elements of special relativity such as time dilation, I derive the same formalism using only three-vectors for the special case of a non-shearing congruence.
7 Gyroscope precession A gyroscope is basically a symmetric body that spins very fast around an axis and is suspended in such a way that there is no torque acting on it. If we move the gyroscope along a circle it will keep pointing in the same direction according to the theory of Newtonian mechanics. According to special relativity however, if we transport the gyroscope very fast along the circle – its spin axis will turn, or in other words precess. As we go to general relativity the situation becomes even more interesting. For instance, if we move the gyroscope along a certain circle around a black hole – the gyroscope spin axis will (automatically) turn in such a manner that the spin axis is always directed along the direction of motion (see Fig. 7.1).
Figure 7.1: A gyroscope moving on a circle around a black hole. The spin axis is the thick bar through the sphere. Despite the fact that we are not affecting the gyroscope by any torque, it still turns.
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7. Gyroscope precession
I consider gyroscope precession in general spacetimes in Paper V using the mathematical formalism of general relativity. I also consider gyroscope precession in a very non-formal manner in Paper VI. Here I use basic principles of relativity and intuition rather than formal mathematics. In particular I show how one can use the special relativistic explanation to explain also the general relativistic effects. The formal paper is a bit more general than the intuitive paper but the results match where comparable.
7.1 A technical note As was the case in the inertial force analysis, the idea rests on introducing an arbitrary congruence of timelike worldlines and expressing the spin precession with respect to this frame of reference. Rather than considering the standard spin vector Sµ , I consider the spin vector we would get if we would stop the gyroscope by a pure boost relative to the congruence. The stopped spin vector obeys simple laws of rotation and is ideally suited for this approach. In Paper V I use a four-covariant formalism starting from the Fermi-Walker transport equation, and derive an effectively threedimensional formalism of spin precession. In the intuitive paper I arrive at the same result, considering a non-shearing reference frame (congruence) using only three-vector formalism together with the equivalence principle and special relativity. I do not use the Fermi Walker transport equation in this paper. In particular I show that one may regard the gyroscope precession in arbitrary stationary spacetimes as a double Thomas precession effect. One part comes from the gyroscope acceleration and the other from the reference frame acceleration (there is also a trivial contribution from any rotation of the reference frame).
8 Optical geometry The mass of a star curves the fabric of space and time. How space is curved is illustrated in Fig. 8.1.
Figure 8.1: Illustrating the curved geometry of a plane through a star. While the velocity of light is everywhere locally the same, clocks near the star will run slow relative to clocks far from the star (for some understanding of this, see section 2 of Paper II). Effectively this means that as we send a light signal from one end of the star to the other (imagine that the star is transparent), it will take a longer time for the light signal than we might think considering only the spatial distance that the light needs to travel. Seen from the outside, there is an apparent slowing down of the light signal. We can account for this by considering a rescaled space, where the extra distance accounts for why the light takes such a long time to reach the other side of the star. This stretched space is known as the optical geometry, see Fig. 8.2. Relative to the optical geometry, photons (particles of light) move along straight spatial lines. This is not generally the case for photons. If we send out a photon horizontally – it will fall away from a straight line just like anything else we might throw in a horizontal direction. The only difference is that photons move so fast that we do not see them fall. But relative to the optical geometry – they do follow straight lines. In [16] M. Abramowicz gives a popular scientific presentation of optical geometry. Together with my collaborator Hans Westman, I present in paper VII a way of 35
36
8. Optical geometry
Figure 8.2: The optical geometry of a plane through a star. It is related to the standard geometry of the plane by a stretching. The darker region lies within the star and the lowest point is the center of the star. Unlike the curved surfaces presented prior to this chapter, this is a visualization of curved space and not of curved spacetime. generalizing the standard theory of optical geometry to include arbitrary spacetimes (so that the optical geometry may change in time). In paper VIII I consider specific applications of the generalized optical geometry. In particular I consider a black hole, including its interior.
8.1 Generalized optical geometry in technical terms In an arbitrary spacetime we may introduce a spacelike foliation specified by a single function t(xµ ). Forming the covariant derivative of this function we get a vector field orthogonal to the foliation. We then introduce a congruence of worldlines parallel to the vector field. Performing a conformal transformation we rescale away time dilation, so that in coordinates adapted to the slices and the congruence we have g˜tt = 1 (here the tilde indicates a rescaled object). The new spatial geometry that we get is the optical geometry. Relative to this geometry a photon moves a unit distance per unit coordinate time. Assuming the congruence to be shearless we show that a a geodesic photon will follow a spatially straight line, and a gyroscope following a straight spatial line will not precess relative to the direction of motion. Also the sideways force required to keep a test particle moving on a straight spatial line will be independent on the velocity. Using a novel measure of spatial curvature introduced in Paper III we show that also for the case of a shearing congruence, photons follow straight lines and the sideways force keeping a test particle moving along a straight line is independent on the velocity. In paper VIII I consider a non-shearing congruence radially falling towards a black hole, allowing us to consider the extended optical geometry across the horizon.
9 Conclusion and outlook Concerning my work with inertial forces, gyroscope precession and optical geometry – I have largely accomplished what I set out to do. At the moment, I see no particular issues here that I would like to further resolve. Concerning the spacetime visualization techniques, I am more or less content concerning the theoretical background. I am however a little curious to find out what can be done in the dual scheme considering time dependent line elements. Also I am bit curious about for instance how the spacetime of a radial line through a collapsing shell of matter would look like in the absolute scheme. Mainly however, my interest concerning the pedagogical techniques lies in the visualization as such. I have plenty of ideas of how one can go beyond the illustrations that I have shown in this thesis to further inspire and teach the physics community and the general community about the marvels of curved spacetime. As I mentioned in the introduction, I hope to use my ideas and methods to write a book about relativity for a general audience. As for my interest in physics I tend towards the foundations of physics. How can one unite general relativity with quantum mechanics? Maybe if we crack this nut we will get a grip on the measurement problem of quantum mechanics, and perhaps even consciousness itself. Or maybe it will be the other way around. In any case, I suspect that a revolution in the way we think about the universe will be necessary to accomplish this fusion. As soon as I have finished this thesis – I will get started on it. It shouldn’t take me very long to crack that nut:)
37
Comments on the Research Papers
10 A spherical interior dual metric This chapter is a comment on Paper I, concerning the shape of spacetime for a radial line inside a planet of constant proper density. It presumes knowledge of the notation of Paper I. We know that in Newtonian theory, a particle that is in free fall around the center of a spherical object of constant density, will oscillate with a frequency that is independent of the amplitude. This would fit well with an internal dual space that is spherical. We saw in Paper I the possibility to choose parameters β and k that produces substantial curvature also in the case of the weak gravity outside our Earth. It is natural to ask whether we really can find parameters such that the internal geometry becomes spherical. In the following sections we will see that this is not generally the case but one can choose parameters such that it is exactly spherical in the weak field limit.
10.1 Conditions for spheres For a sphere of radius R, we introduce definitions of r and z according to Fig. 10.1. Using the Pythagorean theorem it follows that: ! "
dr =− dz
R r
#2
(10.1)
−1
We let z = z(x), where (see Paper I, Eq. (23)): dz $ = c(x) − r !2 dx
here 41
r! =
dr dx
c = gxx
(10.2)
10. A spherical interior dual metric
42
R
r
z Figure 10.1: Definitions of variables for a sphere. The axis of rotation (around which time is directed) is horizontal. We readily find: R2 =
r2 1 − r ! 2 /c
(10.3)
r2 γ 2 − r ! 2 /c
(10.4)
Also if we have a rolled-up sphere (see Fig. 6 of Paper I) such that r = γrsphere and ! r ! = γrsphere this is modified to: R2 =
10.2 The dual interior metric For the Schwarzschild interior dual metric we may show that 1 : 2
r ! /c =
βk 2 x2 4x60 (a0 − β)
and
r 2 = αk 2
a0 a0 − β
(10.5)
Inserting this into Eq. (10.4) we readily find that for the embedding to correspond to a sphere of radius R and rolling parameter γ we must have, for all x < x0 : R2 γ 2 (a0 − β) − R2
βk 2 x2 − αk 2 a0 = 0 6 4x0
(10.6)
Notice that if we divide the expression by αk2 we will see that any change in R2 and γ can be canceled by a corresponding change in α and k respectively. For Eq. (10.6) to be true for all x it must be true for every power in x. We note that a0 has a lot of powers in x which would mean that the factors multiplying a0 would have to vanish. But then the x2 -term in the middle cannot be canceled by anything. Thus the expression cannot be true for all values of x and thus the interior dual metric is not exactly spherical. The first relation can most rapidly be obtained re-juggling Eq. (32) in Paper I (replace < with = and remove the “min”), look also at Eq. (23) for understanding. The second is the square of Eq. (21) in Paper I. 1
10.3. Approximative internal sphere
43
10.3 Approximative internal sphere We can however produce something that is very similar to a sphere. Set γ = 1, corresponding to a non-rolled sphere, and α = (1 − β)/k2 meaning unit radius at infinity. We can expand the two occurrences of a0 in Eq. (10.6) to second order in x2 . Demanding the equation to hold to zeroth, first2 and second order yields after some simplification (Mathematica does fine simplifications): a00 · (R2 − 1) β = R2 − a00
where
a00
% !
1 1 ≡ a0 (0) = 3 1− −1 4 x0
&2
(10.7)
7/4
2x0 k = $ √ √ 3 x0 − 1 − x0
(10.8)
Notice that Eq. (10.8), coming directly from the second order demand, is independent of β. The first relation is nothing but Eq. (21) in Paper I, where one has demanded r = R and α = (1 − β)/k 2 , taken at x = 0. Inserting β and k from Eq. (10.7) and Eq. (10.8) into a plotting program yields pictures with a very spherical appearance. The way it works is that the center of the bulge has the exact radius and curvature of a sphere, then the rest is not exactly spherical. In the Newtonian limit however, where x0 → ∞, we do get a perfect sphere. This is true whichever finite radius R we choose, as is explained below.
10.4 Spheres in the Newtonian limit From Eq. (10.3) we have the necessary relation for a sphere: r2 r!2 =1− 2 c R
(10.9)
Using the specific expressions for β and k given by Eq. (10.7) and Eq. (10.8) we may evaluate both the left and the right hand side of Eq. (10.9) to lowest non-zero order 3 in 1/x0 , to find that the equality holds exactly for all x to this order. Thus in this limit we get an exact sphere. In fact we knew in advance that the equality would hold. The reason is that, a0 expanded to second order in x for which the equality holds, is the same (or in fact a bit more exact) as a0 expanded to first order in 1/x0 4 . So, in the Newtonian limit the interior dual metric is isometric to a sphere for any value of β (so long as R remains finite). 2
It is always satisfied to first order Terms like x2 /x30 are of course treated as' 1/x0 -terms 4 If we instead would have had a0 = 1 − 1 − x3 /x40 which expanded in x to second order is zero, while not being zero expanded to first order in 1/x0 (remember that x3 /x40 is of order 1/x0 ), we could not have used this little trick. 3
44
10. A spherical interior dual metric
An embedding of the Earth spacetime is displayed in Fig. 10.2. Here I have chosen the radius of the central bulge to be twice the embedding radius at infinity. The parameters β and k are given by Eq. (10.7) and Eq. (10.8) with x0 = 7.19 · 108, suitable for the Earth. As can be seen the bulge is quite spherical.
Figure 10.2: An embedding of the dual spacetime of a central line through the Earth. The time per circumference is roughly 84 minutes. Notice the worldlines of the two freefallers on the interior sphere – one static in the center – the other oscillating around the Earth with a period time of roughly 84 minutes.
11 The absolute visualization In this chapter I take the opportunity to comment on some issues concerning the absolute approach to spacetime visualization (Paper II). I will consider only twodimensional applications. I start by giving some intuitive understanding for how changing generators affects the visualization. I then move on to find what the vacuum field equations look like in 2 dimensions assuming a flat absolute geometry (so the freedom lies in the twist of the generators). I will also consider some toy-models of theoretical interest, in particular illustrating spacetimes with timelike loops. I will also give an example of how one may choose generators such that outgoing photons (for instance) follow absolute geodesics for the line element of a radial line through a black hole.
11.1 The perch skin intuition In special relativity one can look at the active Lorentz transformation as a shift of the physical spacetime points along hyperbolas, as illustrated to the left of Fig 15 in Paper II. Alternatively we may view the Lorentz transformation as a two-step stretch and compress process along the light cone coordinates u and v. Stretching by a certain factor along u means compressing by the same factor along v (so areas are preserved) as depicted in Fig. 11.1. This process of stretching and compressing is directly related to the absolute visualization scheme. If we choose generators in the leftmost rhomboidal section of spacetime in Fig. 11.1, such that the generators are rotated a certain angle clockwise from the upwards direction – then the absolute geometry would be that of the surface to the right (think of the generators in the comoving coordinates). Hence changing the generator corresponds to a compression and a stretch along the local lightcone 45
11. The absolute visualization
46
Stretch
And this is related to my skin?
Compress
v
u
Figure 11.1: An active Lorentz transformation corresponds to a stretch and a compression. coordinates. This reminds me of how a perch skin is behaving. Pulling in the skins length-direction makes the skin automatically shrink in stripe direction 1 . We realize that for any embedded surface in the absolute visualization technique, we can locally stretch and compress it to give a different appearance to the same physical spacetime. In other words we can do any deformation that preserves local areas and keeps the null lines at right angles. As an application of this newfound intuition, we consider a deformation of a Lorentzian flat spacetime illustrated by a flat absolute metric with uniformly directed generators – to a new flat absolute metric where the generators are curving, see Fig. 11.2. Stretch
Press
Stretch
u
v
Press
Figure 11.2: A deformation of flat spacetime. All the lines are null except the thick curve which is a generating worldline. Any deformation that preserves the angles between null lines and preserves local areas will leave the Lorentzian spacetime unaffected. 1
As soon as my father starts hooking all those perches that he claims are biting his lure, I can start making more extensive tests of the properties of perch skin:)
11.2. Vacuum field equations for flat absolute metric
47
11.2 Vacuum field equations for flat absolute metric In the absolute visualization scheme we have have a direct visual means of finding the proper distances between nearby events. Maybe we could also build some intuition of how to find out whether there is a proper curvature or not. For this purpose, we study a simple case of flat absolute geometry where the Lorentzian geometry is determined by the direction of the generators, specified by an angle α as depicted in Fig. 11.3. t !
x
Figure 11.3: The direction of the generators (the local Minkowski systems) specified by an angle α. The angle α(t, x) goes clockwise from the t-axis on the plane to the local t-axis of the local Minkowski system. The absolute four-velocity of the generators has the form: uµ = (cos(α), sin(α)) (11.1) We find the Lorentzian metric through gµν = −δµν + 2uµ uν as:
2
−1 + 2 cos (α) g¯µν : 2 cos(α) sin(α)
2 cos(α) sin(α)
−1 + 2 sin2 (α)
(11.2)
We may calculate the Ricci tensor for this using grtensor in Mathematica, but the expressions become quite large. To simplify matters, we choose our t-axis to coincide with the local Minkowski time axis. Then the angle α is small in a region around this point and we may Taylor expand the metric (to second order in α):
1 − 2α2 g¯µν : 2α
2α −1 + 2α
2
(11.3)
Calculating Rµν = 0 and setting α = 0 with grtensor yields a single equation: (∂x α)2 − (∂t α)2 + ∂x ∂t α = 0
(11.4)
11. The absolute visualization
48
So here we have the vacuum field equations 2 , assuming a flat absolute geometry. Notice that Eq. (11.4) holds in a point, where the t-axis is chosen to coincide with the generator. Notice also that the two first terms are first order derivatives whereas the last is a second order derivative. At any point we can thus choose an arbitrary derivative in the local t and x-direction and still have a locally flat spacetime – so long as ∂x ∂t α is given by Eq. (11.4).
11.3 On geodesics and flat metrics As an application of the field equations above, consider the special case of a flat absolute metric in 2D, and geodesic generators (i.e straight lines). Notice that the generators do not necessarily have to be parallel, but could for instance extend from a point and outwards (like spokes on an old cartwheel), or something more complicated. Assuming the generators to correspond to straight lines, we have everywhere: ∇α · n ˆ=0
where
n ˆ = (cos(α), sin(α))
(11.5)
Here α is the angle describing the tilt of the local generator as depicted in Fig. 11.3. In Cartesian coordinates (t, x) where the t-axis is parallel to the generators at the point in question, we have α = 0 and thus Eq. (11.5) yields ∂t α = 0. Differentiating Eq. (11.5) with respect to x, and then setting α = 0 yields: (∂x α)2 + ∂x ∂t α = 0
(11.6)
Comparing with Eq. (11.4) we see that the vacuum field equations are satisfied. Thus for geodesic generators on the flat plane the corresponding Lorentz geometry is flat. For an arbitrary embedded spacetime in the absolute visualization scheme – consider turning the lightcone (the local generator) by 90◦ everywhere while letting the shape of the surface remain. This would exactly correspond to changing the sign of the Lorentz metric. Changing the sign of the metric does not affect the affine connection, and thus not the curvature either. It then follows that also if we have straight lines orthogonal to the generators for a flat absolute geometry – we have a flat Lorentzian geometry 3 . Indeed looking at the rightmost figure in Fig 15 of Paper II we have a good example of a flat absolute metric (embedded as a cone) with straight generators in some parts, and straight orthogonal lines in another part, that is Lorentzian flat.
2
In two dimensions the vacuum field equations imply flat Lorentzian spacetime. . We πcould alsoπ /show this directly assuming that we everywhere have ∇α · cos(α + 2 ), (sin(α + 2 ) = 0. It follows analogously to the above derivation (differentiate with respect to x this time) that also for this case the vacuum equations are automatically satisfied, hence spacetime is flat. 3
11.4. Closed dimensions and timelike loops
49
11.4 Closed dimensions and timelike loops The absolute illustration method is well suited for making spacetime toy models. Such illustrations can be valuable for thought experiments and qualitative reasoning. In this section we illustrate a few spacetimes that are closed in one or two dimensions. In particular, images as that depicted to the left in Fig. 11.5 are useful to illustrate how spacetime can be something much more complex than just space plus time.
Figure 11.4: To the left a spacetime that is closed in space, with the trajectories of two inertial observers. To the right a spacetime that is closed in time.
Figure 11.5: To the left a spacetime that is closed in space in some regions (top and bottom) and closed in time in the middle region. To the right a spacetime that is closed in space and in time.
11. The absolute visualization
50
An observer whose worldline winds as depicted to the left in Fig. 11.5 would as young see himself as old and vice versa. The thought that easily comes to mind is – ’What if he as young kills himself as old?’ or perhaps worse still ’What if he as old kills himself as young?’. Obviously this raises rather interesting questions concerning free will. I will however leave it to the reader to further consider this.
11.5 Warp drive In science fiction terms like warp drive have become standard as a means of transportation that is in some sense faster than light. One might ask to what extent such a transportation is possible within the framework of general relativity. Well, if we have some way of preparing spacetime ’from outside the spacetime’ this is certainly not a problem as depicted in Fig. 11.6.
Figure 11.6: A warp drive spacetime. As always – for any spacetime that we can come up with – there is an energymomentum tensor that via the field equations gives the right local curvature and will automatically obey the local conservation laws of energy and momentum (via the Bianchi identities). Assume now that in our lives everything operates as if we are living through a flat spacetime. Assume further that we have some means of freely manipulating local momentum and energy currents (and that energy may be negative etc). Then if we plan ahead we can send out emissaries that will tweak the energy momentum tensor at some preset time (according to their own clocks) and thus create from within the spacetime a warp bridge like that depicted to the right in Fig. 11.6 or like Fig 17 of
11.6. Finding generators to make a single trajectory an absolute geodesic
51
Paper II. Of course this method of transportaion is no faster than if you would join the rightmost emissary and just go there. It is different however. In the warp method you create an opportunity to travel fast should you so choose at a later time. Of course we have assumed quite much here, the least of which is the ability to make a global smooth tweak of the energy momentum tensor – which is not practically feasible. We could perhaps hope to make a series of discrete explosions or some such. However, in the classical theory I see no problems in principle of the above type of warp drive. Of course if we consider quantum mechanics – it is not at all clear what we can and cannot do for this case. Indeed we would need a theory of quantum gravity to exploit this fully. We could consider a corresponding emissary scenario also for the case of an initially flat spacetime with closed space – to at will create a spacetime as that depicted to the right of Fig. 11.5. Again I will leave it to the reader to consider whatever paradoxes concerning free will that might occur there.
11.6 Finding generators to make a single trajectory an absolute geodesic This section is essentially an appendix that for brevity was cut out from Paper II. The notation is therefore directly related to that of Paper II and references to equations in Paper II will be of the form ’Eq. II.xx’. In this section we assume both the original and the absolute metric to be 2-dimensional, time independent and diagonal. For a geodesic relative to the absolute metric we have (in the coordinates for which it is diagonal): %
dx! dt!
&2
=
g¯tt! (¯ σ ! g¯tt! − 1) ! g¯xx
(11.7)
Here σ ¯ ! is a constant that is fixed for every geodesic. Suppose now that we would want some motion in the original metric, to be a geodesic in the absolute metric. It will then turn out to be practical to express the original motion using the Killing velocity, denoted by w for the motion considered. The corresponding four-velocity, prior to diagonalization, is given analogous to Eq. II.14 as: !
gtt q =± 1 − w2 µ
%
1 −w ,√ gtt −gxx gtt
&
(11.8) !
as Using Eq. II.25, and Eq. II.26 we can transform q µ to q!µ . Then we can express dx dt! q !x which is then given as a function of w. Using this is in Eq. (11.7), together with q !t Eq. II.15 we find after simplification: %
&
1 + v2 w 2 (1 − v 2 )2 ! − σ ¯ g −1 =0 tt (1 + v 2 − 2vw)2 1 − v2
(11.9)
11. The absolute visualization
52
The generating velocity v has to obey this equation to make an original motion, characterized by w(x), a geodesic in the absolute metric. In general it would be tricky to solve this equation for v. We can factorize it, but we still get a fourth order equation in v. There are however cases where we can deal with it analytically.
11.6.1 Considering generators Setting v = w, thus demanding the generators to be absolute geodesics, Eq. (11.9) is reduced to: ¯ ! gtt v2 = 1 − σ
(11.10)
It is easy to show that this corresponds exactly to a geodesic in the original spacetime. Thus we draw the conclusion that the generators of the absolute metric will be geodesics in the absolute spacetime if and only if they are geodesics in the original spacetime. While we here arrived at this conclusion assuming a Killing symmetry among other things, it is a completely general result as is shown in Appendix II.E.
11.6.2 Considering photons Another case where Eq. (11.9) is trivialized is when we consider an outward-moving photon (w = −1). Then the solution is: v=
2 − gtt σ ¯! 2 + gtt σ ¯!
(11.11)
Here σ ¯ ! is an arbitrary constant. Assuming a Schwarzschild line element and choosing v = 0 at infinity yields σ ¯ ! = 2. For this particular example we may insert the v of Eq. (11.11), and the line element of Eq. II.10, into Eq. II.15 to find:
! g¯µν =
1 2
3
1 + (1 − 0
1 2 ) x
4
0 2 1+(1− x1 )2
(11.12)
We see that the absolute metric exists all the way into the singularity. We may however notice that the Killing velocity v, as defined by Eq. (11.11), becomes infinite at x = 1/2. That is however nothing to worry about. It just means that at this point the generators are exactly orthogonal to the Killing field. In Fig. 11.7 we see an embedding of the geometry described by Eq. (11.12). The thick lines with arrows correspond to photons moving outwards. The left one, being inside the horizon, is however guided by the geometry into the singularity 4 . 4
Strictly speaking the photon trajectories should be invisible after having spiraled once around the surface since the spacetime in this embedding is layered.
11.6. Finding generators to make a single trajectory an absolute geodesic
53
Figure 11.7: The absolute spacetime using generators uµ (x) such that outwardmoving photons follow geodesics. The radial parameter x lies in the interval [0.55, 4]. The horizon is located where there is a minimum in the embedding radius. This is necessary if we want the photon to remain on a certain spatial position and at the same time follow a geodesic on the rotational surface. Also it has to be a minimum or a photon could oscillate back and forth around the horizon. While outward-moving photons correspond to geodesics, it is apparent that inward-moving photons do not. To see this we consider an inward-moving photon at the horizon. The trajectory is there directed purely along the surface (no azimuthal variation). A geodesic tangent to the photon trajectory at this point will remain on a fix azimuthal angle, whereas the photon trajectory does not. Thus, without using any mathematics, we may understand that we cannot make both ingoing and outgoing photons correspond to geodesics, while keeping manifest time symmetry (and assuming that we want to embed the horizon). By relaxing the time independency one could hope not only to get inward- and outward-moving photons to follow geodesics, but all free particles. In Appendix II.D, II.F and II.G, we show that this is possible only in a very limited class of spacetimes.
54
11. The absolute visualization
12 Kinematical invariants In Papers III-VIII we employ a congruence of reference worldlines that threads the spacetime1 . The local behavior of the congruence can be described by the covariant derivative of the congruence four-velocity. This derivative can be split into different parts, known as the kinematical invariants of the congruence. They are tensors related to shear, expansion, rotation and acceleration. In this chapter we give some intuition regarding the meaning of these invariants and their relation to the covariant derivative of the congruence four-velocities.
12.1 The definitions of the kinematical invariants The kinematical invariants of a congruence of worldlines of four-velocity uµ are defined as (see e.g [7] p. 566): aµ = uα∇α uµ θ = ∇α u α 1 ρ 1 (P ν ∇ρ uµ + P ρ µ ∇ρ uν ) − θPµν σµν = 2 3 1 ρ (P ν ∇ρ uµ − P ρ µ ∇ρ uν ) ωµν = 2
(12.1) (12.2) (12.3) (12.4)
In order of appearance these objects are denoted the acceleration vector, the expansion scalar, the shear tensor and the rotation tensor. The projection operator P µ α is given by (using the (−, +, +, +) convention): P µ α = g µ α + uµ uα 1
(12.5)
In the more intuitive Papers IV and VI, we do not explicitly talk of reference worldlines threading spacetime, but effectively they are there anyway and this chapter is relevant also for these papers.
55
12. Kinematical invariants
56
Forming P µ α k α yields a vector corresponding to the part of kµ that is orthogonal to the congruence (as is obvious in inertial coordinates locally comoving with the congruence). We may also introduce what we denote as the expansion-shear tensor: 1 (∇ρ uµ P ρ ν + ∇ρ uν P ρ µ ) 2
θµν =
(12.6)
From the normalization of uµ it follows that uα ∇µ uα = 0. Also we know that the covariant derivative of the metric vanishes. Then we can write Eq. (12.4) and Eq. (12.6) in a slightly different but equivalent form: 1 ρ σ P µ P ν (∇σ uρ − ∇ρ uσ ) 2 1 ρ σ P µ P ν (∇σ uρ + ∇ρ uσ ) = 2
ωµν =
(12.7)
θµν
(12.8)
In this form the three-dimensional nature of the rotation tensor and the expansionshear tensor is more obvious. From the definitions it follows that: ∇ν uµ = ωµν + θµν − aµ uν
(12.9)
So here are the definitions of the kinematical invariants. Now let us see if we can understand the physical meaning of these objects. In particular we will focus on the rotation and the expansion-shear tensor.
12.2 The average rotation The rotation tensor is evidently (by its name) connected to rotation of the congruence points. The tensor is however well defined also when the congruence is shearing and deforming, in which case there is no rigid rotation. We may suspect that for this case the rotation tensor is related to an average rotation of the reference congruence points. With this in mind let us derive an expression for the average rotation, and see if the resulting expression is connected to the tensor ωµν . Consider then a collection of moving reference points in three-dimensional Euclidean geometry. Also assume that the velocity at the origin of the coordinates in question is zero momentarily. We start by deriving an expression for the average rotation around the z-axis, considering a circle of radius r in the z = 0 plane. The average rotation of the points along the circle is r−1 times the average velocity in the counter-clockwise direction. Letting u(x, y, z) be the velocity of the reference points, we may express the average rotation as a line integral over the circle: ωz =
1 1 r 2πr
8 ◦
u ∗ dx =
1 1 r 2πr
8 •
(∇ × u) · dS
(12.10)
12.2. The average rotation
57
In the last equality we have used Stokes theorem, turning the line integral over the circle into a surface integral over a circular flat disc with dS = zˆdS. In the limit as the radius of the circle goes to zero, ∇ × u can be considered as constant (to the order necessary) and we can move it out of the integral: ωz *
1 1 (∇ × u) · ˆz r 2πr
8
1 dS = (∇ × u) · zˆ 2
(12.11)
So in the limit as the radius of the circle goes to zero we have: 1 ωz = (∂x uy − ∂y ux ) 2
(12.12)
This is then the average rotation around the z-axis for the motion of the reference points in the z = 0 plane (in the limit where the circle over which we average goes to zero). We realize that also averaging over the the rotation considering non-zero z, Eq. (12.12) gives the local average rotation around the z-axis. Corresponding arguments gives us the average rotation around the other axes: 1 ω = (∂y uz − ∂z uy , ∂z ux − ∂x uz , ∂x uy − ∂y ux ) 2
(12.13)
So here we have the average rotation vector that we set out to find. For some further intuition see Fig. 12.1. y
x Figure 12.1: A shearing velocity field in two dimensions. The reference points along the x-axis are rotating counter-clockwise whereas the points along the y-axis have a clockwise rotation. The average of the corresponding angular velocities (counted positive in the counter-clockwise direction) is ωz , which for this case is zero.
12.2.1 A matrix formulation of rotation Forming ω × x we get the contribution coming from an (average) rotation to the velocity at a point x : ω × x = (ωy z − ωz y, ωz x − ωx z, ωx y − ωy x)
(12.14)
12. Kinematical invariants
58 Now form a matrix ωij : 1 ωij = (∂j ui − ∂i uj ) 2
(12.15)
In matrix form this becomes:
ωij :
0 0 ∂y ux − ∂x uy ∂z ux − ∂x uz −ωz ωy 1 = (12.16) 0 ∂ u − ∂ u 0 −ω ∂x uy − ∂y ux ωz z y y z x 2 ∂x uz − ∂z ux ∂y uz − ∂z uy 0 −ωy ωx 0
It follows that we have: ω ij xj = (ωy z − ωz y, ωz y − ωy z, ωx y − ωy x)
(12.17)
The right hand side of this equation is identical to the right hand side of Eq. (12.14). So we understand that ωij xj , where ω i j is defined by Eq. (12.15), gives the velocity of the congruence point at xi coming from an average rotation. It is not hard to realize that we can form the rotation tensor as defined by Eq. (12.15) from the rotation vector as 2 : ω ij = −*ijk ωk
(12.18)
ijk even permutation of 123 ijk odd permutation of 123 ijk some indices equal
(12.19)
Here *ijk is defined by:
ijk
*
=
+1 : −1 : 0 :
The inverse of Eq. (12.18) is simply 3 : 1 ω i = − *ijk ωjk 2
(12.20)
So now we have some understanding for how the average rotation is related to the derivatives of the velocity field. 1
Multiplying the right hand side of Eq. (12.18) by a factor (−Det(g ij ))− 2 (see [4] p. 98-99 concerning tensor densities) this is a covariant relation where ωij is minus the dual of ωi (see [7] p. 88). 3 We may guess this and then check it. Alternatively we could multiply Eq. (12.18) by #mij , using the fact that #mij #ijk = 2δm k 2
12.3. About deformation
59
12.3 About deformation We can split the velocity derivative tensor into a symmetric and an antisymmetric part. We have then to first order in xi : ui = xj ∂j ui " # 1 j 1 (∂j ui − ∂i uj ) + (∂j ui + ∂i uj ) = x 2 2 = xj (ωij + θij )
(12.21) (12.22) (12.23)
Here we have introduced θij as: 1 θij = (∂j ui + ∂i uj ) 2
(12.24)
Now we would like to show that this object is related to deformations (of any shape connected to the congruence points). It seems obvious that the most general local motion that is non-deforming corresponds to a rigid rotation (as regards the momentary velocities). For this kind of motion we must have (there exists an ω such that): u=ω×x
(12.25)
Obviously in this case the motion due to the average rotation is precisely the motion – and the motion due to the average rotation is uk = ω k j xj . From Eq. (12.23) it then follows that we must have xj θij = 0 for all xj . This can only happen if θij = 0. So it follows that rigid motion implies vanishing θij 4 . Conversely, given that θij = 0, it follows from Eq. (12.23) that ui = ω i j xi . Since ω i j is an antisymmetric tensor it follows from Eq. (12.14), Eq. (12.16) and Eq. (12.17) that this corresponds to a rigid rotation (with rotation vector given by Eq. (12.16)). So we conclude that the congruence is rigid if and only if θij = 0. A simpler argument would be to say that θi j xj gives the extra velocity apart from that coming from an average (best fit) rotation. Hence the congruence is rigid if and only if θij = 0.
12.4 Back to four-dimensional formalism Connecting the three-dimensional ωij and θij to their four-dimensional analogies ωµν and θµν is very simple. In freely falling coordinates locally comoving with the congruence, the spatial parts of ωµν and θµν precisely equals their three-dimensional analogies. Moreover, in these coordinates all time components of ωµν and θµν vanishes due Alternatively we could just evaluate ∂i uj + ∂j ui for ui = #i jk ωj xk . Here ωj is to be treated as a constant (to the necessary order). Using ∂i xj = δij and the antisymmetry of #ijk we readily find that θij = 0. 4
12. Kinematical invariants
60
to the projection. It follows that the congruence is rigid if and only if θµν vanishes, independent of what coordinates we are using. Also in other respects the meanings are analogous. In particular forming ωµ ν ∆xµ (for some vector ∆xµ orthogonal to uµ ) gives that part of the velocity of the congruence point at the position ∆xµ (relative to the inertial system whose origin momentarily comoves with the congruence) that can be seen as coming from an average rotation. Concerning the expansion scalar θ it is not hard to realize that this corresponds to the relative volume increase (of a box of reference points) per unit time. For instance θ = 0.1s−1 implies a momentary relative volume increase rate of 10% per second. Notice that we can have θ = 0 while for instance having expansion in the x-direction and contraction in the y-direction. Having understood the meaning of θµν and θ we understand that the shear tensor σµν , as defined by Eq. (12.3), describes that part of the local congruence motion that is neither due to an (average) rotation nor an (average) expansion. As regards the remaining kinematical invariant aµ , it is simply the acceleration of the congruence.
12.4.1 The four dimensional analogue to the rotation vector In the three-dimensional analysis we saw in Eq. (12.20) how we could relate ω i to ωij : 1 ω i = − *ijk ωjk 2
(12.26)
We would now like to have a corresponding four-covariant expression. In particular we would like a tensorial expression for a four vector corresponding to (0, ω) in freely falling coordinates locally comoving with the congruence. A natural extension of Eq. (12.26) is: ωµ =
1 1 √ uσ *σµγρ ωγρ 2 g
(12.27)
Here g = −Det(gαβ ) (see [4] p. 98-99 concerning tensor densities). Conversely we have in analogy with Eq. (12.18): 1 ω µν = √ uσ *σµνρ ωρ g
(12.28)
So now we have covariant four-tensor equations relating the rotation tensor to the vector describing the local average rotation.
13 Lie transport and Lie-differentiation To form a covariant derivative of a vector defined along a worldline we need a way of transporting the vector, according to some law of transport, from one point to another along the worldline so that we can form the difference between the actual vector and the transported vector (at a single point). This general idea of differentiation is relevant for papers III-VIII – concerning inertial forces and gyroscope precession. There are a few (standard) means of transporting a vector in general relativity. There is the ordinary parallel transport where, relative to freely falling coordinates, the components of the vector does not change as we move it along the worldlines. Then there is the Fermi-Walker transport, describing how the spin four-vector is transported along a worldline. Lastly there is the Lie transport. This transport is defined with respect to a reference congruence, assuming the worldline along which we transport the vector is directed along the congruence. In this chapter we give a little background to the concept of contravariant and covariant Lie differentiation.
13.1 Contravariant Lie differentiation Suppose that we have an arbitrary contravariant vector field uµ . We then choose coordinates such that uµ = δ µ 0 (assuming the vector field to be reasonably well behaved in the region in question). Note that this procedure has nothing to do with geometry or Lorentz structure. The idea is illustrated in Fig. 13.1. The induced time-slices (although strictly speaking what we are doing need not have anything to do with time) are uniquely defined by the procedure, given an arbitrary initial slice (and labeling of this slice). The labeling of the congruence lines (the streamlines of the vector field) is completely arbitrary. In these coordinates we now want to transport an arbitrary vector Kµ along a congruence line such that the coordinate derivative of the vector vanishes, see Fig. 13.2. 61
13. Lie transport and Lie-differentiation
62
Change coordinates
Figure 13.1: Adapting coordinates to a (contravariant) vector field
B
A
Figure 13.2: Transporting a vector in the preferred coordinates We understand that the transport of this vector is uniquely determined by the procedure above (how we label the congruence lines does not matter)1 . Consider now K µ to be a field that is transported into itself as outlined above. We have in the coordinates in question (for the moment we will assume the existence of an affine connection): ∂α uµ = 0 ∇α uµ = Γµαβ uβ
(13.1) (13.2)
0 = uα ∂α K µ = uα ∇α K µ − uα Γµαβ K β
(13.3) (13.4)
Here ∇µ is the covariant derivative. Also we have in the coordinates in question:
= uα ∇α K µ − (∇β uµ )K β
(13.5)
So we have our transport equation in covariant form: 1
u α ∇α K µ = K α ∇α u µ
(13.6)
Also we may understand that what initial arbitrary time slice we are considering does not matter either. If we choose a different slicing and draw these slices in the original coordinates as depicted in Fig. 13.2, they will be tilted – but the tilt must be the same everywhere along a single congruence line. We may thus understand that the time component of the vector in question will be affected, but the transport will not be affected.
13.1. Contravariant Lie differentiation
63
Actually we do not need to assume an affine connection. In the particular coordinates in question we have: α
u ∂α K ∂α u
µ
µ
= 0 = 0
⇒
uα ∂α K µ = K α ∂α uµ
(13.7)
The latter equation is through its form such that it holds in all coordinates given that it holds in one system (do the transformation to another system and we find that the extra, non-tensorial, terms cancel). Assuming then that we have a metrical structure and a covariant derivative, equation Eq. (13.6) follows immediately: Also for a bit of extra intuition, assuming that we have a metrical structure, have a look at Fig. 13.3. Kµ
uµ uµ
K
µ
Figure 13.3: Transporting a vector K µ as seen from freely falling coordinates. It is obvious that the change of K µ should depend on the change of uµ in the direction of K µ . Our first guess from this point of view would likely be: DK µ = K α ∇α u µ (13.8) Dτ This is in fact exactly Eq. (13.6), so the transport equation is very intuitively reasonable. Also, given an arbitrary vector field K µ , we can define the Lie derivative of this field as: Lu K µ = uα ∇α K µ − K α ∇α uµ (13.9) We can also express this in terms of the kinematical invariants of the congruence. We have: (13.10) ∇α uµ = ω µ α + θµ α − aµ uα In particular, for spatial K µ , we have then:
Lu K µ = uα ∇α K µ − (ω µ α + θµ α )K α
(13.11)
13. Lie transport and Lie-differentiation
64
13.2 Covariant Lie differentiation We may perform a similar analysis considering a covariant vector field Kµ . We analogously demand that the components of this vector should be unaltered as we move it along the congruence in the preferred coordinates. Analogously to Eq. (13.7) we have
uα ∂α Kµ = 0
∂α u
µ
= 0
→
uα ∂α Kµ = −Kα ∂µ uα
(13.12)
We note that we have inserted a minus sign here as compared to Eq. (13.7). Whereas the equality holds equally well independently of what sign we choose in the preferred coordinates, the sign is chosen such that the equality holds in any coordinate system. To prove this we evaluate u! α ∂x∂! α Kµ! +Kα! ∂x∂! µ u! α , using the definitions of how the covariant and contravariant objects are related to their non-primed versions. We also use the trick of differentiating a Kronecker delta in the form: ∂ 0 = !µ ∂x
%
∂x! α ∂xρ ∂xσ ∂x!α
&
(13.13)
Expanding this derivative and using the resulting expression, we readily find that uα ∂x∂α K µ + K α ∂x∂α uµ is in fact a tensor (although its individual terms are not). Hence if it vanishes in one system it vanishes in all systems. It follows that a covariant vector field Lie-dragged into itself obeys: uα ∂α Kµ = −Kα ∂µ uα
(13.14)
Note that it is not only the sign that differs from the analogous equation Eq. (13.7) of the preceding section. How the summation runs is also different. For this case the form is less intuitive. Lie differentiation of a covariant field is then given by: Lu Kµ = uα ∇α Kµ + Kα ∇µ uα
(13.15)
Note that here we are using covariant differentiation (to make the derivative manifestly covariant), but the form is such that we could instead use ordinary differentiation. Notice in particular that if we have the Lie derivative of a covariant field we do not get the Lie derivative of the corresponding contravariant field by raising the former with the metric: g µα Lu Kα ,= Lu K µ (13.16) In other words Lie-differentiation does not in general commute with the raising and lowering of indices 2 . If the vector field obeys ∇α uµ = 0, Lie differentiation does however commute with lowering and raising of indices. 2
13.3. Comments
65
13.3 Comments One might consider other transports of the vector in question, not just along the congruence, as depicted in Fig. 13.4
B
A
Figure 13.4: Trying a different transport equation with respect to a reference congruence. In a particular set of coordinates we may here demand that the components of the vector should be unaltered as we transport it. For this case it is however obvious that the labeling of the congruence lines will affect the transport. To gain intuition concerning the difference between the raised covariant transport and the ordinary contravariant transport, let us study a particular example in 2+1-dimensional special relativity. Let Kµ = ∂µ φ where φ = x in inertial coordinates (t, x, y). Now consider another coordinate system that is shearing relative to the inertial coordinates such that x! = x as depicted in Fig. 13.5.
x!µ t=T t
y x
t=0
xµ y x
Figure 13.5: To the left the shearing coordinates (thin lines) relative to the inertial coordinates (thick lines) at a time t = T . To the right the shearing coordinates in a 2+1 spacetime illustration seen relative to the inertial coordinates.
66
13. Lie transport and Lie-differentiation
We have in the inertial coordinates ∂µ φ = (0, 1, 0). Also in the shearing coordinates we have ∂µ! φ = (0, 1, 0). Now consider Lie transporting (with respect to the shearing coordinates) a contravariant vector directed in the x-direction at t = 0 along the congruence. At t = T it will be directed in the x!µ direction (see Fig. 13.5). On the other hand, if we consider a covariant transport of the same vector then in the shearing coordinates (0,1,0) will go to (0,1,0) (per definition), but the latter vector – when raised with the metric – we know has the meaning of the gradient of φ which is directed in the xµ direction. Thus relative to the shearing coordinates the raised covariantly transported vector will rotate. So here we have an intuitive example illustrating the difference between the contravariant Lie transport and the raised covariant Lie transport.
B IBLIOGRAPHY [1] Epstein, L. C. (1994). Relativity Visualized, (Insight Press, San Francisco), ch. 10,11,12 [2] Marolf, D. (1999), Gen. Rel. Grav. 31, 919 [3] W. Rindler (1994), Am. J. Phys 62, 887-893. [4] Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (John Wiley & Sons, U.S.A) [5] Kristiansson, S., Sonego, S., Abramowicz, M.A. (1998). Gen. Rel. Grav. 30, 275 [6] D’Inverno, R. (1998). Introducing Einstein’s Relativity, (Oxford University Press, Oxford), p. 99-101 [7] Misner C.W., Thorne K. S., Wheeler J. A. (1973). Gravitation, (W.H Freeman and Company, New York), 841 [8] Rindler, W. (2001). Relativity: Special, General and Cosmological, (Oxford University Press, Oxford), p. 267-272 [9] Rindler, W. (1977). Essential Relativity: (Springer Verlag, New York), 204-207
Special, General and Cosmological,
[10] Hawking, S.W., Ellis, G.F.R. (1973). The large scale structure of space-time, (Cambridge University Press, Cambridge), 39 [11] Dray, T. (1989) , Am.J.Phys. 58, 822-825 [12] Bini D, Carini P, Jantzen RT, Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, Tsvi Piran, Editor, World Scientific, Singapore, A (1998) p. 376-397 [13] Bini D, Carini P, Jantzen RT, Int. Journ. Mod. Phys. D, 6, No 1 (1997) p. 14 [14] Jantzen RT, Carini P, Bini D, Ann. Phys. 215, No 1 (1992) p. 1-50 67
68
BIBLIOGRAPHY
[15] Marek A. Abramowicz, Jean-Pierre Lasota, Class. Quantum Grav., 14 (1997) p. A23-A30 [16] Marek A. Abramowicz, Scientific American (march) 1993 [17] Marek A Abramowicz, Pawel Nurowski and Norbert Wex, Class. Quantum Grav., 12 (1995) p. 1467-1472 [18] Marek A. Abramowicz, Mon. Not. R. astr. Soc., 256 (1992) p. 710-718 [19] Marek A. Abramowicz, Pawel Nurowski and Norbert Wex, Class. Quantum Grav., 10 (1995) p. L183-L186 [20] Perlick V., Class. Quantum Grav., 7 (1990) p. 1319-1331 [21] George E. A. Matsas, Phys. Rev. D., (2003) 68 [22] Rindler W.,Perlick V., (1990). Gen. Rel. Grav. 22 (9), p 1067-1081 [23] Muller R. A., Am. Journ. Phys. (1992) 60 (4), 313 [24] Gravity Probe B Mission web page, http://einstein.stanford.edu/
Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature Rickard Jonsson1 Submitted: 2000-11-06, Published: July 2001 Journal Reference: Gen. Rel. Grav. 33 1207 Abstract. Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent, or dual, metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles move on the shortest path. Thus one can visualize how acceleration in a gravitational field is explained by particles moving freely in a curved spacetime. Freedom in the dual metric allows us to display, with substantial curvature, even the weak gravity of our Earth. This may provide a nice pedagogical tool for elementary lectures on general relativity. I also study extensions of the dual metric scheme to higher dimensions. KEY WORDS: Embedding spacetime, dual metric, geodesics, signature change
1
Introduction
It is easy to display the meaning of curved space. For instance we may display the spatial curvature created by a star using an embedding diagram. Figs. 1 & 2.
Figure 1: A symmetry plane through a star. 1 Department
of Astronomy and Astrophysics, Chalmers University of Technology, S-412 96 G¨ oteborg, Sweden. E-mail:
[email protected]. Tel +46317723179
1
Figure 2: The embedding diagram.
When it comes to displaying curved spacetime things become more difficult. The acceleration of a free test particle is due to a curvature of spacetime. What does it mean to have curved time, and can we display it somehow? When we create the embedding diagram for the space of a symmetry plane through a star, we make a mapping from our spatial plane onto a curved surface embedded in Euclidean space. This is done so that distances as measured by rulers are the same on the symmetry plane as on the embedding diagram. The difference is that on the symmetry plane there is a metric function giving the true distance between points whereas in the embedding diagram it is the shape of the surface that gives us the distances. If we want to do the same thing for a 1+1 spacetime we immediately run into trouble. We have null distances between points and even negative squared distances. In the Euclidean space, that we are used to embed in, we can never have negative distances.2 Instead of distances, maybe we should study motion. For geodesic lines of freefallers in the ordinary spacetime picture of a Schwarzschild black hole there is nothing special, or singular, with null geodesics for instance. Would it be possible to find a mapping from our coordinate plane to a curved surface such that all the worldlines corresponding to test particles in free fall, are geodesics (moving on the shortest distance) on the curved surface? See Fig. 3. Alternatively, if we can not do it for all particles, can we do it for some One to one mapping
Figure 3: A mapping to a surface where all freefallers take the shortest path. 2 One
can however embed in a Minkowski space, see Sec 11.
2
set of geodesics, like photon-geodesics? Suppose that we have found a mapping onto a curved surface such that all freefallers move between fixed points like a tightened thread, i.e. on the shortest path. Between nearby points on this surface there is then a Euclidean distance as measured with a ruler. This means that we can assign a Euclidean distance for small displacements in our coordinates. Thus, on our original coordinate plane we imagine there to live, not only a Lorentzian3 metric, but also a Riemannian4 metric, that both produce the same geodesics. With this understanding, the problem of finding a surface with the right curvature is reduced to finding a Riemannian metric that produces the right set of geodesics x(t). Once we have found such a geodesically dual metric, if it exists, we can hopefully embed and visualize, a curved spacetime. For some special geometries we know that the dual metric exists. In particular, starting from a flat Minkowski spacetime and using standard coordinates, the geodesic lines on the coordinate plane are just straight lines. This is also the case if we have an ordinary Euclidean metric. In this case we can thus just flip the sign of the spatial part of the Minkowski metric to find a dual metric. The embedding of a two-dimensional Minkowski spacetime will thus simply be a plane.5 Also, in a general Lorentzian spacetime, we can at every point choose coordinates so that the metric reduces to Minkowski, with vanishing derivatives. In this local, freely falling, coordinate system particles move on straight lines. This means that they move on the longest path in the local Lorentzian spacetime. However they also move on the shortest path in the corresponding local Euclidean spacetime. Then we know that there exists a dual metric that produces the right equations of motion at least in every single point. The question is whether we can connect all these single point metrics in a smooth way. Notice that constant time lines for inertial observers in Minkowski, are straight lines. They are also geodesics, moving the longest path, if we change the sign of the whole metric so that spacelike distances becomes positive.6 This means that if we find a dual metric, there will be geodesics that correspond to local time lines of freefallers. We may also consider these lines to be particles moving faster than light, so called tachyons. On the curved surface there is in principle no way to distinguish between 3 Metrics
with both negative and positive distances will be referred to as Lorentzian or simply (+, −). 4 Metrics with only positive distances will be referred to as Riemannian or simply (+, +). 5 Or any embedding that is isometric to a plane, for instance a cylinder. 6 In this article we use the convention that squared timelike distances are positive. Alternative to flipping the sign of the entire metric we can say that the imaginary distance traveled is maximized.
3
timelike and spacelike displacements from the shape of the surface. However, on the surface lives the original Lorentzian metric that tells us the true distances between nearby points. In other words there are small Minkowski systems living on the surface, telling us the proper distance between points. In a sense all we are doing is shaping the manifold. We still need the ordinary metric to get distances right. The difference is that we do not need this function to get the geodesics right. This follows from the shape of the surface.
2
The dual metric in 1 + 1 dimensions
Let us for simplicity, start the analysis with a 1+1 time independent diagonal metric, with Lorentzian signature, and see if we can find a time independent and diagonal dual metric with Euclidean signature.
2.1
Equations of motion
Assume that we have a line element: dτ 2 = a(x) · dt2 + c(x) · dx2
(1)
Using the squared Lagrangian formalism (see e.g. [5]), we immediately get the integrated equations of motion: a
!
dt dτ
"2
+c
!
dx dτ a
"2
= 1
dt = K dτ
(2) (3)
Here K is a constant for every geodesic.7 Now we want to find an equation for x(t). Introducing σ = 1/K 2 for compactness, the result is: !
dx dt
"2
=
a (σ a − 1) c
(4)
Notice that this equation applies to both (+, −) and (+, +) metrics.
7 Notice
that for spacelike geodesics, i.e. tachyons or lines of constant time, we have an imaginary K for the ordinary Schwarzschild metric.
4
2.2
The dual metric equations
The question is now: What other metrics, if any, can produce the same set of geodesics x(t)? Denoting the original metrical components by a0 and c0 and the original constant of the motion for a certain geodesic by σ0 we must have: a0 a (5) (σa − 1) = (σ0 a0 − 1) c c0 This relation must be fulfilled for every x and every geodesic. Notice that σ may depend on σ0 only. We see immediately that we can regain the old metric simply by setting a = a0 , c = c0 and σ(σ0 ) = σ0 . As for other solutions they may appear hard to find at first. We know however that starting from e.g. Minkowski, we must be able to flip the sign on the spatial part of the metric, without affecting the geodesic lines. Let us however rewrite Eq. (5) a bit: ! " ! " c a20 1 c a0 σ= · σ0 + (6) − c0 a 2 a c0 a 2 Now we see more clearly that if this relation is to hold for all x and all σ0 we must have: k1 =
c a20 c0 a 2
1 c a0 − a c0 a 2
k2 =
(7)
We have thus a linear relation between the constants of the motion: σ = k1 · σ0 + k2
(8)
Here k1 and k2 are constants that depend on neither x nor σ0 . From Eq. (7) we may solve for a and c in terms of k1 and k2 : a=
a0 a0 k2 + k1
c=
c0 k1 (a0 k2 + k1 )2
(9)
Defining α = 1/k2 and β = −k1 /k2 this may be rewritten as: a = α c = α
a0 a0 − β
−c0 β
(a0 − β)
(10)
2
(11)
We see that α is a pure scaling constant whereas β is connected to compression along the x-axis as will be discussed later. 5
Now the question is: can we choose α and β so that, assuming a Lorentzian original metric (positive a0 and negative c0 ) we get a Riemannian dual metric? Indeed necessary and sufficient conditions for the dual metric to be positive definite is:8
2.3
0<α
(12)
0 < β < a0
(13)
The Schwarzschild exterior metric
r Introducing dimensionless and rescaled coordinates x = 2MG and correspondingly rescaling Schwarzschild and proper time, the ordinary Schwarzschild metric is given by: " ! "−1 ! 1 1 c0 = − 1 − (14) a0 = 1 − x x
At spatial infinity this reduces to a0 = 1 and c0 = −1. At infinity our new metric is thus reduced to: a∞ = α
1 1−β
c∞ = α
β (1 − β)2
(15)
Let us study the quotient of a∞ and c∞ : 1 a∞ = −1 c∞ β
(16)
We see that for β > 1/2 the quotient is smaller than 1. This implies a stretching in x. When we embed our new metric this will correspond to opening up the photon lines so they become more parallel to the constant time line. In particular, demanding that a∞ = 1 and c∞ = 1 yields α = 1/2 and β = 1/2. Using this particular gauge, from now on denoted the standard gauge, we get from Eq. (10) and Eq. (11) the dual line element: ds2 =
x−1 x3 · dt2 + · dx2 x−2 (x − 1)(x − 2)2
(17)
This metric has positive metrical components from infinity and in to x = 2. We have thus succeeded, and found a Riemannian dual metric, at least on a large section of the spacetime. 8 It
might appear that we would get extra restrictions on these constants from demanding that σ in Eq. (8) must be positive. This constraint turns out to be identical to the constraint of Eq. (13) however.
6
2.4
On the interpretation of α and β
We may rewrite Eq. (10) and Eq. (11) as: a a0 /β = α a0 /β − 1
c −c0 /β = 2 α (a0 /β − 1)
(18)
Then we see that just as α is a rescaling of the new metric – so is β a rescaling of the original metric. We can easily work out the inverse of the relations above to find: c0 −c/α = 2 β (a/α − 1)
a/α a0 = β a/α − 1
(19)
We see that we have a perfect symmetry in going from the original metric to the dual and vice versa, justifying the duality notion. This symmetry would be even more obvious if we would denote β by α0 instead. With hindsight we realize that we must have two rescaling freedoms just like that. A metric that is dual to some original metric must also be dual to a twice as big original metric and vice versa. Notice however that if we make the original space twice as big, then the dual space does not automatically become twice as big. It does however if we double both α and β!
3
The embedding equations
In general if we have a metric with a symmetry, we can embed it as a rotational surface if we can embed it at all. Our task is then to find a radius r(x) and a height z(x) for the embedding of the dual metric. See Fig. 4.
r z
Figure 4: A rotational surface.
7
3.1
Finding r(x)
For pure t-displacements the dual distance traveled is ds = that we must have: √ r=k a
√ a·dt. We realize (20)
Here k is a constant of the embedding only, it does not affect the way we measure distances on the surface – only its shape. See Section 3.4. In terms of the original metric: # a0 r=k· α (21) a0 − β In particular for the Schwarzschild case using the standard gauge (α = 1/2, β = 1/2), and k = 1: # x−1 (22) r(x) = x−2 We see that as x tends to infinity we have a unit radius, whereas approaching x = 2 from infinity the radius blows up. Already now we may understand the qualitative behavior of the embedding diagram. The curious reader may jump immediately to Section 4.
3.2
Finding z(x)
From Fig. 5 using the Pythagorean theorem we find: $ ! "2 dr(x) dz = dx · c(x) − dx
dl =
√
dr · dx dx
c · dx
dz Figure 5: The relation between dz, dr and dl.
8
(23)
Using Eq. (11) and Eq. (21) and defining a#0 = da0 /dx we readily get: $ k 2 αβ 2 a#0 2 −c0 β dz = dx · α − · (a0 − β)2 4 a0 (a0 − β)3
(24)
So: √ % ∆z = α β
3.3
&
$
dx
k2 β a# 20 −c0 − · (a0 − β)2 4 a0 (a0 − β)3
(25)
Embedding criterions
We see from Eq. (25) that there is a limit as to how big the embedding constant k can be lest we get something negative within the root: ' ( 4 a0 (a0 − β) 2 k < min −c0 (26) β (a#0 )2 Let us investigate what this restriction amounts to for the specific cases of the Schwarzschild exterior and interior metric. 3.3.1
The exterior metric
For the exterior Schwarzschild metric, the expression within the brackets of Eq. (26) is smaller the closer to the gravitational source that we are, approaching 0 before we reach the horizon. On the other hand it goes to infinity as x goes to infinity, since a#0 goes to zero here. This means that, for any given k, the embedding works, as we approach infinity. It however only works from a certain point in x and onwards. For the exterior Schwarzschild we have: "−1 ! 1 1 1 (27) a#0 = 2 − c0 = 1 − a0 = 1 − x x x After some minor juggling we then find: # 2x2 1 k< √ 1−β− x β
(28)
Assuming that we are using the standard gauge, β = 1/2, and k = 1 this reduces to a restriction in x: x3 (x − 2) >
1 4
⇒ 9
x > 2, 02988...
(29)
So, using the standard gauge, the dual embedding only exists from roughly two Schwarzschild radii and on towards infinity. Incidentally may insert this innermost x into Eq. (22) and find r ∼ 5.87. Notice that these numbers only apply to the particular boundary condition where we have Pythagoras and unit embedding radius at infinity. By choosing other gauge constants and embedding constants we can embed the spacetime as close to the horizon as we want. Notice that there is nothing physical with the limits of the dual metric and the embedding limit. They are merely unfortunate artifacts of the theory. 3.3.2
The interior metric of a star
Assuming a static, spherically symmetric star consisting of a perfect fluid of constant proper density, we have the standard Schwarzschild interior metric: $ ) # *2 x2 1 1 3 1− − 1− 3 (30) a0 = 4 x0 x0 ! "−1 x2 c0 = − 1 − 3 (31) x0 Here x0 is the x-value at the surface of the star. For the interior star the embedding criterion, Eq. (26), becomes: ' ( 1 4x60 2 min · (a0 − β) (32) k < β x2 The bracketed function can be either monotonically decreasing, have a local minima within the star or even be monotonically increasing depending on x0 and β. Apart from the embedding restriction we have of course the restriction on the metric itself. Since a0 is monotonically increasing, for interior plus exterior metric, the metrical restriction for the entire star becomes: β < a0 (0)
(33)
Using β = 1/2 and the interior a0 in the center of the star this restriction becomes a restriction in x0 : x0 >
9 √ % 2, 838 9 − ( 2 + 1)2
(34)
For any x0 > 2, 837 and β = 1/2 the right hand side of Eq. (32) is always considerably larger than 1 and thus the embedding imposes no extra constraints 10
on x0 in the gauge in question, assuming k = 1. Incidentally we see from Eq. (21) that the radius of the central bulge goes to infinity as x0 approaches its minimal value.
3.4
On the interpretation of the embedding constant k
Recall Eq. (20): r(x) = k
% a(x)
(35)
We see that k determines the scale for the embedding radii. Notice however that the distance to walk √ between two radial circles, infinitesimally displaced, is determined by dl = cdx. So, when we double k we double the radius of all the circles that make up the rotational body while keeping the distance to walk between the circles unchanged. This means that we increase the slope of the surface everywhere. The more slope the bigger the increase of the slope. If we increase k too much the embedding will fail. An example of how different k affects a certain embedding is given in Fig. 6. Now consider Eq. (21): # √ a0 (36) r(x) = k α a0 − β We notice that, while increasing k we can decrease α in such a way that we do not change any r(x). The net effect on the embedding is then to compress the surface in the z-direction, while keeping all radii. This is done in such a way that all distances (dl) on the surface in the z-direction is reduced by the same factor everywhere. This means that where the slope is big we compress a lot in the
Roll tighter
k=1
k<1
Figure 6: Two isometric embeddings of a sliced-open sphere.
11
z-direction. Also since we are rescaling the dual metric, meter- and secondlines on the surface will move closer. Using this scheme we can produce substantial curvature out of something that was originally almost cylindrical. Also, using our β-freedom, we can flip down the photon lines towards the time line to better suit what we humans experience. This way we have a chance of displaying, with reasonable distances and curvatures, why things accelerate at the surface of the Earth. See Section 7.
4
The embedding diagram
Already from Eq. (22) we realize quantitatively how the new dual (+, +) spacetime must look like. See Fig. 7. Notice that time is the azimuthal angle and the whole spacetime is layered (infinitely thin), like a toilet roll. The geometry will approach a cylinder as we go towards spatial infinity. This is as it should be since we want a flat9 spacetime where there is no gravity. Notice that in an ordinary embedding of an equatorial plane of a black hole, the geometry opens up towards infinity and the little hole is at the horizon. Here it opens up towards the horizon and the little hole is towards infinity. So, we have found a dual (+, +) spacetime of Schwarzschild, that can be embedded, where all particles move on geodesics, i.e. shortest distance. We Towards spatial infinity
Observer at rest Out Freely falling observer moves out and back
In
Time
Figure 7: A vision of spacetime. 9A
cylinder is an intrinsically flat geometry.
12
can thus make a real model, say in polished metal, and then find possible geodesics just by tightening a thin thread between pairs of points on the surface. Alternative to tightening threads, one could put a little toy car or motorcycle, on the surface. Starting the car at some point, directed solely in the azimuthal direction, and pushing the car straight forward will result in a spiral inwards. Thus we see how moving straight forward can result in acceleration. Also, if we want the car to stay at a fixed x we notice that we must turn the wheel (assuming an advanced toy car), so that the car is constantly turning left (e.g.), i.e. accelerating upwards. This illustrates in an excellent manner how it is possible for us Earthlings to always accelerate upwards without ever going anywhere. Now that we have understood the name of the game in this embedding scheme we can figure out qualitatively how the embedded spacetime of a line through a star must look like. This is depicted in Fig. 8. For better layout, and also to more naturally connect the embedding diagram to the ordinary Schwarzschild diagram, we have now space in the left-right direction. A particle oscillating around the center of the star is nothing but a thread winding around the central bulge. Notice however that we do not generally expect to have something close to a sphere for the interior embedding. For a non-compact star we would rather expect10 something close to a cylinder, with a long slightly bulged interior star. Also, if we would have a perfect sphere for the the interior, then oscillations around the center of the star would correspond to great circles. This would mean that the period of revolution, as measured in Schwarzschild time, would be independent of the amplitude of the oscillation. This is actually true, for a constant density star, in Newtonian theory. In the full theory, and for more general density Observer oscillating around the center of the star
Constant time line Time
−∞ Observer at rest at the center of the star
Time
+∞
Space
Figure 8: The spacetime of a central line through a star. 10 This
is actually not obvious however – and not even always the case as we will understand later.
13
Flying saucer shaped geometry
Geometry seen from the side with winding geodesics
Small amplitude
Large amplitude
Figure 9: How the shape of the bulge, due to the density distribution, affects the the amplitude dependence of the periods of revolution for freefallers around the center of the star.
distributions, we will not expect a perfect sphere however. Also, more embeddings than the sphere has the focusing feature that makes the period of revolution independent of the amplitude. How a certain density can affect the shape of the interior bulge is depicted in Fig. 9. In this geometry it is obvious that increasing amplitude means increasing period of revolution. This is exactly what may be expected from Newtonian theory if the density is increasing towards the center. We may also consider the opposite situation with decreasing density in the center of the star. Then the central parts of the embedding will be close to cylindrical, and it is easy to imagine that increasing amplitude means decreasing period of revolution. Notice however that if we want to find the exact dependence of x(t), from the embedding diagram, we need also to know how the x’s are distributed on the surface. It is fascinating that we can visualize how density creates spacetime curvature, which in turn affects the geodesics of particles
4.1
Numerically calculated diagrams
Numerically it is no problem to integrate Eq. (25). For the exterior metric a particular result is depicted in Fig. 10. We may as easily get the embedding for the full star. One result of this, where we have omitted the coordinate lines, is depicted in Fig. 11. One may reflect that the embedding is not as bulgy as expected, and still I have chosen a compactness for which it is about as bulgy as it gets for a standard embedding.
14
Figure 10: A standard embedding of the exterior spacetime of a star for 2.03 < x < 3.00. The spatial lines are equidistant in x with a spacing that is one fourth of the spacing between the time lines (for esthetical reasons).
The main reason for the flatness is that as one moves towards the center of the star and the embedding radius is increasing, photon geodesics (and other geodesics) will be tilted further towards the constant time line. This is a direct effect of photons moving the shortest distance on the rotational surface (see Section 6). Also as the radius increases, moving in Schwarzschild time, means moving a longer distance on the surface, remember that the Schwarzschild time is proportional to the azimuthal angle. These two radial effects means that dz/dt, for photon geodesics, increases with increasing r. We therefore understand that we must stretch the bulge in the z-direction to insure that photons do not pass the star too quickly.
Figure 11: A standard embedding of the spacetime of a star with x0 ∼ 3.2.
15
If we want a star with more shape we can increase the k-value. This increases the tilt of the surface everywhere, making the embedding bulgier while photon lines at infinity remains at 45o .
5
The weak field limit
The dual metric and the embedding formulae are rather mathematically complicated, especially for the interior star. To gain some intuition it will prove worthwhile to study the weak field limit, where we can Taylor expand our expressions. In this section we will not use rescaled coordinates, x, but the ordinary radius, denoted by ρ so as not to confuse it with the embedding radius r. Let us use the standard gauge α = 12 , β = 21 and also k = 1 for simplicity. We define: a0 = 1 − ε(ρ)
(37)
Assuming ε(ρ) to be small we may Taylor expand the expression for the embedding radius: # √ ε a0 r = k α ⇒ r %1+ (38) a0 − β 2 Introducing r = 1 + h, we have thus to lowest order h = ε/2. One may show [3] that for a stationary, weak field in general we have: a0 = (1 + 2 · φ)
(39)
Here φ is the dimensionless (using clight = 1) Newtonian potential per unit mass, e.g. GM/ρ for a point mass. We thus conclude that: h = −φ
(40)
So in first order theory, using the standard gauge and k = 1, the height of the perturbation equals exactly minus the Newtonian potential.11 This result is actually not to surprising. See section 6. Incidentally we may also show that: a0 = 1 − ε(ρ) c0 = −(1 + δ(ρ))
⇒
z%
11 If
+
dρ 1 +
δ(ρ) ε(ρ) + 2 2
(41)
we are starting from the mass rescaled metric – the height of the perturbation at any x ∼ z will be the dimensionless Newtonian potential per unit mass divided by the mass of the gravitating system.
16
5.1
Applications
We may use our newly found intuition from Newtonian theory to create a new interesting picture. Suppose that we have a static spherical shell of some mass. Inside the shell we have no forces and thus φ is constant. According to the derivation above we would then have constant h in the interior of the star. See Fig. 12.
Figure 12: The rolling-pin spacetime of a central line through a Newtonian shell of matter. We see that inside the shell the geometry is flat, consistent with having no gravitational forces. I will leave to the reader to figure out what strange spherical mass distribution that could give rise the the embedding diagram depicted in Fig. 13.
Figure 13: The spacetime of a central line through a certain energy distribution.
6
On geodesics on rotational surfaces
Since this paper utilizes geodesics on rotational surfaces – maybe a general note on the subject is in order. Parameterizing any rotational surface with r and ϕ, the metric can be written (in every region of monotonically increasing or decreasing r): ds2 = r2 · dϕ2 + f (r) · dr2
(42)
Using the squared Lagrangian formalism we immediately get the integrated equation of motion: r2
dϕ = const ds 17
(43)
Letting θ denote the angle between the geodesic in question and a purely azimuthally directed line we may rewrite Eq. (43) into: r · cos(θ) = const
(44)
In particular we see that the tilt of a certain geodesic is completely determined by the radius, and that the tilt of the geodesic line increases with increasing radius. By considering a thread tightened on the surface we understand that this is very reasonable. Assuming the rotational surface to be a small perturbation of a cylinder of unit radius r = 1 + h, and assuming the tilt (θ) to be small, we may easily prove from Eq. (44) that to lowest order we have: d2 z dh = dϕ2 dz
(45)
Thus one may verify that, at least for small velocities and gravitational fields, one can explain gravitational attraction by motion on a rotational surface.
7
Displaying the Earth gravity
We would like to display why things accelerate at the surface of the Earth. We want a clearly curved surface where meters and seconds correspond to roughly the same distances as the radius of the cone. We have three parameters that determine the shape and size of the embedded surface. Let us therefore make three demands, exactly at the surface of the Earth: sin Θ0 = 0.8 r0 = 1 ∆τreal = 1s
The angle of slope for the surface (46)
The embedding radius The proper time per circumference
From these requirements it is an easy exercise to find the corresponding values of k, α and β. The results are: ∆τreal cl √ 2π ae0 RG ae0 β = "2 ! k 1+ 2sinΘ0 · x20 1 α = r02 4 2 4x0 sin Θ0 + k 2 k =
18
∼ 5.38 · 109 ∼
ae0 1 + ,4.25 ·-.10−17/ δ
∼ 1.47 · 10−36
(47)
Figure 14: An embedding diagram of the spacetime at the surface of the Earth. We see a freefaller moving up and then down again in perfect agreement with the Newtonian predictions.
Here ae0 = a0 (x0 ) and cl is the velocity of light. Since Matlab only operates at 16 decimals, we cannot cope numerically with the expression for β, Eq. (47). We have to Taylor-expand our expressions for the embedding coordinates z and r. Limiting ourselves to the exterior metric, and introducing ∆x = x − x0 the results are, assuming 0 < ∆x << x0 : √ k α (48) r % 0 ∆x + δ 2 x0 $ & √ 1 k2 1 1 − 4 · ∆x (49) ∆z % α dx ∆x 4x +δ +δ 0 x2 x2 0
0
It is now an easy task to show the embedding, see Fig. 14. I think this picture is really beautiful from a pedagogical point of view. Especially pedagogical would it be to make a real surface in metal, using the outside of the trumpet, 19
with meter and second lines drawn on it. Then you can use tightened threads to find the time it takes a particle initially at rest to fall say 10 meters. Of course there is no action in this, which kids like, but then again it serves to illustrate that there is no action in a spacetime movie, it’s a documentary!
8
Embedding the inside of a black hole
Recall the general formulae for the dual metric of a time independent twodimensional diagonal metric: a=α
a0 a0 − β
c=α
−c0 β
2
(a0 − β)
(50)
Inside a the horizon we have negative a0 and positive c0 . Necessary and sufficient conditions for positive a and c are then: 0<α a0 < β < 0
(51) (52)
We see that the dual metric exists from the singularity outwards to some x limited by our choice of β. Numerically we find that while the metric exists all the way into the singularity, the embedding is restricted. An embedding is displayed in Fig. 15. Notice that moving along the trumpet is now timelike motion. In particular we see that t = const is a timelike geodesic The trumpet is now narrowing towards the singularity, implying acceleration away from the singularity in the sense that a geodesic that was at some point, moving solely in Schwarzschild time, will end up at the horizon. Inside the horizon however, moving only in t, means moving faster than light. Thus it is tachyons (time-lines) that accelerate out towards the horizon. Real observers are however always approaching the singularity. While we might find it unintuitive that the shape of the trumpet implies that they decelerate as they move inwards one must consider also the spacing between lines of constant x. If the distance between x-lines is decreasing as we approach the horizon, we see that it is quite possible to have increasing dx/dt the closer to the singularity that we are, in spite of the trumpet opening up in the “wrong” direction.
9
Comments on the two-dimensional analysis
The dual metric assigns distances between nearby points on the manifold. This is what metrics in general do. Under a coordinate transformation the dual metric will thus transform as a tensor. 20
If we make a coordinate transformation to some wobbling coordinates, the dual metric will appear complicated. However, when we embed it we get the static-looking spacetime depicted in Fig. 11. It’s like the ugly duckling becoming a swan. The difference is that the coordinate lines will now wind and twist on the surface. Notice however that while the embedding of the dual metric is coordinate independent there may in principle exist more ways of embedding it than as a rotational surface. We understand that if the dual metric exists in one coordinate system it exists in all coordinate systems. In particular this means that it is sufficient that there exists coordinates where a and c are time independent, for the dual metric to exist. Also we realize that, starting from the Schwarzschild line element, written in standard coordinates, we are not excluding any possible dual metrics through our choice of coordinates. However we may be excluding dual metrics by our assumption that the dual metric is time independent and diagonal. Another small comment. One may think that the dual metric and the original metric would have the same affine connection, since the affine connection is all that enters the geodesic equation: ν µ d2 xλ λ dx dx + Γ =0 µν dτ 2 dτ dτ
(53)
Explicitly calculating the dual and original affine connections we find however that they are not the same. The explanation is that in the geodesic
Figure 15: An embedding of the spacetime inside a black hole. The singularity is to the left and the horizon is to the right. Schwarzschild time is still the azimuthal angle.
21
equation there are derivatives with respect to proper distance (eigentime). When we change into the dual metric we change the meaning, in a nontrivial way, of the proper distance. Then we see the possibility that different affine connections12 can produce the same x(t). A final comment is in order. The dual metric of Eq. (10) and Eq. (11) is Riemannian from infinity and in to the point where β = a0 . It is however geodesically equivalent to the original Schwarzschild metric also inside this boundary. Here a is negative while c is still positive (until we reach the real horizon where they both flip sign) implying a Lorentzian signature. This signature boundary has nothing to do with coordinates. Coordinate choices will not affect whether there exists negative distances or not. While we can not embed the Lorentzian part, it is however interesting to see that we can have smooth geodesics moving over something as dramatic as a signature change.
10
Extension to 2+1-dimensional spacetimes
I have extended the dual metric analysis to higher dimensions, using techniques very similar to those used in the 1+1-dimensional case. Assuming both the original and the dual metric to be time independent and diagonal, I find that there is in general no dual positive definite metric. The problem is that one needs the extra metrical function (connected to azimuthal distance) to fix x(t) for all values of the angular momentum J0 , but one also needs the extra metrical component to get the azimuthal motion right. It’s simply over-determined. Only for very specific cases of original metrics can we find a dual metric that is positive definite. In particular assuming the original metric to have gtt = const, we find the dual metric simply by flipping the sign of the spatial part. This may be understood without any analysis at all. However, if we restrict ourselves to demand that only those geodesics that correspond to a certain energy (K), shall be geodesics in the new metric – we can find a nontrivial dual metric. In particular studying photons, that have infinite K, the equations for the dual metric are somewhat simplified: λ−1 k3 ad00 − k2 c0 c = · a(1 − k2 a) a0
a =
12 There
(54) (55)
should be some other geometrical object that determines geodesics on the manifold that is the same in both original and dual metric. I do not know of it however.
22
d = λ·
d0 ·a a0
(56)
Here, k3 , k2 and λ are gauge constants much like α and β in the twodimensional analysis. Consider an original metric of the form: dτ 2 = a0 · dt2 + c0 · dx2 − x2 · dθ2
(57)
Let us assume that a0 and c0 reduces to +1 and -1 respectively at infinity. Demanding the dual metric to be flat Euclidean space expressed in polar coordinates at infinity, i.e. a = 1, c = 1 and d = x2 , yields k3 = 0, k2 = 2 and λ = −1. Inserting these constants into Eq. (54)-Eq. (56) gives: a=1
c=
−c0 a0
d=
x2 a0
(58)
This is thus a metric with positive signature that is geodesically dual to the original metric with respect to photons. The spatial part we recognize as the optical geometry, see [4]. For a brief review applicable to this text see Appendix A. When a is constant it is easy to understand that geodesics in spacetime are also geodesics in space. We may thus embed the spatial part of the dual metric to visualize a space where photons move on the shortest path. See Fig. 16.
Figure 16: The optical geometry of a plane through a star with x0 = 1.2. The great circles are closed photon orbits within the star.
23
It is a little bit fascinating that asking for a metric that is geodesically dual with respect to photons, with positive signature, yields the optical geometry, which is normally derived in a completely different manner. Notice however that we knew in advance that the optical geometry (plus time) would be among the possible dual spacetimes. We knew that there existed a three-metric (essentially Fig. 16) for Schwarzschild in which photons moved on geodesics. To this space we knew that we could just add time to create a Riemannian spacetime in which photons would move on geodesics. Thus we knew that the optical geometry (plus time) would be among the possible solutions. It turned out to be the only dual spacetime that reduced to a Euclidean spacetime at infinity. One may also study the dual geometry that springs from other choices of K, but then a doesn’t become constant and the geometrical information doesn’t lie entirely in a spatial metric. Still it is interesting to see that we have a scheme that produces the optical geometry for the particular case of photons. What would be really nice would be if we, using insights and techniques developed in this paper, could generalize the optical geometry to something that makes sense even when the spacetime is not conformally static.
10.1
Comments on the higher dimension analysis
If we introduce freedoms, like off-diagonality and time dependence, it may after all be possible to find a dual metric in higher dimensions. For me it is therefore still an open question whether, as soon as we have a fairly nontrivial metric, we can decide the exact form of the metric up to a global rescaling constant by just studying geodesics? In the two-dimensional analysis it was not so, but in the three-dimensional analysis it was so, assuming time independence and diagonality of the dual metric. In general however I do not know yet.
11
Comparison to other works
In Lewis Carrol Epstein’s book ’Relativity visualized’ [2], there is a similar way of visualizing the acceleration towards a gravitational source. The pictures illustrated are qualitatively very similar to my own, a star is a bulge on a cylinder for instance. In Epstein’s view however the azimuthal angle on the rotational surface is the eigentime experienced by an observer (for instance a freefaller). The Euclidean length of a curve on the surface is the Schwarzschild time elapsed. All freefallers move on geodesics on the surface. In particular photons, which do not experience eigentime, but still travels 24
in Schwarzschild time, are represented by straight lines directed along the rotational surface, without the slightest spiral in the azimuthal direction. The Epstein view is very beautiful in many respects. For instance it naturally explains why it takes infinite Schwarzschild time to reach the horizon but only finite eigentime. What Epstein is embedding is however not strictly a spacetime. A point on his surface is not corresponding to a unique event (in general). To see this consider a photon moving in towards the gravitational source and then bouncing back outwards. In the Epstein diagram the photon returns to the same point that it came from. Thus one point in the Epstein diagram represents two events (at least!) in the physical world. Also in the Epstein view one can not display spacelike distances. Perhaps the biggest advantage of my view, compared to Epstein’s view, is the opportunity to graphically display how gravity on Earth can be explained by spacetime geometry. This is obviously difficult to accomplish with the Epstein view since eigentime and Schwarzschild time are virtually the same thing for us Earthlings. We understand that the two approaches complement each other. It is really fascinating however that two such fundamentally different approaches can produce more or less the same plots! Another way of illustrating curved spacetime is to embed the spacetime in a 2+1 Minkowski space. There one has access to null and negative distances. See the paper [1], by Donald Marolf. The beauty of this scheme is that one can deduce Lorentzian distances between points just from the slope of the surface. Also particles move on geodesics, in the sense of shortest Euclidean distance on the surface.13 These surfaces are not rotational surfaces (in general), and depend on the timelike parameter. In particular Marolf studies the embedding of a Kruskal spacetime of an eternal black hole. The horizons are included in the embedding but not the singularities and the infinities. A lot of physics can be displayed in this type of embedding. For instance one can illustrate how tidal forces become infinite as one approaches the singularity. Fascinating.
13 The
reason that this solution was not included in my analysis is that I only considered time independent dual metrics.
25
12
Summary
Always for a two-dimensional time independent diagonal original metric with a0 positive and c0 negative we can find a dual metric where both a and c are positive. The new dual metric is dual in the sense that it produces the same geodesics as the original metric. a = α· c = α·
a0 a0 − β
0 < β < min{a0 }
0<α
(59)
−c0 β
(60)
(a0 − β)2
Here α and β are gauge freedoms in the dual metric. α is an overall rescaling. β is connected to stretching in the x-direction, while it can also be considered as a rescaling of the original metric. In particular, starting from the exterior Schwarzschild metric, and demanding that the dual metric reduces to Pythagoras at infinity yields α = 12 and β = 12 , the standard gauge. The dual line element may then be written: ds2 =
x3 x−1 · dt2 + · dx2 x−2 (x − 1)(x − 2)2
(61)
In this particular gauge the dual metric stays Riemannian from infinity and in to x = 2. By choosing other gauge constants we can move this boundary arbitrarily close to the horizon. We may embed the dual metric as a rotational surface in Euclidean space. We have then an embedding freedom k. Increasing k means increasing the radius and the slope of the rotational surface everywhere. A schematic embedding of the dual metric of a radial line through a star is depicted in Fig. 17. The spacetime surface is layered, so walking around Observer oscillating around the center of the star
Constant time line Time
−∞ Observer at rest at the center of the star
Time
+∞
Space
Figure 17: The spacetime of a central line through a star.
26
the rotational body one lap means that you come to a new spacetime point. Azimuthal angle on the surface is proportional to the Schwarzschild time. • For non-compact stars, using the standard gauge and k = 1, the difference between the radius of the rotational surface and the radius at infinity, is proportional to minus the Newtonian potential. • We have all in all three parameters, α, β and k, that affects the shape and size of the embedding diagram. In particular this allows us to visualize, with substantial curvature, why, and how fast, our keys fall when we drop them in our office. • The dual metric is geodesically equivalent to the original also in regions where a0 < β. Here it has Lorentzian signature however. • We can embed (parts of) the inside of a Schwarzschild black hole simply by putting β negative in Eq. (59) and Eq. (60). • In 2+1 dimensions we can not generally find a dual metric that is diagonal and time independent. We can however relax the constraints on the dual metric to apply, not to all geodesics, but only photon geodesics. That way we can re-derive the optical geometry. It would be interesting to generalize the dual metric scheme, to include more general original and dual metrics. In particular it would be interesting to study an equatorial plane in a Kerr geometry, and restrict ourselves to photons. Also, using the dual metric scheme, it remains to be seen if we can somehow include the horizon in the embedding, maybe using just a certain set of observers.
13
Conclusions
The ideas presented in this article are probably of minor practical use in calculations and the finding of new physics. Nevertheless they are, I think, of great pedagogical value. They open up our minds to possibilities that we might not have considered earlier. This goes for both professionals in the field, but even more so for those who have never seen a gµν , or even an ηµν . For the experts it is probably the concept of the geodesically dual metric itself that is most interesting. The signature change in the dual metric may as well attract some attention. Also it was nice, though not surprising, to see the optical geometry coming naturally from relaxing the geodesic demands to apply only to photons.
27
For the non-experts it is probably Fig. 14, depicting the spacetime at the surface of the Earth, that has the greatest pedagogical value. Especially useful would it be to construct such a surface, with meter lines and second lines, and threads to pull tight between various spacetime points. Then people get a chance to see how acceleration can be explained by geometry. This I think is very powerful, and something that I have longed for, when giving introductory lectures on general relativity.
A
Review of the optical geometry
Null geodesics are conserved under conformal rescalings of the metric. If we have a manifestly time independent metric, with no cross-terms of dt −1 µ (x ) without affecting the null (e.g. Schwarzschild), we may rescale it by gtt geodesics. The rescaled metric will consist of a unit time-time component, and a spatial three-metric. It is thus a curved space, with time. There is no acceleration or time dilation. For such a metric, a so called ultrastatic metric, it is easy to understand that geodesics in spacetime are also geodesics in space. −1 So, by rescaling the spatial part of, for instance Schwarzschild, with gtt we create a space in which photons moves on geodesics. This rescaled space is known as the optical space.
References [1] Marolf, D. (1999). Gen. Rel. Grav. 31, 919 [2] Epstein, L. C. (1994). Relativity Visualized, (Insight Press, San Fransisco), ch. 10,11,12 [3] Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (John Wiley & Sons, U.S.A), p. 77 [4] Kristiansson, S., Sonego, S., and Abramowicz, M. A. (1998). Gen. Rel. Grav. 30, 275 [5] D’Inverno, R. (1998). Introducing Einstein’s Relativity, (Oxford University Press, Oxford), p. 99-101
28
Addendum to “Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature” Rickard Jonsson1 Abstract. I extend the analysis of Jonsson, R. (2001). Gen. Rel. Grav. 33, 1207, concerning the shape of an embedding of the dual spacetime of a line through a planet of constant proper density. In particular I find that in the non-compact limit, the embedding of the interior of the planet can be chosen to be spherical.
1
A spherical interior dual metric
We know that in Newtonian theory, a particle that is in free fall around the center of a spherical object of constant density, will oscillate with a frequency that is independent of the amplitude. This would fit well with an internal dual spacetime that is spherical. We saw in [1] the possibility to choose parameters β and k that produces substantial curvature of the embedding also for the case of the weakly curved spacetime outside our Earth. It is therefore natural to ask whether we can find parameters β and k such that the internal geometry becomes spherical. In the following we will see that this is not generally the case but one can choose parameters such that it is exactly spherical in the non-compact limit.
1.1
Conditions for spheres
For a sphere of radius R, we introduce definitions of r and z according to Fig. 1. The Pythagorean theorem gives R2 = z 2 + r2 . Differentiating this relation with respect to z and again using the Pythagorean theorem yields: !" # 2 dr R −1 (1) =− dz r For a general surface of revolution described by functions z(x) and r(x), with a corresponding metric on the form ds2 = a(x)dt2 + c(x)dx2 , we have
1 Department
of Theoretical Physics, Physics and Engineering Physics, Chalmers University of Technology, and G¨ oteborg University, 412 96 Gothenburg, Sweden. E-mail:
[email protected]. Tel +46317723179.
1
R
r z
Figure 1: Definitions of variables for a sphere. The axis of rotation (around which time is directed) is the horizontal axis.
according to Eq. (23) of [1]: dz = dx
$ c(x) − r! 2
(2)
r2 1 − r! 2 /c
(3)
Here a prime indicates differentiation with respect to x, as it will henceforth. From Eq. (1) and Eq. (2) we readily find: R2 =
This must hold, for a fixed R, for all x within a certain interval in order for the rotational surface to be spherical in that interval. 2 Note that the term r! /c equals dr2 /ds2 where ds is the infinitesimal 2 distance measured along the rotational curve r(z). Thus r! /c is a measure of the slope of the rotational curve. It is a short exercise to integrate the slope given by Eq. (3) in terms of R and r (also using ds2 = dr2 + dz 2 ) to find that, for a fixed R, Eq. (3) implies a circular rotational curve (with center on the z-axis). Thus Eq. (3), for a fixed R, is a both sufficient and necessary condition for the rotational surface to be spherical.
1.2
The dual interior metric
Taking the square of Eq. 21 in [1] gives: r2 = αk 2
a0 a0 − β
(4)
Taking the derivative of this expression with respect to x and using the expression for c(x) given by Eq. (11) of [1] as well as the explicit expressions 2
for the Schwarzschild interior metric (inside the planet) given by Eqs. (30) and (31) of [1] gives: 2
r! /c =
βk 2 x2 − β)
(5)
4x60 (a0
Inserting Eqs. (4) and (5) into Eq. (3) we readily find that for the embedding of the internal dual metric to correspond to a sphere of radius R we must have, for all x where −x0 < x < x0 : R2 (a0 − β) − R2
βk 2 x2 − αk 2 a0 = 0 4x60
(6)
For Eq. (6) to be true for all x in the interval in question, it must be true for every power in x. We note that a0 has a lot of powers in x which would mean that the sum of the factors multiplying the a0 -terms would have to vanish. But then the x2 -term in the middle cannot be canceled by anything. Thus the expression cannot be true for all values of x in the interval in question, and thus the interior dual metric is not exactly spherical.
1.3
Approximative internal sphere
We can however produce something that is very similar to a sphere. First, to be specific, we let α = (1 − β)/k 2 meaning unit radius at infinity. We then expand the two occurrences of a0 in Eq. (6) to second order in x2 . Demanding the equation to hold to zeroth, first2 and second order in x yields after some simplification (in Mathematica): a00 · (R2 − 1) β = R2 − a00
where
a00
7/4
2x0 k = $ √ √ 3 x0 − 1 − x0
1 ≡ a0 (0) = 4
" % #2 1 3 1− − 1 (7) x0 (8)
Notice that Eq. (8), coming directly from the second order demand, is independent of β. The first relation is nothing but Eq. (21) in [1], where one has demanded r = R and α = (1 − β)/k 2 , taken at x = 0. Inserting β and k from Eqs. (7) and (8) into a program that plots an embedding of the dual spacetime yields pictures with a very spherical appearance. The way it works is that the center of the bulge has the exact radius and curvature of a sphere, then the rest is not exactly spherical. 2 It
is always satisfied to first order.
3
1.4
Spheres in the Newtonian limit
From Eq. (3) we have for a sphere: 2
r! r2 =1− 2 c R
(9)
Using the specific expressions for β and k given by Eqs. (7) and (8), together with Eqs. (4) and (5), we may evaluate both the left and the right hand side of Eq. (9). The left hand side is then a measure of the actual slope of the embedded surface (for the x in question), whereas the right hand side is a measure of the slope the embedding should have (for the x in question) in order to correspond to a sphere. Expanding the expressions on both sides of Eq. (9) to lowest non-zero order3 in 1/x0 , we find that the equality holds exactly for all x. Thus in the limit as x0 → ∞ we get an exact sphere. For the case of the Earth (approximating it to be of constant proper density) we have x0 = 7.19 · 108 . Thus one might guess that the higher order (1/x20 ) differences between the actual slope and the slope of a perfect sphere, are rather small. An embedding of the Earth spacetime is displayed in Fig. 2. Here I have chosen the radius of the central bulge to be twice the embedding radius at infinity, thus choosing R = 2. The parameters β and k are given by Eqs. (7) and (8). The rotational radius r(x) is given by Eq. (21) of [1] and z(x) is given by numerical integration of Eq. (25) of [1]. As can be seen in Fig. 2 the bulge is quite spherical.
1.5
The spacetime circumference
The dual distance around the spacetime rotational surface is given by ∆s = √ and rescaled coordinate 2πr = 2πk a. Letting ∆t denote the dimensionless √ time per circumference, we have also ∆s = a∆t. It follows that: ∆t = 2πk
(10)
Furthermore, the real Schwarzschild coordinate time ∆treal is related to the dimensionless rescaled Schwarzschild time via ∆t = ∆treal /(2M G/c3l ). Here cl is the velocity of light. It follows that around the embedded dual spacetime we have: MG ∆treal = 4πk 3 (11) cl √ 3/2 For large x0 , Eq. (8) can be approximated by k & 2x0 . Also, we have x0 = RE /(2M G/c2l ) where RE is the radius of the Earth. Inserting this into 3 Terms
like x2 /x30 are of course treated as 1/x0 -terms.
4
Figure 2: An embedding of the dual spacetime of a central line through the Earth. Note the worldlines of the two freefallers on the interior sphere. One is at rest in the center, the other is oscillating about the Earth’s center. The time direction is chosen opposite to that illustrated in Fig. 8 of [1], to better connect to ordinary flat spacetime diagrams.
Eq. (11) gives: %
RE3 (12) MG Inserting the appropriate constants yields a circumference of roughly 84 minutes. For the case of a spherical bulge – the period of revolution corresponding to a geodesic circling the spacetime bulge, equals the circumference of the spacetime. It is a short exercise of Newtonian mechanics to show that the period of revolution for an object oscillating about the center of the Earth (within the Earth) is given by (assuming the Earth to be of constant density): % RE3 ∆tNewton = 2π (13) MG ∆treal & 2π
Comparing this with Eq. (12) which holds to good accuracy for non-compact planets such as the Earth, we have a perfect match.
References [1] Jonsson, R. (2000). Gen. Rel. Grav. 33, 1207 5
Visualizing curved spacetime
gravitational time dilation, cosmological expansion, horizons and so on. Rickard M. Jonsson In Sec. II, I give a brief introduction to the concept of Department of Astronomy and Astrophysics, Physics and curved spacetime, using the method of this article, and Engineering Physics, Chalmers University of Technology, consider a few examples of physical interest. This section presumes no knowledge of general relativity. In Sec. III, I and G¨ oteborg University, 412 96 Gothenburg, Sweden explain the method underlying the illustrations thus far, E-mail:
[email protected] and also apply it to a black hole and a hollow star. In Secs. IV-VIII, I present the general formalism, demonSubmitted: 2003-04-23, Published: 2005-03-01 strate how to use it to produce embeddings and investiJournal Reference: Am. Journ. Phys. 73 248 (2005) gate the geodesic properties of the formalism. These secAbstract. I present a way to visualize the concept of tions are of a more technical character. In Secs. IX-XIII, curved spacetime. The result is a curved surface with lo- I apply the formalism to various types of spacetimes. In cal coordinate systems (Minkowski Systems) living on it, Secs. XIV-XVI, I relate this work to other similar apgiving the local directions of space and time. Relative to proaches, and comment on this article. Secs. XVII and these systems, special relativity holds. The method can XVIII include some pedagogical questions and answers. be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea Introduction to curved spaceunderlying the illustrations is first to specify a field of II µ timelike four-velocities u . Then, at every point, one pertime forms a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local Consider a clock moving along a straight line. Special system, the sign of the spatial part of the metric is flipped relativity tells us that the clock will tick more slowly than to create a new metric of Euclidean signature. The new the clocks at rest as illustrated in Fig. 1. positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.
I
Introduction
Einstein’s theory of gravity is a geometrical theory and is well suited to be explained by images. For instance the way a star affects the space around it can easily be displayed by a curved surface. The very heart of the theory, the curved spacetime, is however fundamentally difficult to display using curved surfaces. The reason is that the Lorentz signature gives us negative squared distances, something that we never have on ordinary curved surfaces. However, we can illustrate much of the spacetime structure using flat diagrams that include the lightcones. Famous examples of this are the Kruskal and Penrose diagrams (see e.g. Ref. 1). Such pictures are valuable tools for understanding black holes. In this article I will describe a method that lets us visualize not just the causal structure of spacetime, but also the scale (the proper distances). It is my hope that these illustrations can be of help in explaining the basic concepts of general relativity to a general audience. In exploring the possibilities of this method I use the language and mathematics of general relativity. The level is that of teachers (or skilled students) of general relativity. The resulting illustrations can however be used without any reference to mathematics to explain concepts like 1
Figure 1: A clock moving along a straight line. Relative to the clocks at rest, the moving clock will tick more slowly. Consider two events on the moving clock, separated by a time dt and a distance dx, as seen relative to the system at rest. We can illustrate the two events, and the motion of the clock in a spacetime diagram as depicted in Fig. 2. Time is directed upwards in the diagram. The motion of the clock corresponds to a worldline in the diagram. The proper time interval dτ is the time between the two events according to the moving clock, which is given by2 2 dx dτ 2 = dt2 − . (1) c Here c is the velocity of light. Note that in the limit as the speed of the clock approaches the speed of light we have dx = cdt, and thus from Eq. (1) we have dτ = 0. A clock moving almost at the speed of light will thus almost not tick at all relative to the clocks at rest.
t
B dx
The spacetime of a line through a dense star
As a specific example let us consider the spacetime of a line through a very dense star as depicted in Fig. 4.
dt x Figure 2: A spacetime diagram showing the worldline of the moving clock (the fat line). The two events we are considering are the black dots in the diagram. The shaded area is known as the lightcone. It is customary to choose the axes of the spacetime diagram in such a manner that motion at the speed of light corresponds to a line that is inclined at a 45◦ angle relative to the axes of the diagram. At every point in the diagram we can then draw a little triangle, with a 90◦ opening angle, known as a lightcone. The rightmost edge of the triangle corresponds to a right-moving photon and the leftmost edge corresponds to a left-moving photon. No material objects can travel faster than the velocity of light, which means that the worldlines of objects must always be directed within the local lightcone.
A
Figure 4: A radial line through a very dense star, and an illustration of the curved spacetime for that line. Time is directed around the hourglass shaped surface. Strictly speaking the surface should not close in on itself in the time direction. Rather one should come to a new layer after one circumference as on a paper roll.
Curved spacetime
The circles around the surface correspond to fixed poIn general relativity we have a curved spacetime, which we sitions along the line through the star. The lines directed may illustrate by a curved surface with little locally flat along (as opposed to around) the surface correspond to coordinate systems, known as Minkowski systems, living fix coordinate time (known for this particular case as on it as illustrated in Fig. 3. Schwarzschild time). Consider now two observers, one at rest in the middle of the star and the other at rest far to the left of the star. The worldlines of these observers are circles around the middle and the left end of the spacetime. Obviously the distance measured around the spacetime is shorter at the middle than at the end. This means that the proper time (the experienced time) per turn around the spacetime is shorter in the middle than at the end. From this we may understand that time inside the star runs slow relative to time outside the star. To be more specific consider the following scenario. Figure 3: An illustration of curved spacetime using a Let an observer far outside the star send two photons, curved surface with little Minkowski systems living on separated by a time corresponding to one lap, towards it. The curving line could be the worldline of a moving the center of the star. The corresponding worldlines of the clock. photons will spiral around the surface and arrive at the center of the star still separated by one lap. The points The little coordinate systems on the surface work pre- where the photons arrive at the center will in this illustracisely as the spacetime diagram of Fig. 2. In particu- tion be the same, but they are different points in spacelar the worldlines of moving objects must always be di- time because the surface is layered as in a paper roll. Since rected within the local lightcone. To find out how much a the distance around the central part of the spacetime is clock has ticked along it’s winding worldline, we consider smaller than that towards the ends of the spacetime the nearby events along the worldline and sum up the dτ ’s observer in the center of the star will experience a shorter we get using Eq. (1), where dt and dx are the time and time between the arrival of the two photons than the time space separation between the events as seen relative to between the emission of the two photons, as experienced the local Minkowski system. by the sender. This effect is known as gravitational time
2
D
dilation – and is a consequence of the shape of spacetime. Alternatively we may note that the lines of constant coordinate time are lying closer to each other in the middle of the spacetime surface than at the ends. An observer inside the star will therefore observe that a local clock showing Schwarzschild coordinate time (synchronized with a proper clock far outside the star) ticks much faster than a clock measuring proper time within the star. We may then understand that an observer inside the star will see the Universe outside the star evolving at a faster rate than that experienced by an observer outside the star.
C
Cosmological models
We may use the the same technique that we employed in the previous section to visualize the spacetimes corresponding to various cosmological models (although we are restricted to one spatial dimension). In Fig. 6, a few images of such models are displayed.
Freely falling motion
According to general relativity, an object thrown out radially from the surface of the star, moving freely (so there is no air resistance for instance), takes a path through spacetime such that the proper time elapsed along the worldline of the object is maximized. Consider then two events, at the surface of the star separated by some finite time only. It is easy to imagine that a particle traveling between the two events will gain proper time by moving out towards a larger embedding radius (where the circumference is greater), before moving back to the second event. On the other hand it cannot move out too fast since then it will move at a speed too close to the speed of light – whereby the internal clock hardly ticks at all, see Fig. 5.
Figure 6: Schematic spacetime cosmological models. Notice that time is here directed along the surface and space is directed around the surface. Just like before the local coordinate systems, in which special relativity holds, gives the local spatial and temporal distances. The leftmost illustration corresponds to a Big Bang and Big Crunch spacetime. As we follow the spacetime upwards (i.e forward in time) the circumference first increases and then shrinks. This means that space itself expands and then contracts. The Big Bang is here just a point on the spacetime – where the spatial size of the universe was zero. I will leave it to the reader to describe how space behaves in the two rightmost spacetimes. Using Newtonian intuition one might think of the Big Bang as a giant fire cracker exploding at some point in time. As the fire cracker explodes it sends out a cloud of particles that expands at a great rate relative to a fixed space. In Einstein’s theory it is space itself that expands due to the shape of spacetime. Also unlike in the fire cracker view we cannot in general even talk about a time before the Big Bang in Einstein’s theory. As another application of the illustrations of Fig. 6, consider a set of photon worldlines separated by some small spatial distance shortly after the Big Bang in the leftmost Big Bang model. The worldlines will spiral around the spacetime, always at 45◦ to the local time axis. From this we may understand that they will get further and further separated as the circumference of the universe increases. Thinking of a photon as a set of wave crests that are all moving at the speed of light, we then understand that the wavelength of a photon will get longer and longer as the universe grows larger. This effect is known as the cosmological red shift. We can consider a similar scenario for the gravitational redshift by considering a photon moving along the spacetime of the line through the dense star of Fig. 4.
Figure 5: Three different worldlines connecting two fixed events. The middle worldline corresponds to the actual motion of an object initially thrown radially away from the star and then falling back towards the star. Of the three worldlines this has the largest integrated proper time. To predict the motion of an object that has been thrown out from the star and returns to the same location after a specific amount of time, we can in principle consider different pairs of events (as in Fig. 5), find the worldline that maximizes the integrated proper time. This trajectory corresponds to the motion that we are seeking. Thus we can explain not only gravitational time dilation but also the motion of thrown objects using images of the type shown in Fig. 4 and Fig. 5. 3
III
A simple method
The singularity (where the spacetime curvature becomes infinite) is not visible in the picture. While the The idea allowing us to make a figure like Fig. 4, which is distance as measured along the internal trumpet from an exact representation of the spacetime geometry, is sim- the horizon to the singularity is finite (it has to be since ple. Assume that we have a two-dimensional, Lorentzian, we know that the proper time to reach the singularity time-independent and diagonal metric: once inside the horizon is finite) the embedding radius is infinite at the singularity. Thus, we cannot show the sin2 2 2 dτ = gtt dt + gxx dx . (2) gularity using this visualization. We can however come We then produce a new metric by taking the absolute arbitrarily close by everywhere making the embedding radius smaller. In Fig. 8 we zoom in on the internal gevalue of the original metric components: ometry. d¯ τ 2 = |gtt |dt2 + |gxx |dx2 . (3) The new metric, called the absolute metric, is positive definite and can be embedded in three dimensional Euclidean space as a surface of revolution because gtt and gxx are independent of t. For an observer with fixed x, pure temporal and pure spatial distances will precisely correspond to the absolute distances. There will thus be small Minkowski systems living on the curved surface. Analogous arguments hold if we have x rather than tindependence (as for the cosmological models).
A
Black hole embedding Figure 8: The absolute internal spacetime of a central line through a black hole. Notice the direction of the lightcones. The singularity lies in the (temporal) direction that the lightcones are opening up towards.
As another example of the visualization scheme outlined above, we consider the line element of a radial line through a Schwarzschild black hole. An embedding of the corresponding absolute metric is depicted in Fig. 7.
Note that the singularity is not a spatial point to which we may walk. Once inside the horizon, the singularity lies in the future and it is impossible to avoid it – just like it is impossible to avoid New Years Eve. In this 1+1 dimensional scenario (inside the horizon) the singularity is the time when space expands at an infinite rate. Following a Schwarzschild time line (of fixed azimuthal angle) inside the black hole corresponds to timelike geodesic motion. Imagine then two trajectories directed along two such coordinate lines, starting close to the horizon and extending towards the singularity. The corresponding two observers will be at rest with respect to each other at the start (to zeroth order in the initial separation between them). As they approach the singularity they will however drift further and further apart in spacetime. At the singularity, where the embedding radius is infinite, they will be infinitely separated. We also know that the time it takes to reach the singularity is finite. It is then easy to imagine that if we try to keep the observers together, the force required will go to infinity as we approach the singularity. Hence, whatever we throw into a black hole will be ripped apart as it approaches the singularity. Notice however that there is no gravitational force in general relativity. The shape of spacetime is in this case simply such that a force is needed to keep things together, and in the end no force is strong enough.
Figure 7: An embedding of the absolute spacetime of a central line through a black hole. As before, the azimuthal angle corresponds to the Schwarzschild time. The two points of zero embedding radius correspond to the horizon on either side of the black hole. As we approach these points from the outside, the time dilation becomes infinite. The trumpet-like regions within these points lie within the horizon. Here moving along the surface (as opposed to moving around the surface) corresponds to timelike motion. Photons, however, move at a 45◦ angle relative to a purely azimuthal line, both inside and outside of the horizon. Studying a photon trajectory coming from the outside and spiraling towards the point of zero embedding radius, it is not hard to realize that it will take an infinite number of laps (i.e. infinite Schwarzschild time) to reach that point. 4
B
IV
Thin spherical crust
As a pedagogical example, imagine a hollow massive star, with a radial line through it, as illustrated in Fig. 9.
Generalization to arbitrary spacetimes
The scheme outlined in the preceding section was specific for a particular type of metric expressed in a particular type of coordinates. There is however a way to generalize this scheme. Given a Lorentzian spacetime of arbitrary dimensionality (although we commonly will apply the scheme to two dimensions), the idea is first to specify a field of timelike four-velocities denoted uµ (x) (we will refer to spacetime velocities as four-velocities regardless of dimensionality). We then make a coordinate transformation to a local Minkowski system comoving with the given fourvelocity at every point. In the local system we flip the sign of the spatial part of the metric to create a new absolute metric of Euclidean signature. Notice that the new metric will be highly dependent on our choice of generating four-velocities. The absolute metric together with the field of four-velocities contains all the information about the original spacetime, and allows one to keep track of what is timelike and what is not. We can always do the backwards transformation and flip the local positive definite metric into a Lorentzian (Minkowski) metric. Considering for example the black hole illustrations of the preceding section, the generators (the worldlines tangent to the field uµ ) outside the horizon were simply those of the Schwarzschild observers at rest. Inside the horizon the generators were the worldlines of observers for whom t = const. Notice that the observers located right outside the horizon have infinite proper acceleration. It is then perhaps not surprising that the resulting embedding is singular at the horizon. As we will see in Sec. V we can better resolve the horizon by using the worldlines of freely falling observers as generators.
Figure 9: A line through a thin crust of high mass. The wedge is cut out to obtain a better view of the interior of the star. We know from Birkhoff’s theorem (see e.g. Ref. 3) that, assuming spherical symmetry, the spacetime outside the crust will match the external Schwarzschild solution. On the inside however, spacetime must be Minkowski.4 In Fig. 10 the absolute spacetime of the radial line is displayed.
Figure 10: A schematic picture of the spacetime for a line through a hollow star in the absolute scheme. Notice how, after one circumference in time, we are really at a new layer.
A
A covariant approach
We could carry out the scheme we just outlined explicitly, doing coordinate transformations, flipping the sign of the metric and transforming it back again. There is, however, a more elegant method. We know that the absolute metric, from now on denoted by g¯µν , is a tensor (as any metric), and in a frame comoving with uµ we have
If we were to cut out a square of the interior spacetime it would look just like a corresponding square cut out at infinity. There is thus no way that one, even by finite sized experiments (not just local experiments) within the crust, can distinguish between being inside the star or being at infinity. Even tidal effects are completely absent. If we however were to open up a dialogue with someone on the outside, we would find that the outside person would talk very fast, and in a high pitched tone, whereas our speech would appear very slow and thick to the outside person. The point that one can illustrate is that we do not have to feel gravity for it to be there. Gravity is not about forces pulling things, it is about the fabric of space and time, and how the different pieces of this fabric are woven together.
1 0 g¯µν = 0 0
0 1 0 0
0 0 1 0
0 1 0 = − 0 0 0 1 0
0 -1 0 0
0 0 -1 0
0 1 0 + 2 0 0 0 -1 0
0 0 0 0
0 0 0 0
0 0 . 0 0
(4) Adopting (+, −, −, −) as the metrical signature (as we will throughout the article), we realize that we must have: g¯µν = −gµν + 2uµ uν ,
where
uµ = gµν
dxµ . dτ
(5)
Notice that both sides of the equality are covariant tensors that equal each other in one system, thus the equality 5
To make an embedding of this metric we are wise to first diagonalize it by a coordinate transformation t′ = g ¯tx t + φ(x). Letting dφ dx = g ¯tt the line element in the new coordinates becomes −1 1 1 d¯ τ2 = 1 + dt′2 + 1 + dx2 . (13) x x
holds in every system.5 For later convenience, we may also derive an expression for the inverse absolute metric, defined by g¯µρ g¯ρν = δ µ ν . By a contravariant argument, analogous to the covariant argument above, we find that the inverse absolute metric is given by: g¯µν = −g µν + 2uµ uν .
(6)
This metric is easy to remember since by chance it is the Schwarzschild metric with the minus signs replaced by plus signs (except for the minus sign in the exponent). Notice that nothing special happens with the metrical components at the horizon (x = 1). At the singularity (x = 0) however, the absolute metric is singular. To produce a meaningful picture of this geometry, we must include the worldlines of the freely falling observers used to generate the absolute geometry. Coordinate transforming of the trajectories to the new coordinates t′ , x can be done numerically. The result is depicted in Fig. 11.
That this is indeed the inverse of the absolute metric can be verified directly. It is a little surprising however that we get the inverse of the new metric by raising the indices with the original metric.6 In the new metric, proper intervals will be completely different from those in the original metric. Intervals as measured along a generating congruence line will however be the same; these are unaffected by the sign-flip. Using a bar to denote the four-velocity relative to the absolute metric, it then follows that uµ = u¯µ ,
uµ = u ¯µ .
(7)
Using this in Eq. (5), we immediately find gµν = −¯ gµν + 2¯ uµ u ¯ν .
(8)
Comparing with Eq. (5), we see that there is a perfect symmetry in going from the original to the absolute metric, and vice versa.
V
Figure 11: The absolute freefaller geometry. The dashed line is the horizon. As before the singularity lies outside of the embedding (towards the left).
Freely falling observers as generators
As a specific example of the absolute metric, we again consider the line element of a radial line through a SchwarzNotice how the local Minkowski systems are twisted schild black hole. We set c = G = 1, and introduce di- on the surface. The horizon lies exactly where the genermensionless coordinates, and proper intervals ating worldlines are at a 45◦ angle to a purely azimuthal line. toriginal τoriginal r t= τ= . (9) x= Time dilation is now not solely determined by the 2M 2M 2M local embedding radius, but also by the gamma factor7 The line element then takes the form of the observer at rest relative to the generating observer. −1 For instance an observer at rest at the horizon will be at 1 1 dx2 . (10) a 45◦ angle to the generating observer, corresponding to dτ 2 = 1 − dt2 − 1 − x x an infinite gamma factor, and his clock will therefore not µ As generators (u ) we consider freely falling observers, tick at all during a Schwarzschild lap (one circumference), initially at rest at infinity. Using the squared Lagrangian thus being infinitely time-dilated. Unlike the hour-glass type embeddings of Sec. II B, formalism (see e.g. Ref. 1) for the equations of motion, this explanation of gravitational time dilation requires a we readily find the lowered four-velocity of the generating basic knowledge of special relativistic time dilation. Howfreefallers ever, unlike the illustration of Fig. 7 where there is a cusp √ x right at the horizon, Fig. 11 has the virtue of showing uµ = 1, . (11) x−1 how passing the horizon is not at all dramatic (locally). Spacetime is as smooth and continuous at the horizon as The absolute metric is then according to Eq. (5) everywhere else outside the singularity. g¯µν
1 1+ x = √
2 x x−1
√ 2 x x−1
x(x+1) (x−1)2
.
(12)
6
VI
Symmetry-preserving generators
flat absolute metric. This we can embed as a cylinder or a ′ = C, where C is some arbitrary plane. We simply set g¯tt positive constant. Solving for v yields s In this section we generalize the scheme outlined in the C − gtt preceding section to include arbitrary two-dimensional . (16) v=± 8 C + gtt (2D) metrics with a Killing symmetry, for arbitrary generators that preserve manifest Killing symmetry. In two dimensions, the generating field uµ can (since it is nor- As a specific example we consider a Schwarzschild origimalized) be specified by a single parameter as a function nal line element. We choose v = 0 at infinity, correspondof xµ . A parameter that is well suited to preserve the ing to the generating observers at infinity being at rest, symmetries of the original metric is the Killing velocity which yields C = 1. We also choose the positive sign, v. By this we mean the velocity that a generator uµ ex- corresponding to an in-falling observer (on the outside) periences for a Killing line (a worldline of constant x). to find In other words it is the velocity of a point of constant x 1 . (17) v=√ as seen by the generating observer. The absolute value of 2x − 1 this velocity will be smaller than one outside the horizon, and greater than one inside the horizon. Without loss of This is a completely smooth function at the horizon. We generality we can assume that the original line element see that it remains real for x ≥ 1/2. For other choices is of the form Diag(gtt (x), gxx (x)). The relation between of C we can make the inner boundary come arbitrarily ′ uµ and v is derived in Appendix A. The result is close to the singularity. We notice also that g¯xx = 1/C r and is thus also constant. This means that the constant x1 −v gtt . (14) worldlines will be evenly spaced on the flat surface. Also ,√ uµ = ± 2 1−v gtt −gxxgtt we may, from the expression for v, immediately figure out Using the lowered version of Eq. (14) in Eq. (5) gives how the local generator should be tilted relative to the us the absolute metric as a function of the parameter constant x-worldline on the flat surface. An embedding v. Making a coordinate transformation that diagonalizes for this particular case is displayed in Fig. 12. this metric, analogous to the diagonalization in the preceding section,9 yields after simplification 1 + v2 0 gtt ′ 1 − v2 . g¯µν = (15) 2 1−v 0 −gxx 1 + v2
Notice that if there is a horizon present, where gtt = 0, we have also (1 − v 2 ) = 0. The quotient of these two entities will remain finite and well defined, given that dv/dx 6= 0 and dgtt /dx 6= 0. We see from Eq. (15) that there is much freedom in ′ choosing g¯tt . Since we can choose v arbitrarily close to 1, both inside and outside of the horizon, we can every′ where make g¯tt take an arbitrarily high value. Because ′ the square root of g¯tt is proportional to the embedding radius, there are virtually no limits to what shape the curved surface can be given. To interpret the embedded surface we need also the generating worldlines, relative to the new (diagonalizing) coordinates. How these can be found is derived in Appendix B. While the shape of the embedded surface depends strongly on the choice of generators, the area is independent of this choice. This holds regardless of any assumed symmetries as is explained in Appendix C.
x=1
x=2
Figure 12: A flat embedding of a Schwarzschild black hole. The radial parameter x lies in the interval [0.5, 2.5]. We could equivalently embed this geometry as a cylinder. As we go further to the right (larger x), the lightcones will approach pointing straight up. Notice that for this visualization the curvature of spacetime is manifested solely as a twist of the local Minkowski systems relative to each other. As in the case of Fig. 11, the flat embedding illustrates the smoothness of the spacetime around the horizon.
VIII
Absolute geodesics
We know that the motion of particles in free fall corresponds to trajectories that maximize the proper time. VII Flat embeddings Such trajectories can be found using the absolute scheme, Using Eq. (15) and assuming a time-symmetric and two- as outlined above. The fact that these trajectories are also dimensional original metric, we can produce an absolutely straight, is unfortunately a bit lost in this scheme. There 7
are however ways to manifestly retain at least parts of the original geodesic structure, in the absolute metric. The net value of this discussion turns out to be of more academic than pedagogical value. Below we therefore simply summarize the results derived in the appendices.
The dimensionless constant β lies in the range [0, 1] and is proportional to the charge of the black hole. Just as in Sec. VII, we may find a flat absolute geometry for this line element as depicted in Fig. 13.
• Assuming an original 2D metric with a Killing symmetry, we can demand that some given motion x(t) should be geodesic relative to the absolute metric. For example one can show that there exist generators such that outward-moving photons on a Schwarzschild radial line follow absolute geodesics. For brevity the analysis of this is omitted. • To investigate the general connection between the geodesic structure of the original and the absolute metric, we derive a covariant expression for the absolute four-acceleration in terms of Lorentzian quantities. See Appendix D.
x=1
x=2
Figure 13: A flat embedding of a Reissner-Nordstr¨ om black hole. The dimensionless radial coordinate x lies in the range [0.22, 2.5]. The two internal horizons are marked with the thicker dotted lines. The charge is chosen so that β = 0.95.
• Using the formalism of the preceding point we can show that if the generators are geodesics with reWe see the classic three regions of the Reissner-Nordstr¨ om spect to the original metric they will also be with solution. Thinking of free particles taking a path that respect to the absolute metric, and vice versa. See Appendix E. We also give an intuitive explanation maximizes the proper time we understand that a freely falling observer initially at rest in the innermost region, for this. will accelerate towards the inner horizon. Actually this • In the preceding points we have seen how some becomes clearer still if we form the absolute metric by parts of the geodesic structure can be retained. To simply taking the absolute value of the original metrical completely retain the geodesic structure, as is de- components, as we did in Sec. III. This corresponds to having generators that are orthogonal to the Killing field rived in Appendix F, we must have in the intermediate region, and parallel to the Killing field ▽α uµ = 0. (18) outside this region. See Fig. 14. At any single point in spacetime, this is easily achieved. We just go to an originally freely falling system and in this system choose uµ = √g100 δ µ 0 . Since in this system the metric derivatives all vanish, so will the derivatives of the generators. For a normalized vector field uµ to exist such that Eq. (18) holds everywhere, we must have a so called ultrastatic spacetime – as is derived in Appendix G. By ultrastatic we mean that space may have some fixed shape, but there can be no time dilation. Figure 14: An alternative representation of a Reissner Nordstr¨ om black hole. Notice the direction of the local We conclude that only to a limited extent can we, in the Minkowski systems. Here β = 0.995 and the range is absolute scheme, visualize Lorentz-geodesics as straight [0.425, 0.7]. lines. There are however other visualization methods that are better suited for this, as discussed in Sec. XV.
IX
Charged black hole
All that we have done so far for ordinary black holes, in the absolute scheme, can also be done for charged black holes. The line element of a radial line is then given by (see e.g. Ref. 10) dτ 2 =
−1 1 β2 β2 1 dx2 1 − + 2 dt2 − 1 − + 2 x 4x x 4x
. (19) 8
We notice that the spacetime geometry of the region just inside the inner horizon looks very much like the geometry just outside the outer horizon. Knowing that it takes a finite proper time to reach the outer horizon from the outside, we understand that it must take a finite proper time (while infinite coordinate time) to reach the innermost horizon from the inside. In the embedding there is however apparently no region to which the trajectory may go after it has reached the inner horizon. To resolve this puzzle, we must consider the extended Reissner-
Nordstr¨ om solution. This is in principle straightforward, realize that we can also illustrate a maximally extended as will be briefly discussed at the end of Sec. XI. black hole (Fig. 16).
X
Flat spacetime
The simplest possible spacetime to which we may apply the absolute scheme, is flat Minkowski in two dimensions. Choosing a field of generating four-velocities that is constant, with respect to standard coordinates (t, x), the resulting absolute geometry is flat and can be embedded as a plane. If we choose some more disordered field of four-velocities we can however get an embedding with no apparent symmetries at all. There is however another Figure 16: To the left: A Kruskal diagram of a maximally choice of generators that will produce a regular surface,11 extended black hole. To the right: an embedding of the absolute geometry with generators at fixed radius in the as is illustrated in Fig. 15. exterior regions and at fixed Schwarzschild time in the interior regions (the full drawn lines in the diagram). While all symmetries are preserved in this picture, it is hard to see how one can move between the different regions. Since the generators are null at the horizons, making the absolute distance along these lines zero, all the points along the null lines coming from the Kruskal origin sit at the connecting point in the embedding. Thus a trajectory passing one of the horizons in the diagram will pass through the connecting point in the embedding. However, where it will end up is not evident from the embedding alone. Through a more well behaved set of generators one can remove this obscurity at the cost of loosing manifest symmetry, as will be briefly discussed in Sec. XII. Having seen the absolute version of the extended black hole, we can also figure out how the extended ReissnerNordstr¨ om black hole must look. At all the cusps in the embedding, four locally cone-like surfaces must meet. Otherwise, as is apparent from Sec. X, the spacetime will not be complete. I will leave to the readers imagination the specifics of how to extend the Reissner-Nordstr¨ om embedding depicted in Fig. 14.
Figure 15: To the left: Minkowski spacetime with a certain set (as discussed in the main text) of worldlines (the thick full drawn lines). To the right: The corresponding absolute geometry embedding. Note that the conical surfaces are not closed as one goes around in the space direction, but rather they consist of very tightly rolled layers with no end. In the right and left regions, of the Minkowski diagram, we have chosen so called Rindler observers as generators.12 In the top and bottom regions we are using timelike geodesics converging at the origin as generators. The universe as perceived from this set of observers is known as a Milne universe (in two dimensions).13 It is obvious from the embedding that there is a (Lorentzian) Killing field directed around the conical surfaces. Imagining the corresponding field in the diagram, we realize that it is in fact the Killing field connected to continuous Lorentz transformations.
XII
Other spacetimes
So far in the embedding examples, we have restricted ourselves to Lorentzian spacetimes with a Killing symmetry and also to generators that manifestly preserve this symmetry. The absolute scheme is however completely general. When applying the scheme to the Kruskal black hole, we do not have to let the generators be either parallel or orthogonal to the local Killing field, as we did XI Extended black hole before. Instead we could for instance use geodesic freefallNotice the similarity between the Kruskal diagram of a ers, originally at rest along a t = 0 line in the standard maximally extended Schwarzschild black hole (see e.g. Kruskal coordinates. My best guess is that the correRef. 1) and the Rindler diagram to the left in Fig. 15. sponding embedding would resemble a tortoise shell. Having seen the interior and exterior regions of a (non– We can also consider spacetimes where there is no extended) black hole in the absolute scheme (Fig. 7), we Killing symmetry. As an example one could study a radial line through a collapsing thin shell of matter. (Here 9
there are local Killing fields but no global Killing field.) As a first try, one might choose observers at fixed x as generators. Outside of the shell we would (via the Birkhoff theorem) have a picture similar to Fig. 7. Inside the shell we would have a flat (though it may be rolled up) surface, with straight generating lines. Whether these two pieces can be put together in some meaningful manner I have yet to discover. Maybe one will find that another set of observers will be needed to join the two spacetime Figure 18: A flat spacetime visualization of a radial line regions together. through a star. The lightcones are everywhere, by definition, one proper time unit high, and two proper length units wide. The dashed circle illustrates that when we acXIII Toy models tually go to a specific region, the lightcones will appear While we can use the absolute scheme to produce pictures as they do at infinity. representing exact solutions to Einstein field equations, we can also do the opposite. Suppose that we have a surhourglass embedding there is a shorter distance between face, say a plane, and we specify an angle as a function of two Schwarzschild time lines inside the star than outside. position on the plane. Letting the angle correspond to the From the flat lightcone model we must deduce this fact. direction of the generators, it is straightforward to find Also one looses the visual connection to the concept of the corresponding Lorentzian metric (just do the inverse curved spacetime. transformation of Eq. (6)). We may insert this metric into The flat diagram technique however has the virtue of some program (say Mathematica) and let it calculate the being extendable to 2+1 dimensions. The scheme outcorresponding Einstein tensor and thus, through the field lined in this article can also be used in 2+1 dimensions, equations, also the energy-momentum tensor.14 We may but to produce a faithful image we would need a flat absee the solution as a purely two-dimensional solution. Alsolute spacetime. Then we could embed little lightcones ternatively we may see it as a four-dimensional solution of constant opening angle and size. To demand a Euassuming that we add two dimensions corresponding to clidean absolute spacetime is however quite restrictive, internally flat planes. In Fig. 17 we see an example of and it seems better to allow the lightcones to vary in such a toy model spacetime. apparent width and height. I will leave to the reader’s imagination how this technique could be applied to visualize warp drive, rotating black holes, the Big Bang and so on.
XV
Comparison to other work
There are, to the author’s knowledge, three other distinctly different techniques of visualizing curved spaceFigure 17: A crude illustration of a toy model for warp time using embedded surfaces. drive Marolf15 presents a way of embedding a two-dimensional Lorentzian metric in a 2+1 dimensional Minkowski spacetime (visualized as a Euclidean 3-space). From the look of the spacetime in Fig. 17 we might call L. C. Epstein16 presents a popular scientific visualthe propulsion mechanism ’twist drive’ rather than warp ization of general relativity. The underlying theory rests drive. I will leave to the readers imagination to visualize on the assumption of an original time independent, dia spacetime that more deserves the name warp drive. agonal, Lorentzian line element. Rearranging terms in this line element one can get something that looks like a new line element, but where the proper time is now a XIV Other methods coordinate. The ’space-propertime’ can be embedded as In this article we have seen how one may use curved sur- a curved surface, from which many spacetime properties faces, with local Minkowski systems, to visualize for in- can be deduced. In a previous article17 , I assumed a time independent stance gravitational time dilation. This can also be achieved using a flat diagram, letting the space and time scales be Lorentzian line element. I then found another line eleencoded in the size (and shape) of the local lightcones as ment, also time symmetric, that is positive definite and geodesically equivalent to the original line element. The depicted in Fig. 18. The disadvantage with this technique is that it is more resulting geometry can be embedded as a curved surabstract than the hourglass embedding (Fig. 4). In the face as in Fig. 19. The method can be used to explain 10
straight lines in a curved spacetime, the meaning of forces systems or proper times to give a feeling for how geomas something that bends spacetime trajectories, etc. etry can explain how time can run at different rates at different places, or how space itself can expand. Most importantly we emphasize the point of view that gravity, according to general relativity, is about shapes – not forces and fields.
XVII
Questions for students of general relativity
Here are a handful of questions regarding applications of the absolute scheme. The answers are given in the next section.
Figure 19: Illustration of how straight lines in a curved spacetime can explain the motion of upwards-thrown apples. The lines can be found using a little toy car that is rolled straight ahead on the surface. Each of the three techniques outlined here together with the absolute scheme of this article has different virtues (and drawbacks). Depending on the audience they can all be used to explain aspects of the theory of general relativity.
XVI
1. The hour-glass shaped embedding of Fig. 4 illustrates a spacetime where time “runs slower” in a local region. How would a corresponding embedding look that illustrates how time can run faster in a local region? 2. Can you, using the technique of this article, illustrate a two-dimensional spacetime that is closed in space and time and with no vertices (by a vertex we mean a point from which the Minkowski systems point either outward or inward)? 3. Can a spacetime of the type specified in the preceding question be flat? 4. In exam periods students often need more time to study. Consider as a primary spacetime a flat plane with uniformly directed Minkowski systems. How would you alter this spacetime to ensure that there is sufficient time to study? Include the worldline of the student in need, as well as the worldline of the teacher bringing the exam.
Comments and conclusions
The absolute scheme as presented in this article, can be applied to any spacetime, giving a global positive definite metric. Together with the generating field of fourvelocities, it carries complete information about the original Lorentzian spacetime. While there are mathematical applications of this scheme 5. Imagine an upright-standing cylindrical surface, with (see Ref. 18), we have here focused on its pedagogical upward-directed Minkowski systems. The all-famous virtues. At the level of students of relativity, a study of experiment where one twin goes on a trip and later the mathematical structure itself may have some pedreturns to his brother, can be illustrated by two agogical virtue. In particular it is instructive to see an worldlines on the cylinder, one going straight up alternative representation of the shape of spacetime. and the second going in a spiral around the cylinThe main pedagogical virtue is however that, applied der (intersecting the first one twice). This scenario to two-dimensional spacetimes, the absolute scheme endiffers from the standard one in that no acceleration ables us to make embeddings that illustrate the meaning was needed by either twin for them to still reunite. of a curved spacetime. The same question applies however. Will the twins While such an embedding is not unique, due to the have aged differently? freedom in choosing generators as well as the freedom of the embedding, it gives a completely faithful image of Answers to students questhe true spacetime geometry. With only a basic under- XVIII standing of Minkowski systems, a complete knowledge of tions the embedded parts of the Lorentzian geometry can be deduced from the surface. For instance we can figure out Here are (the) answers to the questions of the preceding how much a clock has ticked along a certain path, or what section. path a thrown apple will take. 1. Instead of a dip in the hour-glass shaped embedding Also, many applications require no knowledge at all of (decrease of the radius towards the middle), we have special relativity. We do not need to mention Minkowski a bulge (increase of radius towards the middle). 11
2. Yes. For instance a torus, with the Minkowski sys- The orthogonal vector v µ can be expressed via tems directed along the smaller toroidal circumfer ence. -1 1 0 v µ = √ ǫµρ gρα uα where ǫµρ = . (22) g 3. Yes. Make a tube out of a paper by taping two op1 0 posite ends together. Flatten the tube, preferably so that the tape is a bit away from the two folds Here g = −Det(gµν ). Through this definition v µ is within that emerges. Roll the flattened tube, so that the 180◦ clockwise from uµ (looking at the coordinate plane tape describes a complete circle and connect the from above, assuming t up and x to the right). Inserting meeting paper ends by some more tape. If the local Eq. (22) into Eq. (20), using Eq. (21), we readily find Minkowski systems on this shape are given a unis form direction, the corresponding Lorentz-geometry 1 − v2 v will be flat. . (23) δ µ ν + √ ǫµρ gρν uν = ± g ξ α ξα 4. Make a sufficiently high and steep bump in the plane, while keeping the direction of the Minkowski This is a linear equation for uµ that can easily be solved. systems (as seen from above the former plane). Make For the particular case of gµν = Diag(gtt , gxx ) and ξ µ = sure that the student’s trajectory passes straight (1, 0) we find over the peak, while the teacher’s trajectory misses r gtt 1 −v µ it. . (24) u =± ,√ 1 − v 2 gtt −gxxgtt 5. Oh yes. What time the twins experience is determined by their respective spacetime trajectories. If So now we have a general expression for the generating the trajectory of the traveling twin is tilted almost four-velocity, expressed in terms of the Killing velocity v. as much as a photon trajectory, he will have aged The ± originates from the ± in the previous expression very little compared to his brother. There is also an for K. article19 that deals with this thought-experiment.
A
Finding uµ as a function of v
B
Vector transformation by diagonalization
Here is a derivation of the general expression for uµ as Under the diagonalization of the absolute metric (as pera function of the Killing velocity v. Let us define v µ as formed in Sec. VI) a general vector transforms according µ a vector perpendicular to u , normalized to −1. Also we to µ denote the Killing field by ξ . The vectors as seen relative µ to a system comoving with u are displayed in Fig. 20. g¯tx x x ′µ t q = q + (25) q ,q . g¯tt u
Using the lowered version of Eq. (14) and the definition of the absolute metric, Eq. (5), we find after simplification
µ
ξµ
−gxx 2v g¯tx √ = . g¯tt 1 + v 2 −gtt gxx
(26)
For the particular case of q µ = uµ , using Eq. (14), we find after simplification r 1 1 − v2 −v gtt √ u′µ = ± . (27) , 1 − v 2 gtt 1 + v 2 −gxx gtt Figure 20: The Killing field in local coordinates comoving with uµ . This expression can be used to find the generating lines in the new coordinates. We simply integrate u′t and u′x We have then numerically, with respect to the parameter τ , to find t′ (τ ) and x(τ ). These lines can then easily be mapped to an ξ µ = (uµ + vv µ )K. (20) embedding of the absolute geometry. Here the variable K may take positive or a negative values. Contracting both sides with themselves, we solve for C Regarding the absolute area K to find r Consider a small square in the coordinates we are usξ α ξα K=± . (21) ing. Then consider two different choices of generators, 1 − v2 vµ
12
u1 µ and u2 µ . Assume also that the coordinate square is small enough that the generating fields (as well as the metric) can be considered constant within the surface. We denote the absolute area of the coordinate square for the two representations by dA1 and dA2 . Knowing that the absolute area is independent of the choice of coordinates we may evaluate each area in the local Minkowski systems x1 µ and x2 µ , comoving with the corresponding generator. In these systems the original square will be deformed, but the absolute area will exactly equal the coordinate area. Since the two Minkowski systems are related via the Lorentz transformation, which preserves coordinate areas, we know that the coordinate areas are equal and thus also dA1 = dA2 . Because the argument applies to arbitrary pairs of generators, it follows that the absolute area is independent of the choice of generators. In N dimensions we can, by a completely analogous argument, show that the N -volume of the absolute metric is independent of the choice of generators.
D
unbarred derivatives to their covariant analog and using the definition of covariant derivatives, Eqs. (28) and (30) we obtain ! ¯ q¯µ D qµ D 1 ¯ τ = p2(uµ q µ )2 − 1 Dτ p2(uµ q µ )2 − 1 D¯ qα qβ (−g µρ + 2uµ uρ ) 2(uµ q µ )2 − 1 ×(uβ ▽α uρ + uρ ▽α uβ − uβ ▽ρ uα ). (31) +2
Here we have a manifestly covariant relation. We can expand this expression, and simplify it somewhat using uµ uµ = 1
¯ qµ D¯ 2 = 2 ¯ 2k − 1 D¯ τ h 1 Dq µ D ρ kq µ − 2 (q ρ q σ ▽σ uρ + uρ q ) 2 Dτ 2k − 1 Dτ −kq α ▽α uµ + uµ q β q α ▽α uβ i (33) +kq α ▽µ uα − 2kuµ q α uρ ▽ρ uα .
First we derive an expression for a general absolute fourvelocity q¯µ , in terms of Lorentzian quantities dxµ dτ dxµ = d¯ τ dτ d¯ τ s gµν dxµ dxν = qµ −gµν dxµ dxν + 2uµ uν dxµ dxν v u =q t µu
E
dxµ dxν dτ dτ
1 = qµ p . 2(uµ q µ )2 − 1
(29)
Using the corresponding definition of the original affine connection, together with the expressions for the absolute metric and its inverse given by Eqs. (5) and (6), we can write this as ¯ µ = Γµ − 2uµ uρ Γραβ + (-g µρ + 2uµ uρ ) Γ αβ αβ × (∂α (uρ uβ ) + ∂β (uρ uα ) − ∂ρ (uα uβ )) . (30) ¯
µ
¯ uµ D¯ ¯ τ = 0. D¯
(28)
Notice that choosing q µ = uµ yields u ¯µ = uµ as we realized before. To covariantly relate the absolute four-acceleration to the Lorentzian quantities we need an expression for the absolute affine connection, i.e. the affine connection for the absolute metric ¯ µ = 1 g¯µρ (∂α g¯ρβ + ∂β g¯ρα − ∂ρ g¯αβ ) . Γ αβ 2
Geodesic generators
Consider a trajectory that everywhere is tangent to the generating field so that q µ = uµ . Also, assume that the µ ρ generators are geodesics Du Dτ = 0, or equivalently u ▽ρ µ µ u = 0. Using the normalization relation u uµ = 1, from which it follows that uµ ▽α uµ = 0, we immediately see that Eq. (31) reduces to
1 −1 + 2uµ uν
k ≡ uµ qµ . (32)
The first two relations follow directly from the normalization of uµ , and the latter is a definition introduced for compactness. The resulting expansion is given by
Covariant relation for the absolute four-acceleration
q¯µ =
u µ ▽α u µ = 0
q¯ Now we evaluate DD¯ ¯ τ in an originally freely falling system where the original affine connection vanishes. Setting all
(34)
Thus if the original generators are geodesic then they are geodesic also relative to the absolute metric. Through the perfect symmetry in transforming from the absolute to the Lorentzian metric and back, we have derived implicitly that if the absolute generators are geodesics they will also be geodesics in Lorentzian spacetime. We conclude that if and only if the original generators are geodesic then they will be geodesic in the absolute spacetime. To get an intuitive feeling for the result we just derived consider a straight generating line (in the absolute sense) on an embedded surface. Any small deviation from this line will introduce negative contributions to the proper time. A rigorous argument is that an infinitesimal variation of a trajectory (with fixed endpoints), around a straight generating line, will to first order in the variation parameter not affect the absolute length of the trajectory. Also we know that the Lorentzian distance along a trajectory is shorter than or equal to the absolute distance (the equality holds if and only if we follow a generator).
13
Hence we cannot gain proper time to first order in the variational parameter as we vary the trajectory. Thus the Lorentz proper time is maximized by the original trajectory. On the other hand, if the absolute generating line is curving relative to the surface, it seems plausible that we could gain proper time by taking a path on the outside of the curving generating line. Thus a non-geodesic absolute generator would imply a non-geodesic Lorentzian generator. The result that the generators are absolute geodesics if and only if they are Lorentzian geodesics, is thus intuitively understandable.
F
For this in turn to hold for all q α it is necessary to have ▽α uµ = 0. This also immediately satisfies the above necessary constraints on antisymmetry and generator geodesics. That it is also sufficient for geodesic equivalence follows directly from Eq. (35). Thus the absolute metric will be geodesically equivalent to the original one, if and only if ▽α uµ = 0.
G
(41)
Proving that ▽µ uν = 0 everywhere implies ultrastatic spacetime
Deriving necessary and sufficient conditions for geodesic equivalence
Assuming ∇µ uν = 0, the Frobenius condition20 u[µ ∇ν uρ] = 0 is trivially satisfied. This means that there exists (locally) a slice for which uµ is normal. Introducing coordinates such that t = const in every slice and letting the To investigate if it is possible to completely retain the spatial coordinates follow the congruence connected to µ original geodesic structure in the absolute metric, we set u , the line element takes the form ¯ q¯µ Dqµ D ¯ τ = 0 and Dτ = 0 in Eq. (33). The resulting equation D¯ is given by 0 gtt . (42) gµν kq µ ρ σ q q ▽σ uρ − kq α ▽α uµ + uµ q β q α ▽α uβ 0= − 2 0 gij 2k − 1 +kq α ▽µ uα − 2kuµ q α uρ ▽ρ uα .
(35)
In this particular system uµ = If this equation is to hold for all directions, q , it must find hold for the particular case q α = uα . Inserting this and using the normalization of uµ , only the second term sur- ▽α uβ ≡ ∂α uβ − Γραβ uρ vives = ... α
uα ▽α uµ = 0.
(36)
√ gtt δ t µ . Then we readily (43)
(44) 1 =√ gtt 1 t δ β (∂α gtt ) − (∂α gtβ + ∂β gtα − ∂t gαβ ) . (45) 2
Thus it is necessary to have geodesic generators to get all the geodesics ’right’. Assuming the generators to be geodesic – the last term in Eq. (35) dies. Multiplying the remaining four terms by qµ , we are after simplification Letting i and j denote general spatial indices and evalleft with another necessary constraint uating this equation for α = t, i and β = t, j (there are −k four different combinations) we readily find + k q α q µ ▽α uµ = 0. (37) 2k 2 − 1 ∂µ gtt = 0 (46) The expression within the parenthesis is zero if and only ∂t gij = 0. (47) if k = ±1. Assuming a future-like convention on both uµ
and q µ we cannot have a negative k, and k = 1 correThus in these particular coordinates, choosing a t-labeling sponds uniquely to uµ = q µ , a direction that we already such that gtt = 1, the metric takes the form considered. Thus, the expression outside the parenthesis must vanish. For this to hold for all directions q α , we 0 must have 1 . (48) gµν = ▽α u µ = − ▽µ u α . (38) 0 gij (x) Using this necessary antisymmetry in Eq. (35) we are left A spacetime where the metric can be put in this form is with called ultrastatic. Thus ▽µ uν = 0 implies an ultrastatic 0 = −kq α ▽α uµ + kq α ▽µ uα . (39) spacetime. The converse, choosing the preferred observers Lowering this with gµν and using the necessary antisym- in the ultrastatic spacetime as observers, is also obviously true. metry again, we obtain q α ▽α uν = 0.
(40) 14
References
[13] W. Rindler, Essential Relativity: Special, General and Cosmological (Springer Verlag, New York, 1977), pp. 204-207.
[1] R. D’Inverno, Introducing Einstein’s Relativity, (Oxford Univeristy Press, Oxford 1998), pp. 99-101.
[14] This may well violate the energy conditions. [2] Eq. (1) defines a so called distance function, or a metric. It can also be used considering events where [15] D. Marolf, “Spacetime Embedding Diagrams for Black Holes,” Gen. Relativ. Gravit., 31, 919-944 dx > cdt. Then dτ 2 is negative which simply means (1999). that it is related to spatial distance rather than temporal distance. A distance function like Eq. (1) cor[16] L. C. Epstein, Relativity Visualized (Insight Press, responds to a flat spacetime, but see Eq. (10) for an San Francisco, 1994), chaps. 10-12. example of a distance function corresponding to a curved spacetime. [17] R. Jonsson, “Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature,” [3] S. Weinberg, Gravitation and cosmology (John Wiley Gen. Relativ. Gravit. 33, 1207-1235 (2000). & sons, New York, 1972), pp. 337-338. [18] S. W. Hawking and G. F. R. Ellis, The large [4] Assuming that there is no black hole inside the crust. scale structure of space-time, (Cambridge University Press, Cambridge, 1973), p. 39. [5] As was pointed out to me by Ingemar Bengtsson, this relation is also used by Hawking & Ellis (Ref. 18), al[19] T. Dray, “The twin paradox revisited,” Am. J. Phys. though for completely different purposes than those 58, 822-825 (1989). of this article. [20] The Frobenius condition in explicit form reads: [6] If we had instead considered a metric of the form uµ (∇ν uρ − ∇ρ uν ) + uρ (∇µ uν − ∇ν uµ ) + uν (∇ρ uµ − −g µν + αuµ uν , where α is some general number, the ∇µ uρ ) = 0. inverse would have been −g µν + α/(α − 1)uµ uν . It is only in the case α = 2 that we can simply raise the indices of the absolute metric with the original metric to get the inverse of the absolute metric. 1
[7] The gamma factor is defined as γ = (1−v 2 )− 2 where v is the relative velocity. With this definition, it follows from Eq. (1) that dτ = dt γ . [8] A geometry has a Killing symmetry if there exists a vector field (called a Killing field) ξ µ such that when we shift our coordinates xµ → xµ + ǫξ µ – the metric has the same form. As an example we can consider a geometry that can be embedded as a curved surface. Then there exists a Killing field directed around the surface (in the azimuthal direction) with a length proportional to the embedding radius. Also, if there are coordinates where the metric is independent of one coordinate, then there is a Killing symmetry with respect to that coordinate. [9] t′ = t + φ(x) where Diag(¯ gtt , g¯xx −
(¯ gtx ) g ¯tt
2
dφ dx
=
g ¯tx g ¯tt
′ gives g¯µν =
)
[10] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (W. H. Freeman and Company, New York, 1973), p. 841. [11] Suggested to me by Sebastiano Sonego. [12] W. Rindler, Relativity: Special, General and Cosmological (Oxford University Press, Oxford, 2001), pp. 267-272.
15
Inertial forces and the foundations of optical geometry Rickard Jonsson Department of Theoretical Physics, Chalmers University of Technology, 41296 G¨ oteborg, Sweden E-mail:
[email protected] Submitted 2004-12-10, Published 2005-12-08 Journal Reference: Class. Quantum Grav. 23 1 Abstract. Assuming a general timelike congruence of worldlines as a reference frame, we derive a covariant general formalism of inertial forces in General Relativity. Inspired by the works of Abramowicz et. al. (see e.g. Abramowicz and Lasota 1997 Class. Quantum Grav. 14 A23-30), we also study conformal rescalings of spacetime and investigate how these affect the inertial force formalism. While many ways of describing spatial curvature of a trajectory has been discussed in papers prior to this, one particular prescription (which differs from the standard projected curvature when the reference is shearing) appears novel. For the particular case of a hypersurface-forming congruence, using a suitable rescaling of spacetime, we show that a geodesic photon is always following a line that is spatially straight with respect to the new curvature measure. This fact is intimately connected to Fermat’s principle, and allows for a certain generalization of the optical geometry as will be further pursued in a companion paper (Jonsson and Westman 2006 Class. Quantum Grav. 23 61). For the particular case when the shear-tensor vanishes, we present the inertial force equation in threedimensional form (using the bold face vector notation), and note how similar it is to its Newtonian counterpart. From the spatial curvature measures that we introduce, we derive corresponding covariant differentiations of a vector defined along a spacetime trajectory. This allows us to connect the formalism of this paper to that of Jantzen et. al. (see e.g. Bini et. al. 1997 Int. J. Mod. Phys. D 6 143-98). PACS numbers: 04.20.-q, 95.30.Sf
1. Introduction Inertial forces, such as centrifugal and Coriolis forces, have proven to be helpful in Newtonian mechanics. Quite a lot of attention has been given to generalizing the concept to General Relativity. In fact the last fifteen years there has been a hundred or so papers related to inertial forces in General Relativity. For an overview see [1]. Many of these articles are related to particular types of spacetimes, and special types of motion. There are also a few that are completely general. This article is of the
Inertial forces and the foundations of optical geometry
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latter kind. The scope is to develop a covariant formalism, applicable to any spacetime, and any motion of a test particle, using an arbitrary reference congruence of timelike worldlines. In view of the already existing bulk of papers we will keep the introductory remarks to a minimum here and just outline the contents of the article. In section 2 we introduce the basic notation of the article. In section 3 we derive a spatial curvature measure for a spacetime trajectory. We do this by projecting the trajectory down along the reference congruence onto the local time slice. We also derive how the time derivative of the speed relative to the congruence is related to the four-acceleration of the test particle. The resulting equations we put together to form a single equation that relates the test particle four-acceleration (and four-velocity) to the spatial curvature, the time derivative of the speed and the local derivatives of the congruence four-velocity. The terms connected to the congruence derivatives can be regarded as inertial forces. We also express the four-acceleration of the particle in terms of the experiences comoving forces, as well as in terms of the forces as given by the congruence observers. In section 4 we introduce a different kind of spatial curvature measure. The new curvature measure is such that when we are following a straight line with respect to this measure, the spatial distance traveled (as defined by the congruence) is minimized (with respect to variations in the spatial curvature). This is in fact not the case for the standard projected curvature when the congruence is shearing. Using the new curvature measure we create a slightly different inertial force formalism. In section 5 we consider general conformal rescalings of spacetime, and how these affect the inertial force formalism. In section 6 we consider a foliation of spacetime into spacelike time slices and a corresponding orthogonal congruence. Given a labeling t of the time slices we rescale away time dilation with respect to t. Relating spatial curvature etc to the rescaled spacetime, but considering the real (non-rescaled) forces, we find an inertial force formalism that is very similar to the already derived formalisms of this paper. We show that a geodesic photon always follows a straight line in the sense of section 4. We also show that it follows a straight line in the projected sense if the congruence is shearfree. These results allows certain generalizations of the optical geometry (for an introduction to optical geometry see e.g. [2]) as will be pursued in a companion paper [3]. In section 7 we show that the fact that a geodesic photon follows a straight line in the new sense relative to the rescaled spacetime follows from Fermat’s principle. In section 8 we introduce two new curvature measures related to geodesic photons, and what we see as straight, and use these in the inertial force formalism. In section 9 we summarize the inertial force formalisms (excepting those related to rescalings) connected to the various introduced curvature measures. In section 10 we rewrite the four-covariant formalism as a three-dimensional formalism, for the particular case of vanishing shear (assuming only isotropic expansion). While fully relativistically correct, in this form the inertial force formalism is very similar
Inertial forces and the foundations of optical geometry
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to its Newtonian counterpart. In section 11 we derive a spacetime transport law of a vector, corresponding to spatial parallel transport with respect to the spatial geometry defined by the reference congruence. In section 12 we consider an alternative approach to inertial forces resting on the transport equation of section 11. In section 13 we connect to the approach of Jantzen et. al. In section 14 we conclude the article. Then follows the appendixes. 2. The basic notation In a general spacetime, we consider an arbitrary reference congruence of timelike worldlines of four-velocity η µ . Each such worldline corresponds to events at a single spatial point in our frame of reference. We can split the four velocity v µ of a test particle into a part parallel to η µ and a part orthogonal to η µ : v µ = γ(η µ + vtµ ).
(1)
Here v is the speed of the test particle relative to the congruence and γ is the corresponding γ-factor. The vector tµ is a normalized spatial vector (henceforth vectors that are orthogonal to η µ will be referred to as spatial vectors), pointing in the (spatial) direction of motion. Projected spatial curvature and direction of curvature we will denote by R and µ n , the latter being a normalized spatial vector. By projected curvature we mean that we project the spacetime trajectory in question down along the congruence onto the local slice1 and evaluate the spatial curvature there. There are also several alternative ¯ and n definitions of curvature and curvature direction. In particular we will us R ¯ µ to denote what we will call the ’new-straight’ curvature and curvature direction, to be introduced in section 4. Throughout the article we will use c = 1 and adopt the spatial sign convention (−, +, +, +). The projection operator2 along the congruence then takes the form P α β ≡ g α β + η α ηβ . We also find it convenient to introduce the suffix ⊥. When applied to a four-vector, as in [K µ ]⊥ , it selects the part within the brackets that is perpendicular to both η µ and tµ . 3. Inertial forces using the projected curvature The objective with this section is to go from the spacetime equations of motion for a test particle and derive an expression for R, nµ and the time derivative of v, in terms 1
If the congruence has no rotation there exists a finite sized slicing orthogonal to the congruence. If the congruence is rotating we can still introduce a slicing that is orthogonal at the point in question. It is easy to realize that whatever such locally orthogonal slicing we choose, the projected curvature and curvature directions will be the same, and are thus well defined. 2 Applying this tensor to a vector extracts the spatial (i.e. orthogonal to η µ ) part of the vector.
4
Inertial forces and the foundations of optical geometry of v and tµ for given forces and congruence behavior. 3.1. The projected curvature and curvature direction
The idea behind the projected curvature with respect to the congruence is illustrated by figure 1. Notice that the time-slice we are depicting is only assumed to be orthogonal to the congruence at the point where the test particle worldline intersects the slice. t
Figure 1. A 2+1 illustration of a projection of a spacetime trajectory onto a time slice, seen from freely falling coordinates, locally comoving with the reference congruence.
Taking the covariant derivative
D Dτ
along the test particle worldline, of (1) we readily
find Dtµ Dv µ = γ 2 [aµ ]⊥ + γ 2 v[tα ∇α η µ ]⊥ + γv . (2) Dτ ⊥ Dτ ⊥ Here aµ is the four-acceleration of the congruence. Now we want to relate the covariant derivative of tµ in (2) to the projected curvature. As concerns the ⊥-part of this we can consider the covariant derivative to stem from a two-step process. First we transport it along the curved projected trajectory, then we Lie-transport it up along the congruence as depicted in figure 13 . Alternatively, we may in the style of [4], consider the worldsheet spanned by the congruence lines that are crossed by the test particle worldline. On this sheet we can uniquely extend the forward vector tµ , defined along the test particle worldline, into a vector field that is tangent to the sheet, normalized and orthogonal to η µ . Considering an arbitrary smooth extension of this field tµ around the sheet, the projected µ curvature can be written as nR = [tα ∇α tµ ]⊥ . We also realize that, as concerns the ⊥part, this field will be Lie-transported into itself (in the η µ direction). Thus we have [η α ∇α tµ ]⊥ = [tα ∇α η µ ]⊥ . Then we we can write Dtµ = [γ(η α + vtα )∇α tµ ]⊥ (3) Dτ ⊥
Letting ds = vdτ0 = vγdτ , we have in freely falling coordinates [dtµ ]⊥ = from which (4) follows immediately. 3
nµ R ds
+ [tα ∇α η α ]⊥ dτ0
Inertial forces and the foundations of optical geometry = γ[tα ∇α η µ ]⊥ + γv
nµ R
5 (4)
Using this together with (2) we get µ 1 Dv µ µ α µ 2n = [a ] + 2v [t ∇ η ] + v . (5) α ⊥ ⊥ γ 2 Dτ ⊥ R So here is a general contravariant expression for the local projected curvature of a spacetime trajectory.
3.2. The speed change per unit time Now we would like a corresponding expression for the speed change per unit time. We have γ = −v α ηα . Differentiating both sides of this expression with respect to the proper time τ along the trajectory readily yields Dv α Dη α dv = − ηα − vα (6) γ 3v dτ Dτ Dτ ! Dv α vα Dη α = − − vtα − γ(ηα + vtα ) (7) Dτ γ Dτ Dv α Dη α − γvtα . (8) = vtα Dτ Dτ In the last equality we used the normalization of η µ and v µ . Notice that the differentiation is along the trajectory in question, so we have Dη α = γ(η ρ + vtρ )∇ρ η α . (9) Dτ Using this in (8) we readily find 1 Dv α dv tα = tα (η ρ + vtρ )∇ρ η α + γ . 2 γ Dτ dτ So here we have a covariant equation for the speed change as well.
(10)
3.3. Putting it together Multiplying (10) by tµ and adding it to (5), we get a single vector equation that relates the four-acceleration to both the speed change and the projected spatial curvature µ 1 Dv α µ Dv µ µ α µ 2n + = [a ] + 2v [t ∇ η ] + v t t + α α ⊥ ⊥ γ2 Dτ ⊥ Dτ R dv (11) tµ tα (η ρ + vtρ )∇ρ η α + γ tµ . dτ This can be simplified to µ 1 µ Dv α dv µ µ α µ µ α ρ 2n P = a + 2v [t ∇ η ] + vt t t ∇ η + γ t + v . (12) α α ρ α ⊥ γ2 Dτ dτ R So here we have a generally covariant relation between the four-acceleration, the projected curvature and the speed change.
Inertial forces and the foundations of optical geometry
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3.4. Experienced forces and the kinematical invariants To make it more clear what an observer performing the specified motion actually experiences, we can rewrite the left hand side of (12) in terms of the experienced forward thrust4 Fk and the experienced sideways thrust F⊥ . This is a simple exercise of special relativity performed in Appendix B. We may then write 1 µ µ = aµ + 2v [tα ∇α η µ ]⊥ + vtµ tα tρ ∇ρ ηα (13) γF t + F m k ⊥ 2 mγ dv nµ + γ tµ + v 2 . dτ R µ µ Here m is a normalized vector perpendicular to t and η µ . We may alternatively express (13) in terms of the kinematical invariants of the congruence, defined in Appendix A. From the definitions follows [5]5 ∇ν ηµ = ωµν + θµν − aµ ην .
(14)
We then readily find h i 1 µ β µ µ µ µ = a + 2v t (ω + θ ) γF t + F m + vtα tβ θαβ tµ (15) β β k ⊥ ⊥ mγ 2 nµ dv + γ tµ + v 2 . dτ R Here we have a covariant expression for the relation between spatial projected curvature and the speed change per unit time in terms of the experienced forces, given the kinematical invariants of the congruence. 3.5. Forces as experienced by the congruence observers It may also be interesting to know what forces are needed to be given, by the observers following the congruence, in order to keep the test particle on the path in question. This again is a simple exercise of special relativity carried out in Appendix C where we readily show that 1 µ Dv α 1 P α = (Fck tµ + Fc⊥ mµ ). (16) 2 γ Dτ γm Here Fck and Fc⊥ are the experienced given forces parallel and perpendicular to the direction of motion. When expressing the forces as given by the congruence observers, it seems reasonable to express the velocity change relative to local congruence time dτ0 , given simply by dτ0 = γdτ . Then (15) takes the form h i 1 (17) Fck tµ + Fc⊥ mµ = aµ + 2v tβ (ω µ β + θµ β ) + vtα tβ θαβ tµ ⊥ mγ nµ dv µ t + v2 . + γ2 dτ0 R 4
By definition the observers forward direction is the direction from which he sees the congruence points coming (assuming he has some way of seeing them). 5 Note that the sign of ωµν is a matter of convention.
Inertial forces and the foundations of optical geometry
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Here we have thus the inertial force equation explicitly in terms of the given forces. As a simple application we may consider a rotating merry-go-round with a railway track running straight out from the center. Suppose that we let a railway wagon move with constant speed along the track. Then (17) gives us the forces on the railway track6 . 3.6. Discussion Looking back at (15) and (17), it is easy to put names to the various terms. On the left hand side we have the real experienced forces, as received and given respectively, in the forward and sidewards direction. On the right hand side we have first three terms that we may call inertial forces7 given that we multiply them by −m: Acceleration : − maµ
(18) h
Coriolis :
− 2mv tβ (ω µ β + θµ β )
Expansion :
− mvtα tβ θαβ tµ .
i
⊥
(19) (20)
From a Newtonian point of view we may be tempted to call the first inertial force ’Gravity’ rather than ’Acceleration’. On the other hand, for the particular case of using points fixed on a a rotating merry-go-round as reference congruence, the term would correspond to what we normally call centrifugal force. To avoid confusion we simply label this term ’Acceleration’. As regards the second term the naming is quite obvious8 . The third term is non-zero if the reference grid is expanding or contracting in the direction of motion. For positive tα tβ θαβ the term has the form of a viscous damping force although for negative tα tβ θαβ it is rather a velocity proportional driving force. The existence of this term illustrates (for instance) that if we are using an expanding reference frame, a real force in the direction of motion is needed to keep the velocity relative to the reference frame fixed. The two last terms of (15) and (17) are µ 2n 2 dv µ t + v . (21) γ dτ0 R These we do not regard as inertial forces, but rather as descriptions of the motion (acceleration) relative to the reference frame. Notice that the formalism is well defined for arbitrary spacetimes and arbitrary timelike congruence lines. 3.7. A note on alternative interpretations Quite commonly the term that we are here denote ’Expansion’, is included with the dv/dτ0 -term (multiplied by −m) and these two terms are collectively denoted the ’Euler’ After we have calculated ω µ β and aµ (for this case θµ β = 0). See section 10.2. Actually exactly what we denote inertial force is subjective to a degree. For instance we could multiply all terms in (15) and (17) by γ and define the inertial forces accordingly. 8 As can be seen from (14) (multiplied by tβ ), the momentary velocities of the congruence points (relative to an inertial system momentarily comoving with the congruence) in the direction of motion is determined by tβ (ω µ β + θµ β ). Selecting the perpendicular part gives a measure of the sideways perpendicular velocities of the reference frame, naturally related to Coriolis. 6
7
Inertial forces and the foundations of optical geometry
8
force. This hides (or at least makes less manifest) the above mentioned feature that a real force is needed to keep a fixed velocity relative to an expanding reference frame. Indeed this lack was one of the original inspirations for making this paper. In section 12 we present an alternative approach to inertial forces resting on the notion of spatial parallel transport (of the relativistic three-momentum relative to the congruence). Then the expansion term arises naturally if we are using a norm-preserving law of spatial parallel transport. Also, quite commonly the last term, when multiplied by -m, in (15) and and (17) respectively, is denoted the centrifugal force. This notation is however not matching the standard definition, where the centrifugal force comes from the acceleration due to the rotation of the reference frame rather than from the motion of the particle relative to the reference frame. See appendix F for further discussion of this. If one interprets (like in e.g. [6]) the two terms related to accelerations relative to the reference frame (when multiplied by −m) as inertial forces – the whole equation takes a form of a balance equation between inertial forces. As interpreted in this article however, the inertial force equation is of the standard type Freal + Finertial = marelative , where the acceleration relative to the reference frame corresponds to the last two terms of (15) and (17). In Appendix F, we briefly review inertial forces in Newtonian mechanics and show that the derived formalism (and interpretation) of this paper is conforming (as far as that is possible) with the standard Newtonian formalism of inertial forces, in the limit of small velocities. We also discuss the possibility to view the terms related to the relative acceleration as inertial forces. For further understanding of the viewpoint that the last two terms are mere descriptions of the motion (acceleration) relative to the reference frame, see also section 12. 4. A different type of curvature radius In the preceding section we used somewhat different techniques in deriving the perpendicular and the parallel parts. One might argue that the derivation of the perpendicular part, i.e. the curvature, was in a sense less local than the derivation of the forward part. The heart of the matter lies in exactly where one measures spatial distances. In figure 2 we illustrate the difference between the on-slice distance d¯ s and the at-trajectory distance ds. To gain some intuition, we consider a finite slice, orthogonal to the congruence at the point where the test particle worldline intersects the slice, and a projection of the worldline down along the congruence onto the slice. The curvature radius, as defined in the preceding section, is such that when it is infinite, the on-slice distance is locally minimized9 . Perhaps it would be more natural however to define a curvature radius 9
Strictly speaking it is necessary for the projected curvature to vanish at the point in question in order for the projected trajectory to minimize the distance on the slice. Note however that the
Inertial forces and the foundations of optical geometry Time
9
Spacetime trajectory ds
d¯ s
Projected trajectory
Figure 2. The difference between at-trajectory distance ds and on-slice distance d¯ s.
such that when it is infinite, the at-trajectory distance is minimized. Obviously these two definitions will coincide if there for instance is a Killing symmetry, and we adapt the congruence to the Killing field. For this case ds = d¯ s, and the two curvature measures will coincide. But in general it is perhaps not so obvious that they will, and indeed we will find that they do not coincide in general. 4.1. Defining a straight line via a variational principle We would now like to introduce a new notion of straight trajectories, as those that minimize the integrated ds. We may formulate the problem of finding trajectories that are straight in the new sense via a variational principle. We thus introduce an action, for an arbitrary spacetime trajectory xµ (λ), connecting two fixed spacetime points s
dxµ dxν dλ. dλ dλ We define a corresponding Lagrangian as ∆s =
Z
Pµν
(22)
s
dxµ dxν . (23) dλ dλ Now we are interested in how a variation xµ (λ) → xµ (λ) + δxµ (λ) affects the action. Analogous (precisely) to the derivation of the Euler Lagrange equations we find (to first order in the change δxµ ) L=
δ∆s =
Pµν
Z "
#
d ∂L ∂L δxµ dλ. − µ µ ∂x dλ ∂ dx dλ
(24)
This expression holds whatever parameterization we choose. In particular choosing the integrated local distance s itself as parameter, the Lagrangian function is unit10 along the trajectory. For this choice of parameter, expanding (24) using (23) is particularly projected curvature, as defined in the previous section, along the test particle worldline will not in general coincide with the spatial curvature of the corresponding point along the projected trajectory (except at the point of intersection). 10 So L = 1, meaning that the absolute derivatives of L vanishes, whereas in general the partial derivatives do not.
10
Inertial forces and the foundations of optical geometry simple, and the result is !#
Z "
1 dxβ ∂Pαβ dxα dxβ d δ∆s = P δxµ ds (25) − 2 µβ 2 ∂xµ ds ds ds ds # Z " dxα dxβ dxρ dxβ d2 xβ 1 (∂µ Pαβ ) − 2(∂ρ Pµβ ) − 2Pµβ 2 δxµ ds (26). = 2 ds ds ds ds ds Also, using
dτ ds
=
1 , γv
it is easy to prove that
d2 xβ 1 d = 2 ds γv dτ
1 dxβ γv dτ
!
= ... = −
1 dv dxβ 1 d2 xβ + . v 3 dτ dτ γ 2 v 2 dτ 2
(27)
Inserting this into (26) we get 1Z δ∆s = 2
"
dxα dxβ dxρ dxβ 1 d2 xβ (∂µ Pαβ ) − 2(∂ρ Pµβ ) − 2Pµβ 2 2 ds ds ds ds γ v dτ 2 # 1 dv dxβ δxµ ds. +2Pµβ 3 v dτ dτ
(28)
Notice that while not explicitly covariant, this expression holds (to first order) whatever coordinates we use11 . In particular it holds using locally inertial coordinates. We can therefore change all ordinary derivatives above to their covariant analogue 12 1Z δ∆s = 2
"
dxα dxβ dxρ dxβ 1 D 2 xβ − 2(∇ρ Pµβ ) − 2Pµβ 2 2 ds ds ds ds γ v Dτ 2 # 1 dv dxβ δxµ ds. (29) +2Pµβ 3 v dτ dτ (∇µ Pαβ )
Now we would like to see how this expression depends on R. The inertial force formula (12) can be written as 1 D 2 xβ α α α β P = η ∇ η + v 2t ∇ η − t t t ∇ η µβ α µ α µ µ α β γ2 Dτ 2 dv nµ + γ2 tµ + v 2 . dτ0 R
Using this in (29), together with δ∆s = −
Z
dxα ds
(30)
= 1v (η α + vtα ), we find after simplification
1 nµ v + tβ ∇µ ηβ + ηµ tβ aβ − tµ tα tβ ∇α ηβ v R i +vηµ tρ tβ ∇ρ ηβ + tβ ∇β ηµ δxµ ds.
(31)
This expression we may now simplify a bit. From (14) (using the antisymmetry of ωµν ) readily follows 2tβ θβµ = tβ (∇µ ηβ + ∇β ηµ ) + ηµ tβ aβ .
11
(32)
This is evident since the original equation (22), and the derivation thus far, holds for arbitrary coordinates. 12 Having done this, we may as a check-up insert the explicit expressions for the covariant derivatives, with the affine connection, and see that the affine connection terms indeed cancel out.
11
Inertial forces and the foundations of optical geometry
Using this in (31), also using tβ θβµ = [tβ θβµ ]⊥ + tµ tα tβ ∇α ηβ , the expression within the brackets of (31) is readily decomposed into an η µ -part a tµ -part and a part that is perpendicular to both tµ and η µ13 δ∆s = −
Z
"
#
1 nµ v + 2[tα θαµ ]⊥ + ηµ vtα tβ ∇α ηβ + tµ tα tβ ∇α ηβ δxµ ds. (33) v R
Now, for the spacetime trajectory to be a solution to the optimization problem, allowing for arbitrary variations δxµ , the expression within the brackets must vanish. Studying the η µ and tµ parts yields tα tβ ∇α ηβ = 0.
(34)
What this means is that for a trajectory to optimize the integrated distance, the trajectory must never pass two close-lying congruence lines in a direction where there is expansion. This is actually quite natural since there is no penalty (increase of ds) in letting the spacetime trajectory follow a congruence line. To minimize the integrated ds, the spacetime trajectory must never cross between two infinitesimally displaced congruence lines unless there is a minimum distance separating them (implying zero expansion). It was to make this point clear that we didn’t just use the Euler Lagrange equations directly before, but kept the expression for the change of the action under the variation. We are however not really interested in minimizing the distance traveled with respect to the spacetime trajectory. In fact we just want to minimize the integrated distance with respect to the spatial curvature. Alternatively we could say that we want to solve the optimization problem with respect to variations perpendicular to tµ and η µ . We see immediately from (33) that this can be accomplished if we have nµ = −2[tα θαµ ]⊥ . (35) v R We thus find that in general when there is a non-zero expansion-shear tensor, the new sense of straightness differs from the projected version. If we have a Killing symmetry, and adapt the congruence to the Killing field, the congruence will necessarily be rigid and thus θαµ = 0. So for a congruence adapted to the Killing field the two curvature measures coincide, as anticipated. We also see that if we have isotropic expansion and no shear, so θµ ν ∝ δ µ ν , the new sense of straightness coincides with the projected version. This is also completely expected. We may also notice that the projected curvature radius depends on the velocity as 1/R ∝ 1/v. The smaller the velocity the greater the spatial curvature (and thus the smaller the curvature radius). This feels quite natural, moving slowly between fixed congruence lines implies more time for expansion and shear effects to kick in, enabling greater detours (in the projected sense). 13
If trying to make sense, by simple thought experiments, of the various terms – keep in mind that while the integral of (33) corresponds to our original integral of (24), the integrands of these two equations are not in general the same (recall the partial integration undertaken in the derivation of Euler Lagrange’s equations).
Inertial forces and the foundations of optical geometry
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4.2. More intuition regarding the new-straight formalism Assume that we have a diagonal θαβ (in inertial coordinates locally comoving with the congruence) where there is a lot of contraction in, say the x-direction, and no expansion or contraction in the y-direction. Consider then, in a 2+1 spacetime, the problem of minimizing the integrated distance while connecting the opposing corners of a spacetime box as illustrated in figure 3. 0.1 1 t
1
1
y
x Figure 3. A box spanned by the generating congruence, seen in coordinates adapted to these. The proper x and y distances of the sides of the box are indicated. At large t, proper distances in the x-direction are smaller than at small t. We understand that a trajectory minimizing the integrated proper distance will in fact curve (relative to the coordinates in question), as illustrated by the thick line.
If there is severe contraction in the x-direction it will towards the end of the trajectory be very cheap to travel in the x-direction. Thus the trajectory should initially start along the y-axis before turning back and at the end almost follow the x-axis. Thus we have some intuitive understanding for why a trajectory that is straight in the new sense has a projected curvature (in general)14 . ¯ 4.3. Defining a new curvature measure R Now we know what kind of motion that is straight in the new sense. Notice that the projected curvature radius of such a line depends on both the spatial direction tµ and the velocity v. For any given tµ and v we can however define a new curvature radius and a direction of curvature, for a general trajectory of the tµ and v in question, by how fast and in what direction the trajectory deviates from a corresponding line that is straight in the new sense. See figure 4. Let nk0 and R0 denote projected curvature direction and radius respectively for the projection of a trajectory that is straight in the new sense, and let ds denote proper 14
Strictly speaking we may at least understand that such a trajectory (as depicted in figure 3) can be shorter than a coordinate straight trajectory connecting the opposite corners of the box.
Inertial forces and the foundations of optical geometry General trajectory
13
New-straight
dxk
On-slice straight
Figure 4. A projection onto the local slice of a new-straight trajectory, of a certain tµ and v, and a corresponding projection of another trajectory of the same tµ and v. The deviation between the lines can be used to define a new curvature and curvature direction. We may understand that to lowest non-zero order in time, the on-slice deviation is the same as the at trajectory deviation (thinking in 2+1 spacetime). So the projected deviation should do nicely as a measure also of the at-trajectory deviation.
distance along a curve. To lowest non-zero order the deviation is given by nk ds2 nk0 ds2 − . (36) R 2 R0 2 Letting a bar denote curvature direction and curvature in the new sense, we define (see 2 k figure 4) dxk = n¯R¯ ds2 . Using this together with (35) for nk0 and R0 in (36) we readily find nµ 2 β n ¯µ = + [t θβµ ]⊥ . (37) ¯ R R v So here is the new curvature measure in terms of the projected curvature. dxk =
4.4. The inertial force formalism using the new curvature measure Using (37) in the inertial force equation (15), we immediately find the corresponding equation for the new curvature measure α 1 ¯µ 2n µ Dv µ β µ α β µ 2 dv µ (38) t + v P = a + 2vt ω + vt t θ t + γ α β αβ ¯. R mγ 2 Dτ dτ0 We see that the inertial force expression in fact is a bit cleaner with the new representation of curvature. The difference lies in the Coriolis term, the second term on the right hand side, which contains no shear-expansion term now. Notice that while we introduced the concept of curvature in the new sense easily enough, it is a bit more abstract than the projected curvature which can be defined via a projection onto a single locally well defined spatial geometry. It would appear that no such geometry applies to the new sense of curvature15 . For every fixed speed v, we know however what is straight in every direction, and that is sufficient to define a curvature. 15
The argument goes like this. Suppose that we have some spatial geometry on the local slice such that trajectories that are straight in the new sense, and of a certain v, when projected down along η µ are straight relative to the spatial geometry. Consider then trajectories with a different velocity v, that are also straight in the new sense. These will (assuming a non-zero [tβ θβµ ]⊥ ) according to (35) have another projected curvature (relative to the standard spatial geometry). They will thus deviate (to second order) from the corresponding projected trajectories of the previous velocity. Thus these cannot also be straight relative to the spatial geometry in question. We could in principle consider projecting along some other local congruence down to a local slice
Inertial forces and the foundations of optical geometry
14
Certainly the new definition of curvature is in some sense more ’local’ than the projected one. It feels like a better match with the forward part (connected to dv/dτ ) now. 4.5. A joint expression For brevity it will prove useful to have a single expression that incorporates both the projected and the new-straight formalisms. We therefore let the suffix ’s’ denote either ’ps’ standing for projected straight, or ’ns’ standing for new-straight. Introducing Cps = 1, Cns = 0 we can then write: h i 1 µ β µ µ µ µ = a + 2v t (ω + C θ ) γF t + F m + vtα tβ θαβ tµ β s β ⊥ k ⊥ mγ 2 nµ dv µ t + v2 s . + γ2 dτ0 Rs µ µ ¯ and analogously n ≡ n and nµ ≡ n Here Rps ≡ R, Rns ≡ R ¯µ. ps ns
(39)
4.6. A comment on another alternative A line that is spatially straight in the new sense is such that the distance taken relative to the congruence is minimized (with respect to variations in the spatial curvature). One could alternatively consider optimizing the arrival time for a particle moving with a fixed speed, relative to the congruence, from one event (along some congruence line) to another congruence line. Considering for instance a static black hole (where there is time dilation) we understand that to optimize the arrival time it is beneficial to travel where there is relatively little time dilation (hence moving out and then back relative to a straight line). We may understand that a trajectory that is straight in the timeoptimizing sense is curving inwards relative to a line that is straight in the projected sense. We will not pursue the issue further here, but we will comment on it again in section 7. 5. General conformal rescalings In a series of papers Abramowicz et. al. (see e.g [2, 7, 8, 6, 9, 10]) investigated inertial forces in special and general cases using a certain conformally rescaled spacetime. In this section we consider how a general rescaling of the spacetime affects the inertial force formalism. In section 6 we will apply this formalism to the particular rescaling of Abramowicz et. al. Study then an arbitrary rescaling of spacetime g˜µν = e−2Φ gµν . Relative to the rescaled geometry we can express the rescaled four-acceleration of the test particle in terms of the rescaled curvature, the rescaled rotation tensor etc. Letting a tilde on such that all the trajectories of a certain tµ (but different v) that are straight in the new sense get the same projected trajectory. To have two effective congruences (to achieve the goal of a unique spatial geometry) seems, at least at first sight, quite contrived and we will not pursue the idea further here.
15
Inertial forces and the foundations of optical geometry
an object indicate that it is related to the rescaled spacetime, we just put tilde on everything in the joint expression (39) for both the projected and the new-straight formalisms. Notice that v and γ are unaffected by the rescaling16 and we may omit the tilde on them h i ˜ 2 xα 1 ˜µ D µ µ ˜µ β ) + v t˜α t˜β θ˜αβ t˜µ ˜β (˜ = a ˜ + 2v t ω + C θ (40) P β s α ˜τ2 ⊥ γ2 D˜ ˜ µs 2 dv ˜µ 2n +γ . t +v ˜s d˜ τ0 R While we have rescaled the spacetime, we are still interested in knowing what a real observer experiences in terms of forward and sideways thrusts. Then we need to relate the four-acceleration relative to the rescaled spacetime to the four-acceleration of the non-rescaled spacetime. In Appendix D we show how the four-acceleration transforms under conformal transformations. The result is given by (D.14) 2 µ ˜ 2 xµ dxµ dxρ ˜ D 2Φ D x ˜ ρ Φ. ∇ρ Φ − g˜µρ ∇ (41) = e − 2 2 ˜ Dτ d˜ τ d˜ τ D˜ τ µ
2 µ
We know that dx = γ(˜ η µ +v t˜µ ) and DDτx2 = m1 γFk tµ + F⊥ mµ , as derived in Appendix d˜ τ B. Also using t˜µ = eΦ tµ and m ˜ µ = eΦ mµ , we can rewrite (41) as ˜ 2 xα 1 Φ 1 ˜µ D µ ˜µ + F⊥ m − γF t ˜ P = e α k ˜τ2 γ2 mγ 2 D˜ ! 1 ˜ µρ ˜ µρ ˜ ρ˜ µ ˜ [P ∇ρ Φ]k + 2 [P ∇ρ Φ]⊥ + v(˜ η ∇ρ Φ)t˜ . γ
(42)
This we may now insert into (40) to get an expression for the real experienced forces, in terms of the motion relative to the rescaled spacetime, the rescaled expansion etc. The results follow immediately. Here is the rescaled version the inertial force expression eΦ µ ˜ ρ Φ]k + 1 [P˜ µρ ∇ ˜ ρ Φ]⊥ + v(˜ ˜ ρ Φ)t˜µ ˜µ + F⊥ m =a ˜µ + [P˜ µρ ∇ γF t ˜ η ρ∇ k 2 mγ γ2 h i ˜ µs 2n β µ µ α ˜β ˜ ˜µ 2 dv ˜µ ˜ ˜ ˜ t +v +2v t (˜ ω β + Cs θ β ) + v t t θαβ t + γ . (43) ˜s ⊥ d˜ τ0 R We notice that under general conformal rescalings, the inertial force formalism contains extra terms, making it more complicated in general.
6. Optical rescalings for a hypersurface-forming congruence Now study the special case of a timelike hypersurface-forming congruence. The congruence must then obey ωµν = 0. Such a congruence can always be generated by introducing a foliation of spacetime specified by a single scalar function t(xµ ). We simply form ηµ = −eΦ ∇µ t (recall that we are using the (−, +, +, +)-signature) where the scalar field Φ is chosen so that η µ is normalized. 16
One can say that space is stretched as much as time, or that spacetime angles (and hence velocities) are preserved under a conformal rescaling.
16
Inertial forces and the foundations of optical geometry
Now consider a rescaling of spacetime by a factor e−2Φ . It follows that for displacements along the congruence we have dt = d˜ τ 17 . For this particular choice of Φ it is easy to prove, as is done in Appendix E, that a ˜µ = 0. This is also easy to understand. The rescaling, apart from stretching space, removes time-dilation (lapse). Then it is obvious, from the point of view of maximizing proper time, that the congruence lines are geodesics in the rescaled spacetime. In the optically rescaled spacetime the congruence is still orthogonal to the same slices, hence ω ˜ µ ν vanishes. 6.1. The inertial force formalism in the rescaled spacetime Before considering the effect of the rescaling, let us for comparison first have a look at the non-rescaled inertial force equation, for the congruence at hand. From (E.7) in Appendix E, we know that aµ = P α µ ∇α Φ. Using this together with ωαβ = 0 in (39), we are left with h i 1 µα α β µ β µ µ µ = P ∇ Φ + vt t θ t + C 2v t θ γF t + F m α αβ s β ⊥ k ⊥ mγ 2 µ n dv µ t + v2 s . (44) + γ2 dτ0 Rs Now consider a congruence and a conformal rescaling as described in the beginning of the section. Equation (43) is then simplified to 1 Φ µ µ ˜ ρ Φ]k + 1 [P˜ µρ ∇ ˜ ρ Φ]⊥ ˜ = [P˜ µρ ∇ γF t + F m ˜ e (45) k ⊥ 2 mγ γ2 h i ˜ ρ Φ + t˜α t˜β θ˜αβ )t˜µ + Cs 2v t˜β θ˜µ β + v(˜ η ρ∇ ⊥
n ˜µ dv ˜µ t + v2 s . +γ ˜s d˜ τ0 R 2
While we loose the manifest connection to experienced four-acceleration, we can further simplify this by dividing the parallel parts of both sides by γ 2 yielding !
1 Φ Fk ˜µ 1 ˜ ρ Φ + v (˜ ˜ ρ Φ + t˜α t˜β θ˜αβ )t˜µ e η ρ∇ t + F⊥ m ˜ µ = 2 P˜ µρ ∇ 2 mγ γ γ γ2 h i n ˜µ dv ˜µ t + v2 s . + Cs 2v t˜β θ˜µ β + ˜s ⊥ d˜ τ0 R
(46)
Comparing this inertial force equation with its non-rescaled analogue given by (44), we find that excepting tildes, γ-factors and a factor eΦ , the only difference lies in the ˜ ρ Φ = ∂Φ -term within the expansion term. The occurrence of appearance of a η˜ρ ∇ ∂t the extra term is quite natural considering that any time derivative in the spacetime rescaling will act as a spatial expansion. If we so wish, we can alternatively express (46) in terms of the non-rescaled kinematical invariants, see (D.6)-(D.10) (while still keeping the rescaled spatial curvature), thus effectively considering a rescaled space rather than a rescaled spacetime. Contracting both sides of ηµ = −eΦ ∇µ t by η µ = d˜ τ = e−Φ dτ we get d˜ τ = dt. 17
dxµ dτ
µ
Φ dt yields 1 = eΦ dx dτ ∇µ t = e dτ . Using
Inertial forces and the foundations of optical geometry
17
6.2. The projected curvature As a particular example we consider motion along a line that is straight in the projected sense. The perpendicular part of (46) then becomes (recall that Cps = 1) h i eΦ ˜ ρ Φ + 2vγ 2 t˜β θ˜µ β . F⊥ m ˜ µ = P˜ µρ ∇ (47) ⊥ m In particular, when the rescaled congruence is rigid (so θ˜αβ = 0), as in conformally static spacetimes (using a suitable rescaling and a corresponding congruence) the experienced comoving sideways force is independent of the velocity along the straight line. This is a well known result of optical geometry. Now we see also how this somewhat Newtonian flavor is broken (for the curvature measure at hand) in general when the rescaled shearexpansion tensor is non-zero. 6.2.1. Geodesic photons For geodesic particles the left hand side of (46) vanishes. dv = 0, and the In particular, for a geodesic photon, the forward part yields simply d˜ τ0 perpendicular part yields simply h i n ˜µ 0 = 2v t˜β θ˜µ β + v 2 . (48) ˜ ⊥ R We see that the projected curvature vanishes for the free photon if we have θ˜µ β t˜β ∝ t˜µ . (49) ˜ ˜ µ β t˜β ∝ t˜µ . We Knowing that θ˜µ β = σ˜ µ β + 3θ P˜ µ β we see that (49) is equivalent to σ know that (may easily show that) σ˜ µ β η˜β = 0. Knowing also that σ ˜µν is a symmetric tensor it follows that in coordinates adapted to the congruence, only the spatial part of σ ˜µν is nonzero. Also, for (49) to hold for arbitrary spatial directions t˜i , we must have σ ˜ i j ∝ δ i j . Knowing also that the trace σ ˜ α α always vanishes, it follows that σ ˜ µ ν must vanish entirely. If a tensor vanishes in one system it vanishes in all systems. Thus we conclude that for photons to follow optical spatial geodesics in the (standard) projected meaning, the congruence must (relative to the rescaled space) be shearfree (and also rotationfree). It is not hard to show (see Appendix D) that we have σ˜µν = e−Φ σµν and ω ˜ µν = e−Φ ωµν . Thus also relative to the original spacetime geometry must the shear (and rotation) vanish. This result will be used in a companion paper [3] on generalizing the theory of optical geometry.
6.3. The new sense of curvature As a particular example we consider motion along a line that is straight in the new sense. The perpendicular part of (46) then becomes F⊥ µ h ˜ µρ ˜ i ˜ = P ∇ρ Φ . (50) eΦ m ⊥ m Notice in particular the absence of γ factors in this expression. In a rescaled spacetime, with the new definition of curvature, the perpendicular part works just like in Newtonian gravity (up to a factor eΦ ). The experienced sideways force is independent of the velocity, even when the congruence is shearing (unlike when using the projected curvature).
Inertial forces and the foundations of optical geometry
18
6.3.1. Geodesic photons For geodesic particles the left hand side of (46) vanishes. dv = 0, and the In particular, for a geodesic photon, the forward part yields simply d˜ τ0 perpendicular part yields simply µ n ¯˜ = 0. (51) ˜¯ R So a geodesic photon follows a line that is spatially straight in the new sense. This results will also be used in a forthcoming paper on generalizing the theory of optical geometry. 7. Fermat’s principle and its connection to straightness in the new sense, in rescaled spacetimes Fermat’s principle (see [11] for a formal proof) tells us that a geodesic photon traveling from an event P to a nearby timelike trajectory Λ will do this in such a way that the time (as measured along Λ) is stationary18 . In particular any null trajectory minimizing the arrival time is a geodesic. By introducing any spacelike foliation of spacetime, and a corresponding futureincreasing time coordinate t, optimizing the arrival time at Λ is equivalent to optimizing the coordinate time difference δt. In particular, assuming a hypersurface-forming generating congruence, we may introduce an orthogonal foliation and a corresponding time coordinate t. After rescaling the spacetime (to take away time dilation), coordinate time, velocity and spatial distance are related simply by d˜ s = vdt. The total coordinate time δt needed for a particle (not necessarily a photon) moving with constant speed v from P to Λ can then be expressed as Z Z Z 1 1 d˜ s. (52) d˜ s= δt = dt = v v What this says is quite obvious: no time dilation and constant speed means that time is proportional to distance. In particular for a photon, having fixed speed v = 1, the coordinate time taken is minimized if and only if the integrated local (rescaled) distance is minimized. This in turn can hold only if the curvature in the new sense vanishes. So it in fact follows19 from Fermat’s principle that a geodesic photon20 has zero curvature 18
By stationary we mean that it is a minimum or a saddle point with respect to variations in the set of all null trajectories connecting P to Λ. As an example we may consider a 2+1 spacetime where the spatial geometry is that of a sphere, and there is no time-dilation. Then a geodesic photon can take the long way around (following a great circle), rather than the short, in going from P to Λ. This would be a saddle point rather than a minimum. 19 Strictly speaking, what we have shown is that any null geodesic that minimizes the arrival time, ˜¯ It seems safe to assume that any sufficiently small (but for the P and Λ in question, has vanishing R. finite) section of any null geodesic must correspond to minimizing the arrival time for some P and Λ (consider the equivalence principle). Since the argumentation holds for arbitrary P and Λ, it then ˜¯ follows that any null geodesic has vanishing R. 20 Logically, we have here always referred to a photon geodesic with respect to the standard spacetime, which is precisely what we are after. We may however note that, for the particular spacetime
Inertial forces and the foundations of optical geometry
19
in the new sense relative to the optically rescaled spacetime. This is a verification of our earlier result (51) that was derived without reference to Fermat’s principle. Note also that in the rescaled spacetime any trajectory (not only null trajectories) of constant speed that is minimizing the spatial distance is also minimizing the arrival time. Hence the time-optimizing curvature measure as discussed briefly in section 4.6 is identical (up to a pure rescaling) to the new-straight curvature in the optically rescaled spacetime. The connection between Fermat’s principle, null geodesics and straight lines in the optical geometry was realized, for conformally static spacetimes, a long time ago. Now we see that with the new definition of curvature the connection holds in any spacetime. 8. Other photon related curvatures Besides the already discussed curvature measures, and their relation to geodesic photons, it is not hard to come up with a couple of more approaches with different virtues and set-backs. 8.1. Curvature relative to that of a geodesic photon The new sense of curvature has the virtue that, in the optically rescaled spacetime, geodesic photons follow spatially straight lines. On the other hand expressions like ’follow the photon’ loose their meaning in the sense that two spacetime trajectories, cutting the same congruence lines and hence taking the ’same’ spatial trajectory (as seen in coordinates adapted to the congruence) need not have the same measure of curvature. We could try to keep the cake, while also eating it, by using a modified version of the projected curvature. We project the trajectory onto the local slice, but we define the curvature – not via the deviation from a straight line on the slice – but via the deviation from the projected trajectory of a geodesic photon. This definition of curvature can be applied without the restriction to a hypersurface-forming congruence. Also, regardless of rescalings photons will per definition follow straight lines. From (15) it immediately follows that a geodesic photon obeys h i nµ (53) 0 = [aµ ]⊥ + 2 tβ (ω µ β + θµ β ) + . ⊥ R We then introduce the new curvature as h i n′µ nµ µ β µ µ = + [a ] + 2 t (ω + θ ) . (54) β β ⊥ ⊥ R′ R transformation we are considering, there is no need to distinguish between null geodesics relative to the standard and the rescaled spacetime. A null worldline is a geodesic relative to the standard spacetime if and only if it is a geodesic relative to the rescaled spacetime. Indeed this follows from Fermat’s principle (which in turn is very reasonable considering the equivalence principle) since neither null-ness, nor whether a null trajectory corresponds to a stationary arrival time or not, are affected by the conformal transformation. It can also readily be shown using (41) considering vanishing rescaled 2 four-acceleration, evaluating ddtx2 in originally freely falling coordinates and then letting γ → ∞.
20
Inertial forces and the foundations of optical geometry Inserting this back into (15), and writing aµ = aµk + aµ⊥ , yields
h i 1 µ 1 µ µ µ β µ µ = a + γF t + F m a + 2v(1 − v) t (ω + θ ) k ⊥ β β ⊥ k ⊥ mγ 2 γ2 ′µ dv µ n + vtα tβ θαβ tµ + γ 2 t + v2 ′ . (55) dτ0 R Here we have then a formalism that works for an arbitrary congruence, where geodesic photons always have zero spatial curvature – by definition. If we want we can (as usual) form a single aµ -term by multiplying the perpendicular part by γ 2 . As an example, we see that for vanishing rotation and shear, the sideways force on a particle following a straight line (i.e. following a geodesic photon) is independent of the velocity.
8.2. The look-straight based curvature In [8] a ’seeing is believing’ principle is discussed. In a static spacetime (using the congruence generated by the Killing field as congruence), following a line that is seen as straight means that the experienced comoving sideways force will be independent of the velocity. Also a geodesic photon will follow a trajectory that looks straight. When there for instance is rotation of the local reference frame we may however realize that the path taken by a geodesic photon in fact will not look straight. We may however ask if it is possible to define a curvature, in more general cases than the static one (using the preferred congruence), that rests on what we see as straight? Indeed, as is illustrated in figure 5, we already have the necessary formalism to do this easily. Test particle worldline
tµ Geodesic photon Figure 5. A 2+1 illustration, in freely falling coordinates, of a test particle following a string of congruence points (dashed worldlines), momentarily seen as aligned. The congruence points are those that are touched by an incoming (from below in time) null geodesic in the direction −tµ . In fact knowing that the upwards and downwards (in time) projection of a geodesic photon passing the slice are the same (as is obvious in coordinates adapted to the congruence), we may understand that the projected curvature of the test particle will equal the projected curvature of a geodesic photon in the −tµ -direction.
The figure illustrates that the projected curvature of a set of points that at some time was seen as aligned in a direction tµ , in fact corresponds to the projected curvature
Inertial forces and the foundations of optical geometry
21
of a geodesic photon emitted in the direction opposite to tµ (i.e. the −tµ direction). From (15) we immediately find (let tµ → −tµ , set v = 1, let Fk = F⊥ = 0 and select the perpendicular part only) h i nµ (56) 0 = [aµ ]⊥ − 2 tβ (ω µβ + θµ β ) + . ⊥ R This is then the projected curvature of those congruence lines that are momentarily seen as straight in the tµ direction. We define a new curvature as h i n′′µ nµ µ β µ µ = + [a ] − 2 t (ω + θ ) . β β ⊥ ⊥ R′′ R Using this in (15), and writing aµ = aµk + aµ⊥ , we get
(57)
h i 1 µ 1 µ µ µ β µ µ + = a γF t + F m a + 2v(1 + v) t (ω + θ ) k ⊥ β β k ⊥ mγ 2 γ2 ⊥ ′′µ n dv µ t + v 2 ′′ . (58) + vtα tβ θαβ tµ + γ 2 dτ0 R So here we have the inertial force expression when we describe our motion in terms of what we see as straight. As always we may form a single aµ -term if we want by dividing the parallel terms by γ 2 . We may notice that the latter definition of curvature (57) matches the definition (54) of the preceding section (curvature relative to geodesic photon), for arbitrary tµ , if and only if h
tβ (ω µ β + θµ β )
i
⊥
= 0.
(59)
This obviously holds when we have a rotationfree and shearfree congruence. Also, using an argumentation similar to that under (49), this is also necessary for (59) to hold. Notice however that (59) holds if we have (only) an isotropic expansion. 8.2.1. A comment on what looks curved The curvature as introduced in section 8.2 is good measure for the curvature as seen by a congruence observer. For the test observer that moves relative to the congruence we must also consider beaming, making small angular displacements from the forward direction shrink. 8.3. General comments In standard optical geometry, the optical curvature radius of a spatial line that we look upon is related to the curvature radius that we experience by locally looking at the line, via a factor eΦ . The latter two definitions (sections 8.1 and 8.2), for the particular case of a static spacetime with a Killing-adapted congruence, however both correspond exactly to the curvature that we see. The interesting thing with the standard optical curvature radius is however that it is related to a global spatial geometry. Take a trajectory, project it down onto the slice and the rescaled spatial geometry gives us the curvature. For our two latter photon-related curvatures there is in general (so far as I can see) no such global geometry (to which the curvature radius is directly related), even in the static
Inertial forces and the foundations of optical geometry
22
case. In this sense they are more abstract than the standard optical curvature radius. On the other hand the look-straight definition (in particular) is very operationally well defined regardless of there being a geometry connected to it. Lines that are seen to have a certain curvature have that very curvature, by definition. Actually, in this sense the standard optical curvature is not locally well defined operationally21 . Looking back at the joint inertial force expression (46) for optical rescalings utilizing the projected and the new-straight curvature measures respectively, and comparing this with the latter two results, (55) and (58), we see that for a rotationfree and shearfree congruence, a geodesic photon has zero spatial curvature in all the four different formalisms. Notice however that for a rotating congruence, we cannot do the rescaling scheme (at least not without modification), since there is no well defined slicing. This excludes some of the simplest and most interesting systems where one can have use of inertial forces – such as rotating merry-go-rounds and stationary observers near a rotating object (like a Kerr black hole). The latter two definitions can however be used also for these cases. 9. Summary of the curvature measures The perpendicular part of the inertial force equation (excepting those related to rescalings) as presented in this article is of the form µ 1 Dv µ µ 2 ns = [X ] + v . (60) ⊥ s γ 2 Dτ ⊥ Rs Here the index s may stand for either ’ps’, ’ns’, ’rp’ or ’ls’, corresponding to the various curvature measures as listed in order below. For these curvature measures we have [Xsµ ]⊥ as Projected Straight:
aµ⊥ + 2v (tα ω µ α + [tα θµ α ]⊥ )
(61)
New-Straight:
aµ⊥
(62)
α
µ
+ 2vt ω α 1 µ a + 2v(1 − v) (tα ω µ α + [tα θµ α ]⊥ ) Relative Photon: γ2 ⊥ 1 µ a + 2v(1 + v) (tα ω µ α + [tα θµ α ]⊥ ) . Look Straight: γ2 ⊥ The parallel direction of the inertial force equation is given by 1 Dv µ dv = [Xsµ ]k + γ tµ . 2 γ Dτ k dτ And here [Xsµ ]k = aµk + vtα tβ θαβ tµ .
(63) (64)
(65)
(66)
This part is the same for all the above curvature measures. Notice that all the four different physical ways of describing the motion relative to the reference frame yield precisely the same inertial force formalisms when using an inertial congruence (indeed there are no inertial forces then). 21
We can never figure out what Φ is through local experiments, only its gradient can be deduced.
Inertial forces and the foundations of optical geometry
23
9.1. A comment on the different ways of defining inertial forces As presented in this article, as is also standard for inertial forces in Newtonian mechanics, the final equation is of the form Freal +Finertial = marelative . For a given physical scenario, the real force is fixed, whereas the relative acceleration, and hence the inertial forces, depend on what reference frame (congruence) we are using. Furthermore, as we have illustrated, there is more than one plausible way to define a spatial curvature for the motion of a test particle, when the reference frame is shearing22 . This effectively means that there is more than one plausible way of describing the acceleration relative to the reference frame – hence even for a fixed reference frame there is more than one way of defining the inertial forces. As concerns the photon related approaches, they also conform to the standard Newtonian formalism for non-shearing congruences in the limit of small velocities. We may however note that they have somewhat of a less fundamental geometrical nature to them – being more of a practical and physical nature. Consider specifically the second photon related formalism connected to what an observer comoving with the reference frame in question actually experiences visually. The apparent (inertial) forces of this formalism together with the real forces (focusing on the perpendicular part), are precisely the (apparent) forces that will make a test particle deviate from what the observer sees as straight. We understand that if we let the concept of apparent (inertial) forces be wide enough to incorporate alternative (physical) ways of measuring the apparent motion of a test particle – then there is room for even more definitions of inertial forces. Apart from the above mentioned alternative prescriptions, there is also a certain level of freedom concerning γ-factors, in part connected to the fact that there are two types of forces (given and received), but see also the footnote in section 3.6. 10. Three-dimensional formalism, assuming rigid congruence We can rewrite the four-covariant inertial force formalism thus far as a purely threedimensional formalism. For brevity let us consider a non-shearing (isotropically expanding) congruence23 , so [tα θµ α ]⊥ = 0. Then the projected straight and the newstraight formalisms are identical. Since [θµ α tα ]⊥ = 0 for all directions tµ then, in freely falling coordinates locally comoving with the congruence, the spatial part of θµ ν must be proportional to δ i j . Also, in the coordinates in question the time components of θµ ν 22
Note incidentally that the distinction between the projected and the new type of curvature measure can be made also in non-relativistic mechanics. 23 If we want to consider a shearing congruence in three-dimensional formalism, that is in principle no problem at all. We just define θij = 12 (∇i uk + ∇k ui ). Here uk is the velocity of a reference point, seen relative to a freely falling frame locally comoving with the congruence. Also ∇i is understood to be covariant derivative with respect to the local spatial metric and lowering of indices are made using the local spatial metric. The latter can be defined, as the spatial metric on a geodesic slice (i.e. an instant in a freely falling system) orthogonal to the congruence at a single point (the point in question), without the existence of global orthogonal slices. Then we could let θµ α tα → θ · ˆ t where θ denotes the three-dimensional shear-expansion matrix.
Inertial forces and the foundations of optical geometry
24
vanishes. Defining (as is standard) θ = θα α , we have thus θi j ∝ θ3 δ i j . We may then write tα tβ θαβ = 3θ (being a scalar expression this holds in general coordinates). Looking back at for instance (15), we have then µ 1 θ µ dv µ µ β µ µ µ 2n = a + 2vt ω + v γF t + F m t + γ t + v . β ⊥ k mγ 2 3 dτ R
(67)
In coordinates locally comoving with the congruence24 we have aµ : (0, a), nµ : (0, n) and and vtµ : (0, v). To avoid confusion with the acceleration of the test particle, let us define g = −a. Also we let25 ω µ α tα → ω × ˆt and τ → τ0 /γ (recall that τ0 is local time along the congruence). The three-dimensional analogue of (67) is then simply 1 θ dv ˆ n ˆ ˆ γF t + F m ˆ = − g + 2ω × v + v + γ 2 (68) t + v2 . ⊥ k 2 mγ 3 dτ0 R Notice that this formalism is defined irrespective of whether there exists any global slices orthogonal to the reference congruence. For instance we can apply it to calculate the real forces on a particle orbiting outside of the ergosphere of a Kerr black hole, using the stationary (non-rotating) observers as our reference congruence. Multiplying the first three terms of (68) by −m they can be seen as the inertial forces Acceleration, Coriolis and Expansion. The forces Fk and F⊥ are the experienced (comoving) perpendicular and parallel forces respectively. If we want to consider the given forces Fck and Fc⊥ , assuming that that observers following the congruence push (or pull) the object in question, we have from Appendix C that Fck = Fk and γFc⊥ = F⊥ . Indeed defining Fc = Fckˆt + Fc⊥ m, ˆ the inertial force equation becomes even simpler
n ˆ θ dv ˆ Fc t + v2 . = −g + 2ω × v + v + γ 2 mγ 3 dτ0 R
(69)
Notice that while (68) and (69) are fully relativistically correct they are very similar to their Newtonian counterpart(s) (just set γ = 1, see also Appendix F). Notice however that τ0 is local time in the reference frame. Considering for instance a static black hole 1 ) 2 dt, where t is the global (Schwarzschild time). Also space will we have dτ0 = (1 − 2M r of course in general be curved unlike in (standard) Newtonian theory.
24
For any specific global labeling of the congruence lines (i.e. any specific set of spatial coordinates adapted to the congruence) we can locally choose a time slice orthogonal to the congruence so that e.g. aµ : (0, a). This then uniquely defines a at any point along the test particle trajectory. 25 Let ω µ = 12 √1g ησ ǫσµγρ ωγρ , where g = −Det[gαβ ] and ǫσµγρ is +1, −1 or 0 for σµγρ being an even, odd or no permutation of 0, 1, 2, 3 respectively. Then we can define ω through ω µ = (0, ω) in coordinates locally orthogonal to the congruence. Strictly speaking, what we mean by the cross product a × b of two three-vectors a and b is 1 g − 2 ǫijk aj bk where the indices have been lowered with the local three-metric (assuming local coordinates orthogonal to the congruence), and g is minus the determinant of this metric. Notice that in general (for congruences with rotation) there are no global time-slices that are orthogonal to the congruence. The local three-metric corresponding to local orthogonal coordinates is however well defined everywhere anyway.
Inertial forces and the foundations of optical geometry
25
10.1. Applying the three-formalism to a rotating platform As a simple application of the three-dimensional formalism we consider coordinates attached to a rotating platform in special relativity. Let ω0 be the counterclockwise angular velocity of the platform, and r be the distance from the center (this distance is obviously the same whether we are corotating with the platform or not). We understand that the circumference of a circle of fixed r relative to the platform will (length contraction) be greater than the corresponding circumference, as measured on the ground, by a factor γ = γ(ω0 r). The spatial metric in the corotating cylindrical coordinates can thus be written as r2 dϕ2 + dz 2 . (70) ds2 = dr 2 + 1 − ω02r 2 /c2 Here c is the velocity of light. Note that this metric is well defined despite the fact that there are no time slices globally orthogonal to the reference congruence in question. For circular motion relative to an inertial frame – the proper acceleration, as follows from (68), is given by γ 2 v 2 /r = γ 2 ω02 r. We understand that relative to the rotating platform we have g=
ω02r ˆ r. 1 − ω02 r 2 /c2
(71)
A gyroscope orbiting with a counterclockwise angular velocity ω0 around a circle of radius r with respect to inertial coordinates, will Thomas-precess (see e.g. [12]) with a clockwise angular velocity ωgyro = (γ − 1)ω0 . Adding this rotation to the rotation of the reference frame and multiplying by a factor γ to take time dilation into account, it follows that with respect to an observer corotating with the platform, the gyroscope will precess with a clockwise angular velocity given by γ(ω0 + (γ − 1)ω0) = γ 2 ω0 . We have thus the local rotation of the platform (as experienced by a locally comoving inertial observer) ω0 ˆ z. (72) ω= 1 − ω02r 2 /c2 Now we have the necessary tools for making calculations with respect to this reference frame. 10.2. Radial motion on the rotating platform As a particular example we consider a wagon moving along a radially directed rail (fixed ϕ) on the rotating platform with constant velocity v. We are interested in what force that will act on the rail from the wagon. We note that this force is precisely minus the given force by the rail, thus Fonrail = −Fc . Letting m be the rest mass of the wagon (we assume the rotational energy of the wheels to be negligible), we have then from (69) for this simple case Fonrail = g − 2ω × v. (73) mγ
Inertial forces and the foundations of optical geometry
26
Here γ = γ(v). Using (71) and (72), assuming the wagon to move outwards from the center so that v = vˆ r, this can be written as ω0 Fonrail = mγ (ω0 rˆ r − 2v ϕ) ˆ . (74) 1 − ω02 r 2 /c2 Here is thus the force from the wagon on the rail. Note that the equation applies to r < c/ω0 . 10.3. A few comments on the three-dimensional formalism For typical applications where the reference congruence lines are integral curves of a timelike Killing field, we can directly use (67) and (69) respectively as equations of motion, for specified forces, to find the resulting spatial path26 . The path can be expressed in terms of the the test particle proper time since dτ = dτ0 /γ = ds/(vγ). For the most general case however (still assuming a non-shearing reference congruence), if we want to integrate the three-dimensional equations of motion – we need to introduce a global time parameter. In other words we need to introduce time slices in spacetime and associate with each slice a parameter t. In general g, ω, θ and spatial distances between adjacent congruence lines will be functions of this time parameter as well as of the spatial position. Notice however that irrespective of whether the time slices are orthogonal to the congruence or not (in general they cannot be globally orthogonal to the congruence) spatial distances are always measured proper orthogonal to the congruence lines27 . Note also that even for the stationary case, if we want to make predictions of coincidences (like whether two particles will collide or not) we need the global time parameter. For a static spacetime (like a Schwarzschild black hole), adapting the reference congruence to the static observers, there is a very simple such global time t where dt = f (x)dτ0 for some function f (x). Note, however, that as local equations, (67) and (69) are directly applicable, without introducing a global time, to answer questions like for instance what perpendicular forces one gets if one follows the path of a geodesic photon. 11. A general derivation of a vector transport equation from the inertial force formalism Jantzen et. al. have also developed a covariant inertial force formalism, see e.g [13]. They are employing various covariant differentiations of vectors defined along a spacetime trajectory. These types of covariant differentiation can readily be defined if we have a means of transporting a vector along the trajectory in question. The general idea is simple, and illustrated in figure 6. 26
We are of course assuming that g, ω (for this case θ = 0) and the spatial geometry are known as functions of spatial position, i.e. in terms of the labeling of the congruence lines. 27 Thus the spatial geometry is not defined as the spatial geometry on the slice related to the global time parameter – consider for instance the example in section 10.1. Note also that the comoving coordinates used when introducing the bold face three-notation have nothing to do with the time slices related to the global time.
Inertial forces and the foundations of optical geometry
27
B
A
Figure 6. Vector differentiation along a timelike spacetime trajectory. The full drawn arrows correspond to the vector defined along the trajectory, for instance the momentary forward direction tµ . The dashed arrow at B is the transported version of the vector at A. Forming the difference between the vectors at B and dividing by the proper time dτ along the trajectory from A to B gives us our derivative.
In particular one may define a spatial curvature and curvature direction by how fast (and in what direction) the forward direction deviates from a corresponding transported vector. The idea is that the transport law should somehow correspond to a spatial parallel transport with respect to the spatial geometry defined by the congruence. That way, the definition of spatial curvature and curvature direction is analogous to the definition in standard Riemannian three-dimensional differential geometry. In the approach of this article we started from the other end by deriving the spatial curvature measures of the various physical meanings, and we will now derive corresponding vector transports and vector differentiations. 11.1. Rigid congruence For the case of a rigid congruence28 the matter of spatial transport is quite intuitively reasonable. The idea is illustrated in figure 7. It is easy to show that in the coordinates (xk , t) of a freely falling system, locally comoving with the congruence, the velocity of the congruence points (assuming vanishing θµ ν ) is to first order in xk and t given by v k = ω k j xj + ak t.
(75)
Knowing that the velocity of the congruence is zero to lowest order, we need not worry about length contraction and such. It is then easy to realize that the proper spacetime transport law of a vector k µ , orthogonal to η µ , corresponding to standard spatial parallel transport is Dk µ = γω µ α k α + bη µ . (76) Dτ 28
The congruence may rotate and accelerate but it may not shear or expand.
Inertial forces and the foundations of optical geometry
28
Figure 7. A 2+1 illustration of transporting a spatial vector along a worldline, seen from freely falling coordinates locally comoving with the congruence. As the coordinates attached to the grid rotate due to ω µ α , so should the vector in order for it to be proper spatially transported.
Here b can easily be determined from the orthogonality of k µ and η µ29 . 11.2. General congruence Now let us consider a congruence with non-zero expansion-shear tensor. Here there is no fixed (rigid) spatial geometry. How then to define a spacetime generalization of spatial parallel transport? While we have no fixed spatial geometry, we still have a spatial curvature measure (of several types) given the spacetime trajectory. Suppose then that we transport a vector along a timelike worldline with vanishing spatial curvature (whichever curvature measure we choose). If the initial vector pointed in the tµ -direction it seems natural that the parallel transported vector should keep pointing in the tµ direction. Also, if the trajectory curves relative to a corresponding trajectory of vanishing spatial curvature, but the initial vector still pointed in the tµ direction, the transported vector should deviate from the forward direction in the same manner as it would for a fixed geometry. We also demand of the parallel transport that the norm of the vector should be constant and it should remain orthogonal to the congruence, given that it was originally orthogonal to the congruence. Then the derivation, as concerns parallel transport of a vector momentarily parallel to tµ , is straightforward as we illustrate in the coming two subsections. 11.2.1. The standard contravariant derivative of the forward direction Using (2), (60) and (14) we readily get
Dtµ Dτ
⊥
γ γ µ nµs − γtα ω µ α − γ[tα θµ α ]⊥ − aµ⊥ . = [Xs ]⊥ + γv v Rs v
We have k µ ηµ = 0 which means that α yields b = k α Dη Dτ . 29
Dkα Dτ ηα
(77)
α + k α Dη Dτ = 0. Contracting (76) with ηµ then readily
Inertial forces and the foundations of optical geometry
29
So here is the perpendicular (spatial) part of how the forward direction is propagated, given the spatial curvature radius30 . Notice that Xsµ depends on what curvature measure we are using (see (61)-(64)). 11.2.2. The relation between spatial transport and spatial curvature Suppose now that we have some vector tµk that momentarily is equal to the forward direction vector tµ . Suppose further that we have some (as of yet undefined) parallel transport defined for tµk . Then we can define a curvature measure for a trajectory, with respect to the transport in question, as nµ Dtµ γv v = Rv Dτ
−
⊥
"
Dtµk Dτ
#
.
(78)
⊥
Here the subscript ’v’, stands for ’vector transport related curvature’. The definition is analogous to how one defines (may define) ordinary spatial curvature using ordinary spatial parallel transport. The γ is included since we have τ and not τ0 on the right µ nµ s , hence demanding a parallel hand side. Using (77) and (78), making the ansatz Rnvv = R s transport of the momentarily parallel vector to be such that the two types of curvature measures coincides, readily gives "
Dtµk Dτ
#
⊥
=
γ µ γ [Xs ]⊥ − γtα ω µ α − γ[tα θµ α ]⊥ − aµ⊥ . v v
(79)
In particular for the projected and new-straight formalisms (see (61) and (62)) this yields Projected Straight: New-Straight:
"
Dtµk
#
Dτ ⊥ # " Dtµk Dτ
= γtα ω µ α + γ[tα θµ α ]⊥
(80)
= γtα ω µ α − γ[tα θµ α ]⊥ .
(81)
⊥
We then define the transport equation for any vector kkµ momentarily parallel to tµ correspondingly Projected Straight: New-Straight:
"
Dkkµ
#
Dτ ⊥ " # Dkkµ Dτ
= γkkα ω µ α + γ[kkα θµ α ]⊥
(82)
= γkkα ω µ α − γ[kkα θµ α ]⊥ .
(83)
⊥
An alternative (but equivalent) way of deriving these transport laws is to demand that the parallel transport, along a trajectory with in general non-zero spatial curvature, of a vector momentarily equaling the forward direction vector should be the same as the transport of the forward direction of a line that is straight with respect to the curvature measure in question. 30
The ⊥-sign on the left hand side is really only necessary for the projection onto the slice, not to take away components in the tµ -direction (since the normalization of tµ tells us that there are no µ tµ -component in Dt Dτ ).
Inertial forces and the foundations of optical geometry
30
Note that in the absence of shear, these definitions match (76). We may however note that if we instead had considered for instance the look-straight curvature, the corresponding transport would not have matched (76), even for pure rotation. 11.2.3. Spatial parallel transport of a general vector While the just derived transport laws are sufficient for the purposes of the inertial force formalism, we may be curious to know whether we could find a transport law for general vectors, that corresponds to (80) and (81) for the particular case of a vector momentarily parallel to the forward direction. Indeed we can, although how we do it is quite subjective. Let us however demand that, considering momentarily spatial vectors, the transport should be norm-preserving, and preserving orthogonality to η µ . Also we demand that any pair of parallel transported vectors should have a fixed relative angle (in particular vectors that were initially orthogonal should remain orthogonal). In 2+1 dimensions it is obvious, concerning spatial vectors, that these considerations completely determine the parallel transport. In 3+1 dimensions there is however a freedom of (spatial) rotation around the spatial direction of motion. Here we may however take guidance from (76), and demand that in the absence of shear we should get a transport corresponding to (76). Indeed this is not generally doable as was commented upon at the end of the preceding section (11.2.2), although it will turn out to be for the case of the projected and the new-straight curvatures. Let us assume that the parallel transport should be formulated in terms of tensors, in likeliness with (76). The tensors that we have to work with are aµ , ω µ α , θµ α , tµ , v, η µ , nµ and R. From these tensors we can of course in principle form other tensors. To insure fixed norm and angles, the transport must effectively be a spatial rotation relative to freely falling coordinates locally comoving with the congruence 31 . Given any two orthogonal spatial vectors dµ and eµ we can form a rotation tensor as dµ eα − eµ dα 32 . For brevity we define dµ ∧ eα ≡ dµ eα − eµ dα . Several rotation tensors of this type can of course be added together to form a net rotation tensor. There are possibly several ways to match the above criterias but the one we present below seems quite natural as concerns the new-straight and the projected curvature measures. Looking at the different tensors available and (81) and (80) it is easy to find general transport laws, that obeys the just outlined requirements. The spatial parts of our transport laws are given below "
Dk µ Projected Straight: Dτ
31
#
= γk α ω µ α + γk α ([tβ θµ β ]⊥ ∧ tα )
(84)
⊥
The argument is similar to that in section 11.1, where length contraction will not enter. Also there will of course be a η µ term entering to insure orthogonality. 32 Forming (dµ eα − eµ dα )k α , for a spatial vector k α , amounts to forming d(e · k) − e(d · k) in (spatial) bold-face notation. This is a so called vector triple product and equals (e × d) × k. Thus dµ eα − eµ dα is a rotation tensor.
Inertial forces and the foundations of optical geometry New-Straight:
"
Dk µ Dτ
#
= γk α ω µ α − γk α ([tβ θµ β ]⊥ ∧ tα ).
31 (85)
⊥
Defining the transport law in such a way that a vector originally orthogonal to α the congruence remains orthogonal to the congruence, we can add a term η µ k α Dη Dτ (analogous to what we did in section 11.1) to the right hand side of (84) and (85). That way we may remove the ⊥ sign on the left hand side (which was anyway there only for projection, not for orthogonality to tµ ), and express the full transport equations. Note that rather than k α ω µ α we might for instance have tried k α (tβ ω µ β ∧ tα ). These would both give the right transport equation when k µ = tµ momentarily, while in general being different for other vectors k µ . The latter rotation version would introduce no rotation at all around the direction of motion (the rotation vector is given by ω × t, where ω is the rotation three vector corresponding to the rotation tensor ω µ α ), as seen from an inertial system. This is however not really what we want. For a static rotating grid it seems obvious that the parallel transport should coincide with standard spatial parallel transport. Hence if the congruence rotates around the direction of motion (seen from an inertial system) so should a parallel transported vector. We should thus use k α ω µα rather than k α (tβ ω µ β ∧ tα ). As regards the θµ α -term, what we want is not as clear. The way that we have chosen gives the minimal rotation needed (seen from an inertial system) to get the transport right. Note that the ambiguity in rotation around the spatial direction of motion, for parallel transport of a general vector, has no impact on the discussion of inertial forces. Here we are always concerned with rotation of vectors momentarily in the forward direction, for which case there is no ambiguity. The general transport laws can however be used in other contexts. In particular one may use them when developing a relativistic three-dimensional formalism of gyroscope precession relative to a given reference frame. In such a formalism, see [12], the occurrence of for instance terms of the type γk α ([tβ θµ β ]⊥ ∧ tα ) follows naturally, independent of what spatial parallel transport we consider. Thus the form of (84) and (85) fits well with the formalism of three-dimensional relativistic gyroscope precession. 11.2.4. Covariant differentiation along trajectory Having derived the transport laws, the corresponding covariant differentiations along a trajectory follows immediately. Including the η µ -component as discussed under (85) we simply get Dk µ Dηα Dps k µ = − γk α ω µ α − γk α ([tβ θµ β ]⊥ ∧ tα ) − η µ k α Dps τ Dτ Dτ µ µ Dns k Dk Dηα = − γk α ω µ α + γk α ([tβ θµ β ]⊥ ∧ tα ) − η µ k α . Dns τ Dτ Dτ
(86) (87)
α is readily given by (9) and (14). Notice however that for the purposes of the Here Dη Dτ inertial force formalism presented here, only the projected part of these equations is of importance and only when applied to a vector momentarily parallel to the forward direction.
32
Inertial forces and the foundations of optical geometry 12. Reformulating the inertial force formalism
Consider a rigid Cartesian reference system that rotates and possibly accelerates in Newtonian mechanics. The law of motion can then be expressed relative to the reference system as (see also Appendix F) F 1 dv n = − F inertial + t + v 2 . (88) m m dt R Here v and R are the velocity and spatial curvature relative to the reference system, analogous to the approach of the preceding section. Alternatively we could express (88) as 1 1 dp F = − F inertial + . (89) m m m dt Here p ≡ mv is the three-momentum relative to the reference system. The question arises if we could do something similar in the general relativistic scheme? Indeed we already have the necessary tools to transport relativistic three. When only concerned with momentum, and do a differentiation corresponding to dp dt inertial forces, there is however a more direct way (allowing some overlap with the preceding section) as will be presented below. 12.1. The reformulation, with the corresponding transport in implicit form Let us introduce the relativistic three-momentum relative to the congruence as p¯µ ≡ P µ α pα (the bar here has nothing to do with the bar indicating new-straight curvature and curvature direction). For the particular case of special relativity, for an inertial congruence, (12) then gives us µ dv µ 1 D p¯µ 2n =γ t +v . (90) mγ 2 Dτ dτ R By analogy with this relation we now define a covariant differentiation of threemomentum along a curve as33 µ dv µ 1 Ds p¯µ 2 ns = γ t + v . (91) mγ 2 Ds τ dτ Rs For a general inertial force equation of the form µ 1 dv µ µ µ µ 2 ns = X + γ γF t + F m t + v . (92) k ⊥ s mγ 2 dτ Rs we can thus write alternatively 1 Ds p¯µ 1 µ µ µ = X + γF t + F m . (93) ⊥ k s mγ 2 mγ 2 Ds τ Here we have then a reformulation of the inertial force formalism, although the transport equation connected to the derivative is left implicit. We can however derive it from the µ
µ
p¯ Considering that DDτ has an η µ component, one might think that also DDssp¯τ should have it. The latter is however intended to be a differentiation between two infinitesimally different vectors that are exactly orthogonal to η µ after some infinitesimal time. It is then easy to show that it should not contain any explicit η µ component. 33
Inertial forces and the foundations of optical geometry
33
above formalism, analogous to the derivation of the preceding section. We do this in the following section. 12.2. Re-deriving the transport equation We have by definition µ D p¯µ D p¯k Ds p¯µ ≡ − . (94) Ds τ Dτ Dτ Here p¯µk is understood to be a vector that is momentarily parallel to p¯µ and then ’parallel’ transported with respect to the congruence (and the curvature measure in question). This we can now use to derive the transport equation. First we write (93) as α 1 1 Ds p¯µ µ Dp µ P = X + . (95) α s mγ 2 Dτ mγ 2 Ds τ Using the definitions of p¯µ and P µ ν together with (94) it is then easy to show that µ
1 Dpk = Xsµ − aµ − vtα ∇α η µ + η µ tα (η ρ + vtρ )∇ρ ηα . (96) 2 mγ Dτ Using (61), (62) and (66) together with (9) and (14) we readily find the projected version of this equation for the projected and new-straight formalisms respectively D p¯αk 1 µ P α = vω µα tα + v [θµ α tα ]⊥ (97) Projected: 2 mγ Dτ D p¯αk 1 µ P α = vω µα tα − v [θµ α tα ]⊥ . (98) New-straight: 2 mγ Dτ These are a perfect match with (82) and (83) (substituting k µ → p¯α ). Note that for this particular type of transport there are no ambiguities, since the vector we are transporting is momentarily parallel to the direction of motion. 13. The Jantzen et. al. approach revisited Jantzen et. al. (see e.g. [14]), are using four different definitions of covariant differentiation along curve. In the language of this article, assuming the vector in question to be momentarily orthogonal to η µ34 , the definitions are35 Dfw k µ Dτ Dcfw k µ Dτ Dliek µ Dτ Dlie♭ k µ Dτ
Dk β Dτ Dk β = P µβ − γω µ α k α Dτ β Dk = P µβ − γ (ω µ α + θµ α ) k α Dτ Dk β = P µβ − γ (ω µ α − θµ α ) k α . Dτ
= P µβ
(99) (100) (101) (102)
If the vector has a time component we should add a term γk α ηα aµ on the right hand side of (101). Note in particular that they are using a different convention regarding the sign of ω µ α , here we are however using the convention of this article. 34
35
Inertial forces and the foundations of optical geometry
34
The subscripts are short for ’Fermi-Walker’, ’Co-rotating Fermi-Walker’, ’Lie’ and ’covariant Lie’. Note that while ’fw’ really stands for Fermi-Walker (99) is not the standard Fermi-Walker derivative. Defining p¯µ ≡ mP µ α v α , we have v µ = γη µ + m1 p¯µ and thus 1 µ Dv α 1 µ D 1 ¯α α µ Dp P = P (γη ) + P . α α α γ2 Dτ γ2 Dτ mγ 2 Dτ
(103)
We have also Dη µ = v α ∇α η µ Dτ = γ(η α + vtα )(θµ α + ω µ α − aµ ηα ) µ
= γa + γv(θ
µ
µ
+ ω α ).
α
(104) (105) (106)
This we may use in (103) together with (99)-(102) (subsequently), substituting k ρ → p¯ρ . Letting ’tem’ denote ’fw’,’cfw’,’lie’ or ’lie♭’, we immediately retrieve the result of Jantzen et. al. 1 µ Dv α 1 Dtem p¯µ µ P = + Xtem . (107) α 2 2 γ Dτ mγ Dτ Here µ Xtem = aµ − vHtem µ α tα .
(108)
Here in turn, Htem µ α is given by Hfw µ α = − ω µ α − θµ α Hcfw
µ
α
= − 2ω
µ
α
−θ
µ
(109) α
(110)
Hlieµ α = − 2ω µα − 2θµ α
(111)
Hlie♭ µ α = − 2ω µα .
(112)
Already here we have the inertial force formalism. In the coming subsection we will compare the two formalisms. Jantzen et. al. has also considered an inertial force formalism in terms of curvatures as experienced by the comoving observer [15]. The idea is essentially to study how fast, and in what direction, the incoming congruence points are changing their velocity relative to a comoving reference frame of gyroscopes. 13.1. Comparing the formalisms We can write (95) as 1 Ds p¯µ 1 Dv µ = + Xsµ . mγ 2 Dτ mγ 2 Ds τ
(113)
Looking at (61), (62) (we skip the relative photon and look-straight curvature measures) and (66) we have Xsµ as µ Projected Straight: Xps = aµ + 2v (tα ω µ α + [tα θµ α ]⊥ ) + vtα tβ θαβ tµ (114)
New-Straight:
µ Xns = aµ + 2vtα ω µ α + vtα tβ θαβ tµ
(115)
Inertial forces and the foundations of optical geometry
35
We may compare these two equations with (111) and (112). We see that as regards the perpendicular part, the new-straight formalism of this article corresponds to the lie♭-formalism and the projected straight formalism corresponds to the lie-formalism36 . The corresponding parallel parts are however not equal. How one deals with the parallel part is to a large degree a matter of taste. In this article we have defined parallel transport in such a way that the norm of the parallel transported vector is preserved. This is a natural definition if we want to connect directly to changes in the local speed v. Consider for instance an isotropically expanding universe with a particle moving along a straight line (here all the curvature measures coincide) with constant local speed (this requires a forward thrust) relative to the preferred congruence. With parallel transport as defined in this article we have p¯µ = 0. In this view the forward thrust cancels the fictitious expansion force. then DDτ The philosophy regarding the perpendicular part of the transport equations are also a little different. We have here considered transport equations that by definition are not altering the angles between transported vectors, which is not generally the case in the approach of Jantzen et. al. Again this is a matter of taste, and it has no impact at all on the discussion of inertial forces since we are anyway only interested in the transport of a vector locally aligned with the forward direction. The biggest difference in our approaches is that we have here started from various physically defined curvature measures, and derived an inertial force formalism from this. Only after this was done have we considered the notion of spatial parallel transport with respect to the congruence. Jantzen et. al. on the other hand start from various transport equations and derive the formalism and curvature measures from this. Considering the new-straight formalism the Jantzen et. al. approach is not really applicable. While the curvature connected to the lie♭-transport in fact corresponds to the new-straight curvature, the connection appears coincidental. The physical meaning of this curvature (related to minimizing the local integrated distance) has not previously been discussed (to the author’s knowledge). Neither has any formalism previously been presented (again to the author’s knowledge) employing this curvature measure explicitly. 14. Summary and conclusion The inertial force formalism as developed here was initially inspired by the works of Abramowicz et. al. who have employed a rescaled version of space(time) to study inertial forces. We have here extended the formalism of inertial forces in rescaled spacetimes to include arbitrary hypersurface-forming congruences (applicable to any spacetime). A generalization has earlier been studied in [6] using a different philosophy, but see [13] for criticism. We find that the inertial force formalism is very similar in the rescaled and the standard spacetime and that the difference lies mainly in how the γ-factors enter. The main part of this article has turned out to be more connected to the work of Jantzen et. al. A novelty with the approach of this paper is that we are starting from 36
The latter is expected considering the way the Lie derivative entered the derivation of section 3.
Inertial forces and the foundations of optical geometry
36
various physically defined spatial curvature measures, and are using these to describe the local motion of a test particle, and derive a corresponding inertial force formalism. In particular we introduce a new curvature measure that we denote new-straight curvature. This measure is defined in such a way that, even when we have a shearing congruence, following a straight line with respect to the new curvature measure means taking the shortest path relative to the spatial geometry defined by the congruence (which is actually not the case for the standard projected curvature). This provides us with a natural way of extending the optical geometry, to include the most general hypersurfaceforming reference frames, while keeping the most basic features. Indeed we show that as regards photons, the new-straight curvature is strongly connected to Fermat’s principle. These considerations and others will be further commented upon in a companion paper [3] on generalizing the optical geometry. We have also considered a pair of more unorthodox curvature measures, the curvature relative to that of a geodesic photon and the look-straight curvature. Likely these will have even less practical import than the projected and the new-straight curvature measures, but they serve as examples of the variety of different curvature measures, and corresponding inertial force formalisms, that one may introduce. They also illustrate how one may apply the inertial force formalism to answer some particular questions in physics. From the derived curvature measures, we have derived spacetime transport laws for vectors, along a test particle worldline, corresponding to spatial parallel transport with respect to the congruence. These transport laws can for example be used to derive an expression for how a gyroscope precesses relative to the reference congruence. We have not in this paper spent much time on explaining for instance why the sideways force increases by a γ 2 -factor if we follow a straight line in a static spacetime. For such considerations we refer to a companion paper [16]. There we rely on simple principles such as time dilation and the equivalence principle and derive the relativistic three-dimensional form of the inertial force equation (68) using no four-covariant formalism at all. While this paper is considerably more formal in its approach, we have tried to employ an (in the author’s mind) more accessible mathematical notation than that employed by Jantzen et. al. The explicit three-formalism as presented for shearfree (but isotropically expanding, accelerating and rotating) reference frames is, to the author’s knowledge, also novel. Appendix A. The kinematical invariants of the congruence The kinematical invariants of a congruence of worldlines of four-velocity η µ are defined as (see e.g. [5]) aµ = η α ∇α ηµ θ σµν
α
= ∇α η 1 1 = (∇ρ ηµ P ρν + ∇ρ ην P ρ µ ) − θPµν 2 3
(A.1) (A.2) (A.3)
Inertial forces and the foundations of optical geometry
37
1 (∇ρ ηµ P ρν − ∇ρ ην P ρ µ ) . (A.4) 2 In order of appearance these objects denote the acceleration vector, the expansion scalar, the shear tensor and the rotation tensor. We will also employ what we may denote the expansion-shear tensor 1 (A.5) θµν = (∇ρ ηµ P ρ ν + ∇ρ ην P ρ µ ) . 2 ωµν =
Appendix B. Rewriting f µ in terms of experienced (comoving) forward and sideways forces Consider a freely falling frame, locally comoving with η µ , with a particle moving relative to this frame. In the coordinates of the inertial frame, the particle is acted upon by a force f µ . This force may be decomposed as f µ = f 0 η µ + fk tµ + f⊥ mµ .
(B.1)
Here mµ is a normalized spatial vector orthogonal to tµ . The corresponding four-force in a system locally comoving with the particle, with velocity vtµ , is related to f µ simply via the Lorentz transformation. We may then align the first spatial coordinate with the direction of motion, and the second with the direction of the perpendicular force (mµ ). Denoting the components of the corresponding decomposition in the comoving system by (capital) F , using the fact that F 0 = 0, the Lorentz-transformation gives us 0
= γ(f 0 − vfk ) 0
(B.2)
Fk = γ(fk − vf )
(B.3)
F⊥ = f⊥ .
(B.4)
From the first and second equation above follows that fk = γFk . Using (B.1), we have then P µ α f α = γFk tµ + F⊥ mµ .
(B.5)
Here Fk is the experienced forward thrust (by a comoving observer), and F⊥ is the experienced sideways thrust. Note that while we proved the equality in a certain system, both sides are tensorial and thus it holds in any coordinate system. Appendix C. Expressing the four-acceleration in terms of the forces given by the congruence observers Letting pµ = mv µ denote the four-momentum of a particle we have 1 Dpµ Dv µ = Dτ m Dτ γ Dpµ . = m Dτ0
(C.1) (C.2)
38
Inertial forces and the foundations of optical geometry
Here τ0 is local time along the congruence. Looking at the right hand side in the coordinates of an inertial system locally comoving with the congruence, we see that the spatial part expresses momentum transfer per unit time i.e. force. So denoting the given forces parallel and perpendicular to the direction of motion by Fck and Fc⊥ we have by definition Dpα ≡ Fck tµ + Fc⊥ mµ . (C.3) P µα Dτ0 Hence we have 1 µ Dv α 1 P α = (Fck tµ + Fc⊥ mµ ). (C.4) 2 γ Dτ γm We may note by comparison with (B.5) that Fck = Fk and Fc⊥ = F⊥ /γ. Appendix D. Conformal transformations, covariant differentiation and the rescaled kinematical invariants Consider a conformal transformation g˜µν = e−2Φ gµν . Let k µ be a general vector field and k˜ µ = eφ k µ its rescaled analogue. We have then ˜ µ k˜ν = ∂µ (eΦ k ν ) + Γ ˜ ν eΦ k α . ∇ µα
(D.1)
Evaluated in a system in free fall relative to the original spacetime (so ∂µ → ∇µ ), we have ∗ 1 µρ ˜µ = g˜ (∇α g˜ρβ + ∇β g˜ρα − ∇ρ g˜αβ ) (D.2) Γ αβ 2 ∗ = ... h
∗
i
= − g µ α (∇β Φ) + g µ β (∇α Φ) − g µρ (∇ρ Φ)gαβ .
(D.3)
Using this in (D.1), evaluated in a freely falling system relative to the original spacetime, we readily find
˜ µ k˜ν = eΦ ∇µ k ν − g ν µ k α ∇α Φ + kµ g νρ ∇ρ Φ . ∇
(D.4)
This holds in originally freely falling coordinates. Since both sides are tensorial it holds in any coordinates. A corresponding expression for a covariant vector k˜µ = e−Φ kµ is given by ˜ µ k˜ν = e−Φ (∇µ kν − gµν k α ∇α Φ + kµ ∇ν Φ) . ∇
(D.5)
Now let us apply this to the congruence invariants. The invariants are defined according to (A.1)-(A.5). Using (D.5) and (D.4), assuming a (−, +, +, +) metric, we readily find the corresponding rescaled analogues a ˜µ = e2Φ (aµ − P µα ∇α Φ) θ˜ = eΦ (θ − 3η α ∇α Φ) −Φ
σ ˜µν = e
σµν
ω ˜ µν = e−Φ ωµν θ˜µν = e−Φ (θµν − Pµν η α ∇α Φ) .
(D.6) (D.7) (D.8) (D.9) (D.10)
39
Inertial forces and the foundations of optical geometry
It may also be convenient to know how the covariant derivative of a vector defined along a curve transforms. Suppose then that we have a vector k µ defined along a trajectory of four-velocity v µ . Let k˜µ = eΦ k µ and v˜µ = eΦ v µ . Considering an arbitrary smooth ˜k ˜µ ˜ α k˜µ , and ˜α ∇ extension of the vector k µ around the worldline, we can write37 DD˜ ˜τ = v apply (D.4) ˜ k˜µ D ˜ α k˜µ = v˜α ∇ (D.11) ˜ D˜ τ = ... ! µ Dk = e2Φ − (v µ k ρ − v α kα g µρ )∇ρ Φ . (D.12) Dτ In particular, considering k µ = v µ , we get the transformation of the four-acceleration µ ˜ vµ D˜ 2Φ Dv µρ µ ρ =e − (g + v v )∇ρ Φ . (D.13) ˜τ Dτ D˜ Equivalently we may write (D.13) as ˜ 2 xµ D D 2 xµ dxµ dxρ ˜ ˜ ρΦ (D.14) ∇ρ Φ − g˜µρ ∇ − = e2Φ 2 2 ˜ Dτ d˜ τ d˜ τ D˜ τ So here is how the four-acceleration with respect to the rescaled spacetime is related to the four-acceleration with respect to the standard spacetime. Appendix E. The acceleration of the generating observers in optical geometry We have ηµ
= − eΦ ∇µ t
(E.1)
η α ηα = − 1.
(E.2)
From the normalization follows that η α ∇µ ηα = 0. 0 = η α ∇µ ηα α
(E.3) Φ
= − η ∇µ (e ∇α t) α
Φ
(E.4) Φ
= − η (e ∇µ Φ∇α t + e ∇µ ∇α t)
(E.5)
= − ∇µ Φ − eΦ η α ∇µ ∇α t.
(E.6)
This will useful when we evaluate the four-acceleration below Dηµ = η α ∇α ηµ Dτ = − η α ∇α (eΦ ∇µ t)
(E.7) (E.8)
= − η α (eΦ ∇α Φ∇µ t + eΦ ∇α ∇µ t)
(E.9)
α
= ηµ η ∇α Φ + ∇µ Φ
(E.10) µ
We could just do an analogous derivation to that leading to (D.4) but for Dk Dτ . Using the trick of extending the vector around the trajectory we can however use the already derived formalism and save a little time. 37
40
Inertial forces and the foundations of optical geometry = (η α ηµ + δ α µ )∇α Φ
(E.11)
= P α µ ∇α Φ.
(E.12)
We notice that the right hand side is orthogonal to η µ , as it must. While the above derivation by itself had nothing to do with rescalings of spacetime, we can still in ˜ µ t. principle consider an optically rescaled spacetime g˜µν = e−2Φ gµν , where η˜µ = −∇ Then just setting tildes on everything above (for the case Φ = 0) it immediately follows ˜ η˜µ that DD˜ ˜ τ = 0. This is very intuitively reasonable, because in the rescaled spacetime there is no time dilation, thus being at rest must maximize the proper time. Therefore, ˜ η˜µ in the rescaled spacetime we have DD˜ ˜ τ = 0. If we use this, then (E.12) follows from (D.6). Appendix F. A note on the Newtonian analogue In typical inertial force applications in the Newtonian theory, one assumes a rigid reference frame that has an acceleration A0 of the origin and a rotation ω that may ˙ Following e.g. [17] we have change over time (non-zero ω). ˙ × r}′ −mω × (ω × r′ ) = ma′ F − mA0 −2mω × v}′ −m ω | {z {z | FCor
Ftrans
|
{z
Fcentrif
(F.1)
}
Here F is the real force and a prime means that the quantity is connected to the reference frame in question (which is not inertial in general). In particular a′ is the acceleration relative to the reference frame. From now on we will however let a′ = arel and drop the primes to conform with the notation of this article. In the above expression FCor is the Coriolis force, Fcentrif is the centrifugal force and Ftransv is the transverse force. While the former two forces have standard names, the latter does not appear to have a universally accepted name (as pointed out in [18]) – in fact in [18] it is denoted the Euler force. Considering motion along a special path of local curvature direction n ˆ and curvature radius R with respect to the reference frame, we can alternatively express the relative acceleration as n ˆ dv (F.2) t + v2 arel = ˆ dt R Here ˆ t is the (normalized) direction of motion with respect to the reference frame. Using (F.2) the inertial force equation then takes the form dv ˆ n ˆ (F.3) t + v2 dt R The proper (relative to an inertial frame) acceleration of a certain point r′ fixed relative to the reference frame can be found from (F.3). Note that the Coriolis force and the relative acceleration (the right hand side of (F.3)) are zero for this case, thus we have F − mA0 + FCor + Ftransv + Fcentrif =
mareference = mA0 − Ftransv − Fcentrif
(F.4)
Inertial forces and the foundations of optical geometry
41
Using this in (F.3) and moving terms around, also using the explicit form of the Coriolis force, we readily find 1 dv n ˆ F = areference + 2ω × v + ˆ (F.5) t + v2 m dt R Modulo the lack of expansion-shear terms (obviously since we are assuming a rigid reference frame) and factors of γ, the Newtonian formalism (in this form) precisely corresponds to the relativistic analogue (15), or equivalently (17). In general relativity there does not in general exist extended rigid reference frames, and there cannot be a general analogue of the Newtonian version in the form of (F.1). Indeed we may understand that any instance of r′ should vanish for the general case. Setting r′ = 0 in (F.1) we note that we reproduce (F.5) (since then A0 = aref ). Thus, as far as it is at all possible for general spacetimes, (15) and (17) conform precisely with the standard Newtonian formalism. Appendix F.1. A two-step point of view in Newtonian mechanics Consider in Newtonian mechanics a rigid non-inertial reference frame. For this case we have n ˆ dv (F.6) t + v2 Faparent = ˆ dt R Here Faparent is the sum of the real and the inertial forces. Relative to the reference frame, henceforth denoted the base reference frame, we may choose a new reference frame that may rotate and accelerate relative to the base reference frame. Then we may treat Faparent just like we treated the real force F above, to define a new frame of reference and introduce apparent forces with respect to that frame. In particular we note that for any particle motion relative to the base reference frame – we can always, as velocity and acceleration are concerned, consider the particle to momentarily move on a circle with accelerating angular velocity. In a rigid coordinate system with origin at the center of the circle in question, and with angular frequency and acceleration to match the particle motion, the particle is at rest and has zero acceleration (momentarily). In these coordinates there is centrifugal force and a transverse (Euler) ˆ t respectively. force whose magnitude and direction are given by −mv 2 Rnˆ and −m dv dt These two ’extra’ inertial forces will then precisely balance the real and the inertial forces as expressed relative to the base reference frame. Notice however that it is only in this double reference frame sense that it makes sense denote the relative acceleration (multiplied by −m) as inertial forces. Note also that by this philosophy, for a rotating base reference frame, we would get two types of centrifugal forces38 . 38
Note the difference between the two forces however – the first (standard) centrifugal force can be seen as a field – living in the base reference frame independent of the test particle motion. The second is a force defined at a single point and dependent on the motion of the particle. Note also as concerns general relativity – the very name “centrifugal” seems to indicate that there is somewhere a center of some relevance for the motion – in general relativity there can naturally be no such a center for general spacetimes.
Inertial forces and the foundations of optical geometry
42
For particular applications, such as a static black hole, using a static reference frame, the only inertial force is due to the acceleration of the reference frame – which one may connect in a Newtonian sense to gravity. In standard Newtonian mechanics gravity is not an inertial force, but a real force – hence the original base reference frame has a certain Newtonian inertial flavor to it (modulo curved space, time dilation etc.). From this point of view, the extra reference frame needed to denote the acceleration relative to the base reference frame as inertial forces is “almost” the first reference frame. Likely this philosophy (or something similar) is underlying the ideas by those authors (see e.g. [6, 10]) who denote the terms related to the relative acceleration as inertial forces (when multiplied by −m). In this article we are in any case considering just a single reference frame, and are allowing for acceleration relative to that frame. References [1] Bini D, Carini P and Jantzen RT 1998 Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity Tsvi Piran, Editor, World Scientific, Singapore, A 376-397 [2] Abramowicz MA and Lasota J-P 1997 Class. Quantum Grav. 14 A23-A30 [3] Jonsson R and Westman H 2006 Generalizing optical geometry Class. Quantum Grav. 23 61-76 [4] Foertsch T, Hasse W and Perlick V 2003, Class. Quantum Grav. 20 4635-4651 [5] Misner CW, Thorne KS and Wheeler JA 1973 Gravitation (New York: Freeman) p 566 [6] Abramowicz MA, Nurowski P and Wex N 1995 Class. Quantum Grav. 12 1467-1472 [7] Abramowicz M A 1992 Mon. Not. R. Astr. Soc. 256 710-718 [8] Abramowicz MA 1993 Scientific American no 3 (March) 266 pp 26-31 [9] Abramowicz MA, Nurowski P and Wex N 1995 Class. Quantum Grav. 10 L183-L186 [10] Abramowicz MA 1990 Mon. Not. R. Astr. Soc. 245 733-7466 [11] Perlick V 1990 Class. Quantum Grav. 7 1319-1331 [12] Jonsson R 2005 A covariant formalism of spin precession with respect to a reference congruence Class. Quantum Grav. 23 37-59 [13] Bini D, Carini P and Jantzen RT 1997 Int. Journ. Mod. Phys. D 6 143-198 [14] Jantzen RT, Carini P and Bini D 1992 Ann. Phys. 215 1-50 [15] Bini D, de Felice F, Jantzen RT 1999 Class. Quantum Grav. 16 2105-2124 [16] Jonsson R 2006 An intuitive approach to inertial forces and the centrifugal force paradox in general relativity Am. Journ. Phys. 74 905 [17] Fowles GR and Cassidy GL 2005 Analytical Mechanics (Belmont, CA: Thomson-Brooks/Cole) p 197 [18] Lanczos C 1970 The variational principles of mechanics, (New York: Dover) pp 100-103
An intuitive approach to inertial forces and the centrifugal force paradox in general relativity Rickard M. Jonsson Department of Theoretical Physics, Physics and Engineering Physics, Chalmers University of Technology, and G¨ oteborg University, 412 96 Gothenburg, Sweden E-mail:
[email protected] Submitted: 2004-12-09, Published: 2006-10-01 Journal Reference: Am. Journ. Phys. 74 905
Figure 1: Rockets orbiting a static black hole. The solid arrows correspond to the force (the rocket thrust) necAbstract. As the velocity of a rocket in a circular oressary to keep the rocket in circular orbit. Inside of the bit near a black hole increases, the outwardly directed photon radius (the dashed circle), the required force inrocket thrust must increase to keep the rocket in its orcreases as the orbital velocity increases. bit. This feature might appear paradoxical from a Newtonian viewpoint, but we show that it follows naturally from the equivalence principle together with special rel- centrifugal force is directed inward inside of the photon ativity and a few general features of black holes. We also radius and directed outward outside of the photon radius. derive a general relativistic formalism of inertial forces for This reversal of the direction of the fictitious centrifureference frames with acceleration and rotation. The re- gal force is described by the formalism of optical geomesulting equation relates the real experienced forces to the try (see Appendix A) in which the phenomena has been time derivative of the speed and the spatial curvature of discussed.3, 4, 5, 6, 7, 8 the particle trajectory relative to the reference frame. We Our purpose is not to explain the velocity depenshow that an observer who follows the path taken by a dence of the rocket thrust by analogy with Newtonian free (geodesic) photon will experience a force perpendic- theory, and we will use neither gravitational nor centrifuular to the direction of motion that is independent of the gal forces. Instead we will use the basic principles of relobservers velocity. We apply our approach to resolve the ativity to explain how the real force required to keep an submarine paradox, which regards whether a submerged object moving along a specified path depends on the vesubmarine in a balanced state of rest will sink or float locity of the object. when given a horizontal velocity if we take relativistic efWe start by illustrating how the fact that the rocket fects into account. We extend earlier treatments of this thrust increases with increasing orbital speed (sufficiently topic to include spherical oceans and show that for the close to the black hole) follows naturally from the equivcase of the Earth the submarine floats upward if we take alence principle (reviewed in Appendix B). We do so by the curvature of the ocean into account. first considering an idealized special relativistic scenario
I
of a train moving relative to an (upward) accelerating platform. We then consider a more general but still effectively two-dimensional discussion of forces perpendicular to the direction of motion for motion relative to an accelerated reference frame in special relativity. In a static spacetime, the reference frame connected to the static observers behaves locally like an accelerating reference frame in special relativity and the formalism can therefore be applied also to this case. We then illustrate how to apply the formalism of this paper to the submarine paradox.9 We ask whether a submarine with a density such that it is vertically balanced when at rest, will sink or float if given a horizontal velocity and relativistic effects are taken into account. Next we generalize the formalism of forces and curvature of spatial paths to include three-dimensional cases, forces parallel to the direction of motion, and rotating reference frames. The acceleration and rotation of the reference frame will be shown (as in Newtonian mechanics) to introduce terms that can be interpreted as iner-
Introduction
Consider a rocket in a circular orbit outside the event horizon of a black hole.1 If the orbit lies within the photon radius, the radius where free photons can move on circular orbits,2 a greater outward rocket thrust is required to keep the rocket in orbit the faster the rocket moves. However, outside of the photon radius the outward thrust decreases as the orbital speed increases just as it would for a similar scenario in Newtonian mechanics (the thrust will be inward directed for sufficiently high speeds, see Fig. 1). Analogous to the situation in Newtonian mechanics we can introduce in general relativity a gravitational force that is velocity independent. This force is fictitious (unlike in Newtonian mechanics). We can also introduce a velocity dependent (fictitious) centrifugal force that together with the gravitational force balances the real force from the jet engine of the rocket. By this definition, the
1
tial (fictitious) forces. By using the equivalence principle, order in δt the formalism can be applied to arbitrary rigid reference δh = aδt2 /2 (1a) frames in general relativity. We verify the results by com2 ′ ′ ′ paring with Ref. 10. (1b) δh = a δt /2 Although this paper is primarily aimed at readers ′ δh = δh. (1c) with a background in general relativity, the main part assumes only an elementary knowledge of special rela- From these equations follows that tivity together with a knowledge of a few basic concepts δt 2 of general relativity. Some of the more important cona′ = a = aγ 2 . (2) cepts such as curvature of a spatial path, spatial geomeδt′ try, geodesics, and the equivalence principle are reviewed Thus the proper acceleration, that is, the acceleration as in Appendix B. Sections IX–XI are more specialized. observed from an inertial system momentarily comoving with the apple, is greater than the acceleration of the platform by a factor of γ 2 . The force required to keep the II The train and the platform apple of rest mass m at a fixed height relative to the train 2 We consider the special relativistic description of a train is thus given by F = mγ a. To further clarify the main idea, we can also consider moving relative to a platform with proper upward acceleration a.11 The force required by a man on the train to a similar scenario where there are two apples on a horihold an apple at a fixed height increases as the train speed zontal straight line which accelerates upward relative to increases (assuming nonzero acceleration of the platform) an inertial system, as depicted in Fig. 3. as illustrated in Fig. 2. a (a)
(b) v
Figure 3: Two apples on an upward accelerating line (the solid line). The apples were initially at the position of the unfilled apple, one at rest and the other moving horizontally to the right. Both apples have to move up the same amount for a given coordinate time. But the one that moves horizontally has less proper time to do it. It must therefore experience a greater acceleration.
Figure 2: A train on a platform with a constant proper acceleration a upward. (a) The train is at rest; (b) the train is moving relative to the platform. The force required of a man on the train to keep an apple at a fixed height is higher when the train moves than when it is at rest relative to the platform. To understand this effect we consider the accelerating train as observed from two inertial systems. The first system S is a system in which the platform is momentarily (t = 0) at rest. The second system S ′ is comoving with the train at the same moment. The two systems are related to each other by a Lorentz transformation of velocity v, where v is the velocity of the train relative to the platform along the x-axis. Relative to S the apple moves to the right and accelerates upward with acceleration a. Consider now two physical events at the apple, one at t = 0 and one at t = δt, as observed from S. The time separation as observed in S ′ is (to lowestpnonzero order in δt) given by δt′ = δt/γ, where γ = 1/ 1 − v 2 /c2 and c is the velocity of light. In the following we will use c as the unit of velocity so that v = 1 for photons.12 The height δh separating the two events as observed in S equals the corresponding separation δh′ relative to S ′ . If we denote the upward acceleration relative to S ′ by a′ , we have to lowest nonzero
It follows from the equivalence principle (see Appendix B) that a flat platform on Earth (neglecting the Earth’s rotation) behaves like a flat platform with proper upward acceleration g in special relativity. Hence for a sufficiently flat platform, the force required to hold an apple at rest inside a moving train on Earth would increase as the velocity of the train increases.
III
The static black hole
Let us apply the reasoning of Sec. II to circular motion around a static black hole. A schematic of the exterior spatial geometry of an equatorial plane through a black hole is depicted in Fig. 4 (also see Appendix B). A local static reference system outside of the black hole will behave as an accelerating reference frame (train platform) in special relativity (again see Appendix B). Locally, the scenario is thus identical to that in Sec. II, except that the path along which the object in question moves (a circle in the latter case) is not straight in general 2
a circular orbit (given by F = mγ 2 arel ) increases as the orbital velocity increases. In brief, if an object moves it has less time (due to time dilation) to accelerate the necessary distance upward needed to remain at a fixed height (that is, fixed radius). Thus we can understand that that close to the horizon a greater outward force is needed to keep an object in orbit the faster the object moves.
g Figure 4: A freely falling frame (the grid) accelerating relative to the spatial geometry of a black hole.13 We consider circular motion along the dashed line. The bottom edge of the depicted surface corresponds to the horizon. At this edge the embedding approaches a cylinder and the circle at the horizon is thus straight in the sense that it does not curve relative to the surface.
IV
A more quantitative analysis
To understand where the transition from a more Newtonianlike behavior occurs, we need a more detailed analysis. Because the reference frame connected to the static observers around the black hole locally behaves like an accelerating reference frame in special relativity, we first consider this special relativistic case. (although circles can in fact be straight, see Appendix Let v be the velocity of a particle relative to the accelB). Instead, the circular motion corresponds locally to erated reference frame and let g be the acceleration of an letting the object in question follow a slightly curved path inertial frame, momentarily at rest relative to the referrelative to the accelerating platform (see Fig. 5). ence frame, which falls relative to the the reference frame. ˆ (discussed Also assume that the direction of curvature n in Appendix B) of the particle trajectory relative to the reference frame lies in the same plane as that spanned by v and g. In this way we consider an effectively twodimensional scenario as depicted in Fig. 6.
Figure 5: A zoom-in on the circular motion observed from a static system (with proper upward acceleration). The trajectory curves slightly downward, which will decrease the upward acceleration of the object relative to a freely falling system. In the limit that the acceleration of the freely falling frames is infinite, we can disregard the small curvature.
v g⊥
It is a well known property of Schwarzschild black holes that the proper acceleration of the local static reference frame goes to infinity as the radius approaches the radius of the event horizon. In other words the acceleration of a freely falling inertial frame (where special relativity holds, see Appendix B) which falls relative to the static reference frame, goes to infinity as the radius approaches the radius of the horizon. Furthermore we know that there is a maximum velocity v = 1 for material objects. Thus the perpendicular acceleration relative to the properly accelerated reference frame due to the curvature of the path remains finite (it is given by v 2 /R) for nonzero R; R is non-zero for the circular motion in question. It follows that the acceleration arel , relative to a freely falling frame, of an object in circular motion is dominated by the acceleration of the freely falling frame in the limit where the radius of the circle approaches the radius of the event horizon. Thus in this limit we can neglect the curvature of the path and from the reasoning in Sec. II we conclude that the force required to keep an object in
g Figure 6: A particle moving along a trajectory of curvature R relative to the accelerating reference frame. The thick line is freely falling and is initially (t = 0) aligned with the dashed line. Concerning forces perpendicular to the direction of motion, only the perpendicular part of the acceleration g is relevant. The perpendicular acceleration of a particle moving on a curve of radius R, as observed from the accelerated reference frame, is given by v 2 /R. In other words, the proper spatial distance δs from the particle to a straight line fixed to the accelerated reference frame and aligned with the particle initial (t = 0) direction of motion is given by δs = (δt)2 v 2 /(2R) to lowest nonzero order in δt; the latter is the local time as measured in the acceler3
ated reference frame. Because the inertial (freely falling) system is initially at rest with respect to the accelerating reference frame, the time as measured by a grid of ideal clocks in the freely falling system is identical to time (to first order in δt) relative to the accelerating reference frame. The same goes for distances (length contraction does not kick in until the two frames have an appreciable relative velocity). Consider a straight line fixed to the freely falling system that at t = 0 coincides with the previously mentioned line fixed to the reference frame. The separation between the two lines is (to lowest nonzero order in δt) given by g⊥ (δt)2 /2. Here g⊥ is the part of the acceleration of the freely falling system that is perpendicular to the initial direction of motion, as observed from the accelerating reference frame. It follows that observed from the freely falling system, where special relativity holds, the particle will have an acceleration perpendicular to the direction of motion given by g⊥ − v 2 /R. In analogy to our earlier reasoning, the perpendicular acceleration as observed in a system comoving with the particle is greater by a factor of γ 2 , and the perpendicular force F⊥ (as experienced in the particle’s own system) is thus given by F⊥ = mγ 2 (g⊥ − v 2 /R).
To make this fact more transparent, we consider the curvature vector of a geodesic photon given by Eq. (4) (set F⊥ = 0 and v = 1) n ˆ phot = g⊥ . Rphot
ˆ rel /Rrel as the curvature vector relative to We introduce n the trajectory of a geodesic photon: n ˆ n ˆ phot n ˆrel = . − Rrel R Rphot
(6)
ˆ rel /Rrel gives how quickly a particle This definition of n trajectory deviates from a geodesic photon trajectory in analogy to how n ˆ /R gives how quickly the particle trajectory deviates from a straight line. We substitute Eqs. (5) and (6) in Eq. (3) and find F⊥ n ˆrel . = −g⊥ + γ 2 v 2 m Rrel
(7)
Equation (7) also holds for more general three-dimensional scenarios as will be shown in Sec. IX. Within the photon radius, a photon trajectory departs inward relative to a locally tangent circle. Thus the relative curvature direction n ˆ rel of a circle within the photon radius is directed outward. From Eq. (7) it then follows that the faster an object orbits the black hole, the greater the outward force must be. However, outside of the photon radius, a photon trajectory departs outward from a locally tangent circle. Thus the relative curvature of the circle is directed inward. It follows from Eq. (7) that outside of the photon radius, the outward force required to keep the rocket in orbit will decrease as the velocity increases. Thus we see that the effective centrifugal force reversal for circular motion occurs exactly at the photon radius.
(3)
To clarify any sign ambiguities we rewrite Eq. (3) in terms of vectors: ˆ n F⊥ = −γ 2 g⊥ + γ 2 v 2 . (4) m R Equation (4) relates the perpendicular part of the force (as observed in the particle’s own reference system) to the spatial curvature of the particle trajectory relative to a reference frame with proper acceleration −g. Although Eq. (4) was derived for an effectively two-dimensional scenario, it holds also in three dimensions as we will see in Sec. IX. From the equivalence principle it follows that Eq. (4) applies also to motion around a black hole. For this case the curvature vector is defined relative to the spatial geometry connected to the static observers, as explained in Appendix B.
V
(5)
Following the geodesic photon
Inspired by the reasoning of Abramowicz et al.,3 we now consider motion along the spatial trajectory of a geodesic photon (a photon whose motion is determined by gravity alone, see Appendix B). For a geodesic particle we have F⊥ = 0, and thus according to Eq. (3), g⊥ = v 2 /R. For a geodesic photon whose path curvature we denote by Rphot , we have thus g⊥ = 1/Rphot (because v = 1 for photons). For a particle following the path of a such a photon (so 1/R = g⊥ ) we have according to Eq. (3), F⊥ = mγ 2 (g⊥ − v 2 g⊥ ), which simplifies to F⊥ = mg⊥ . Thus, the perpendicular force required to make a particle follow the trajectory of a geodesic photon is independent of the velocity of the particle.
Figure 7: Trajectories of geodesic photons (dashed curves) relative to circles around a black hole. Inside the photon radius a circle curves outward relative to a locally tangent photon trajectory.
4
VI
The difference between the given and the received force
write Fc⊥ = F⊥ /γ.
Hence the given force is smaller than the received force by a factor of γ. Because the received force required to keep an object (like an entire train) moving along a straight horizontal line relative to a vertically accelerating reference frame is proportional to γ 2 (as discussed in Sec. II), it follows that the force required by the rail to support the train scales with a factor of γ. In Sec. V we showed that the perpendicular received force is independent of the velocity for an object that follows the trajectory of a geodesic photon. Now we ask if there is a corresponding path for which the perpendicular given force is velocity independent. The analogue of Eq. (4) for the given force is
Before considering a more general analysis, we will distinguish between two types of forces. The perpendicular force that we have discussed is the force as observed in a system comoving with the object in question. Consider now a situation where the observers connected to the reference frame in question (like the accelerating platform we have considered previously) provide the force that keeps the object on its path. How is this force, which we will refer to as the given force, related to the previously considered force, which we will refer to as the received force, that is, the force as observed in a system comoving with the object? For instance, we might be interested in the magnitude of the vertical force by the rail that is needed to support a train moving with a relativistic speed along the track. Unlike in Newtonian theory, this given force will be different from the force as observed in a system comoving with the train. For forces perpendicular to the direction of motion, the relation between the given and the received force can be understood by considering a simple model of force exertion (a more formal derivation is given in Ref. 10). Assume that the force on the object is exerted by little particles bouncing elastically on the object. Every bounce gives the object a certain impulse (see Fig. 8). (a)
v
(8)
n ˆ Fc⊥ = −γg⊥ + γv 2 . m R
(9)
We now require Fc⊥ to be the same for an object moving with speed v as that of an object at rest. For v = 0, Eq. (9) gives Fc⊥ /m = −g⊥ . We substitute this result into Eq. (9) and find n ˆ γ 1 g⊥ = g⊥ = 2 1− . R v γ γ+1
(10)
This curvature depends on the velocity.14 Thus considering the given force (the force as observed from the accelerating reference frame), there is no path for which the perpendicular force is independent of the velocity.
(b)
VII
The relativistic submarine
As an application of our discussion we consider a submarine submerged in water with a density such that it remains at rest.9 If we take relativistic effects into account, but disregard the more subtle aspects of fluid dynamics such as viscosity and turbulence, will the submarine sink or float when it is given a horizontal velocity?
Figure 8: A simple model where small particles bounce elastically on the object. (a) The scenario as observed from a system where the impulse giving particle has no horizontal velocity. (b) The corresponding scenario as observed from a system comoving with the object.
A
If we give an object moving relative to an inertial system S an impulse perpendicular to the direction of motion, the object will in its own reference system receive the same impulse because a Lorentz transformation does not affect the perpendicular part of the momentum change. On the other hand, the proper time of the object runs slower by a factor of γ compared to local time in S. Hence the bouncing particles will bounce more frequently by a factor of γ as observed from the reference frame of the object. Because force equals transferred impulse per unit time, it follows that the received force, perpendicular to the direction of motion, is greater than the corresponding given force, by a factor of γ. We let F⊥ denote the perpendicular received force and Fc⊥ the perpendicular given force (to conform with the notation of Ref. 10) and
A flat ocean in special relativity
We first consider a special relativistic scenario where the flat bottom of the ocean has a constant proper upward acceleration. In Ref. 9 accelerated (Rindler) coordinates are used to find out whether the submarine sinks or floats, after several pages of calculation. By using our earlier reasoning, we can readily find the answer without any calculations. The received force needed to keep the submarine at the same depth increases by a factor of γ 2 as demonstrated in Sec. II. The given force needed to keep it at a constant depth thus increases by a factor of γ because it is smaller than the received force by a factor of γ as explained in Sec. VI. However, the submarine is length contracted, so the actual given force from the water pressure (or rather the differences of
5
(a)
(b)
submarine is independent of the velocity. Although every single column of water yields a received force that is smaller than the force given by that column (by a factor of γ), there are γ 2 times more columns contributing to the net force as observed from the submarine frame, than observed from the rest frame of the Figure 9: (a) An idealized (rectangular) submarine sub- water (see Fig. 11). Thus consistent with the reasoning of merged in water at rest relative to the water. (b) As the Sec. VI, the net received force is greater than the given submarine moves, it will be length contracted and thus force by a factor of γ. the given force from the water will decrease by a factor (a) (b) of γ. v v v
v water pressure at the top and bottom of the submarine) will decrease by a factor of γ (see Fig. 9). Therefore the given force decreases by a factor of γ, whereas it should increase by a factor of γ in order for Figure 11: (a) Observed from the water system the subthe submarine to remain at a fixed depth. Thus the sub- marine is length contracted by a factor of γ. (b) Observed marine will sink (see Fig. 10). from the submarine the water columns are length contracted and thus denser by a factor of γ.
B
A real spherical ocean
It is easy to generalize the above discussion to apply to a submarine in the ocean of a spherical planet. Locally the scenario is almost identical to the one we have discussed, assuming that the submarine is small compared to the size of the planet. The question of whether the submarine floats or sinks amounts to whether the submarine, when given an azimuthal velocity, departs outward or inward from a circle locally tangent to the direction of motion. The answer follows from our previous discussion. As argued in Sec. A, the force as experienced in a system comoving with the submarine, is independent of the velocity for this case. It then follows from Eq. (7) that the submarine will have zero curvature relative to a geodesic photon and will thus follow the path of a geodesic photon.
Figure 10: A submarine submerged in a balanced state of rest in a flat ocean with proper upward acceleration, will sink due to relativistic effects if it starts moving horizontally. Now let us analyze the situation from the submarine. Due to length contraction the actual given force is decreased by a factor of γ, as we have argued. The received force is γ times the given force. Thus, the received force is independent of the velocity.15 This force is not sufficient to keep the submarine at the same depth. The experienced force would have to increase by γ 2 for that. The submarine thus sinks. To understand why the received force on the submarine is independent of the velocity, we can also look at the water at the molecular level. Assume that the water molecules are moving along columns fastened to the ocean bottom (a very crude model). Assume also that the particles elastically bounce back down the same column (without interfering with the up-moving water molecules) when they hit the hull of the submarine (and analogously for the water molecules on top of the submarine). The impulse given by a single molecule is the same as when the submarine was at rest. However, as observed from the moving submarine the columns of water molecules are length contracted by a factor of γ. There are thus more columns under the submarine (and above) in the submarine frame, when the submarine moves than when it is at rest. On the other hand, due to time dilation, how often a molecule from a single column hits the hull is decreased by a factor of γ (consider a clock fixed to the column just where the column intersects the submarine hull). The effects thus cancel. It follows that the received force on the
Figure 12: Submarines moving at different depths in the ocean of an imaginary very dense planet. The dashed line is the photon radius. The submarines outside the photon radius will float upward if they are given an azimuthal velocity; the opposite holds within the photon radius.
6
IX
So, outside the photon radius (the radius where photons would move on circular orbits if there were no refraction effects from the water) the submarine will float upward, at the photon radius it will remain at the same depth, and inside the photon radius it will sink. The scenario is illustrated in Fig. 12. Consider the Earth, which is not sufficiently dense to have a photon radius. If we take into account the Earth’s curvature, it follows that when given a horizontal velocity, the submarine will not sink after all but rather float upward.
VIII
Consider a reference frame with a proper (upward) acceleration. Given the curvature and curvature direction of the path taken by a test particle relative to the reference frame, we want to express the perpendicular acceleration of the test particle relative to an inertial system S in which the reference frame is momentarily (t = 0) at rest. In Fig. 14 we illustrate how the trajectory will deviate from a straight line (directed along the particle initial direction of motion) which is fixed to S, and thus falls relative to the accelerated reference frame. From this deviation we can find the perpendicular acceleration relative to S, analogous to the two-dimensional discussion in Sec. IV.
The weight of a box with moving particles
As another application of our discussion, we consider the weight of an object whose internal components move. In general relativity, if we for instance heat an object, it will become heavier. In other words, a greater upward force is required to keep the object at rest (on Earth) when the object is warm (molecules moving faster) than when it is cold. Although not directly related to the main topic of this article (inertial forces), we can give a simple explanation. Consider a black box containing two balls connected by a rod of negligible mass which is suspended in such a way that the balls can rotate in a horizontal plane. If they rotate, the upward force needed to keep a single ball in the horizontal plane as observed from the balls’ reference system is mgγ 2 , where m is the rest mass of the ball. The given force is smaller by a factor of γ and is hence given by mgγ. So the weight of the box is greater when the internal particles move than when they are at rest (see Fig. 13).
(a)
Generalizing to three dimensions
z δx1
δx3
δx2 Freely falling line y x
g
Figure 14: Deviations from a straight line relative to the (properly) accelerated reference system. The z-direction is chosen to be antiparallel to the local g. The plane in which we study the deviations is perpendicular to the momentary direction of motion (the dashed line) and the three vectors lie in this plane. The solid curve is the particle trajectory as observed in the (properly) accelerated reference system. The thick line is a freely falling line that was aligned with the dashed line (and at rest relative to the reference frame) at the time when the particle was at the origin.
(b)
Figure 13: A black box (transparent for clarity) containing a pair of balls that (a) are at rest and (b) are moving. If we let δt denote a small time step and use the defThe force needed to hold the box at a fixed height on initions introduced in Fig. 14, we have to lowest nonzero Earth is greater when the balls are moving than when order in δt: they are at rest. The force is proportional to the total relativistic energy of the box. n ˆ v 2 δt2 δx1 = (11a) R 2 δt2 (11b) δx2 = g⊥ For vertical or arbitrary motion, this type of reason2 ing is not as powerful, and we can instead make a more δx3 = δx1 − δx2 , (11c) formal proof using four-vectors and conservation of fourmomentum. ˆ are the curvature and curvature direcwhere R and n tion of the spatial trajectory relative to the accelerated reference frame. Let arel⊥ be the acceleration of the test particle perpendicular to the direction of motion relative 7
t=0
to the freely falling frame. By using δx3 = arel⊥ δt2 /2 and Eq. (11), we find arel⊥ = −g⊥ + v 2
n ˆ . R
t = δt
v
(12)
We denote the received perpendicular force by F⊥ . According to our previous reasoning, we have F⊥ = mγ 2 arel⊥ , and thus 1 n ˆ F⊥ = −g⊥ + v 2 . (13) 2 mγ R Equation (13) relates the experienced perpendicular force and the curvature relative to the accelerating reference system. We note that the only difference from its Newtonian analogue is the factor of γ 2 on the left-hand side. In analogy to the two-dimensional discussion in Sec. V, we may introduce a curvature relative to that of a geodesic photon as n ˆ n ˆ phot n ˆrel = . (14) − Rrel R Rphot
v + δv δu
Figure 15: An object moving relative to an accelerated reference frame. At t = 0 the velocity of the particle is v. In a time δt the reference frame is accelerated to a velocity δu, and the velocity of the particle relative to the accelerated reference frame is v + δv. S ′ moving with velocity δu relative to S (along the x-axis) follows from the Lorentz transformation (see for example, Ref. 17, p. 31) w − δu w′ = . (16) 1 − wδu
ˆ phot /Rphot (setting F⊥ = 0 If we use Eq. (13) to find n and v = 1) and substitute the expression for n ˆ/R from If we substitute w = v + δvs and w′ = v + δv in Eq. (16) Eq. (14) into Eq. (13), we obtain and do a Taylor expansion to first order in δu and δvs , we obtain F⊥ n ˆrel δv = δvs − (1 − v 2 )δu. (17) . (15) = −g⊥ + γ 2 v 2 m Rrel We also know (see for example, Ref. 17, p. 33), that the We see that Eqs. (4) and (7), which were previously deproper acceleration α of the object, that is, the accelerrived only for effectively two-dimensional scenarios, are ation as observed in a system comoving with the object, also valid for arbitrary three-dimensional scenarios. For is related to the acceleration dvs /dt relative to S by the case where the observers at rest in the accelerating reference frame provide the pushing needed to keep the dvs . (18) α = γ3 particle on track, we obtain the given force as before by dt dividing the received force by a factor of γ. If we denote the received forward thrust by Fk , we have Fk = mα. We use this relation in Eqs. (17) and (18), take the limit where δt is infinitesimal, together with a = X Parallel accelerations du/dt and find Now that we know how the spatial curvature depends on a 1 dv the perpendicular force, it would be useful also to know Fk − 2 . (19) = 3 dt mγ γ how forces in the forward direction affect the speed v of the particle relative to the accelerated reference frame. Consider now a more general case where the accelWe could derive this relation using four-velocities,16 but eration of the reference frame need not be aligned with for simplicity, we will use only standard results that follow the direction of motion. It is easy to realize (or at least from the Lorentz-transformation. guess) that the acceleration of the reference frame perTo determine dv/dt, where v is the local velocity relapendicular to the direction of motion will not affect the tive to the accelerating reference frame, we must take into 18 We let g = −a, where a is the local speed derivative. account that the derivative implies that we are comparing acceleration of the reference frame relative to an inertial the velocity at two different times, relative to two differsystem in which the reference frame is momentarily at ent systems (effectively) because the reference system is rest, and write accelerating. Consider a scenario where the acceleration a of the reference frame is aligned with the direction of gk 1 dv Fk + 2 . (20) = motion. Relative to an inertial system S in which the ref3 dt mγ γ erence frame is at rest at t = 0, the reference frame gains a velocity δu = aδt after a time δt. We denote by δvs the Here gk is minus the part of the reference frame accelvelocity difference of the particle relative to S from t = 0 eration that is parallel to the particle’s direction of motion. Thus we now have a general expression for the speed to t = δt (see Fig. 15). The relation between an arbitrary object’s velocity change relative to the accelerating reference system. Note w (along the x-axis) as observed from S and the corre- that t is the local time relative to the reference frame (so sponding velocity w′ as observed from an inertial system dt = γdτ ). 8
in the comoving system S ′ than in S by a factor of γ. These two factors of γ cancel each other, and we conclude From the form of Eqs. (20) and (13), we see that we that the given and the received force in the direction of can combine them into a single vector relation. Let ˆt be motion are the same. One can easily give a formal proof a normalized vector in the forward direction of motion of this fact (see for example, Ref. 10). (to conform with the notation of Ref. 10). By multiplyWe can now express Eq. (21) in terms of the given 2ˆ ing Eq. (20) by γ t and adding the resulting equation to forces. We let Fck denote the given force in the direction Eq. (13), we can form a single term g (by adding the g⊥ of motion and write and gkˆ t terms) and obtain 1 v2 2 dv ˆ ˆ 2 (γF t + γF m) ˆ = −g + γ t + n ˆ. (24) c⊥ ck dv v 1 mγ 2 dt R (γFkˆ t + F⊥ m) ˆ = −g + γ 2 ˆ t+ n ˆ. (21) 2 mγ dt R We define Fc = Fckˆ t + Fc⊥ m ˆ and write Eq. (24) as ˆ Here m ˆ is a unit vector perpendicular to t. We thus have v2 dv Fc an expression for the spatial curvature and the speed t+ n = −g + γ 2 ˆ ˆ. (25) derivative in terms of the received forces. Note that g mγ dt R may be interpreted as an inertial (fictitious) force; we If we compare (25) with Eq. (21), we see that the formalwill discuss this interpretation in Sec. XI. We have previously considered a rocket in circular or- ism is a bit cleaner if we consider the given force rather bit with constant speed around a black hole. Now we con- than the received force. sider a rocket in radial motion with constant speed outward from a black hole. From the parallel part of Eq. (21) XI Rotating reference frame we find Fk = mgγ, (22) Suppose that we would also like to consider stationary
A
Combining the force equations
spacetimes, such as the spacetime of a rotating (Kerr) black hole. For this case we have a spatial geometry defined by the stationary (Killing) observers. In this case through frame dragging, the local reference frame connected to the stationary observers is not only accelerating, but also rotating. Consider (in special relativity) a reference frame that rotates around its origin relative to an inertial system S. For simplicity, we consider motion along a straight line that passes the origin and is fixed to the rotating frame. The particle is assumed to be at the origin at t = 0. This scenario is depicted in Fig. 16. Let δx denote the perpendicular separation from the particle to a line that is fixed in the inertial system S and that at t = 0 was aligned with the rotating line. We let δu denote the velocity of the line fixed to the rotating system at the position of the particle after a time δt. To lowest order in δu we have
where g is the magnitude of the acceleration of the local freely falling frames (g is a function of the radius that can readily be found from the spacetime metric). Here there are no reversal issues. However, we can see that (unlike in Newtonian theory), a greater thrust is needed to keep a constant speed the faster the rocket moves.
B
The given parallel force
If we would like an expression of the type Eq. (21) for the parallel given force, we need to know how the given force along the direction of motion is related to the received force along the direction of motion. We can make an argument similar to the one we made in Sec. VI. Let S denote a certain rest system, and let S ′ be a system in a standard (non-rotated) configuration relative to S, which comoves with the object in question along the x-axis of S. Consider the force parallel to the direction of motion to be mediated by (very light) particles bouncing elastically on the object. For simplicity let us assume that in a system comoving with the object, each bouncing particle is reflected in such a way that the energy of the bouncing 0 particle is unaffected by the bounce (so ∆p′ = 0). If we consider motion along the x-axis and use the fact that the change of momentum four-tensor transforms according to the Lorentz transformation, we have
δx = δu δt.
(26)
The position of the particle after a time δt is vδt (to lowest order in δt), where v = vˆ t. We thus have δu = ω×(vδt). We use this relation in Eq. (26) and obtain δx = ω × vδt2 . In the limit where the time step is infinitesimal, the perpendicular acceleration coming from the rotation is x ′x ′0 arel⊥ = 2ω × v. (27) (23) ∆p = γ(∆p + v ∆p ). | {z } For the low reference frame velocities that occur during 0 the short time δt, the effects of length contraction and x Thus the received impulse ∆p′ is smaller than the given time dilation will not enter the expressions for the perimpulse ∆px by a factor of γ. On the other hand, due to pendicular deviations (to lowest nonzero order). Theretime dilation the frequency at which these impulses are fore we can add the effect of rotation to the effects of curreceived (assuming several bouncing particles) is greater vature and acceleration. The generalization of Eq. (12) is 9
XII
ω
Discussion
On the left-hand side of Eq. (29) there are real forces as experienced in a system comoving with the object in question. On the right-hand side the first two terms multiplied by −m may be interpreted as inertial forces
δx
Acceleration:
mg,
Coriolis: − 2mω × v.
Figure 16: A particle (the black dot) moving along a rotating straight line (depicted at two successive time steps – the dashed and the solid line), as observed relative to an inertial system S. Relative to S the particle trajectory (the dotted line) curves. thus
n ˆ . (28) R Here arel⊥ is the perpendicular acceleration of the test particle relative to the inertial system S. Because the changes in the reference frame velocity (as observed from the inertial system in question) are perpendicular to the direction of motion, the derivative of the speed will not be affected by the rotation. Thus we can write the generalization of Eq. (21) as arel⊥ = −g⊥ + 2ω × v + v 2
v2 1 2 dv ˆ ˆ (γF t + F m) ˆ = −g + 2ω × v + γ t + n ˆ . (29) ⊥ k mγ 2 dt R Equation (29) relates the real received forces to both the curvature and the speed change per unit time relative to the accelerating and rotating reference frame. Note that although g is minus the acceleration of the reference frame, ω is the rotation vector of the reference frame. As an application we consider a person walking on a straight line through the center of a rotating flat merrygo-round (in special relativity). The perpendicular force experienced as he/she passes the center (where g is zero) is given by Eq. (29) as F⊥ = 2ω0 vγ 2 .
(30)
Here ω0 is the angular frequency of the merry-go-round. Apart from the γ 2 factor, Eq. (30) is the same as the corresponding equation in Newtonian mechanics. For points other than the central point we must consider that the proper rotation ω (as measured by an observer riding the merry-go-round at the point in question) is different from the rotation ω0 as observed from the outside.19
(31a) (31b)
We might be tempted to denote the first term by “gravity” rather than “acceleration,” but if we consider a rotating merry-go-round as a reference frame, this term would correspond to what is commonly called the centrifugal force. To avoid confusion we therefore label this term “acceleration.” For the second term the name Coriolis is obvious in analogy with the standard notation for inertial forces in non-relativistic mechanics. Note that what we call an inertial force is ambiguous. For example, we could multiply the perpendicular part of Eq. (29) by γ. By defining F = Fkˆ t +F⊥ m, ˆ we could then simplify the left-hand side of Eq. (29) to F/mγ. However, because of the γ-multiplication we would need to express the g-term as a sum of a parallel and a perpendicular part (with different factors of γ), thus creating two different acceleration terms. There is is thus more than one way of expressing Eq. (29), and identifying inertial forces, that reduce to the Newtonian analogue by setting γ = 1. We do not regard the last two terms on the righthand side of Eq. (29) as inertial forces, but rather as descriptions of the motion (acceleration) relative to the frame of reference. There are alternative interpretations; see Ref. 10 for further discussion. Note that dt is the local time (for the local reference frame observers) and is related to the proper time dτ for the particle in question by dt = γdτ . Equation (29) is identical to the more formally derived corresponding expression in Ref. 10. We have considered accelerating and rotating reference frames, but not shearing or expanding reference frames. The extension is straightforward for an isotropically expanding reference frame, but for brevity we refer to Ref. 10. In summary, we have seen how we can derive a formalism of inertial forces that applies to arbitrary rigid reference frames in special and general relativity. Apart from factors of gamma, the formalism is locally equal to its Newtonian counterpart. We have also applied the insights and formalism of this paper to various examples, such as moving trains and submarines.
A
A comment on static spacetimes, index notation, and the optical geometry
For the purposes of this article it is not necessary to discuss a formalism known as optical geometry. However, because the latter is the inspiration for this article and the
10
formalisms are very similar, a comment is in order. The index formalism (which distinguishes between covariant and contravariant vectors) is vital for the comparison. Suppose that we have a static spacetime with the line element ds2 = −e2Φ dt2c + gij dxi dxj . (A1) We denote coordinate time by tc so as not to confuse it with the local time of the reference frame which we denote by t. Also, Latin indices are spatial indices running from 1–3. It is easy to show (see for example, Ref. 10, Appendix E) that the acceleration of the freely falling frames for a line element of this form is given by g = −∇Φ. We can equivalently write this relation as g k = −g kj ∇j Φ. For later convenience we define F⊥k = F⊥ mk , where mk is a normalized spatial vector. If we use these results and definitions, we can rewrite Eq. (15) as
Note that because the covariant derivative acts on a scalar ˜ j = ∇j Φ (in contrast to a vector for example), we have ∇ j 2Φ j ˜ (although ∇ Φ = e ∇ Φ). By comparing Eq. (A5) with the more general (and more formally derived) corresponding equation in Ref. 10, we have a perfect match.22 In covariant form (lower indices with g˜ij ) Eq. (A5) becomes slightly more compact: ˜k F⊥ Φ ˜ k Φ]⊥ + γ 2 v 2 n e m ˜ k = [∇ . ˜ m R
(A6)
Because m ˜ k = e−Φ mk , the left-hand side of Eq. (A6) can be expressed as F⊥k /m. On the other hand, the left-hand side of Eq. (A5) can be written as F⊥k e2Φ /m. Expressed in these forms, but using the boldface vector notation, the right-hand sides of Eq. (A5) and Eq. (A6) are identical and the left-hand sides differ by a factor e2Φ . We hence understand the hazard of using the bold face vector notak F⊥ k kj 2 2 nrel . (A2) tion, at least if we use vectors that naturally “belong” to m = [g ∇j Φ]⊥ + γ v m Rrel two different metrics in the same expression. As Eqs. (A5) and (A6) are written, only vectors belonging to the optiHere ⊥ means that we should select the part perpendicucal geometry are used, and we could use vector notation lar to the spatial direction of motion tk . For a line element after all. such as Eq. (A1), the optical geometry (see for example, The parallel part of Eq. (21) in index notation (for Ref. 20 although a different sign convention for Φ is used) the line element in question and a static reference frame) is given by a rescaling of the standard spatial geometry takes the form −2Φ g˜ij = e gij . (A3) dv 1 Fk tk = [g kj ∇j Φ]k + γ 2 tk . (A7) mγ dt −Φ We thus stretch space by a factor e to create a new spatial geometry. We may consider both metrics to live Here we have dt = eΦ dt . We use the latter relation, c on the same (sub)manifold. Relative to the rescaled ge- rewrite the tensors in terms of their rescaled analogues, ometry, the curvature of a given spatial (coordinate) tra- multiply the entire expression by e2Φ , and add it to Eq. (A5). jectory is in general different from that relative to the The result is standard spatial geometry. In particular, the spatial tra eΦ Fk ˜k ˜k jectories of geodesic photons are straight with respect to ˜ j Φ+γ 2 dv t˜k +γ 2 v 2 n . (A8) t +F⊥ m ˜ k = g˜kj ∇ the optically rescaled space. It follows that the curvature ˜ m γ dtc R and curvature direction with respect to the rescaled (optical) space gives how fast (with respect to the distance Equation (A8) is the inertial force formalism in terms along the trajectory) and in what direction a trajectory of the optical geometry. Again it agrees with the corredeviates from that of a geodesic photon. This curvature sponding equation of Ref. 10.23 and curvature direction thus correspond to the relative curvature and curvature direction introduced in Sec. V B Some basic concepts and Sec. IX, except that the deviation and the distance along the trajectory are now rescaled. The optical spatial This appendix is included for readers with little or no ˜ the optical curvature direction n curvature R, ˜ k , and the background in differential geometry or Einstein’s theory optically normalized direction of the perpendicular force of gravity. m ˜ k for a certain (coordinate) trajectory are related to Curvature. Consider a curved path on a plane. At any Rrel , nkrel , and mk by21 point along the path we can find a circle that is precisely tangent to the path and whose curvature matches that of the path (see Fig. 17). At any point along the curve n ˜ k = eΦ nkrel (A4b) we can thus introduce a curvature direction n ˆ , a unit k Φ k vector, and a curvature radius R as shown. The greater m ˜ =e m . (A4c) the curvature, the smaller the curvature radius. For paths If we use g˜ij = e2Φ g ij , we may rewrite Eq. (A2) as (mul- that are not in a plane we can locally match a circle to every point along the path and define the curvature tiply the entire expression by e2Φ ) direction and curvature radius analogously. Note that the k n ˜ F⊥ Φ k curvature direction n ˆ is always perpendicular to the path. ˜ j Φ]⊥ + γ 2 v 2 . (A5) e m ˜ = [˜ g kj ∇ ˜ m R ˜ = e−Φ Rrel R
(A4a)
11
R n ˆ
Figure 17: A path on a plane always corresponds locally to a circle as far as direction and curvature are concerned. Spatial geometry. Consider a symmetry plane through a black hole. For the purposes of this article we may illustrate the black hole as a black sphere (see Fig. 18).
Figure 18: A symmetry plane through a black hole.
If we could walk around on the plane and measure distances, we would notice that the distances would not match those we would expect from a flat plane. Rather, the apparent geometry would be as that depicted in Fig. 19. In particular, we would note that as one walks outward from the surface of the black hole, the circumference would initially hardly change. Although the geometry of the curved surface corresponds to the geometry of the symmetry plane, the symmetry plane neither curves upward nor downward in reality. Distances on the plane are as if the plane curves as depicted in Fig. 19.
Figure 19: Sketch of the apparent geometry of a symmetry plane through a black hole. The innermost circle is at the surface of the black hole. Straight lines as geodesics. On a curved surface we can determine if a line is straight or curved at a certain point by looking at the line. We position our eye somewhere on an imagined line extending from the point in the direction of the normal to the surface, and look down along this imagined line at the surface. If the line on the surface looks straight, it is straight. If the line looks curved, it
is curved. A line that everywhere, as seen from the local normal, looks straight, is known as a geodesic. For a spherical surface like the surface of the Earth, the equator is a geodesic. For a line that is not straight, we can introduce a curvature direction and a curvature radius by considering how fast and in what direction the line deviates from a corresponding straight line on the surface, analogous to the definition for flat surfaces. In Einstein’s theory of relativity, the motion of particles whose motion is determined by gravity alone corresponds to geodesics in curved spacetime. For the purposes of this article it is sufficient to know that a geodesic particle is a particle that is free to move as gravity alone dictates. Examples are a dropped apple or a flying cannonball (assuming that we neglect air resistance). In general relativity there is no gravitational force, but there are forces such as air resistance. These forces cause objects to deviate from the motion determined by gravity. The equivalence principle can be formulated as follows: At any point in space and time we can introduce freely falling coordinates relative to which special relativity holds. As an example we consider an elevator whose support cables have just snapped at the topmost level of a high building. An observer dropping a coin inside the elevator will note that the coin will float in front of him. If he tosses the coin, he will note that the coin moves away from him on a straight line with constant speed just as it would if the elevator was in outer space where there is no gravity and special relativity holds. We can alternatively say that being in an elevator at rest on Earth is equivalent to being in an accelerated elevator in outer space (see Fig. 20). (a)
(b)
Figure 20: Dropping an apple inside an elevator on Earth gives the same motion relative to the elevator as dropping it inside a (properly) accelerated elevator in outer space. In both cases we can introduce an inertial (fictitious) gravitational force – but there is (in either case) no real gravitational force (in Einstein’s theory). It is a standard technique of Einstein’s general theory of relativity to first understand how a scenario will work relative to a freely falling frame where everything is simple, and then express the result with respect to the coordinates that really interest us. These freely falling frames are however not falling relative to a flat spatial geometry. For the particular case of a symmetry plane of a static black hole (see Fig. 18), we can imagine the freely falling frames (a coordinate grid in this case) to be falling
12
relative to the curved geometry depicted in Fig. 19. How fast the freely falling frames accelerate depends on the position (the radius). At spatial infinity the acceleration is zero and at the horizon it is infinite. The idea is illustrated in Fig. 21.
[4] Marek A. Abramowicz and Jean-Pierre Lasota, “A note of a paradoxical property of the Schwarzschild solution,” Acta Phys. Pol. B5, 327-329 (1974). [5] Marek A. Abramowicz, Brandon Carter, and JeanPierre Lasota, “Optical reference geometry for stationary and static dynamics,” Gen. Relativ. Gravit. 20, 1173–1183 (1988). [6] Marek A. Abramowicz and A. R. Prasanna, “Centrifugal force reversal near a Schwarzschild blackhole,” Mon. Not. R. Astr. Soc. 245, 720–728 (1990). [7] Marek A. Abramowicz, “Centrifugal force: A few surprises,” Mon. Not. R. Astr. Soc. 245, 733–746 (1990).
Figure 21: A coordinate system accelerating (falling) relative to the curved spatial geometry of a black hole. The depicted freely falling frame coordinate lines are geodesics24 on the curved surface. With respect to the falling coordinate grid a free particle, that is, a particle whose motion is determined by nothing but gravity, will move in a straight line. This law of motion applies to all free particles, including free photons. Because the freely falling system is accelerating relative to the spatial geometry, the paths of free particles will curve relative to the spatial geometry. The fact that the spatial geometry is curved does not complicate the analysis as far as this paper is concerned. The point is that locally we can always consider the geometry to be flat. Living on a small patch of the curved surface is like living in an accelerated reference system in special relativity. It is only when we consider circles around the black hole that we need to think about the spatial geometry to determine the correct curvature of the circular path. For instance, due to the curved spatial geometry, the innermost circle (at the surface of the black hole) is not curved at all.
References [1] For the purposes of this article we may consider the event horizon to be an (invisible) sphere. If one ventures inside of this sphere, one cannot come back out again. [2] There is a radius where free photons, that is, photons whose motion is determined only by gravity, can move on circular orbits around a black hole. The circumference of this circle is 1.5 times the circumference of the surface of the black hole (the event horizon). [3] Marek A. Abramowicz, “Relativity of inwards and outwards: An example,” Month. Not. Roy. astr. Soc. 256, 710-718 (1992).
13
[8] Bruce Allen, “Reversing centrifugal forces,” Nature 347, 615–616 (1990). [9] George E. Matsas, “Relativistic Archimedes law for fast moving bodies and the general-relativistic resolution of the ‘submarine paradox’,” Phys. Rev. D 68, 027701-1–4 (2003). [10] Rickard Jonsson, “Inertial forces and the foundations of optical geometry,” Class. Quantum Grav. 23, 1–36 (2006). [11] The proper acceleration of an object is the acceleration measured relative to an inertial system (a freely falling system) momentarily comoving with the object. [12] Strictly speaking we are using geometrized units in which c = 1. In these units time has the same dimensions as distance. If we want to express distances and times in terms of standard units, we should replace any instance of v by v/c, where c is the velocity of light in standard units. [13] The embedded geometry (t = 0 and θ = π/2 in Schwarzschild coordinates) corresponds √ to√a section of a parabola (see Ref. 25), z = 2 RG r − RG , revolved around the vertical (z) axis (RG is the radius at the event horizon). [14] In the derivation we divided by v, thus assuming ˆ /R will do. v 6= 0. For v = 0 any n [15] Strictly speaking the argument only holds exactly as long as the velocity is purely horizontal. [16] Consider a 1 + 1 dimensional scenario. Let u be the velocity of the reference frame relative to an inertial system in which the reference frame is momentarily at rest, and let η µ = (γ(u), γ(u)u) be the corresponding four-velocity. Let vs be the velocity of the test particle relative to the inertial system in question, and let uµ = (γ(vs ), γ(vs )vs ) be the corresponding four-velocity. Let v be the velocity of the test particle relative to the reference frame. We
have γ(v) = −η µ uµ (using the (−, +, +, +) convention). If we differentiate both sides of this expression by d/dt and use the fact that u = 0 and 4 2 du v = vs momentarily, we find v dv dt γ = −γ v dt + dvs 4 v dt γ . This result corresponds to Eq. (17) (multiply Eq. (17) by γ 4 , divide by δt, and take the limit where δt is infinitesimal). Also we know that the µ du µ proper acceleration α is given by α2 = − du dτ dτ . µ If we differentiate u = (γ(vs ), γ(vs )vs ) with reµ 3 dvs spect to τ , we find du dτ = γ dτ (v, 1). It follows s that α = γ 3 dv dt , which is Eq. (18). [17] Wolfgang Rindler, Introduction to Special Relativity (Clarendon Press, Oxford, 1982), 2nd ed. [18] It is true in the Newtonian limit that accelerations of the reference frame perpendicular to the direction of motion do not affect the local speed derivative. We can also reason this strictly relativistically knowing a little about time dilation and simultaneity. We could also understand it using fourtensors knowing that γ = η µ uµ , where η µ is the reference frame four-velocity and uµ is the particle four-velocity. Consider a particle moving in the xdirection and consider the reference frame to reach a velocity δvy in the y-direction after a time δt. To first order in δt we have η µ : (1, 0, δvy , 0) and uµ : (γ0 , vx , 0, 0). Here γ0 is the value of γ at t = 0. We see that to first order γ = η µ uµ = γ0 , and thus a perpendicular acceleration of the reference grid does not affect the local speed derivative. Similarly an infinitesimal perpendicular velocity of the particle will have no first order effect on the speed. Any one-dimensional reasoning of how the speed derivative is related to force parallel to the direction of motion thus holds also when there are perpendicular effects. [19] The difference between the proper rotation and the rotation as observed from outside of the merry-goround, for the non-central points, is not only due to time dilation, but also to relativistic precession (rotation) effects. We can also use the formalism of this paper for these points assuming that we correctly express the proper local reference frame rotation ω, that is, the rotation as experienced by an observer at rest relative to the reference frame at the points in question. [20] Marek A. Abramowicz and Jean-Pierre Lasota, “A brief story of a straight circle,” Class. Quantum Grav. 14, A23–A30 (1997). [21] It is easy to make a formal proof of how the factors should enter. For contravariant vectors one can reason it out instead. Consider two coordinate points separated by an infinitesimal vector dxk . Assume that we stretch space by a factor e−Φ . A coordinate vector that has the same norm (length) rela14
tive to the stretched space that dxk had relative to the standard space would have to be componentwise smaller by a factor e−Φ . Hence given a normalized vector relative to the standard space, we obtain a corresponding normalized vector relative to the rescaled space by dividing by the stretching factor e−Φ . [22] Here we consider the perpendicular part of the spatial part of Eq. (44) of Ref. 10 and set θ˜αβ = 0 as is appropriate for this case. ˜ α Φ = 0 and identify τ˜0 = tc in [23] Set θ˜αβ = 0, η˜α ∇ Eq. (44) of Ref. 10. [24] Actually they are not all exact geodesics. They cannot be in general (while at the same time being orthogonal) when the surface is curved. But at the center of coordinates they are orthogonal, and the curvature of the coordinate lines vanishes, which is sufficient for the type of arguments made in this article. [25] Charles W. Misner, Kip S. Thorne, and John A. Wheeler, Gravitation (W. H. Freeman, New York, 1973), p. 615, Eq. (23.34b).
A covariant formalism of spin precession with respect to a reference congruence Rickard Jonsson Department of Theoretical Physics, Chalmers University of Technology, 41296 G¨ oteborg, Sweden E-mail:
[email protected] Submitted 2004-12-10, Published 2005-12-08 Journal Reference: Class. Quantum Grav. 23 37 Abstract. We derive an effectively three-dimensional relativistic spin precession formalism. The formalism is applicable to any spacetime where an arbitrary timelike reference congruence of worldlines is specified. We employ what we call a stopped spin vector which is the spin vector that we would get if we momentarily make a pure boost of the spin vector to stop it relative to the congruence. Starting from the Fermi transport equation for the standard spin vector we derive a corresponding transport equation for the stopped spin vector. Employing a spacetime transport equation for a vector along a worldline, corresponding to spatial parallel transport with respect to the congruence, we can write down a precession formula for a gyroscope relative to the local spatial geometry defined by the congruence. This general approach has already been pursued by Jantzen et. al. (see e.g. Jantzen, Carini and Bini 1997 Ann. Phys. 215 1), but the algebraic form of our respective expressions differ. We are also applying the formalism to a novel type of spatial parallel transport introduced in Jonsson (2006 Class. Quantum Grav. 23 1), as well as verifying the validity of the intuitive approach of a forthcoming paper (Jonsson 2007 Am. Journ. Phys. 75 463) where gyroscope precession is explained entirely as a double Thomas type of effect. We also present the resulting formalism in explicit three-dimensional form (using the boldface vector notation), and give examples of applications. PACS numbers: 04.20.-q, 95.30.Sf
1. Introduction In special and general relativity the spin of a gyroscope is represented by a four-vector S µ . Assuming that we move the gyroscope without applying any torque to it (in a system comoving with the gyroscope), the spin vector will obey the Fermi transport equation Duα DS µ = uµ Sα . (1) Dτ Dτ
A covariant formalism of spin precession with respect to a reference congruence
2
Here uµ is the four-velocity of the gyroscope. For a trajectory in a given spacetime, and a spin vector specified at some point along this trajectory, we can integrate (1) to find the spin at any point along the trajectory. The Fermi transport equation is however deceivingly simple since we have not inserted explicitly the affine connection coming from the covariant differentiation. Also, even when we have a flat spacetime and inertial coordinates (so that the affine connection vanishes) the equation is more complex than you might think. As an example we consider motion with fixed speed v along a circle in the xy-plane, with an angular frequency ω. Letting the groscope start at t = 0 at the positive x-axis, we get a set of coupled differential equations dS x = γ 2 v 2 ω sin(ωt) (S x cos(ωt) + S y sin(ωt)) (2) dt dS y = − γ 2 v 2 ω cos(ωt) (S x cos(ωt) + S y sin(ωt)) (3) dt dS z = 0, S0 = v · S (4) dt and S is the spatial part of S µ . For initial conditions (S x , S y , S z , S 0) = Here v = dx dt (S, 0, 0, 0) the solutions (see [1] p. 175-176) can be written as S x = S (cos[(γ − 1)ωt] + (γ − 1) sin[ωγt] sin[ωt]) y
S = S (sin[(1 − γ)ωt] − (γ − 1) sin[ωγt] cos[ωt]) z
S = 0,
0
S = −SRωγ sin[ωγt]
(5) (6) (7)
Looking at S x and S y , we note that (written in the particular form above) the first terms in respective expression corresponds to a rotation around the z-axis, but then there is also another superimposed rotation with time dependent amplitude. To find this solution directly from the coupled differential equations that are the Fermi equations, seems at least at first sight quite difficult, even for this very symmetric and simple scenario. To get a simpler formalism we may consider, not the spin vector S µ itself, but the spin vector we would get if we momentarily would stop the gyroscope (relative to a certain inertial frame) by a pure boost (i.e. a non-rotating boost). This object we will call the stopped spin vector. While being a four-vector it is effectively a threedimensional object (having zero time component in the inertial frame in question) and we will show that the spatial part of this object undergoes pure rotation with a constant rate for the example of motion along a circle in special relativity. Knowing that there is a simple algebraic relation between the stopped and the standard spin vector, the stopped spin vector can be used as an intermediate step to easily find the standard spin vector. There is however also a direct physical meaning to the stopped spin vector, apart from being the spin vector we would get if we stopped the gyroscope. The stopped spin vector directly gives the spin as perceived in a comoving system, see section 4.10 for further discussion on this. In this article we will also consider more general reference frames than inertial ones. For instance we will consider a rotating and accelerating reference frame. This allows
A covariant formalism of spin precession with respect to a reference congruence
3
us to apply the formalism, via the equivalence principle, to describe in a simple threedimensional manner how a gyroscope orbiting for instance a rotating black hole will precess relative to the stationary observers. In figure 1 we illustrate how a gyroscope spin vector precesses relative to a vector parallel transported with respect to the spatial geometry. B
A
Figure 1. A schematic illustration of how an orbiting gyroscope will precess relative to the spatial geometry of a black hole. The full drawn arrow is the stopped spin vector (stopped with respect to the stationary reference observers) of the gyroscope at two different points along the orbit. The dashed arrow is a vector coinciding with the gyroscope spin vector at A and then parallel transported to B with respect to the spatial geometry. For an intuitive explanation of why the gyroscope precesses relative to the spatial geometry even though there are no torques acting on it, see [2].
Given a reference congruence of timelike worldlines, we first derive a general spacetime transport equation for the stopped spin vector (stopped relative to the congruence in question). We then consider a spacetime equation corresponding to spatial parallel transport with respect to the spatial geometry defined by the congruence. For the case of a rigid congruence, we easily derive such a transport law. Considering a shearing congruence we use the formalism derived in [3]. Having both the transport equation for the stopped spin vector and the equation for parallel transport, we can put them together and thus get an equation for how fast the stopped spin vector precesses relative to the local spatial geometry connected to the reference congruence. As is the case for the inertial congruence, we will see that the precession corresponds to a simple law of three-rotation. The general scheme as outlined here has already been pursued by Jantzen et. al. (see [4]), although the angle of approach and the algebraic formalisms are different. The explicit use of the three-dimensional formalism of this paper also appears novel. This article is complementary to a companion paper [2], where the formalism of relativistic spin precession in three-dimensional language is derived in a very intuitive manner. This paper verifies, through a more formal derivation, the result of [2] for the particular case of a rigid congruence as assumed in [2]. 2. The stopped spin vector Let us denote the local four-velocity of our reference congruence by η µ . We introduce a stopped spin vector S¯µ as the spin vector that we get if we make a pure boost of the
A covariant formalism of spin precession with respect to a reference congruence
4
spin vector such that it is at rest with respect to the local congruence line. In figure 2 we illustrate in 2+1 dimensions how the two spin vectors are related to each other. uµ
ηµ
Sµ
S¯µ
Figure 2. A 2+1 illustration of the relation between the spin vector S µ and the stopped spin vector S¯µ . Through the stopping, the tip of the spin vector can in two dimensions be seen as following the hyperbola connected to the Lorentz transformation down to the local slice. Notice that the stopped spin vector is not in general simply the spatial (projected) part of the standard spin vector (the thin dotted arrow).
It follows readily from the Lorentz transformation that we get the stopped vector by removing the η µ -part of S µ , and shortening the part parallel to the spatial direction of motion by a γ-factor. Note that the resulting stopped vector is not in general parallel to the spatial part of S µ . Letting tµ be a normalized vector orthogonal to η µ in the direction of motion, we can express the stopped spin vector as "
!
#
1 − 1 tµ tα S α . S = δ α + η ηα + γ ¯µ
µ
µ
(8)
Here we have adopted the spatial sign convention (−, +, +, +) as we will throughout the article. Knowing a little about Thomas precession we may guess that for the simple case of motion along a circle in an inertial frame as discussed earlier, there is a simple law of three-dimensional rotation for this object. Indeed in the following discussion we will show this, and at the same time consider the effects of rotation coming from having non-inertial reference frames (connected to η µ ). We also need an explicit expression for the standard spin vector in terms of the stopped spin-vector S¯µ . The relationship between the two vectors follows readily from the Lorentz-transformation: = S¯α K µ α
Sµ K
µ
α
= [δ
µ
α
(9) µ
µ
+ γvη tα + (γ − 1)t tα ] .
(10)
This we may now insert into the Fermi transport equation to derive an expression for the stopped spin vector.
A covariant formalism of spin precession with respect to a reference congruence
5
3. Covariant derivation of the transport equation for the stopped spin-vector In this section we consider gyroscope transport relative to an arbitrary reference congruence η µ . For a spin vector S µ transported along a worldline of four-velocity uµ , we have the Fermi transport law Duρ DS µ = uµ S ρ . (11) Dτ Dτ Using (9) in (11) readily yields Duρ DK µ α D S¯α µ . (12) K α = S¯α uµ K ρ α − Dτ Dτ Dτ We need now the inverse of K µ α to get an explicit transport equation for the stopped spin vector. Through a general ansatz1 , we find ! 1 −1 ν ν − 1 tν tµ − vη ν tµ . (13) K µ=δ µ+ γ That this is indeed the inverse of K µ α is easy to verify2 . So we have D S¯ν DK µ α ν µ ρ Duρ α ¯ K −1 µ . (14) =S u K α − Dτ Dτ Dτ Here we have the desired expression. In Appendix A we expand and simplify this to find γv ¯α µ D D D S¯µ µ µ S t = (uα + ηα ) − tα (u + η ) Dτ γ+1 Dτ Dτ ⊥ ⊥ Dηα + η µ S¯α . (15) Dτ By the perpendicular sign ⊥ we here mean that we should select only the part orthogonal D means covariant differentiation along the gyroscope to both tµ and η µ . Note that Dτ worldline. Equation (15) then tells us how the stopped spin vector deviates from a parallel transported vector relative to a freely falling system. In fact we notice from the antisymmetric form of (15) that (excepting the η µ term) it corresponds to a spatial rotation (see section 4.2 for a more detailed argument). That seems very reasonable since it insures that the norm of the stopped spin vector will be constant (consider the rotation with respect to a freely falling system locally comoving with the congruence). We also see that only if uµ + η µ changes along the gyroscope worldline, with respect to a freely falling system, do we get a net rotation relative to this freely falling system. Introducing the wedge product defined by aα ∧ bβ ≡ aα bβ − bα aβ and the projection operator P µ α = δ µ α + η µ ηα , we can put (15) in a more compact form D γv ¯α µ D S¯α S t ∧ = (uα + ηα ) . (16) P µα Dτ γ+1 Dτ ⊥ ν
ν
We have K −1 ρ K ρ α = δ ν α . The ansatz is of the form K −1 α = δ ν α +atµ tα +btµ ηα +cη µ tα +dη µ ηα . 2 In defining K µ α we are free to add terms containing ηα , since these anyway die when multiplied 1 α ¯ (uµ + η µ )(uα − ηα ) we would get the inverse by S . If we instead would have defined K µ α = δ µ α + γ+1 µ 1 K −1 α = δ µ α + γ+1 (uµ + η µ )(ηα − uα ) . Here the perfect symmetry in S µ , η µ and S¯µ , uµ is transparent. There however does not appear to be any particular advantages of this gauge. 1
A covariant formalism of spin precession with respect to a reference congruence
6
Incidentally we may note that, as regards tµ -components within the bracketed expression, we do not need the ⊥ sign. Any tµ components within the bracketed expressions will cancel due to the anti-symmetrization as is easy to see. We however keep the ⊥ sign to indicate orthogonality to η µ . The simple form of (16) appears to be novel. 4. Application to flat spacetime, and inertial congruences While we have yet to put the formalism in its final form, some applications and discussion may be useful already at this point for the simple case of an inertial reference congruence in special relativity. 4.1. Employing the spatial curvature of the gyroscope trajectory As a particular example, consider a flat spacetime with an inertial congruence. For this case it is not hard to show, see e.g [3], that the spatial curvature of a trajectory depends on the four-acceleration as nα Duα = γ 2v2 . (17) Dτ ⊥ R Here R is the spatial curvature3 and nµ is a normalized four-vector, orthogonal to the inertial congruence η µ , pointing in the direction of spatial curvature. Using this in (16) we get ¯α nα µ DS µ α ¯ P α . (18) = γv(γ − 1)S t ∧ Dτ R As we will see in the following section, this differential equation corresponds to a threedimensional rotation. 4.2. Three-dimensional formalism, for flat spacetime and an inertial congruence Choosing inertial coordinates adapted to the inertial congruence in question so that ¯ tµ = (0, ˆt) and nµ = (0, n S¯µ = (0, S), ˆ) we get from (18) " # ¯ n ˆ n ˆ ¯ ˆ dS ¯ ˆ = γv(γ − 1) t(S · ) − (S · t) . (19) dτ R R The expression within the brackets is a vector triple product and we may write it as a double cross product. Letting v = vˆt we get ! ¯ dS n ˆ ¯ = γ(γ − 1) × v × S. (20) dτ R
3
As is illustrated in [3] there are plenty of ways to define spatial curvature measures in general, but for an inertial congruence most of these coincide with the standard projected curvature that we here assume.
A covariant formalism of spin precession with respect to a reference congruence
7
Rather than using τ we could use local time τ0 (the time as experienced by observers at rest relative to the inertial congruence in question) in which case we get a gamma factor less on the right hand side. ! ¯ n ˆ dS ¯ = (γ − 1) × v × S. (21) dτ0 R This is the famous Thomas precession, in stopped spin vector three-formalism. Introducing Ω as the precession vector, around which the stopped spin vector rotates, we can alternatively write (21) as ¯ dS ¯ =Ω×S (22) dτ0 ! n ˆ Ω = (γ − 1) ×v . (23) R Looking at (22) component-wise, it is a set of coupled differential equations, just like the standard Fermi equations. Unlike the Fermi-equations however, the new equations correspond to a simple law of rotation (precession). 4.3. The circular motion revisited As a specific example we may consider, as in the introduction, the precession of a gyroscope transported at constant speed v around a circle of radius R in the z = 0 plane. Assuming a motion with a clockwise angular velocity ω = v/R, the counterclockwise angular velocity Ω for the precession of the stopped spin vector is then according to (23) given by Ω = (γ − 1)ω.
(24)
Consider then for instance the net precession after one lap. The local time per lap is simply 2π/ω and hence the net precession angle (in radians) around the plane normal is given by 2π(γ − 1). If the circular motion is counter-clockwise, the precession is clockwise and vise versa. 4.4. Re-deriving the solution for the standard spin vector We can also trivially find the solution for the standard (projected) spin vector for the case of circular motion with constant speed with initial conditions as listed in the example in the introduction. We know that the standard (projected) spin vector is related to the stopped spin vector through a lengthening of the stopped spin vector in the forward direction of motion ˆ t by a γ-factor. We have then ¯ + (γ − 1)(S ¯ ·ˆ S=S t)ˆ t.
(25)
Using the notation of the previous subsection we have then trivially for the case at hand ¯ = S cos(Ωt)ˆ S x − S sin(Ωt)ˆ y ˆ t = − sin(ωt)ˆ x + cos(ωt)ˆ y.
(26) (27)
A covariant formalism of spin precession with respect to a reference congruence
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Using these expressions in (25) we immediately get the desired solution. Using elementary rules for manipulating the trigonometric functions we can write it in the form of (5)-(7). If we are interested also in S 0 , it is given by the orthogonality of the standard spin vector and the four-velocity as S 0 = S· v. Note that by use of the stopped spin vector formalism there is effectively no differential equation solving involved for this simple case. 4.5. A special relativistic theorem of spin precession for planar constant velocity motion For motion in a circle with constant velocity, the Fermi equation can be solved without use of the stopped spin vector formalism, although the solution is a bit complicated. What about if we consider motion with constant velocity along some other curve, say a part of a parabola or some more irregular curve? Then the Fermi equation would likely appear to be very complicated to solve analytically in the general case. Using the method with the stopped spin vector the solution can however trivially be found for arbitrary curves. First let us state a small theorem that we will then easily prove. The stopped spin vector of a gyroscope transported with constant speed v along a smooth curve in a spatial plane in a flat spacetime will rotate a net clockwise angle around the normal of the plane given by ∆αprecess = (γ − 1)∆αcurve where ∆αcurve is the net counterclockwise turning angle of the tangent direction of the curve. Note that the parameter ∆αcurve may be larger than 2π. For a simple closed curve (one that is not crossing itself), assuming the gyroscope to be transported once around the curve, we have ∆αcurve = 2π. This theorem is easily proven by dividing an arbitrary smooth curve into infinitesimal segments within which we can consider the local curvature radius to be constant. Letting ω denote the counter-clockwise angular velocity of the forward direction of motion ˆ t (so ω = dαcurve /dt), we have according to (24) the clockwise angular velocity as Ω = (γ − 1)ω. Thus the net angles of the gyroscope precession and the turning of the forward direction of the curve, along the segment in question, are related through dαprecess = (γ − 1)dαcurve . Adding up the precession contributions from all the segments of the curve we get ∆αprecess = (γ − 1)∆αcurve .
(28)
Thus the theorem is proven. Note that while the motion is assumed to be in a plane, the spin vector may point off the plane. 4.6. Some consequences of the theorem We can draw a conclusion from the above proven theorem (also knowing that there is a simple algebraic relation between the stopped and the standard spin-vector) that can be expressed in terms of the standard spin vector, without reference to the stopped
A covariant formalism of spin precession with respect to a reference congruence
9
spin vector. Consider then a smooth simple closed curve and let a certain point along this curve be the initial position for the gyroscope. For given initial spin vector, initial direction of motion4 and constant speed v, the final spin vector (after one lap around the loop) is independent of the shape of the loop as illustrated in figure 3.
Final projected spin vector Initial projected spin vector
Figure 3. Illustrating that for a fixed initial direction of motion, fixed initial spin vector, and fixed constant speed v – the final spin vector after one lap around any simple smooth closed curve is independent of the shape of the curve.
But of course, the theorem is stronger than this. Given an arbitrary, not necessarily closed but smooth curve along which we transport the gyroscope with constant velocity, we can trivially find the standard spin vector at any point along the curve. We take the spatial part of the initial spin vector and shorten the part parallel to the direction of motion by a γ-factor to form the initial stopped spin vector. For any given curve x(λ) we then calculate the initial direction of the curve together with the direction of the curve at the point in question. Then, modulo a winding number times 2π 5 , we can trivially find the corresponding ∆αcurve and thus through (28) the corresponding stopped spin vector at the point in question. Lengthening the parallel part of the stopped spin vector by a factor γ, we get the spatial part of the standard spin vector at the point in question. If we are interested in the zeroth component of the standard spin vector it is given by S 0 = S · v. Thus solving a possibly very complicated differential equation is reduced to performing a few algebraic steps6 . 4.7. More complicated motion For motion in a plane where the velocity is not constant, the procedure is analogous to that described in section 4.6 except that we need to integrate (a single integral which may or may not be complicated to solve analytically) to find dαprecess . For the most general motion, not necessarily confined to a plane and with a speed that may vary, it
4
One cannot in general keep the standard (unlike the stopped) spin vector fixed while altering the initial direction of motion of the gyroscope since the standard spin vector must be orthogonal to the gyroscope four-velocity. 5 The only non-trivial part of calculating the turning angle lies in finding out the number of turns taken by the curve since for a curve x(λ) we only get the turning angle ∆αcurve up to a term 2πn, where n is an integer, from the local quantity dx dλ 6 Again modulo the winding number mentioned earlier. For many cases, like for instance for a parabola, this however presents no problem at all.
A covariant formalism of spin precession with respect to a reference congruence
10
is however not just a matter of ordinary integration7 . Given an arbitrary motion x(τ0 ) along a smooth curve we can however solve a differential equation, given by (22) and ¯ Likely this differential equation will be simpler to solve than the Fermi (23), for S. equation. 4.8. A comment on the relation between the intrinsic angular momentum, the projected spin vector, the gyroscope axis and the stopped spin vector To gain further intuition on the meaning of the stopped spin vector it may be useful to explore how it is related to other vectors of physical interest connected to the gyroscope spin. In particular we may consider the gyroscope intrinsic angular momentum, and the momentary direction of the gyroscope axis as perceived in the reference system in question (where the observers are integral curves of η µ ). Consider then a gyroscope moving along a straight line in the xy-plane in special relativity (using inertial coordinates) with constant speed. The gyroscope axis is assumed to lie in the plane of motion and to be tilted somewhere between the forward and the sideways direction. In 2+1 dimensions we can easily visualize the worldsheet of the gyroscope central axis as well as various vectors of interest, see figure 4.
t The worldsheet of the gyroscope axis
x The standard spin vector
The projected spin vector The stopped spin vector The gyroscope axis direction
y Figure 4. A sketch in 2+1 dimensions of vectors related to a spinning gyroscope.
We note that there are (at least) three different spatial directions of relevance for the gyroscope. It is easy to realize (length contraction) that the direction of the gyroscope axis is simply related to the direction of the stopped spin vector through a 7
One could for instance represent a finite precession (rotation) by a vector whose direction determines the axis of rotation and whose norm determines the angle (in radians) of the precession. It is however easy to realize that for a a finite such rotation (like the net rotation after some finite stretch along a trajectory) followed by an infinitesimal rotation around some other axis – one cannot in general simply add the two corresponding rotation vectors (to first order) to form a new rotation vector. Of course there are examples of non-planar motion, like motion along a helix for instance, where the precession vector remains in the same direction for which case it is a simple matter of integration after all to find the net rotation of the stopped spin vector.
A covariant formalism of spin precession with respect to a reference congruence
11
gamma factor. Given any of these directions the other two can thus easily be found. Furthermore one can show, at least for an idealized scenario as considered in Appendix B, that the the intrinsic angular momentum, that we will denote SL , is in fact given by S/γ. The various vectors involved are illustrated in figure 5. ˆ t
S
SL
¯ S
γ 1
X
1 γ
Figure 5. The three different directions in question are simply related through a stretching by a gamma factor in the direction of motion. In this illustration a gamma factor of 2 was assumed, with motion in the upwards direction (ˆ t). Note that the depicted norm of the gyroscope axis vector X is arbitrary.
4.9. Four vectors, four differential equations Consider a spatial vector X that connects the base of the gyroscope to the tip of the gyroscope, as perceived in the reference system connected to η µ . We understand that this vector evolves according to a simple rule of rotation given by (21) modulated by a contraction by a factor of γ in the direction of motion. It is a short exercise to show that this means that X in fact obeys a rather compact differential equation dv dX = −γ 2 [X · v] . (29) dτ0 dτ0 We can perform a corresponding analysis for the projected spin vector to find8 "
#
dS dv = γ2v S · . dτ0 dτ0 The equations for the stopped spin vector can be written in the form ! ¯ dS γ − 1 dv ¯ = × v × S. dτ0 v2 dτ0
(30)
(31)
From (30), letting S = γSL , we readily find "
#
dv dv dSL = γ 2 v SL · − γ 2 v SL . dτ0 dτ0 dt 8
(32)
This also follows readily from the standard Fermi equations for the case of inertial coordinates in special relativity.
A covariant formalism of spin precession with respect to a reference congruence
12
Comparing the four differential equations we see that they are all quite compact, although the equation for the stopped spin vector, corresponding to a pure rotation, is more likely to be simple to solve (as we have seen for the example of motion on a circle). 4.10. A comment on the meaning and purpose of the stopped spin vector One might argue that the object of physical interest is the intrinsic (spin) angular momentum of the gyroscope which is given by S/γ, or perhaps the observed direction of the gyroscope central axis. From this point of view the stopped spin vector is in a sense a means to an end. By using the stopped spin vector as an intermediate step we can find the solutions to otherwise quite complicated differential equations for the objects of physical interest. From a mathematical point of view this is certainly sufficient to motivate the use of the stopped spin vector. There is however more to the stopped spin vector than this. In particular we note that the stopped spin vector directly gives us the spin as perceived in a comoving system. For instance, if the stopped spin vector is at a 45◦ angle with respect to the forward direction – so it will be with respect to a system comoving with the gyroscope9 . This is contrary to the standard spin vector which only gives the spin direction with respect to the comoving system after a Lorentz transformation. Consider the following example. A gyroscope is suspended inside a satellite such that no torque is exerted on the gyroscope as seen from the satellite. The satellite is assumed to be orbiting along some predetermined smooth simple closed curve, on a plane in special relativity10 , using it’s jet engines to stay on the path. Suppose then that we wish to measure, from the satellite, the precession angle of the gyroscope (as predicted by relativity) after a full orbit (or maybe several full orbits). We note that the direction of the gyroscope relative to the satellite itself is not a good measure11 . Assuming that we have a couple of fixed stars, we can however use the direction of these stars (as perceived from the satellite) as guidelines to set up a reference system within the satellite12 For this scenario the stopped spin vector is exactly the physical object that we are interested in. It exactly represents the gyroscope direction relative to the star-calibrated reference system of the satellite. Thus if the stopped spin vector turns a certain angle, that is precisely the turning angle of the gyroscope relative to the 9
If the stopped spin vector has certain components with respect to a set of base vectors adapted to the reference congruence in question, then those components precisely corresponds to the components of the standard spin vector with respect to a boosted version (a pure boost to comove with the gyroscope) of the base vectors just mentioned. This viewpoint is mentioned in [1] p. 1117, although they do not consider general spacetimes and velocities. 10 The general argument works also for gyroscopes orbiting the earth in a general relativistic treatment. More on this in section 10. 11 The satellite may have had an initial rotation from the start or the jet-engines may give it one. Also, even if it would have zero proper rotation then the gyroscope would keep its direction relative to the satellite and thus would not turn at all relative to the satellite. 12 We also assume that the satellite has some way of knowing when it is at its initial position (so it knows when to calibrate its coordinates with respect to the stars).
A covariant formalism of spin precession with respect to a reference congruence
13
star-calibrated reference system of the satellite. While we are here focusing on spinning gyroscopes, it should also be noted that the formalism of the stopped spin vector is immediately applicable to describe the resulting rotation of any object which has zero proper (comoving) rotation. In conclusion, the stopped spin vector may be used as an intermediate step to simplify the calculation of the evolution of the intrinsic angular momentum (spin) of a gyroscope, or the perceived direction of the gyroscope axis. The stopped spin vector is however also of direct physical importance since it gives us the spin as perceived in a comoving system. So far we have only given examples that apply to flat spacetime, and inertial reference frames. As we will see in the following sections the stopped spin vector can be just as useful also for curved spacetimes and non-inertial reference frames. 5. Spatial parallel transport The transport equation (16) tells us how the stopped spin vector deviates from a vector that is parallel transported with respect to the spacetime geometry. This by itself is however not really what we are after if the reference congruence is non-inertial. To get a truly three-dimensional formalism, we in stead want an expression telling us how fast the stopped spin vector deviates (rotates) from a vector that is parallel transported with respect to the spatial geometry determined by the congruence. As is demonstrated in [4] and in [3], it is possible derive a spacetime transport law corresponding to a spatial parallel transport. For the simple, and perhaps most useful, case of a rigid congruence13 the issue is sufficiently simple that we will briefly review it in the coming subsection. 5.1. Rigid congruence Suppose then that we have a rigid congruence with nonzero acceleration aµ , nonzero rotation tensor ω µ ν but with vanishing expansion-shear tensor θµ ν 14 . In figure 6 we show an illustration of the spacetime transport of a vector orthogonal to the congruence. It is easy to show that in the coordinates of a freely falling system (t, xk ), locally comoving with the congruence, the velocity of the congruence points (assuming vanishing θµ ν ) is to first order given by v k = ω k j xj + ak t.
(33)
Knowing that the velocity of the congruence is zero to lowest order, relative to the inertial system in question, we need not worry about length contraction and such. It 13
The congruence may rotate and accelerate but it may not shear or expand. The kinematical invariants of the congruence are defined as (see [1] p. 566): The expansion scalar θ = ∇α η α , the acceleration vector aµ = η α ∇α η µ , the shear tensor σµν = 12 (∇ρ ηµ P ρ ν + ∇ρ ην P ρ µ ) − 1 1 ρ ρ 3 θPµν and the rotation tensor ωµν = 2 (P ν ∇ρ ηµ − P µ ∇ρ ην ). Furthermore we employ what we 1 ρ denote the expansion-shear tensor θµν = 2 (P ν ∇ρ ηµ + P ρ µ ∇ρ ην ). 14
A covariant formalism of spin precession with respect to a reference congruence
14
Figure 6. A 2+1 illustration of transporting a spatial vector along a worldline, seen from freely falling coordinates locally comoving with the congruence. As the reference coordinates rotate due to ω µ α , so should the vector in order for it to be proper spatially transported.
is then easy to realize that the proper spacetime transport law of a spatial vector k µ corresponding to standard spatial parallel transport is Dk µ = γω µ α k α + bη µ . (34) Dτ Here b can easily be determined from the orthogonality of k µ and η µ15 . Here we have then a spacetime transport equation corresponding to spatial parallel transport, for the case of a non-shearing (non-expanding) congruence. 5.2. Including shear and expansion For a more complicated congruence that is shearing and expanding, it is not quite so obvious how to define the spatial parallel transport. Indeed as discussed in e.g [4] and [3], there are several ways of doing this. We will here follow the approach of [3], and consider two different such parallel transports. These transports are connected to two different ways of defining a spatial curvature for a test particle worldline, with respect to the congruence µ 1 Duµ µ µ α µ α 2 nps = [a ] + 2v(ω t + [θ t ] ) + v (35) Projected: ⊥ α α ⊥ γ 2 Dτ ⊥ Rps µ 1 Duµ µ µ α 2 nns New: . (36) = [a ] + 2vω t + v ⊥ α γ 2 Dτ ⊥ Rns Here Rps and nµps are the curvature and the curvature direction that we get if we project the the spacetime trajectory down along the congruence onto a local timeslice (orthogonal to the congruence at the point in question). The suffix ’ps’ stands for ’Projected Straight’. The curvature Rns and the curvature direction nµns are defined
D From the orthogonality k µ ηµ = 0 follows (differentiate Dτ along the gyroscope worldline) that Dη Dη µ µ ηµ = −k Dτ . Contracting both sides of (34) by ηµ gives b = k µ Dτµ .
15 µ
Dk Dτ
A covariant formalism of spin precession with respect to a reference congruence
15
with respect to deviations from a certain (new) notion of a spatially straight line. The latter is defined as a line that with respect to variations in the projected curvature, leaves the integrated spatial distance (as defined by the congruence) unaltered (to first order in the variation). As it turns out, a straight line with respect to this definition, has in general a non-zero projected curvature when the congruence is shearing. The suffix ’ns’ stands for ’New-Straight’. This particular curvature is connected to Fermat’s principle, and optical geometry [3, 5]. For brevity we let the suffix ’s’ denote either ’ps’, or ’ns’. Introducing Cps = 1, Cns = 0 we can then express both curvatures jointly as µ 1 Duµ µ µ α µ α 2 ns = [a ] + 2v(ω t + C [θ t ] ) + v . (37) ⊥ α s α ⊥ γ 2 Dτ ⊥ Rs From these two curvature measures one can introduce corresponding equations for spatial parallel transports [3]. A joint expression for the parallel transport of a vector k µ is given by Dk µ Dηα = γk α ω µ α + γ(2Cs − 1)k α (θµ β tβ ∧ tα ) + η µ k α . (38) Dτ Dτ α Here Dη is the covariant derivative along the (gyroscope) worldline in question. Notice Dτ that for vanishing shear expansion tensor, the two transports both correspond to (34). Having defined two types of parallel transport according to (38), we can define corresponding covariant differentiations along a curve as Ds k µ Dηα Dk µ = − γk α ω µ α + (2Cs − 1)(θµ β tβ ∧ tα ) − η µ k α . (39) Ds τ Dτ Dτ These derivatives then tells us how fast a vector deviates from a corresponding parallel transported vector (momentarily parallel to the vector in question). Substituting k µ → S¯µ and using (16) we get the equations for how fast the stopped spin vector precesses relative to a spatially parallel transported vector (of the two types). First we however rewrite (16). 6. Rewriting the stopped spin vector transport equation We saw in the preceding section how the kinematical invariants of the congruence entered α α + Du ), naturally in the definition of spatial parallel transport. We can also expand ( Dη Dτ Dτ in the transport equation (16) for the stopped spin vector, in terms of the kinematical invariants of the congruence. First of all we have Dηα = uρ ∇ρ ηα = γ(η ρ + vtρ )∇ρ ηα . (40) Dτ Also we know that (see e.g [1] p. 566) ∇ρ ηα = ωαρ + θαρ − aα ηρ .
(41)
Using (40), we have then Dηα = γv (ωαρ tρ + θαρ tρ ) + γaα . Dτ
(42)
A covariant formalism of spin precession with respect to a reference congruence
16
Using this together with (37) in (16), also adding the proper η µ -term enabling the removal of the projection operator in (16), we readily find " γv ¯α µ D S¯µ S t ∧ γ(γ + 1)aα + γv(2γ + 1)ωαρ tρ = Dτ γ+1 # n Dηα sα + γv(2γCs + 1)θαρ tρ + γ 2 v 2 + η µ S¯α . (43) Rs Dτ Notice that we have omitted the perpendicular signs (⊥) on θαρ tρ and aα since these objects are already orthogonal to η µ and any tµ components die due to the antisymmetrization. 7. The rotation of the stopped spin vector relative to a parallel transported vector Now it is time to put together the results of the preceding two sections. What we want is the net rotation of the stopped spin vector relative to a spatially parallel transported vector. Using (43) and (39) (setting k α = S¯α ), we then readily find " Ds S¯µ α ¯ = S γ 2 v(tµ ∧ aα ) + (γ − 1)(2γ + 1)(tµ ∧ ωαρ tρ ) − γω µ α Ds τ # nsα 2 µ ρ µ . (44) + (2γ Cs − 1)(t ∧ θαρ t ) + γv(γ − 1) t ∧ Rs Here Cps = 1 and Cns = 0. So this gives us how fast a gyroscope stopped spin vector deviates from a corresponding (spatially) parallel transported vector. In particular considering the expression in a freely falling system locally comoving with the congruence, we understand that the expression within the brackets on the right hand side is simply the effective rotation tensor relative to the spatial geometry. It could be practical with an expression corresponding to (44) but where the proper four-acceleration is explicit. Using (16), (39) and (42) we readily find " # γ +1 Duα γv ¯α µ Ds S¯µ ρ ρ S t ∧ + γaα + γvωαρ t + (2γCs −1) = θαρ t Ds τ γ +1 Dτ ⊥ γv − γω µ α S¯α . (45) Notice that the expression for the four-acceleration here (naturally) is independent of what curvature measure that we use. Still (45) depends on what curvature measure we are using (manifesting itself in the occurrence of Cs ) assuming non-zero θαρ tρ , since the transport laws for the two types of spatial parallel transport differs. 8. Three-dimensional formalism, assuming rigid congruence We can rewrite (44) and (45) as purely three-dimensional equations. For any specific global labeling of the congruence lines (i.e. any specific set of spatial coordinates adapted to the congruence) we can locally choose a time slice orthogonal to the congruence so
A covariant formalism of spin precession with respect to a reference congruence
17
¯ This then uniquely defines the three-vector S ¯ at any point along the that S¯µ = (0, S). gyroscope trajectory. Analogous to what we did in going from (18) to (20), for a set of µ µ µ α µ ¯ 16 vectors S¯ , t and k orthogonal to the congruence, we let S¯ t ∧ kα → k × ˆt × S. Also we let ω µ α tα → ω × tˆ17 . For simplicity, let us assume that the congruence has vanishing shear and expansion.18 For this case the two different approaches to spatial curvature radius coincide and we will drop any instances of subscripts ’ns’ or ’ps’. 2 Introduce then agyro = ddτx2 , where x and τ0 are the inertial coordinates of a system 0 locally comoving with the congruence19 . Also denoting the acceleration of the reference congruence relative to an inertial system locally comoving with the reference congruence by aref , we get from (45)20 " # ! ¯ 1 γ3 DS ¯ − γω × S. ¯ agyro + (aref + ω × v) × v × S = (46) Dτ γ+1 γ This is a perfect match with the result of the intuitive derivation performed in [2]. Analogously we may study (44) for the particular case of vanishing shear, thus considering a rigid congruence. The three-dimensional version of this equation then becomes " ¯ DS = γ 2 v(aref × ˆ t) − γω + (γ − 1)(2γ + 1)(ω × ˆt) × ˆ t Dτ !# n ˆ ˆ ¯ ×t × S. (47) + γv(γ − 1) R We may simplify this expression a bit by introducing ω = ω k + ω ⊥ , where k and ⊥ means parallel respectively perpendicular to ˆ t. Also using v = vˆt we readily find ! ! " ¯ DS 1 2 ω⊥ = γ (aref × v) − γ ω k + 2γ − Dτ γ !# n ˆ ¯ ×v × S. (48) + γ(γ − 1) R Again this is a perfect match with the intuitive formalism of [2].
16
Strictly speaking, what we mean by the cross product a × b of two three-vectors a and b is 1 [Det(gij )]− 2 ǫijk aj bk where the indices have been lowered with the local three-metric (again assuming local coordinates orthogonal to the congruence). Notice that in general (for congruences with rotation) there are no global time-slices that are orthogonal to the congruence. The local three-metric corresponding to local orthogonal coordinates is however well defined everywhere anyway. For a shearing (expanding) congruence it will however be time dependent (whatever global time slices we choose). 17 Letting ω µ = (0, ω) in coordinates locally comoving with the congruence, we have ω µ = 1 √1 σµγρ ωγρ , where g = −Det[gαβ ] and ǫσµγρ is +1, −1 or 0 for σµγρ being an even, odd or 2 g ησ ǫ no permutation of 0, 1, 2, 3 respectively. 18 This incidentally implies that the ’orthogonal’ three-metric mentioned in a previous footnote is time independent. 19 Working in another set of spatial coordinates agyro naturally transforms as a three-vector. 20 Notice that D/Dτ corresponds to covariant differentiation with respect to the three-metric.
A covariant formalism of spin precession with respect to a reference congruence
18
8.1. The rotation vector relative to the reference observers. Letting τ0 = γdτ denote local time for an observer comoving with the congruence we can write (46) and (48) respectively as ¯ DS ¯ = Ω × S. (49) Dτ0 Here Ω is given by (46) !
γ 1 γ2 ω⊥. (agyro × v) + (aref × v) − ω k − 2 − Ω= γ+1 γ+1 γ
(50)
This form is practical if the gyroscope is freely falling, in which case agyro = 0. Alternatively we can get Ω from (48) !
!
1 n ˆ Ω = γ(aref × v) − ω k − 2γ − ω ⊥ + (γ − 1) ×v . γ R
(51)
This form is practical if the gyroscope follows some predetermined path while being acted on by forces. 8.2. A note on the gyroscope axis and the projected spin vector From the simple relation (see section 4.8) between the stopped spin vector and the projected spin vector and the gyroscope axis respectively, we can use the law of rotation for the stopped spin vector to derive corresponding differential equations for S and X "
#
!
dv 1 dS = γ2v S · + Ωek × [S]⊥ + Ωe⊥ × [S]k + γ[S]⊥ dτ0 dτ0 γ ! dX 1 2 dv =− γ [X · v] + Ωek × [X]⊥ + Ωe⊥ × γ[X]k + [X]⊥ . dτ0 dτ0 γ
(52) (53)
Here we have for brevity introduced !
1 ω⊥. Ωe = γ(aref × v) − ω k − 2γ − γ
(54)
dv entering (52) and (53) is the velocity derivative relative to the reference Note that the dτ 0 frame (not relative to a freely falling frame). We note that these differential equations are more complicated than the ones for the stopped spin vector. We conclude that if we are interested in S or X, it is likely wise to first solve the equation for the stopped spin vector and then (as in section 4.4) use the result to find S or X.
9. Motion along a straight line in static geometry As a first example of how one may use the derived formalism, consider a train moving along a straight spatial line in some static geometry. In fact, to be specific, we may consider the train to be moving relative to an upwards accelerating platform in special relativity. On the train we have suspended a gyroscope so that there are no torques acting on it in the comoving system. We thus consider gyroscope motion along a straight
A covariant formalism of spin precession with respect to a reference congruence
19
line, with respect to a non-rotating but accelerating reference frame. Letting g = −aref and τ0 = γτ , (48) is immediately reduced to ¯ dS ¯ = −γ (g × v) × S. (55) dτ0 We understand that a gyroscope initially pointing in the forward direction will tip forward as depicted in figure 7. t=0
t = dt v g
Figure 7. A train moving relative to a straight platform with proper upward acceleration. A gyroscope with a torque free suspension will precess clockwise (for positive v).
Note that the stopped spin vector with respect to the platform corresponds precisely to the spin vector as perceived relative to the train. For example, if the stopped spin vector points 45◦ down from the horizontal direction, the gyroscope as seen from the train points 45◦ down from the horizontal direction. To express the gyroscope precession with respect to coordinates comoving with the train we therefore just let τ0 → γτ in (55) and we have the precession explicitly in terms of the time τ on the train. Relative to the train, the gyrocope thus precesses at a steady rate given by Ωrelative train = γ 2 vg. This means that the train in fact has a proper rotation, but more on this is given in [2]. We can parameterize the gyroscope trajectory by the distance s along the platform rather than the time τ0 . Then (55) can be expressed as ¯ dS ¯ = −γ g × ˆt × S. (56) ds Assuming the train velocity to be low, the tipping angle per distance traveled is thus independent of the velocity. We have simply dα/ds ≃ g. Thus on a stretch of length δs we get a net rotation δα δα ≃ gδs.
(57)
If we want to express δs and g and in SI units we must divide the right hand side by c2 (expressed in SI units). Setting δs = 103 m and g = 9.81 m/s2 we get simply δα =
9.81 · 103 ≈ 1 · 10−13 (rad). (3 · 108 )2
(58)
This is quite a small angle, and we understand that the relativistic effects of gyroscope precession for most cases here at Earth are small. Notice how simple this calculation was in the stopped spin vector three-formalism.
A covariant formalism of spin precession with respect to a reference congruence
20
10. Axisymmetric spatial geometries, and effective rotation vectors The equations (46) and (48) both describe how the gyroscope rotates with respect to a coordinate frame that is parallel transported with respect to the spatial geometry. Suppose then that we consider motion in the equatorial plane of some axisymmetric geometry. As a specific example we may want to know the net rotation of the gyroscope relative to its initial configuration after a full orbit (not necessarily a circular orbit). We must then take into consideration that a parallel transported frame in general will be rotated relative to its initial configuration after a complete orbit. We can however introduce a new reference frame, that rotates relative to local coordinates spanned by ˆ r and ϕ, ˆ in the same manner as a parallel transported reference frame does on a plane. In other words, if we for instance consider a counterclockwise displacement (δϕ,δr), then relative to the local vectors ˆ r and δ ϕ, ˆ the new reference frame should turn precisely δϕ clockwise. Such a reference frame would always return to its initial configuration after a full orbit. To find the rotation of the new reference frame with respect to a parallel transported frame, we first investigate how a vector that is parallel transported with respect to the curved axisymmetric geometry rotates relative to the local coordinates spanned by ˆ r and ϕ. ˆ 10.1. The rotation of a parallel transported vector relative to ˆ r and ϕ ˆ The line element for a two-dimensional axisymmetric spatial geometry can be written in the form21 ds2 = grr dr 2 + r 2 dϕ2 .
(59)
As depicted in figure 8 we can imagine an embedding of the geometry, where we cut out a small section and put it on a flat plane. What we want is an expression for how much a vector that is parallel transported, for example along the depicted straight dashed line, rotates relative to the local coordinates ˆ r and ϕ. ˆ We understand that the rotation angle corresponds to the angle δα as depicted. Using the notations introduced in figure 8 we have simply R0 δα = rδϕ (R0 + ds)δα = (r + dr)δϕ. √ Eliminating R0 and using ds = grr dr it follows readily that δϕ δα = √ . grr 21
(60) (61)
(62)
Note that if we consider for instance a Kerr black hole, where we (in standard representation) have dϕdt-terms, we cannot simply select the spatial terms (without dt) to get the spatial line element. There is however an effective spatial geometry also for this case. We may derive the form of this geometry by for instance sending photons back and forth between nearby spatial points and checking the proper time that passes.
A covariant formalism of spin precession with respect to a reference congruence replacemen
α + δα
21
(r + δr)δϕ α δs rδϕ R0
δα
Figure 8. Cutting out a section of a certain dϕ and dr of the embedded geometry (to the left) and putting it on a flat plane (to the right). Note that r is the circumferential radius, and R0 is the radius of curvature for a circle at the r in question (not to be confused with the R of the trajectory along which we are parallel transporting the vector)
So this tells us how a parallel transported vector turns relative to the local ˆ r and ϕ, ˆ for a small displacement in ϕ and r. 10.2. The new reference frame, and the effective rotation vector On a flat plane, the corresponding expression to (62) is of course simply δα = δϕ.
(63)
A reference frame that with respect to ˆ r and ϕ ˆ rotates as if we had a flat geometry would then according to (62) and (63) rotate relative to a parallel transported reference frame with an angular frequency (never mind the sign for the moment) ωspace
dϕ = dτ0
!
1 −1 . √ grr
Note also that we have dϕ rdϕ 1 1 1 ˆ = |v × ˆ r|. = = |v · ϕ| dτ0 dτ0 r r r
(64)
(65)
Thinking about the sign for a second, we realize that with respect to the ’would-be-flat’ reference frame, a parallel transported reference frame will have a rotation given by ω space
1 = r
!
1 − 1 v ׈ r. √ grr
(66)
Knowing that infinitesimal rotations can simply be added (to lowest order), using (66) together with (51) and letting g = −aref , we get the gyroscope rotation relative to the
A covariant formalism of spin precession with respect to a reference congruence
22
’would-be-flat’-grid as22 Ωeffective
!
1 n ˆ × v − γ(g × v) − ω k − ω ⊥ 2γ − = (γ − 1) R γ ! 1 1 + −1 v ׈ r. √ r grr
!
(67)
We can integrate this equation to find the net precession of a gyroscope transported along any path in the axisymmetric geometry. 10.3. Comments on the integrability As a particular application of (67), we can consider the net precession of a gyroscope transported along some closed orbit. Since the ’would-be-flat’ reference frame returns to its original configuration after a full turn, we just integrate the effects of the infinitesimal rotations following from (67) to calculate the net turn. Notice however that to do this straightforwardly, we need Ωeffective in the coordinate base of the reference frame (i.e. the would-be-flat frame). In general we however only have Ωeffective in the coordinates adapted to the stationary observers. For most cases where we would be interested in motion in an axisymmetric geometry, like motion in the equatorial plane of a Kerr black hole, this however presents no problem. Then all rotations are in the plane of motion and the rotation vector Ωeffective is constant (in the zˆ-direction) in the coordinate basis of the reference frame. Notice that the τ0 implicitly entering these equations in the Ωeffective is the proper time for a stationary observer. If we instead want to express the precession in global (Schwarzschild) time, we just multiply by the time dilation factor. Even assuming all rotations to be in the plane of motion, we must still in general integrate to get the net precession of the gyroscope23 . For the particular case of circular motion with constant speed, assuming the time dilation (i.e. the lapse), ω and g · ˆ r to be independent of ϕ (as is the case for the equatorial plane of a Kerr black hole), there is however no need to integrate at all since all the terms of (67) are constant. The result follows immediately, assuming that we know ω, g and grr . Incidentally it follows from √ (60) and (61) that R = r grr for circular motion. 10.4. Comment regarding g, ω and grr The reference background (fixed to the stars) around a spinning planet, like the Earth, is both accelerating and curved. Also there is frame dragging due to the planet rotation, 22
If the geometry in question contains regions where the circumferential radius has a minimum (in 2 dimensions one may call these regions necks from the appearance of an embedding of such regions), r, which by definition is one can modify (67) a little by introducing a ± sign in the √g1rr -term. If ˆ taken to point away from the center of symmetry, points in the direction of increasing r, we choose the positive sign, otherwise the negative sign should be chosen. Note that √g1rr is zero for radii where the sign changes, so there is no discontinuity in Ωeffective . 23 Parameterizing the trajectory by some parameter λ, we understand that time dilation, R, ω, v and g all depends on λ. Assuming all rotations to be in the plane of motion it is effectively a single (scalar) integral (of the net rotation angle around the z-axis).
A covariant formalism of spin precession with respect to a reference congruence
23
giving a non-zero rotation of the stationary reference observers. If we have the spacetime metric, we can easily find ω, g and grr . If we do not have an exact spacetime metric however, as is the case for a spinning planet, we need some approximate method (like the Post-Newtonian approximation) to estimate ω, g and grr . Once this is done, assuming the approximation to be valid, (67) gives an accurate description of the precession even considering relativistic speeds. In the case of a rotating (Kerr) black hole, we do know the metric, and the precession relative to the stationary observers can readily be calculated. Notice that within the ergosphere , there are no stationary (timelike) observers. Still we can in principle use the formalism of this paper also within the ergosphere. To do this we consider coordinates that rigidly rotate around the black hole sufficiently fast to be timelike in the region in question. Indeed for the particular case of circular motion there is a paper [6], that uses this technique. 10.5. Free orbit at large radii from a Schwarzschild black hole As a simple example, consider a freely falling gyroscope (agyro = 0) orbiting in the equatorial plane of a Schwarzschild black hole. Using the static observers as our reference congruence, we have ω = 0. Then it follows from (50) that we have γ aref × v + ω space . (68) Ωeffective = γ+1 The Schwarzschild line element in the equatorial plane is given by dτ 2 = (1 + 2φ) dt2 − (1 + 2φ)−1 dr 2 − r 2 dϕ2 .
(69)
Here φ = − Mr . For convenience we now consider large r, so that M/r is small. We have then the acceleration of the freely falling frames g ≃ ∂φ (counted positive in the inwards ∂r direction). It follows readily, using (66), that for this case we have 1 q 1 + 2φ − 1 v × ˆ r (70) ω space = r φ ≃ ·v ׈ r (71) r ≃ − g · v ׈ r. (72) For the large r in question the velocities are low and we may set γ ≃ 1. Using (72) together with aref = −g and g = −gˆ r in (68) gives 1 (73) Ωeffective ≃ − (g × v) − g × v 2 3 = − (g × v). (74) 2 This result was derived by W. de Sitter in 1916 (although in a quite different manner than that described here, see [1] p. 1119). We may note that one third of the net effect comes from the spatial geometry. Using a little bit of Newtonian mechanics it is easy to derive that for a satellite orbiting the Earth at a radius R ≃ REarth , inserting the
A covariant formalism of spin precession with respect to a reference congruence
24
proper factor of c to handle SI-units, (74) amounts to 3 GM = 2 2 2c R
s
GM ≃ 1.3 · 10−12 rad/s R arcsec rad ≃ 8.3 . (75) ≃ 4.0 · 10−5 year year Knowing that the exterior metric of the Earth is approximately Schwarzschild, we have then an approximation of the effective rotation vector for a gyroscope orbiting the Earth. We can refine this approximation by considering an appropriate non-zero ω, as discussed earlier. Note that, as discussed in section 4.10, the derived precession is the precession with respect to a star-calibrated reference system on the satellite. In [1] p. 1117-1120, a similar explicitly three-dimensional formalism of spin precession is derived. It is only valid in the Post-Newtonian regime however. The precession given by (67) is however exact (assuming an ideal gyroscope). For instance, considering the above example of free circular motion in a static geometry, we can easily calculate the exact expressions for g and v, and thus express the gyroscope precession rate arbitrarily close to the horizon. Ωeffective
11. Summary and conclusion We have seen how we in a covariant manner can derive an effectively three-dimensional spin precession formalism in a general spacetime. In particular the simple form of (16) seems novel. In [1] p. 1117 a similar approach is taken where they consider only the standard spin vector, but expressed relative to a boosted set of base vectors. They however only apply it to the post-Newtonian regime. As mentioned earlier, Jantzen et. al. ([4, 7, 8]) have already pursued the same general idea, although the specific approach and final form of the results differ. In particular they have not employed the explicit 3-dimensional formalism. While the general formalism is derived assuming a general congruence, it seems to have its greatest use as a simple three-dimensional formalism assuming a non-shearing congruence. Then we have a fixed spatial geometry and the spatial parallel transport is unambiguous. For this particular case, the derived three-dimensional formalism verifies the result of the intuitive derivation of [2]. We have also given examples of how the three dimensional formalism can be used to easily find results of physical interest. Appendix A. Simplifying (14) In the expansion of the second term within the brackets of (14) there will according to µ Dηµ Duµ . These can be rewritten in terms of and since (13) be terms of the type Dt Dτ Dτ Dτ we have uµ η µ uµ = γ(η µ + vtµ ) → tµ = − . (A.1) γv v
A covariant formalism of spin precession with respect to a reference congruence
25
Dealing with Du rather than Dt is convenient since the former readily can be expressed Dτ Dτ in terms of spatial curvature and velocity changes relative to the congruence, see [3]. µ γv Also Du has a direct physical relevance. Using the identity γ−1 = γ+1 , it is then easy Dτ γv µ to derive an alternative form of K α 1 K µα = δµα + (uα − γηα ) (η µ + uµ ) . (A.2) γ+1 µ
µ
Using this in the second term within the brackets of (14) we have µ
1 D DK α = (uα − γηα ) (η µ + uµ ) . {z } | {z } Dτ Dτ γ + 1 | γvtα
(A.3)
(γ+1)ηµ +γvtµ
As we expand this expression there will be terms containing η µ , ηα and tµ tα . These we will disregard for the following reasons. Terms containing ηα will anyway die when multiplied by S¯α (as they are in (14)). Terms containing η µ we will disregard since we for the moment only are interested in P µ α S¯α = (g µα + η µ ηα )S¯α . When we have a neat expression for this we can find the η µ -part a posteriori using the orthogonality of S¯α and ηα . We will disregard terms of the type tµ tα since we know that these must cancel anyway for S¯α to stay normalized (as we know it must by construction of the Fermi transport and the relation to the stopped spin vector)24 . Note however that in ν principle, we should contract with the inverse K −1 µ 25 , before disregarding the terms of the described types (see (14)). The form of the inverse is however such that we can carry out the effective cancellations prior to applying the inverse26 . We then readily find Duα Dηα Duµ Dη µ DK µ α eff γv µ . (A.4) −γ t + tα + = Dτ γ+1 Dτ Dτ ⊥ Dτ Dτ ⊥ By the perpendicular sign ⊥ we here mean that we should select only the part orthogonal eff to both tµ and η µ . By = we indicate that the equality holds excepting terms of the type η µ , ηα and tµ tα . In an analogous manner we readily find for the first term within
¯α S ¯ α momentarily, = 0. For the particular case where S¯α = St From normalization follows that S¯α DDτ µ it follows that any net term of the form at tα in the right hand side of (14) must vanish. Since the parameter a does not depend on S α it follows that it must vanish entirely. The point is that the form of (A.3) is such that, when expanded it can be written as a sum of tensors of the type Aµ Bα . Letting the suffix ⊥ indicate that only the part orthogonal to both η µ and tµ should be selected, each such term can be written in the form (tµ tρ Aρ + [Aµ ]⊥ )(tα tσ Bσ + [Bα ]⊥ ). Adding up the resulting terms of the type tµ tα (including the terms of this type coming from the first term within the brackets of (14)) into a single term atµ tα we know that a must be zero. µ 25 Note from (13) that the effect of contracting K −1 α with a contravariant vector is that it shortens the tµ -component of the vector by a γ-factor, while the rest of the on-slice (orthogonal to η µ ) part of thee vector is unaffected. 26 If the inverse had contained for instance terms of the type tν ηµ – we could not cancel η µ terms directly within the brackets of (14). That the inverse is not containing any such terms is a benefit of the particular gauge choice in choosing K ν µ – where we had a freedom to include any terms containing ηµ . 24
A covariant formalism of spin precession with respect to a reference congruence
26
brackets of (14)
Duα Duρ eff . (A.5) = γvtµ u K α Dτ Dτ ⊥ Now use (A.4) and (A.5) in (14). Shorten the tµ components by a γ factor (according to the effect of the inverse), and neglect the η µ -term. We readily find D S¯α γv ¯α µ D D P µα S t = (uα + ηα ) − tα (uµ + η µ ) . (A.6) Dτ γ+1 Dτ Dτ ⊥ ⊥ Now that we have this compact expression we may also find the η µ term that we earlier S¯α α ηα = −S¯α Dη which gives omitted. From orthogonality, S¯α ηα = 0, follows that DDτ Dτ γv ¯α µ D D S¯µ D µ µ S t = (uα + ηα ) − tα (u + η ) Dτ γ+1 Dτ Dτ ⊥ ⊥ Dηα + η µ S¯α . (A.7) Dτ So here we have the transport equation for the stopped spin vector (giving the rotation relative to inertial coordinates). µ
ρ
Appendix B. A note concerning the intrinsic angular momentum As an idealized scenario we consider a special relativistic gyroscope which we model as an isolated system of point particles undergoing four-momentum conserving internal collisions. Following the discussion in [9] p. 87-90, we define the angular momentum tensor with respect to the spacetime origin as Lµν =
X
xµ pν − xν pµ .
(B.1)
Here the summation runs over events xµ and four-momenta pµ for the various particles at a particular time slice t = const. The (Pauli-Lubanski) spin vector can be written as 1 (B.2) Sµ = ǫµνρσ Lνρ V σ 2 Here V µ is the four-velocity of the center of mass and ǫµνρσ is the Levi-Civita tensor (density) where ǫxyz0 = 1. Furthermore we introduce an angular momentum four-vector hµ := (0, h), where h is the standard (relativistic) angular momentum three-vector, with respect to our reference coordinates. Defining η µ as a purely time directed normalized vector with respect to the reference coordinates, we can write 1 hµ = ǫµνρσ Lνρ η σ . (B.3) 2 Letting v be the velocity of the center of mass, γ the corresponding gamma factor and setting (0, v) := vtµ with respect to the reference coordinates, we can decompose the four-velocity of the center of mass as V µ = γ(η µ + vtµ ). Using this in (B.2) together with (B.3), it follows that 1 Sµ = γhµ + γv ǫµνρσ Lνρ tσ . (B.4) 2
A covariant formalism of spin precession with respect to a reference congruence
27
It is a short exercise to show that in three-formalism this amounts to h = S/γ + rc × p.
(B.5)
Here h is the net angular momentum of the system of point particles, rc is the center of mass (center of energy), γ is the gamma factor for the velocity of the center of mass, p is the net relativistic three-momentum and S is the spatial part of the spin vector. Note that the intrinsic angular momentum is not given by S but by S/γ. Note incidentally also that there is a difference between the center of mass and the proper center of mass (see [9] p. 87-90). As pointed out e.g. in [10], the gyroscope center of mass does not in general lie on the gyroscope central axis. A real gyroscope moving under the influence of forces is neither (simply) consisting of point particles nor is it isolated. A more detailed analysis would likely assume a general energy momentum tensor T µν and allow for external forces acting on the elements of the gyroscope. For the purposes of this article the simple derivation outlined above will however suffice. References [1] Misner CW, Thorne K S and Wheeler J A 1973 Gravitation (New York: Freeman) [2] Jonsson R 2007 Gyroscope precession in special and general relativity from basic principles Am. Journ. Phys. 75 463-471 [3] Jonsson R 2006 Inertial forces and the foundations of optical geometry Class. Quantum Grav. 23 1-36 [4] Jantzen R T, Carini P and Bini D 1992 Ann. Phys. 215 1-50 [5] Jonsson R and Westman H 2006 Generalizing optical geometry Class. Quantum Grav. 23 61-76 [6] Rindler W and Perlick V 1990 Gen. Rel. Grav. 22 1067-1081 [7] Bini D, Carini P and Jantzen RT 1997 Int. Journ. Mod. Phys. D 6 1-38 [8] Bini D, Carini P and Jantzen RT 1997 Int. Journ. Mod. Phys. D 6 143-198 [9] Rindler W (1991) Introduction to special relativity (Oxford: Oxford University Press) [10] Muller R A 1992 Am. Journ. Phys. 60, 313
Gyroscope precession in special and general relativity from basic principles Rickard M. Jonsson Department of Theoretical Physics, Physics and Engineering Physics, Chalmers University of Technology, and G¨ oteborg University, 412 96 Gothenburg, Sweden E-mail:
[email protected] Submitted: 2004-12-09, Published: 2007-05-01 Journal Reference: Am. Journ. Phys. 75 463
Figure 1: A gyroscope transported around a circle. The vectors correspond to the central axis of the gyroscope at different times. The Newtonian version is on the left, the Abstract. In special relativity a gyroscope that is sus- special relativistic version is on the right. pended in a torque-free manner will precess as it is moved along a curved path relative to an inertial frame S. We small. Thus to obtain a substantial angular velocity due explain this effect, which is known as Thomas precession, to this relativistic precession, we must have very high by considering a real grid that moves along with the gyro- velocities (or a very small circular radius). scope, and that by definition is not rotating as observed In general relativity the situation becomes even more from its own momentary inertial rest frame. From the interesting. For instance, we may consider a gyroscope basic properties of the Lorentz transformation we deduce orbiting a static black hole at the photon radius (where how the form and rotation of the grid (and hence the free photons can move in circles).3 The gyroscope will gyroscope) will evolve relative to S. As an intermediate precess as depicted in Fig. 2 independently of the velocity. step we consider how the grid would appear if it were not length contracted along the direction of motion. We show that the uncontracted grid obeys a simple law of rotation. This law simplifies the analysis of spin precession compared to more traditional approaches based on Fermi transport. We also consider gyroscope precession relative to an accelerated reference frame and show that there are extra precession effects that can be explained in a way analogous to the Thomas precession. Although fully relativistically correct, the entire analysis is carried out using three-vectors. By using the equivalence principle the formalism can also be applied to static spacetimes in general relativity. As an example, we calculate the pre- Figure 2: A gyroscope transported along a circle at the cession of a gyroscope orbiting a static black hole. photon radius of a static black hole. The gyroscope turns so that it always points along the direction of motion.
I
Introduction How a gyroscope precesses for these examples can be derived using four-vectors and Fermi transport.4 Although the Fermi approach is very general, it typically results in a set of coupled differential equations that are rather complicated and do not provide much physical insight (see Appendix A). In the following we will take a different approach. We start by discussing why there is Thomas precession in special relativity. We also derive the exact relation, Eq. (1), using only rudimentary knowledge of special relativity. We then consider gyroscope precession with respect to an accelerated reference frame within special relativity. We show that if the gyroscope moves inertially, but the reference frame accelerates perpendicularly to the gyroscope direction of motion, the gyroscope will precess relative to the reference frame. As an application where both the reference frame and
In Newtonian mechanics a spinning gyroscope, suspended such that there are no torques acting on it, keeps its direction fixed relative to an inertial system as we move the gyroscope around a circle. However, in special relativity the gyroscope will precess, meaning that the direction of the gyroscope central axis will rotate, see Fig. 1. If we denote the circle radius by R and the gyroscope velocity by v we can express the angular velocity Ω of the gyroscope precession as1 v (1) Ω = (γ − 1) . R ! Here γ = 1/ 1 − v 2 /c2 and c is the speed of light. Henceforth, unless otherwise stated, we will for convenience set c = 1. The precession given by Eq. (1) is known as Thomas precession.2 In particular we note that for v " 1 and γ # 1, the right-hand side of Eq. (1) tends to be very 1
the gyroscope accelerates we will consider a gyroscope on an object with respect to the object’s initial rest frame a train that moves along an upward accelerating platform means giving the object the velocity of the boost and as shown in Fig. 3. length contracting the object along the direction of motion. At times we will use the term pure boost to stress the non-rotating aspect of the boost. We will also assume v that any real push of an object (such as the gyroscope grid), works like a pure boost relative to the momentary inertial rest frame of the object. We will next illustrate how three consecutive physical boosts of a grid, where the boosts are each non-rotating as Figure 3: A train at two consecutive times moving with observed from the momentary rest frame of the grid, will velocity v relative to a platform that accelerates upward. result in a net rotation. The result can be formally derived A gyroscope with a torque free suspension on the train by making successive Lorentz transformations (multiplywill precess clockwise for v > 0. ing matrices), but from the derivation in the next section we can also understand how the rotation arises. The net resulting rotation is the key to the Thomas precession as If we neglect the Earth’s rotation, an ordinary plat- presented here. form on Earth behaves just like an accelerated platform in special relativity (the equivalence principle),5 so the B The effect of three boosts result can be applied to every day scenarios. The equivalence principle also allows us to apply the analysis to a Consider a grid in two dimensions, initially at rest and gyroscope orbiting a black hole. non-rotating relative to an inertial system S. We then Sections II–VI assume knowledge of special relativity. perform a series of boosts of the grid as sketched in Fig. 4. Knowledge of general relativity is assumed in Secs. VII– Note especially what happens to the thick bar and its end IX. points.
II
The gyroscope grid
Although spinning gyroscopes are the typical objects of interest when discussing relativistic precession effects, we will in the following consider a grid (say of metal) that we call the gyroscope grid. This grid is by definition not rotating as observed from its own momentary inertial rest frame. The central axis of an ideal gyroscope with a torque-free suspension is, by definition, also non-rotating as observed from its own momentary inertial rest frame. It follows that the axis of an ideal gyroscope, which is transported together with the grid, will keep its direction fixed relative to the grid. Thus, the precession of an actual gyroscope (assuming it behaves like an ideal gyroscope) follows from the behavior of the gyroscope grid. The use of the grid will also allow us to put the effects of precession due to the gyroscope grid acceleration on an equal footing with the precession effects that come from the acceleration of the reference frame.
A
(d) (c) (a)
(b)
Figure 4: (a) The grid at rest with respect to S. (b) The grid after a pure boost with velocity v to the right, relative to S. Note the length contraction. (c) The grid after a pure upward boost relative to a system S ! that moves with velocity v to the right. (d) The grid after a pure boost that stops the grid relative to S. After the first boost by a velocity v to the right, the grid is at rest relative to another inertial system S ! . The grid is then given a pure upward boost relative to S ! by a velocity δu! . Relative to the original system S, the ˆ up and to the right. grid will then move in a direction n Through the upward boost the originally vertical grid bars remain vertical relative to S ! ; thus they will also remain vertical relative to S, as follows from the Lorentz transformation. However, the originally horizontal bars will become rotated. To understand this rotation, consider all of the events along a horizontal bar just as the bar starts moving upward relative to S ! . These events are all simultaneous relative to S ! , but relative to S the rearmost event (the leftmost event) will happen first (relativity of simultaneity). Thus the leftmost part of the bar
The boost concept
In special relativity, a Lorentz transformation to a new set of coordinates, which are non-rotated relative to the original coordinates, is known as a boost of the coordinates.6 Equivalently a physical boost of an object can be performed. As seen by an observer at rest in a certain inertial reference frame, a physical boost of an object by a certain velocity is equivalent to performing a boost of the observer’s reference frame by minus the velocity (while not physically affecting the object). In particular, boosting
2
will have a head start (upward) relative to the rightmost part, and the bar will therefore become rotated. Finally, we stop the grid, in other words we make a pure boost in the −ˆ n direction, so that the grid stops relative to S. The effect will be to remove the length contraction in the ˆ direction, that is, to stretch the grid in the n ˆ direction. n Through this stretching we understand that the originally vertical grid bars will rotate clockwise. Because none of the boosts deform the grid as observed in the grid’s own momentary rest frame, it follows that the entire final grid will be rotated clockwise relative to the original grid.
C
δu. It follows that, to first order in δu, Eq. (2) reduces to δθ =
δu , v
δα + δθ =
γδs , L0
δθ =
δs . L0
(3)
From Eq. (3) we find δα =
δu (γ − 1). v
(4)
Here δα is the resultant clockwise angle of rotation of the grid, after the three consecutive boosts. The result also applies to a grid that was initially rotated by a certain angle relative to the grid we considered above. To see this, suppose that we perform the three boosts simultaneously on the two grids. Because the boosts are all non-rotating as observed in the momentary rest frame of the grids, the relative angle between the grids must be preserved. Thus Eq. (4) gives the angle of rotation resulting from the three boosts in question, regardless of the initial rotation of the grid. For an infinitesimal boost in a general direction relative to S ! , only the upward directed part of the boost contributes to the rotation.7 Thus Eq. (4) holds also for this case if we interpret δu in Eq. (4) as the part of the infinitesimal velocity change that is perpendicular to the direction of motion.
Calculating the precise turning angle
The upward boost by a velocity δu! relative to S ! yields an upward velocity δu = δu! /γ (time dilation) as observed from S. Consider now two points separated by a distance L0 along an originally vertical bar of the grid, as measured in the grid’s own frame. As observed in S ! , the distance between the points after the upward boost is, due to length contraction, given by L = L0 /γ(δu! ). This distance is also the distance between the points as observed in S, as follows from the Lorentz transformation. Also, the velocity of the points after the upward boost is vˆ x + δuˆ y as observed in S. When we stop the grid, the length expansion (that is, the removal of length contraction) will shift the topmost point relative to the lowest point, resulting in a rotation D The uncontracted grid by an angle δα as depicted in Fig. 5. From the definitions For a grid in motion relative to a certain specified referin Fig. 5 it follows that ence frame (for example an inertial frame), we now introduce what we call the uncontracted grid. This grid is y obtained by imagining the real grid without length conγδs traction along the direction of motion. The idea is illusδs trated in Fig. 6. Stretch direction δθ L δu v L0 δθ δα
Figure 6: The real grid (black thin lines) and the corresponding imagined uncontracted grid (grey thick lines) before and after a large upward boost.
x
Figure 5: The stretch-induced tilt of the two points (the filled circles) due to the final stopping of the grid. The From Fig. 4, we note that the imagined uncontracted distance between the points prior to the stretching, as grid, immediately before and after the infinitesimal upmeasured in the direction of motion, is denoted by δs. ward boost, is identical (in form and rotation) to the initial and the final actual grid respectively. From the discussion at the end of Sec. II C it therefore follows that for any real grid moving on a plane that receives an inδu δs γδs tan δθ = , sin δθ = . (2) finitesimal boost (non-rotating as observed in the grid’s , sin(δα + δθ) = v L0 L own momentary inertial rest frame) by a velocity δu perpendicular to the direction of motion, the corresponding From now on we will assume that δu is infinitesimal. Beuncontracted grid will rotate an angle given by Eq. (4) cause the γ-factor entering the relation between L and as L0 depends on (δu)2 , we have L = L0 to first order in δu δα = (γ − 1). (5) v 3
Henceforth we will always describe the gyroscope grid in which uses the Fermi transport equation for the spin vecterms of the uncontracted grid. If we have found the evo- tor of the gyroscope, is also comparatively complicated lution of the uncontracted grid for a particular path, we (see Appendix A and Ref. 8 for further details). can always find the observed real grid by length contracting the uncontracted grid in the momentary direction of G Comments on the uncontracted grid motion. Although we may think of the uncontracted grid as a mathematically convenient intermediate step in finding E Circular motion the actual grid, there is more to this concept. As follows Consider a gyroscope grid moving with velocity v along a from its definition, the uncontracted grid corresponds dicircle of radius R. During a time step δt, the grid receives rectly to the grid as experienced in a system that moves an infinitesimal boost perpendicular to the direction of with the grid and that is related to the reference frame motion. In the inertial frame of the circle the perpendic- in question by a pure boost. Consider a special relativistic scenario of a gyroscope ular velocity change is given by grid suspended in a torque free manner inside a satellite. v2 (6) The satellite uses its jet engines to move along a smooth δu = δt. R simple closed curve on a plane. We want to measure from The corresponding uncontracted grid will rotate an angle the satellite the precession angle of the gyroscope grid according to Eq. (5) during the boost. At the next time after a full orbit. If we assume that there are a couple of step there is a new boost and a new induced rotation. It suitably placed fixed stars, we can use their direction as follows that there is an ongoing precession of the uncon- observed from the satellite at the initial and final point tracted grid as depicted in Fig. 7. The angular velocity of the orbit (which coincide), as guidelines to establish a reference system within the satellite. For this scenario the of rotation is given by Eqs. (5) and (6) as uncontracted grid is the physical object in which we are interested, because it’s orientation precisely corresponds δα v = (γ − 1) . (7) to the orientation of the actual gyroscope grid relative δt R to the star calibrated reference frame of the satellite. In Thus we have derived the Thomas precession given by particular, if the uncontracted grid has rotated a certain Eq. (1). Note that Eq. (7) describes how fast the imagined angle after the full orbit, so has the actual gyroscope grid uncontracted grid rotates. as measured from the satellite.
III
Boosting the reference frame
Now let us consider the effect of a boost of the reference frame rather than of the gyroscope grid. To make the analogy with the discussion in Sec. II clearer, we consider a real grid as a reference frame. We assume that the Figure 7: A gyroscope grid at successive time steps. Both reference frame initially is at rest relative to an inertial the grid and the gyroscope are depicted as they would be system S, and is then boosted upward so that it is at rest observed if they were uncontracted. with respect to another inertial system S 0 , see Fig. 8. The gyroscope grid is assumed to move with constant speed v to the right as observed in S.
F
The mathematical advantage of the uncontracted grid
S
S0
δuref
S!
We have shown that the uncontracted grid evolves acv cording to a simple law of rotation. The central axis of a gyroscope, if it were not length contracted along the direction of motion, obeys the same simple law of rotation. The actual axis of a gyroscope, however, changes its length over time, and its angular velocity would not be as Figure 8: Boost of the reference frame (of which the desimple as that given by Eq. (7). A differential equation for picted thin grid is a small part) upward by a velocity the evolution of the actual axis, would hide the simple dy- δuref . The velocity of the gyroscope grid is maintained. namics of a rotation and a superimposed length contraction, which would complicate the analysis. Similarly, the Relative to the gyroscope system, the reference frame standard approach to calculating gyroscope precession, initially moves to the left, and is then (due to the boost) 4
given an upward velocity δuref /γ (time dilation). Because velocity change δugyro of the gyroscope, using the identity the reference frame moves relative to the gyroscope sys- (γ − 1) = γ 2 v 2 /(γ + 1), can be written as tem, the reference frame is length contracted along the γ2 direction of motion. However, we can imagine the refer(δugyro × v). (9) δα = γ +1 ence frame without the length contraction. Analogous to the discussion in Sec. II, the uncontracted reference frame The corresponding vector analog of Eq. (8) for a velocity will rotate during the boost, as depicted in Fig. 9. change δuref of the reference frame is given by ! S γ δα = (δuref × v). (10) γ+1 Note that the cross product selects only the part of the velocity change that is perpendicular to the relative direction of motion. Consider now an infinitesimal boost of both the gyroscope and of the reference frame. If we assume that we start by boosting the gyroscope, which gives a velocity Figure 9: Relative to the gyroscope system, the reference change δugyro , this boost yields a rotation according to frame (thin lines), of which we illustrate a certain part, Eq. (9). Subsequently boosting the reference frame by a rotates during the boost. The reference frame is depicted velocity δuref yields another rotation given by Eq. (10), as it would appear if it was not length contracted relative but v should be replaced by v + δugyro . However, to first to the gyroscope system. order in δuref and δugyro this replacement does not affect Eq. (10). Because infinitesimal rotations can be added (to first order in the magnitude of the rotations), it then The net counterclockwise angle of rotation for the ref- follows that the net rotation is given by Eqs. (9) and (10) erence frame is found by substituting δu by δuref /γ into as Eq. (5): γ2 γ δuref γ − 1 δα = (δugyro × v) + (δuref × v). (11) . (8) δα = γ+1 γ+1 γ v Note that removing the length contraction of the uncontracted reference frame relative to the gyroscope grid yields the same relative configuration as removing the length contraction of the gyroscope grid relative to the reference frame. It follows that the upward boost of the reference frame yields a clockwise rotation of the uncontracted gyroscope grid, described by Eq. (8), relative to the reference frame. This relative rotation is precisely the rotation in which we are interested. Note in particular that an upward boost of the reference frame yields a clockwise relative rotation just as an upward boost of the gyroscope grid yields a clockwise relative rotation.
Now consider a continuously accelerating reference frame and gyroscope grid. Relative to an inertial system in which the reference frame is momentarily at rest, we have δugyro = agyro δt and δuref = aref δt for a time step δt. We substitute these relations into Eq. (11) and obtain the net angular velocity vector Ω = δα/δt for the gyroscope grid rotation relative to the reference frame as Ω=
γ [γagyro + aref ] × v. γ+1
(12)
Because we are interested in how the gyroscope grid rotates relative to the accelerating reference frame, it can be useful to express the motion relative to the reference frame. Consider a path with local curvature radius R and IV Three dimensions ˆ , fixed to the reference frame. In Apcurvature direction n pendix B we show that for motion along this path we have In the two-dimensional reasoning of Secs. II and III, the (just like in Newtonian mechanics) induced rotation occurred in a plane spanned by the velocity vector and the vector for the velocity change. For ˆ n more general three-dimensional motion and velocity changes, [agyro ]⊥ = [aref ]⊥ + v 2 . (13) R the induced rotation should still occur in a plane spanned by these two vectors. The axis of rotation can therefore Here agyro and aref refer to the accelerations relative to be expressed in terms of the cross product of these two an inertial frame in which the reference frame is momenvectors. Let us introduce δα as a vector whose direction tarily at rest. The notation ⊥ denotes the part of the indicates the axis of rotation and whose magnitude cor- acceleration that is perpendicular to the direction of moresponds to the angle of rotation for the uncontracted tion for the gyroscope grid. Because of the cross product gyroscope grid relative to the reference frame. Also, let in Eq. (12), the perpendicular part of the acceleration is v be the velocity vector of the gyroscope relative to the the only part that matters for Ω. If we substitute agyro reference frame. The three-vector analog of Eq. (5) for a 5
from Eq. (13) into Eq. (12) and simplify the resultant corresponds to the grid as experienced by an observer on the train (as discussed in Sec. II G). Thus we obtain expression, we find the angular velocity relative to the train by multiplying # "n ˆ × v + γ(aref × v). (14) the right-hand side of Eq. (16) by γ to account for time Ω = (γ − 1) R dilation.9 Relative to the train the gyroscope thus preThe first term on the right-hand side of Eq. (14) has the cesses at a steady rate Ω0 (clockwise as depicted) given same form as the standard Thomas precession term given by by Eq. (1). The second term corresponds to both direct Ω0 = γ 2 vg. (17) effects of rotation from the reference frame acceleration If we assume the train velocity to be low and introduce and to the indirect effects of this acceleration, because the 2 us to express g and v in acceleration of the gyroscope grid relative to an inertial the proper factor of c to enable 2 SI units, we have Ω ≈ vg/c . For a train with a velocity 0 frame depends on the reference frame acceleration in this of 50 m/s and a platform acceleration corresponding to formulation. Equation (14) matches the formally derived that of a dropped apple on the Earth, we obtain Eq. (51) of Ref. 8. m 50 m s × 9.81 s2 Ω0 ≈ ≈ 5 × 10−15 rad/s. (18) m (3 × 108 s )2 V Applications In this section we discuss applications of the derived for- This special relativistic scenario mimics a train moving malism, Eq. (14), for gyroscope precession relative to an along a straight platform on the Earth (neglecting the Earth’s rotation). It follows that precession effects due to accelerating reference frame. gravity are small for everyday scenarios on the Earth. Because a torque-free gyroscope precesses relative to A Motion along a horizontal line the train, it follows that the train has a proper rotation, Consider a special relativistic scenario of a train moving meaning that the train rotates as observed from its own along a horizontal line relative to a platform that contin- momentary inertial rest frame. This rotation can be unually accelerates upward relative to an inertial frame (see derstood without reference to the gyroscope precession. The heart of the matter lies (as is often the case) in siFig. 10). multaneity. Let S be an inertial system where the rail of the continuously accelerating platform is at rest momenv tarily (at t = 0). As observed in S, the horizontal straight rail will first move downward (when t < 0), decelerate to be at rest at t = 0, and then accelerate upward. Consider now all the events along a section of the rail, when the rail Relative to the train’s momentary inertial Figure 10: A train with a gyroscope moving relative to an is at rest in S. ! rest frame S , which moves with velocity v to the right accelerating platform observed at two successive times. as observed from S, the rightmost of the events along the rail will occur first. Thus relative to S ! , when the On the train a gyroscope is suspended so that there rail at the rear end of the train has no vertical motion, are no torques acting on it as observed from the train. For the rail at the front end (and thus the train’s front end) the special case of motion along a straight line relative to will already have an upward velocity. Hence a train movthe reference frame (the platform in this case), we have ing as depicted in Fig. 10 has a proper counterclockwise 1/R = 0. If we define g = −aref as the local acceleration rotation. By this reasoning we can verify the validity of of an object dropped relative to the platform, Eq. (14) Eq. (17). We also understand that as observed from S ! , the rail is not straight but is curved as depicted in Fig. 11. reduces to Ω = γ(v × g). (15)
Note that both the gyroscope and the platform reference frame accelerate with respect to an inertial frame, and hence we expect two precession effects. Both of these effects are included in the single term on the right-hand v side of Eq. (15). Let Ω denote the clockwise precession rate, v the velocity to the right, and g the the downward acceleration of dropped object relative to the platform. Figure 11: A sketch of the rail and the train observed from an inertial system where the train is momentarily Then Eq. (15) gives at rest. Ω = γvg. (16) Note that the uncontracted grid, whose rotation with respect to the platform reference frame is given by Eq. (16), 6
B
VI
Following the geodesic photon
As another application we now study the precession of a gyroscope that follows the spatial trajectory of a free photon.10 From Eq. (B2) it follows that the trajectory of a free photon (set v = 1 and aparticle = 0) as observed relative to an accelerating reference frame satisfies ˆ n = g⊥ . R
We have seen how the basic principles of special relativity can be used to derive a simple but exact three-vector formalism of spin precession with respect to an accelerating reference frame. The precession is given by Eq. (14) as # "n ˆ × v + γ(aref × v), Ω = (γ − 1) R
(19)
(22)
ˆ /R is the curvature of the gyroscope path relwhere n ative to the accelerated reference frame. Recall that Ω describes the rotation of a gyroscope axis as we imagine it without length contraction along the direction of motion. In the following, knowledge of general relativity is assumed.
Here g⊥ is the part of g that is perpendicular to v (recall that g = −aref ). If we substitute the curvature given by Eq. (19) into Eq. (14), we find that the gyroscope grid angular velocity is given by Ω = v × g⊥ .
Conclusions
(20)
Consider now a normalized vector ˆt directed along the VII Axisymmetric spatial geomespatial direction of motion. The time derivative of ˆt reltries and effective rotation vecˆ /R. It ative to the reference frame satisfies dˆt/dt = v n follows that ˆt rotates with an angular velocity Ωˆt = tors ˆ /R. If we substitute the curvature n ˆ /R = g⊥ given v×n by Eq. (19), into this expression for Ωˆt, we obtain In a static spacetime such as that of a Schwarzschild black hole, the global static reference frame locally corresponds Ωˆt = v × g⊥ . (21) to the accelerated reference frames we have considered in special relativity. If we integrate the infinitesimal rotaIf we compare Eqs. (21) and (20), we see that ˆt rotates with the same angular velocity as the gyroscope grid. tions from Ω given by either Eqs. (12) or (14), we can find the net rotation of a gyroscope that is transported It follows that a gyroscope transported along a spatial along a given spatial path. Note, however, that Ω detrajectory of a free photon will keep pointing along the scribes how the gyroscope grid rotates relative to a frame direction of motion if it did initially (see Fig. 12). that is parallel transported with respect to the local spatial geometry associated with the reference frame. Thus directly integrating the effects of rotation from Ω gives the rotation relative to a frame that is parallel transported with respect to the global spatial geometry. In Fig. 13 we illustrate a section of the spatial geometry of an equatorial plane of a static black hole.
Figure 12: A free photon will in general follow a curved path relative to an accelerated reference frame. A gyroscope transported along such a path will keep pointing along the path if it did so initially. If we imagine a static reference frame outside the event horizon of a static black hole, then locally this reference frame behaves just like an accelerated reference frame in special relativity (the equivalence principle). Hence a gyroscope outside of a black hole that follows the path of a free photon, such as a circle at the photon radius, will not precess relative to the forward direction of motion, as depicted in Fig. 2 in the introduction.
Figure 13: Sketch of the spatial geometry of a symmetry plane outside a black hole. The local static reference frame shown (the square grid) has a proper acceleration outward. For a sufficiently small such reference frame it works just like an accelerated reference frame in special relativity. Suppose then that we consider motion in the equatorial plane of some axisymmetric geometry. For instance, 7
to the plane of motion) in the coordinate basis of the would-be-flat reference frame. For a counterclockwise motion the clockwise angular velocity of precession is then ˆ = −ˆr and g = −gˆr) given by Eq. (25) as (with n # v v" 1 Ωeff = (γ − 1) − γgv + −1 . (27) √ R r grr
we might be interested in the the net rotation of a gyroscope (grid) after a closed orbit around the center of symmetry (not necessarily a circular orbit). We need then take into consideration that a parallel transported frame will be rotated relative to its initial configuration after a complete orbit due to the spatial geometry. To deal with this complication we introduce a new reference frame that rotates relative to the local coordinates, spanned by the polar vectors ˆr and ϕ, ˆ in the same manner as a parallel transported reference frame does on a plane. In other words, if we consider a counterclockwise displacement (δϕ,δr), then relative to the local vectors ˆr and ϕ, ˆ the new reference frame should rotate δϕ clockwise. Such a “would-be-flat” reference frame always returns to its initial configuration after a full (closed) orbit. The line element for a two-dimensional axisymmetric spatial geometry can be written in the form
It is easy to show that the curvature radius of a circle at a certain r, for a geometry of the form of Eq. (23), is given √ by R = r grr . If we substitute this result into Eq. (27), we find # v" γ Ωeff = − 1 − γgv. (28) √ r grr For a general spherically symmetric static spacetime, the line element of a radial line can be written in the form dτ 2 = gtt (r)dt2 − grr (r)dr2 .
(29)
Note that grr is positive as defined here (to match the definition in Sec. VII). From Eq. (29) it is easy to derive ds2 = grr dr2 + r2 dϕ2 . (23) the local acceleration of a freely falling particle momenWith respect to such a geometry it is easy to show that tarily at rest. The result is ∂gtt 1 the angular velocity of a parallel transported frame rela. (30) g= √ tive to a “would-be-flat” frame is given by8 2gtt grr ∂r # So here we have an explicit expression for the g which 1" 1 − 1 v × ˆr. (24) enters the expression for Ωeff in Eq. (28). We are now ω space = √ r grr ready to consider a specific example. Because infinitesimal rotation vectors can be added (to lowest order), it follows from Eq. (24) and Eq. (14) that A The Schwarzschild black hole the gyroscope grid rotation relative to the would-be-flat For a Schwarzschild black hole (using standard coordiframe is given by nates and c = G = 1) we have # # "n ˆ 1" 1 gtt = (1 − 2M/r) (31a) − 1 v × ˆr. × v − γ(g × v) + Ωeff = (γ − 1) √ R r grr −1 grr = (1 − 2M/r) . (31b) (25) Alternatively we could express Ωeff in terms of the gyro- We substitute these two expressions into Eq. (30) and scope acceleration agyro relative to a local freely falling find M (inertial) frame momentarily at rest relative to the static g= ! . (32) 2 r 1 − 2M/r reference frame. If we use Eq. (12) and add the rotation due to the spatial geometry as described by Eq. (24), we If we use Eq. (32) and Eq. (31b) in Eq. (28), we obtain find $ % v γv Ωeff = ! (33) 1 − 3M/r − . " # 2 r r 1 − 2M/r 1 γ 1 γ (agyro ×v)− (g×v)+ √ −1 v׈r. Ωeff = γ+1 γ +1 r grr Equation (33) gives the precession rate as a function of r (26) and v. For constant velocity v we obtain the net rotation Note that the time t implicitly entering in Eqs. (25) and after a full orbit by multiplying the precession rate by (26) through Ωeff = dα/dt is the local proper time for the local orbital period 2πr/v. Thus we have αper-lap = a static observer. We obtain the net induced rotation Ωeff 2πr/v. If we use this result together with Eq. (33), we of a gyroscope in closed orbit by integrating the effects obtain for counterclockwise motion the clockwise angle of of the infinitesimal rotations given by either Eq. (25) or precession per lap Eq. (26). (1 − 3M/r) αper-lap = γ! − 1. (34) 2π 1 − 2M/r
VIII
Circular orbits in static spherically symmetric spacetimes
In particular, for the photon radius (where geodesic photons can move on circles) at r = 3M , we obtain a rotation angle of −2π per orbit, independently of the velocity. This For circular motion in a spatial symmetry plane of a static result is precisely what we would expect from the discusspherically symmetric spacetime, the direction of the ro- sion in Sec. V B. Equation (34) is equivalent to Eq. (39) tation vector Ωeff is constant (directed perpendicularly of Ref. 11.12 8
vector (S x , S y , S z ) is in the xy plane (so S z = 0) and let the gyroscope start at t = 0 on the positive x-axis. For a free (geodesic) gyroscope in circular motion around Solving the Fermi equation is then (effectively) reduced 2 a static black hole we have, according to Eq. (13), v /R = to solving two coupled differential equations (see Ref. 8): √ g where R = r grr . By also using grr given by Eq. (31b) and g given by Eq. (32), we find the γ factor for free dS x = γ 2 v 2 ω sin(ωt)(S x cos(ωt) + S y sin(ωt)) (A2a) circular motion: dt ! dS y 1 − 2M/r = −γ 2 v 2 ω cos(ωt)(S x cos(ωt) + S y sin(ωt)). (A2b) γ= ! . (35) dt 1 − 3M/r For the initial conditions (S x , S y ) = (S, 0) the solutions Note that γ becomes infinite for r = 3M as it should. If can be written as2 we use Eq. (35) in Eq. (34), we obtain S x = S [cos((γ − 1)ωt) + (γ − 1) sin(ωγt) sin(ωt)] (A3a) αper-lap ! (36) = 1 − 3M/r − 1. 2π S y = S [sin((1 − γ)ωt) − (γ − 1) sin(ωγt) cos(ωt)] . (A3b) Equation (36) is an exact expression for the net precession angle per full orbit for an ideal gyroscope in free The first term on the right-hand side of Eq. (A3a) and circular motion around a static black hole. If we assume Eq. (A3b) respectively, corresponds to a rotation about the gyroscope to be freely “floating” within a satellite, the z-axis, but there is also another superimposed rotaanalogous to the discussion of Sec. II G, Eq. (36) gives tion with a time dependent amplitude. To find this sothe rotation relative to a star-calibrated reference system lution directly from the coupled Fermi equations seems of the satellite. Equation (36) matches Eq. (37) of Ref. 11. rather difficult, even for this very symmetric and simple scenario.
B
IX
Geodesic circular motion
Relation to other work
B
The standard approach to calculating gyroscope precession in special and general relativity is to solve the Fermi equation for the spin four-vector of the gyroscope. Even for simple applications in special relativity, such as circular motion, the resulting equations can, however, be quite complicated (see Appendix A). In general relativity, the classical approaches to gyroscope precession are based on approximations that assume “weak” gravity and small velocities (see e.g Refs. 4 and 13). The derived formalisms can therefore not be applied accurately to, for example, a gyroscope orbiting close to a black hole. Other approaches, such as that in Ref. 11, are exact but specific to circular motion. The approach of this paper, which is exact (assuming an ideal gyroscope) and applies to arbitrary motion relative to a static reference frame, is strongly linked to the more formal approaches in Refs. 8 and 14.
A
Suppose that we have an upward accelerating reference frame. A test particle moves with velocity v along a path, fixed to the reference frame, with the local curvature R ˆ . We would like to express the and curvature direction n part of the test particle’s acceleration that is perpendicular to the particle’s momentary direction of motion, relative to an inertial system in which the reference frame is momentarily at rest. For this purpose, we consider how the test particle will deviate from a straight line fixed to the inertial system and directed in the momentary direction of motion of the test particle. For the small relative velocities between the inertial system and the reference frame that we will consider here, we need not differentiate between the length and time scales of the two systems. Consider a short time step δt after the particle has passed the origin. To lowest order with respect to δt, the perpendicular acceleration relative ˆ /R. From Fig. 14 to the reference frame is given by v 2 n we have to lowest nonzero order in δt
The Fermi approach to circular motion
ˆ v 2 δt2 n R 2 δt2 δx2 = g⊥ 2 δx3 = δx1 − δx2 . δx1 =
In special and general relativity the spin of a gyroscope is represented by a four-vector S µ . The Fermi transport law for S µ is given by DS µ Duα = uµ Sα . Dτ Dτ
Curvature and acceleration
(A1)
(B1a) (B1b) (B1c)
Here g⊥ is the acceleration of the inertial system relative Here uµ is the four-velocity of the gyroscope. As a spe- to the reference frame (we have g = −[a ] ) in the ⊥ ref ⊥ cial relativistic application we consider motion with fixed direction perpendicular to the direction of motion. We speed v along a circle in the xy-plane with an angular know that δx = [a δt2 3 particle ]⊥ 2 to lowest order in δt. If frequency ω. We assume that the spatial part of the spin 9
z
S ! moves along the x-axis of S, and their spatial origins coincide at t = t! = 0. The systems are then related by the standard Lorentz transformation. Two inertial systems S and S ! are related by a boost assuming that the same rotation of S and S ! is required to put the two systems into standard configuration.
δx3
δx1
δx2 Freely falling line y x
g
Figure 14: Deviations from a straight line relative to a reference frame that accelerates in the z-direction. The plane shown is perpendicular to the momentary direction of motion (along the dashed line), and all the three vectors lie in this plane. The solid curving line is the particle trajectory. The thick line is the line that is fixed to the inertial system in question, and is thus falling relative to the reference frame. we substitute this expression for δx3 into Eq. (B1c) and take the infinitesimal limit, it follows that [aparticle ]⊥ = [aref ]⊥ + v 2
ˆ n . R
(B2)
Equation (B2) gives the acceleration aparticle of a test particle, relative to an inertial system in which the reference frame is momentarily at rest, for given path curvature relative to the reference frame and given acceleration aref of the reference frame.
References [1] This angular velocity describes how the gyroscope axis precesses if we imagine that the length contraction along the direction of motion is removed. We will discuss this point further in the text. [2] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, New York, 1973), pp. 175–176. [3] There is a radius where free photons, that is, photons whose motion is determined by nothing but gravity, can move on circles around a black hole. The circumference of this circle is 1.5 times the circumference of the surface of the black hole (the event horizon). [4] Reference 2, pp. 1117–1120. [5] R. D’Inverno, Introducing Einstein’s Relativity (Oxford University Press, Oxford, 1998), p. 129. [6] Two inertial systems S and S ! are said to be in standard configuration if their spatial axes are aligned, 10
[7] Consider an infinitesimal boost in a general direction between the x and y axis of S ! . Because relativistic rotation effects vanish at low speeds (according to Eq. (4) δα # vδu/2), the boost is, to first order in the velocity change, equivalent to first boosting the grid in the x-direction and then in the y-direction. The forward x-boost has no impact on the relative rotations. It follows that, to first order in the velocity change, the y-part of the boost contributes to the rotation as if there were no simultaneous forward x-boost. [8] R. Jonsson, “A covariant formalism of spin precession with respect to a reference congruence,” Class. Quantum Grav. 23, 37–59 (2006). [9] Note that g should still be the acceleration of a dropped object relative to the platform. In fact, as observed relative to the train, the acceleration of a dropped object will be greater than g for nonzero train velocities. [10] M. A. Abramowicz, “Relativity of inwards and outwards: An example,” Month. Not. Roy. Astr. Soc. 256, 710–718 (1992). [11] Wolfgang Rindler and Volker Perlick, “Rotating coordinates as tools for calculating circular geodesics and gyroscopic precession,” Gen. Rel. Grav. 22, 1067–1081 (1990). √ [12] Replace ωr in Eq. (39) of Ref. 11 by v gtt where gtt = 1 − 2M/r. [13] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley & Sons, New York, 1972), pp. 233–238. [14] R. T. Jantzen, P. Carini and D. Bini, “The many faces of gravitoelectromagnetism,” Ann. Phys. (NY) 215, 1–50 (1992).
Addendum to “Gyroscope precession in special and general relativity from basic principles”
−ω term, to account for the mentioned (non-relativistic) rotation, to find Ω=
Rickard M. Jonsson Department of Theoretical Physics, Physics and Engineering Physics, Chalmers University of Technology, and G¨ oteborg University, 412 96 Gothenburg, Sweden
γ ([γagyro + (aref + ω × v)] × v) γ+1 −ω.
(1)
By agyro we here mean the acceleration of the gyroscope relative to the inertial frame S. In the non-rotating case n ˆ + [aref ]⊥ . Now that we have rotawe had [agyro ]⊥ = v 2 R E-mail:
[email protected] n ˆ tion this is modified to [agyro ]⊥ = v 2 R + [aref ]⊥ + 2ω × v. This is a simple Coriolis effect that is easy to derive analAbstract. I extend the reasoning of Rickard Jonsson, ogous to the proof in Appendix B of Ref. 1. Using this in Am. Journ. Phys 75 463, (2007) to include also rotating Eq. (1) yields reference frames. γ !" 2 n ˆ Ω= γv + aref (γ + 1) γ+1 R # $ 1 Rotating reference frame +(2γ + 1)ω × v × v − ω. (2) Suppose that the reference frame is accelerating with acceleration aref and rotating (rigidly) with angular velocity So here is the precession rate of the gyroscope grid (in the ω, relative to an inertial frame S momentarily comoving stopped sense) relative to an accelerating and rotating with a certain point of the reference frame at t = 0. At reference frame. Comparing Eq. (1) and Eq. (2) with the this time the gyroscope (grid) is assumed to pass the more formally derived results of Ref. 2 (Eqs. 50 and 51), point in question. The scenario is illustrated in Fig. 1. we find a perfect match. To apply this formalism to for instance rotating black S holes, we must take the spatial geometry into account a B analogous to the case for static black holes, see Ref. 2. ω
References [1] R. Jonsson, “Gyroscope precession in special and general relativity from basic principles,” Am. Journ. Phys. 75, 463 (2007) [2] R. Jonsson, “A covariant formalism of spin precession with respect to a reference congruence,” Class. Quantum Grav. 23, 37-59 (2006).
Figure 1: A gyroscope grid (thick lines) moving relative to a rotating and accelerating reference frame (thin lines). Notice that due to the rotation, the part of the reference frame situated at the gyroscope grid at t = δt has an extra velocity apart from that coming from the acceleration of the reference frame. This Coriolis related velocity will give an extra contribution to the relative precession, analogous to the already discussed effect of pure accelerations of the reference frame, apart from the obvious non-relativistic effects of the rotation. Let us denote the part of the reference frame situated at the gyroscope grid at t = δt by B. Relative to S, B has a small velocity already at t = 0. At t = δt, it has to lowest order the velocity aref δt + ω × vδt. To lowest order in δt we may consider B to have been at rest in S at t = 0, but rotated by an angle ωδt around ω ˆ . It is then accelerated to a velocity aref δt + ω × vδt, hence getting an extra turn from the Thomas precession as seen from a system comoving with the gyroscope grid at t = δt. With this insight one can modify Eq. 12 of Ref. 1, to include the rotation. We just replace aref by aref +ω×v and add a 1
Generalizing Optical Geometry Rickard Jonsson1 and Hans Westman2 1 Department
of Theoretical Physics, Chalmers University of Technology, 41296 G¨ oteborg,Sweden. E-mail:
[email protected] 2 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada. E-mail:
[email protected] Submitted 2004-12-10, Published 2005-12-08 Journal Reference: Class. Quantum Grav. 23 61 Abstract. We show that by employing the standard projected curvature as a measure of spatial curvature, we can make a certain generalization of optical geometry (Abramowicz and Lasota 1997 Class. Quantum Grav. 14 A23). This generalization applies to any spacetime that admits a hypersurface orthogonal shearfree congruence of worldlines. This is a somewhat larger class of spacetimes than the conformally static spacetimes assumed in standard optical geometry. In the generalized optical geometry, which in the generic case is time dependent, photons move with unit speed along spatial geodesics and the sideways force experienced by a particle following a spatially straight line is independent of the velocity. Also gyroscopes moving along spatial geodesics do not precess (relative to the forward direction). Gyroscopes that follow a curved spatial trajectory precess according to a very simple law of three-rotation. We also present an inertial force formalism in coordinate representation for this generalization Furthermore, we show that by employing a new sense of spatial curvature (Jonsson 2006 Class. Quantum Grav. 23 1) closely connected to Fermat’s principle, we can make a more extensive generalization of optical geometry that applies to arbitrary spacetimes. In general this optical geometry will be time dependent, but still geodesic photons move with unit speed and follow lines that are spatially straight in the new sense. Also, the sideways experienced (comoving) force on a test particle following a line that is straight in the new sense will be independent of the velocity. PACS numbers: 04.20.-q, 95.30.Sf
1. Introduction General Relativity is a theory about curved spacetime. Nevertheless we can often gain insight by splitting spacetime into space and time [1]. To do this we may introduce a foliation of spacetime into spatial hypersurfaces, henceforth referred to as time slices, and study the geometry on these slices. In general this spatial geometry will be time dependent, where time is a parameter that we associate with the different slices.
2
Generalizing Optical Geometry
For the particular case of conformally static spacetimes, Abramowicz et. al. [2]-[7], have demonstrated that it can be fruitful to study a certain rescaled version of the spatial geometry, known as the optical geometry. This rescaled geometry (which is static) has several features that in general the non-rescaled spatial geometry lacks: • A photon moves with unit speed (with respect to the preferred time coordinate). • A photon corresponding to a spacetime null geodesic follows a spatial geodesic.
• An observer following a spatial geodesic will experience an acceleration (force), perpendicular to the direction of motion, that is independent of the velocity. • A gyroscope following a spatial geodesic will not precess relative to the forward direction of motion.
The optical geometry allows us to explain, in a pictorial manner, several interesting features of black holes [1]. For example a gyroscope orbiting close to a black hole will precess (relative to the forward direction of motion) in the opposite direction from that of a gyroscope in orbit far away from the hole. Another feature is that a rocket orbiting the black hole on a circle near the horizon will require a higher outwards directed rocket thrust the faster it orbits the hole, contrary to the situation far from the hole. The optical geometry, with the above features, has however only been successfully constructed in conformally static spacetimes. The question then arises if one can generalize it to incorporate a larger class of spacetimes. We will show that this is indeed possible. This article rests in part on the results of papers [8] and [9]. Where appropriate we will briefly review the necessary formalism of those papers. 1.1. The basic notation In a general spacetime, consider a reference congruence of timelike worldlines of fourvelocity η µ . We can split the four-velocity v µ of a test particle into a part parallel to η µ and a part orthogonal to η µ v µ = γ(η µ + vtµ ).
(1)
Here v is the speed of the test particle relative to the congruence and γ is the corresponding γ-factor. The vector tµ is a normalized spatial vector (orthogonal to η µ ), pointing in the (spatial) direction of motion. We will also use the kinematical invariants of the congruence defined for a timelike vector field η µ as [10]3 aµ = η α ∇α ηµ
θ 3
= ∇α η
α
(2) (3)
As concerns ωµν and σµν , the projection operators in the definition given in [10] enters in a slightly different way than as presented here. For normalized vector fields η µ the definitions are equivalent, but the form presented here is more useful if the vector field in question is not normalized. Note also that the sign on ωµν is a matter of convention.
Generalizing Optical Geometry
3
1 1 σµν = P ρ ν P κ µ (∇ρ ηκ + ∇κ ηρ ) − θPµν (4) 2 3 1 (5) ωµν = P ρ ν P κ µ (∇ρ ηκ − ∇κ ηρ ) . 2 In order of appearance these objects denote the acceleration vector, the expansion scalar, the shear tensor and the rotation tensor. We will also employ what we may denote the expansion-shear tensor 1 θµν = P ρν P κ µ (∇ρ ηκ + ∇κ ηρ ) . (6) 2 Throughout the article we will use c = 1 and adopt the spatial sign convention (−, +, +, +). The projection operator4 along the congruence then takes the form P α β ≡ δ α β + η α ηβ . Vectors that are orthogonal to η µ , will be referred to as spatial vectors. We will also find it convenient to introduce the suffix ⊥. When applied to a four-vector, like [K µ ]⊥ , it selects the part within the brackets that is perpendicular to both η µ and tµ5 . 2. Generalizing the optical geometry A key feature of optical geometry is that we have a space that accounts for the motion of geodesic photons. To define a space, given a spacetime, we specify a congruence of timelike worldlines (generated by a normalized vector field η µ ). Every worldline corresponds to a single point in the space. To completely account for the behavior of photons we need also to introduce a (global) time in which the position of photons in the spatial geometry is evolved. This is done by introducing time slices. These time slices can be defined by a single function t(xµ ) (where the slices are defined by t = const)6 . It is easy to realize that if we want the velocity7 of photons to be independent of direction we must have the time slices orthogonal to the congruence. This means that, given t(xµ ), the local direction of the congruence is uniquely determined by ηµ = −eΦ ∇µ t.
(7)
The function Φ can be determined by demanding η µ ηµ = −1. Then we can (in principle) integrate (7) to find the congruence lines uniquely. Photon geodesics are invariant under conformal rescalings of the metric. Thus without affecting the spacetime properties with respect to geodesic photons, we can Applying this operator to a vector extracts the part of the vector that is orthogonal to η µ . 5 Hence [K µ ]⊥ = P µ α (K α − K β tβ tα ). 6 We can introduce spacelike time slices within a finite region around any point in an arbitrary spacetime. In globally hyperbolic spacetimes such slices can be globally defined. 7 We have not introduced a three-metric yet so we cannot really talk about velocities. Still given two nearby congruence lines A and B, we can compare the coordinate time (t) it takes a photon to move from A to B and vice versa. If the time slice is not orthogonal to the congruence then tAB will not in general equal tBA for infinitesimally displaced congruence lines. Thus the velocity would be dependent on the spatial direction of motion. 4
Generalizing Optical Geometry
4
rescale the metric around every spacetime point with a factor e−2Φ . Letting a tilde denote objects related to the rescaled spacetime, we have g˜µν = e−2Φ gµν . The conformal rescaling effectively removes time dilation (lapse) so that dt = d˜ τ0 , where d˜ τ0 is the proper time along the congruence in the rescaled spacetime. Relative to the rescaled s space, photons move with unit speed d˜ = 1. Thus the first point in the list of features in dt the introduction (photons move with constant speed with respect to coordinate time), we can always achieve. What about the second point? Under what conditions will photons follow straight spatial lines, and indeed what do we mean by following a straight line if the spatial geometry is time dependent? 3. Generalizing the optical geometry using the projected curvature As regards what is spatially straight, most likely the first thing that comes to mind (at least it was for the authors of this article), is to consider a projection of the null trajectory in question down along the congruence to the local slice8 . If the spatial curvature of the projected curve vanishes – then we say that the trajectory is straight. In [8] a general formalism of inertial forces in terms of the projected curvature is derived using an arbitrary congruence of timelike worldlines. The (projected) four-acceleration of the test particle in question can be decomposed as (see section 1.1 concerning notation) µ dv µ 1 µ Dv α µ α µ µ α ρ 2n P α = a + 2v [t ∇α η ]⊥ + vt t t ∇ρ ηα + γ t + v . (8) γ2 Dτ dτ R Here R is the projected spatial curvature radius and nµ is a normalized spatial vector (orthogonal to both tµ and η µ ), pointing in the direction of projected spatial curvature (the principal normal). The left hand side of (8) can be expressed in terms of the forces acting on the test particle. We have [8] 1 1 Dv α (9) γFk tµ + F⊥ mµ . = P µαf α = P µα Dτ m m Here mµ is a normalized spatial vector orthogonal to both tµ and η µ . F⊥ and Fk are respectively the forces perpendicular and parallel to the direction of motion, as experienced in a system comoving with the test particle in question. The right hand side of (8) can be expressed in terms of the kinematical invariants of the congruence, through the identity [10] ∇ν ηµ = ωµν + θµν − aµ ην . Then (8) takes the form [8] h i 1 µ β µ µ µ µ = a + 2v t (ω + θ ) γF t + F m + vtα tβ θαβ tµ (10) β β ⊥ k ⊥ mγ 2 µ dv µ 2n +γ t +v . dτ R On the right hand side we have first three terms that enter as inertial forces (if we multiply them by −m), and the last two terms describe the motion (acceleration) relative to the reference congruence. 8
If the congruence has no rotation there exists a finite sized slicing orthogonal to the congruence. If the congruence is rotating we can still introduce a slicing that is orthogonal at the point in question. It is easy to realize that whatever such locally orthogonal slicing we choose, the projected curvature and curvature directions will be the same, and are thus well defined.
5
Generalizing Optical Geometry
Following [8] one can form a corresponding rescaled version of (8) by putting a tilde on everything in (8). Next one finds the general relation between rescaled and non-rescaled four-acceleration, the result is given by given by (A.2). Setting a ˜µ = 0 and ω ˜ µν = 0 as is appropriate for the congruence and rescaling at hand, also using t˜µ = eΦ tµ and m ˜ µ = eΦ mµ and using (9) one readily gets [8] !
1 1 Φ Fk ˜µ ˜ ρ Φ + v (˜ ˜ ρ Φ + t˜α t˜β θ˜αβ )t˜µ e η ρ∇ (11) t + F⊥ m ˜ µ = 2 P˜ µρ ∇ 2 mγ γ γ γ2 h i dv ˜µ n ˜µ + 2v t˜β θ˜µ β + t + v2 . ˜ ⊥ d˜ τ0 R Recall that a tilde implies that the object is related to the rescaled spacetime. As concerns γ and v we have however omitted the tilde since these are the same as their non-rescaled analogues. Note also that Fk and F⊥ are the real (non-rescaled) comoving forces. For a discussion of how to interpret this expression in terms of inertial forces we refer to [8], and the discussion in section 6.1 of this paper. One sees from (11) (set the left hand side to zero and v = 1) that the projected optical curvature of a geodesic photon vanishes, for all spatial directions, if and only if [t˜β θ˜µ β ]⊥ = 0 for all t˜µ . We also readily see that the sideways (perpendicular) experienced force9 F⊥ is independent of the velocity when following a trajectory whose projected curvature vanishes, if and only if [t˜β θ˜µ β ]⊥ = 010 . As discussed in [8], and reviewed in Appendix B, this holds for all directions t˜µ if and only if the congruence is shearfree11 . Note that for the standard static optical geometry, we have θ˜µν = 0, thus the congruence is trivially shearfree. Incidentally, for this case (11) can be written as eΦ m
Fk µ ˜µ ˜ ρ Φ + v(˜ ˜ ρ Φ)t˜µ + γ 2 dv t˜µ + γ 2 v 2 n . (12) ηρ∇ t˜ + F⊥ m ˜ µ = P˜ µρ ∇ ˜ γ d˜ τ0 R !
We conclude that for shearfree congruences we can always manage the first three points of the list of optical geometry features given in the introduction. Now what what about the fourth point, concerning gyroscope precession? 3.1. Gyroscope precession The spin vector S µ of an ideal gyroscope transported without any torque acting on it in a comoving system, along a trajectory of four-velocity v µ , obeys the Fermi-Walker equation DS µ Dv α = v µ Sα . (13) Dτ Dτ 9
Notice that the force that we are referring to here is the force as received in a system comoving with the test-particle. If we on the other hand consider the reference congruence observers to be providing the sideways force, this force is in fact smaller than the received force by a γ-factor and is hence not independent of the velocity, see section 7. q 10 ˜ β Φ]⊥ . ˜ ρ Φ]⊥ [P˜ νβ ∇ For this case the sideways force is given by F⊥ = me−Φ g˜µν [P˜ µρ ∇ The shear tensor in the rescaled spacetime is given by σ ˜µν = e−Φ σµν , so the shear tensor in the rescaled spacetime vanishes if and only if it does so in the non-rescaled spacetime. 11
6
Generalizing Optical Geometry
Introducing S˜µ = eΦ S µ and v˜µ = eΦ v µ , using the orthogonality of S µ and v µ , it is a quick exercise (carried out in Appendix A) to show that this implies ˜ vα ˜ S˜µ D˜ D = v˜µ S˜α . (14) ˜τ ˜τ D˜ D˜ Thus a spin vector which is Fermi-Walker transported relative to the non-rescaled spacetime is Fermi-Walker transported also relative to the rescaled spacetime if we just rescale the spin vector itself. In particular, a gyroscope initially pointing in the forward direction tµ precesses relative to the forward direction in the standard spacetime if and only if it does so in the rescaled spacetime. In [9], a general formalism of gyroscope precession with respect to a reference congruence is discussed. Here one considers the spin vector that one would get if one were to momentarily stop (by a pure boost) the gyroscope with respect to the reference congruence. This spin vector is called the stopped spin vector, denoted by S¯µ and related to S µ through "
S¯µ = δ µ α + η µ ηα +
!
#
1 − 1 tµ tα S α . γ
(15)
While S¯µ is orthogonal to the congruence it is not simply the projected part of the standard spin vector (in general both the norm and the spatial direction of these two objects differ). The reason for using S¯µ rather than S µ is that S¯µ (unlike the projected part of S µ ) obeys a simple law of (three-dimensional) rotation D γv ¯α µ Dηα D S¯µ S t ∧ = (vα + ηα ) + η µ S¯α . (16) Dτ γ+1 Dτ Dτ ⊥ Here the last term insures that orthogonality to the congruence is preserved. Note also D means covariant differentiation along the gyroscope worldline. The contraction that Dτ with the wedge product (defined as k µ ∧ bα ≡ k µ bα − k µ aα ) corresponds to the three-rotation12 . Furthermore one considers a spacetime analogue of standard spatial transport Dηα Dk µ = γk α ω µ α + γk α (θµ β tβ ∧ tα ) + η µ k α . (17) Dτ Dτ The transport as defined here is norm preserving, and also preserves angles between transported vectors13 . Considering a spatially straight line (1/R = 0), and a vector momentarily aligned with the forward direction, the transport is defined such that the vector remains in the forward direction (analogous to standard spatial transport). Given (16) and (17) one can form an equation for how fast the stopped spin vector deviates from a corresponding (spatially) parallel transported vector. For the case of a ¯ tµ = (0, ˆ Choose inertial coordinates adapted to the the congruence so that S¯µ = (0, S), t) and ¯ n ˆ n ˆ ¯ ˆ dS ¯ ˆ n = (0, n ˆ). Then (16) amounts to dτ = γv(γ − 1) t(S · R ) − R (S · t) . The expression within the brackets is a vector tripleproduct and we may write it as a double cross product. Letting v = vˆ t we ¯ n ˆ ¯ which is a simple equation of three-rotation. × v × S, get ddτS = γ(γ − 1) R 13 The general idea of the transport equation is most readily understood considering a rigid (nonshearing and non-expanding) reference congruence. When the reference frame rotates – so does the parallel transported vector. 12
µ
7
Generalizing Optical Geometry
congruence with vanishing rotation (as is appropriate for the hypersurface orthogonal congruence we are here considering) one finds [9] " Dps S¯µ α = S¯ γ 2 v(tµ ∧ aα ) + (2γ 2 − 1)(tµ ∧ tβ θαβ ) + Dps τ # n α + γv(γ − 1) tµ ∧ . (18) R Here the suffix ’ps’ is short for ’projected straight’14 . What (18) tells us is how the stopped spin vector precesses relative to a frame that is spatially parallel transported with respect to the reference congruence. We can of course also consider the analogue of (18) for the rescaled spacetime. There the congruence acceleration vanishes and we are left with ˜¯µ ˜ ps S ˜α D ˜¯α (2γ 2 − 1)(t˜µ ∧ t˜β θ˜ ) + γv(γ − 1) t˜µ ∧ n . (19) =S αβ ˜ ps τ˜ ˜ D R In particular, demanding that the gyroscope should remain pointing in the forward direction tµ as it follows a spatially straight line in the rescaled spacetime, the left hand side of (19) must vanish and we get 0 = t˜α (t˜µ ∧ t˜β θ˜αβ ).
(20)
This equation can be simplified to 0 = [t˜β θ˜µ β ]⊥ .
(21)
As mentioned earlier, (21) holds for all directions t˜µ if and only if the congruence is shearfree. So a gyroscope (initially directed in the forward direction) will not precess relative to the forward direction, when transported along a spacetime trajectory whose spatial projection has vanishing curvature relative to the rescaled spacetime, if and only if the congruence is shearfree. When the congruence is shearfree, (19) is simplified to ˜¯µ ˜ ps S α D n ˜α µ ˜ ¯ ˜ . = S γv(γ − 1) t ∧ ˜ ps τ˜ ˜ D R
(22)
Comparing with (18) (setting aµ = 0 and θµν = 0 corresponding to an inertial congruence), we see that if we consider the gyroscope precession with respect to the rescaled spacetime, there is only standard Thomas-precession (see e.g. [9, 16]). Note that while (22) is a four-vector relation, it is effectively a three-dimensional ˜ equation since all the terms are orthogonal to η µ . Letting (0, v ˜) = v t˜µ , (0, n ˆ) = n ˜ µ and µ ˜ ˜¯ in coordinates locally comoving with the the reference congruence we can ¯ =S (0, S) express (22) in manifest three-form as (see footnote on previous page for details) ˜ ˜ ¯ n ˆ DS ˜ ¯ = (γ − 1) ×v ˜ ×S ˜ Dt R !
14
(23)
In the coming section we will consider a different derivative connected to a different notion of straightness, that will get the suffix ’ns’.
8
Generalizing Optical Geometry
So here we see how the (rescaled and stopped) spin vector precesses relative to a corresponding frame that is parallel transported with respect to the optical geometry. Note the absence of explicit factors eΦ and how very simple this law of precession is. Considering gyroscope precession relative to some curved (rescaled) spatial geometry we must take into account that a parallel transported frame will in general be rotated relative to its initial configuration if we transport it along some closed spatial trajectory. For motion in the equatorial plane of some axisymmetric static optical geometry this can easily be dealt with by introducing a reference frame that ˜ˆ in the same manner as a parallel rotates relative to a local frame spanned by ˜ ˆ r and ϕ, transported reference frame does on a plane. Such a reference frame always returns to its initial configuration after a full orbit. In [9] the effective rotation relative to this new frame of reference is derived. If the line element can be written on the form d˜ s2 = g˜r˜r˜d˜ r 2 + r˜2 dϕ2 the effective rotation vector can be written as ! ! ˜ ±1 1 n ˆ ˜ effective = (γ − 1) √ −1 v ˜ ט ˆ r. ×v ˜ + Ω ˜ r˜ g˜r˜r˜ R
(24)
(25)
We have here included a ± sign. If ˜ ˆ r, which is assumed to be pointing away from the center of symmetry, points in the direction of increasing r˜ we have the positive sign, otherwise we should use the negative sign15 . Note that (25), for the particular case of motion in an axisymmetric spatial geometry, gives the precession relative to the ’would-be-flat’ reference frame in terms of the parameter time t16 . 3.2. Conclusion as regards the standard projected curvature We have seen that the optical geometry, as presented here, retains all of the features listed in the introduction given that the shear-tensor of the congruence in question vanishes. This corresponds to a larger set of spacetimes than the conformally static spacetimes, see section 8. We have also seen that gyroscopes precess according to a very simple law of rotation with respect to the optical geometry. 4. Generalizing the optical geometry using a different curvature measure While vanishing projected curvature is likely to be the first notion of spatial straightness that comes to mind, it is perhaps not the most natural for all cases. In [8], a novel definition is proposed via a variational principle. Let ds be the distance traveled for a test particle as seen from the local congruence observers (ds = γvdτ ). Parameterizing 15
For the standard optical geometry of a black hole there is a neck (minimum value of r˜) at the photon radius. For this geometry we should use the positive sign outside of the photon radius and the negative sign inside of this radius. Note that √g˜1 = 0 at the photon radius, so there is no discontinuity r ˜r ˜ ˜ effective . in Ω 16 This time equals the local time of the congruence observers in the rescaled spacetime. In the case of standard optical geometry for a Schwarzschild black hole, it is simply the Schwarzschild time.
Generalizing Optical Geometry
9
the trajectory by λ, the integrated distance δs along a trajectory connecting two fixed spacetime points can be written as δs =
Z
ds
(26)
s
dxµ dxν dλ. (27) dλ dλ One can show [8] that in order for this action to be stationary (minimized) with respect to variations of the trajectory that are perpendicular to η µ and tµ , the projected curvature must obey nµ v = −2[tα θαµ ]⊥ . (28) R Trajectories that are obeying this relationship are thus minimizing17 the spatial distance traveled and are said to be straight in the new sense or simply new-straight (for want of a better name). Thus the two notions of straightness differ (in general) if and only if there is shear. Notice that the curvature relation of (28) is velocity dependent. For more details and some intuition of why the two notions of straightness differ see [8]. One may introduce a new curvature measure from how fast a trajectory deviates from a corresponding (same v and tµ ) new-straight trajectory. Denoting the µ ¯ respectively, the corresponding curvature direction and curvature radius by n ¯ and R inertial force formalism [8], in an optically rescaled spacetime (analogous to the outline in the preceeding section) takes the form ! v ˜α ˜β ˜ 1 ˜ µρ ˜ 1 Φ Fk ˜µ µ ˜ ρ Φ)t˜µ e P ∇ Φ + (t t θαβ + η˜ρ ∇ (29) = t + F m ˜ ρ ⊥ 2 2 2 mγ γ γ γ ˜¯ µ dv ˜µ 2n + . t +v ˜¯ d˜ τ0 R It follows immediately that a geodesic photon (set the left hand side to 0 and v = 1) ˜¯ = 0) in the new sense relative to the rescaled spacetime. Also, has zero curvature (1/R the sideways force on a massive particle following a straight line is independent of the velocity. Notice that this holds independent of whether the congruence is shearing or not. So in fact, with the new sense of curvature, for any spacetime and any spacelike foliation (and corresponding rescaling), a geodesic photon follows a spatially straight line, and the sideways force on an object following a spatially straight line is independent of the velocity. =
Z
Pµν
4.1. Gyroscope precession In [9], a spin precession formalism connected to the new-straight curvature measure is presented, analogous to (18) above for the projected curvature measure. Relative to the rescaled spacetime (set aµ = 0, ω µν = 0 and put tilde on everything) we have from [9] !# " ˜¯µ ˜ ns S ˜¯ α α n D µ β µ ˜ . (30) = S¯ −(t˜ ∧ t˜ θ˜αβ ) + γv(γ − 1) t˜ ∧ ˜ ns τ˜ ˜¯ D R 17
Strictly speaking, the distance traveled with respect to the congruence observers is stationary with respect to variations perpendicular to η µ and tµ if the curvature obeys (28).
10
Generalizing Optical Geometry
Here ’ns’ stands for new-straight18 . While the term containing the expansion-shear tensor is simpler in this equation, compared to (18), it is not vanishing. Thus, analogous to the discussion in section 3.1, a gyroscope initially directed in the forward direction will remain in the forward direction, as we move along an arbitrary line that is straight in the new sense relative to the rescaled spacetime, if and only if the shear-tensor of the congruence vanishes. 4.2. Fermat’s principle and the new-straight curvature As discussed in [8], the new-straight curvature relative to the rescaled spacetime is closely related to Fermat’s principle. Indeed for a photon in the rescaled spacetime, the coordinate time it takes for a photon to go from a certain event along one spatial point (congruence line) to another spatial point (congruence line) is given by δt =
Z
dt =
Z
d˜ s.
(31)
Fermat’s principle states that a null trajectory is a geodesic if and only if it extremizes R δt19 . Also, by definition d˜ s is extremized20 if and only if the curvature in the new sense (with respect to the rescaled spacetime) vanishes , which according to (31) means that δt is extremized. It follows that any null geodesic has vanishing spatial curvature in the new sense relative to the rescaled spacetime. The connection between Fermat’s principle and straight lines in the optical geometry was realized, for conformally static spacetimes, a long time ago. With the new definition of curvature the connection holds in any spacetime. 4.3. Conclusion regarding the new sense of curvature in relation to optical geometry We have seen that with the new sense of curvature, photons move along spatial geodesics for any slicing and corresponding rescaling in any spacetime. Also the sideways force on a particle following a spatially straight line will be independent of the spatial velocity. Gyroscopes following straight lines will however precess (in general) relative to the forward direction. 5. Some comments on rescalings and other transformations We have in this article used conformal rescalings more or less without motivation. In principle one can imagine other transformations. In particular we can consider a transformation of just the spatial geometry, and do this in such a manner that no α
˜¯ . ˜¯ have nothing to do with the bar in S ˜ Note that the bar in n ¯ α and the bar in R 19 We here naturally refer to the part of the null trajectory that connects the two congruence lines in question. By extremizes we mean that we may have a minimum or a saddlepoint in δt with respect to (null-preserving) variations around the trajectory in question. R 20 Strictly speaking d˜ s is extremized with respect to variations perpendicular to the trajectory. For the case of null-preserving variations, perpendicular variations are however (to the necessary order) the only allowed variation. 18
Generalizing Optical Geometry
11
matter what shear or rotation the reference congruence has, all the projected geodesic photons get vanishing spatial curvature. But in fact, this is not generally doable. To see this, consider the projection of a left-moving and a right-moving geodesic photon when the congruence rotates (think in 2+1 inertial coordinates). The projected curvature direction (relative to the standard on-slice geometry) will be opposite for the two directions as illustrated in figure 1. Right-moving photon Spatial geodesic Left-moving photon Figure 1. Photons moving in opposite spatial directions will in general have different projected curvatures when the congruence is shearing or rotating. This means that the motion of photons cannot generally correspond to projected geodesics of any threegeometry for a rotating or shearing congruence.
The two projected trajectories are aligned at the origin, but are then separating. A geodesic aligned with the two trajectories can however not split in two. Hence there simply exists no three-geometry relative to which the projections of general geodesic photons will correspond to spatial geodesics if the congruence is rotating. The same argument applies when the congruence is shearing. So whatever transformation of the spatial metric we are considering, using the standard projected curvature, the optical congruence would have to be rotationfree and shearfree for geodesic photons to follow spatial geodesics. But why conformal rescalings? Well, as we argued before, we need a congruence and a corresponding orthogonal slicing to get an isotropic speed of light (with respect to coordinate time), no matter what spatial metric we come up with. Also, for the chosen labeling of the time-slices, if we want the speed of light (with respect to the s = 1), where d˜ s is the transformed transformed) spatial metric to be unit everywhere ( d˜ dt spatial distance (no matter what transformation we are considering), then a rescaling of spatial distances by a factor e−Φ is in fact the only option. That geodesic photons follow projected straight lines relative to the rescaled space, we may see as a pure bonus when there is no shear. Notice however that even if we would relax the unit speed requirement, we would still need a shearfree congruence to get photons to follow straight spatial lines in the projected sense. In summary, as concerns the unit speed requirement, the rescaling is the only option. Concerning the requirement that the projection of a null geodesic should correspond to a spatially straight line, no other transformation would make a better job21 . Also, using the new-straight curvature, we get all of the features concerning photons that we want using the rescaling scheme. 21
In other words no other transformation would be applicable to a larger set of spacetimes (and congruences) than those already considered for the generalized (in the projected sense) optical geometry.
12
Generalizing Optical Geometry
Notice that (as we have argued above) the fact that photon geodesics are unaffected by the conformal rescaling, is not of any direct importance. Considering the projected curvature, we can in principle imagine some other spacetime transformation that does affect null geodesics, but that gives a vanishing projected curvature (in the transformed spacetime) for a photon that is a geodesic in the original spacetime. However, for conformally static spacetimes, assuming that we rescale away time dilation (lapse), the fact that photon geodesics are unaffected by the rescaling trivially means that the projected spatial curvature of a geodesic photon must vanish22 . Notice also that whatever slicing we choose in a general spacetime, we get a corresponding rescaled spacetime with a line element of the form BlockDiag[−1, gij (t, x)], where the motion of free photons corresponds to null geodesics. In the rescaled spacetime the only degrees of freedom are those of the spatial metric. So from this point of view, we can always have a space that accounts for the behavior of photons. Indeed the formalism of the new-straight curvature can be seen as a certain way of describing how that space dictates the motion of geodesic photons. 6. The generalized optical geometry in coordinates, assuming vanishing shear In order to get a better feeling for the properties of spacetimes that admits a hypersurface-forming shearfree vector field, it is instructive to carry out an analysis in coordinates adapted to the chosen time slices and the corresponding orthogonal congruence. In such coordinates, the rescaled metric takes the following form g˜µν =
"
−1 0 0 hij
#
.
(32)
In a spacetime of the form (32) the only non-zero elements of the affine connection are ˜ 0 = 1 ∂t hij (33) Γ ij 2 ˜ k0i ˜ ki0 = 1 hkl ∂t hli = Γ (34) Γ 2 ˜ k = 1 hkl (∂i hlj + ∂j hil − ∂l hij ) ≡ γ˜ k . Γ (35) ij ij 2 Using these relations it is easy to evaluate the shear-expansion tensor in the coordinates in question 1 ˜ ˜ j η˜i ) = ... ˜j − ∇ θ˜ij = (∇ iη 2 1 = ∂t hij . (36) 2 Here Latin indices run from 1 to 3. The other components of θ˜αβ are zero. As mentioned before, the shear tensor vanishes if and only if [θ˜µ α t˜α ]⊥ = 0 for all t˜µ . Hence, using (36) 22
Thinking of geodesics as maximizing the proper time (null geodesics being a limiting case), it is obvious that in the absence of time dilation (lapse), shear and rotation, a geodesic must locally take the shortest spatial path, thus having vanishing projected curvature.
13
Generalizing Optical Geometry
(raising the first free index to get θ˜i j = 12 hik ∂t hkj ), necessary and sufficient conditions for the congruence shear to vanish (in the coordinates in question) is hik ∂t hkj ∝ δ i j .
(37)
∂t hlj ∝ hlj .
(38)
Multiplying both sides by hli yields What this means is that when moving in time only, all the components of hij must increase with the same factor, for every fixed x. The most general form for hij is then ¯ ij (x). hij = e2Ω(t,x) h (39) ¯ ij (x) is independent of Here Ω is an arbitrary well behaved function of x and t, and h the time coordinate. Using this in (32) yields
g˜µν =
0
0
−1
2Ω(t,x) ¯
e
hij (x)
.
(40)
So, in coordinates adapted to the time slices and corresponding congruence, the line element takes the above form if and only if the congruence is shearfree. The conformally static spacetimes is a subset (set Ω = 0) of the spacetimes described by conformal rescalings of (40). Incidentally one may use the above coordinate formalism to verify the necessary and sufficient condition of vanishing shear in order for (null) geodesics in the rescaled spacetime to correspond to projected straight lines23 . 6.1. The inertial force formalism in coordinates ¯ ij (x) in (36) we find Inserting hij = e2Ω(t,x) h θ˜ij = (∂t Ω)hij .
(41)
Using this relation in (29) we readily find the optical inertial force formalism in coordinates adapted to the congruence, assuming vanishing shear ! 2 eΦ Fk ˜k kl k 2 dv ˜k k 2v ˜ n ˜ k . (42) = h ∇l Φ + v t ∂t (Φ + Ω) + γ t + F⊥ m ˜ t +γ ˜ m γ dt R Setting ∂t Ω = 0, yields the inertial force equation in the standard optical geometry for conformally static spacetimes. Also setting ∂t Φ = 0 yields the inertial force equation in standard optical geometry for static spacetimes. Multiplying (42) by m and and shifting the first two terms on the right hand side to the left, we may identify two different types of inertial forces as Gravity
: − mhkl ∇l Φ
Expansion : − m(∂t Φ + ∂t Ω)v k . 23
The three spatial equations for a geodesic in the rescaled spacetime, in the coordinates in question, 2 k k dxi dxj ˜ k dt dxi . Since γ˜ k is the affine connection on the slice, it is takes the form of dd˜τx2 + γ˜ij ij d˜ τ d˜ τ = −2Γi0 d˜ τ d˜ τ ˜ k ∝ δ k i , in order for the projected trajectory to correspond to a spatial obvious that we must have Γ i0 geodesic. According to the above discussion, this holds if and only if the congruence shear vanishes.
Generalizing Optical Geometry
14
The term ’Gravity’ is introduced by analogy to the Newtonian sense of gravity. From the point of view of general relativity, this term would simply be called acceleration (referring to the congruence acceleration). Strictly speaking, the term ’Expansion’ refers to expansion in the non-rescaled spacetime24 . For positive ∂t (Φ + Ω) the term has the form of a viscous damping force although for negative ∂t (Φ + Ω) it is rather a velocity proportional driving force. The last two terms of (42) are a representation of the motion (acceleration) relative to the reference congruence, and are not regarded as inertial forces. For further discussion of these types of interpretations, see [8]. Note however that precisely what we call an inertial force is ambiguous up to factors of γ. We could for instance multiply the entire equation, or just the parallel part of the equation, by γ and thus introduce γ-factors in the inertial forces. 7. A note on given and received forces In the formalism thus far presented, we have expressed forces in terms of what is experienced by an observer comoving with the test particle in question. These forces are in general different from the forces needed to be given by the congruence observers, in order to make the test particle move as specified by the curvature, curvature direction and the time derivative of the speed. In [8] the relationship between the given forces Fc⊥ and Fck (where ’c’ stands for congruence) and the received (comoving) forces F⊥ and Fk is derived F⊥ γ = Fk
Fc⊥ =
(43)
Fck
(44)
These relations can be used to get the given forces in any of the different formulations of this article ((11),(29) and (42)). For instance we may consider a railway track in some static geometry. The perpendicular force exerted by the track (as seen from the rest frame of the track) on the train is given by Fc⊥ = Fγ⊥ . Note that if the track has vanishing optical curvature, this force (unlike the comoving force) has a velocity dependence. Indeed it is easy to find [11] that there is no way to lay out a railway track (no curvature) such that the sideways given force is velocity independent. 8. A note on uniqueness In the scheme of section 4 (employing the novel curvature measure), we can do optical geometry in a finite region around any point in any spacetime. There are always more than one possible such geometry (corresponding to different choices of slices). The optical geometry as generalized in section 3 (using the projected curvature) is however more restrictive and the standard optical geometry is more restrictive still. In 24
It is easy to show, for instance comparing (42) with (11) and using the formulas for how the kinematical invariants are transformed as listed in [8], for the particular case of vanishing shear, that we have ∂t (Φ + Ω) = eΦ 13 θ.
Generalizing Optical Geometry
15
the coming two subsections we comment on the similarities and differences between the standard and the generalized optical geometry of section 3. 8.1. The standard optical geometry Standard optical geometry is defined for conformally static spacetimes. These are defined as spacetimes that admits a timelike hypersurface forming conformal Killing field. Mathematically this amounts to that there must exist a scalar field f and a vector field ξ µ that obeys25 (∇µ ξν + ∇µ ξν ) = f gµν
Pµ ρ Pν σ (∇σ ξρ − ∇ρ ξσ ) = 0
(45) (46)
Here Pµ ρ = δµ ρ + −ξ1σ ξσ ξµ ξ ρ . The first equation is the conformal Killing equation. The second equation corresponds to setting ωµν as defined in (5), to zero26 . For the particular case when f = 0 we have a Killing field rather than a conformal Killing field. Given a field ξ µ that obeys (45) and (46), we define e2Φ = −gαβ ξ α ξ β . After a rescaling g˜µν = e−2Φ gµν we get our optical geometry. Notice however that for any solution (ξ µ , f ), we can form another solution as (αξ µ , αf ) for some constant α. Thus the definition of e2Φ is not unique. For simple cases, like a Schwarzschild black hole, where we have asymptotic flatness, we can however choose α so that ξ µ is normalized at infinity. There can however be a larger freedom still in ξ. For instance in flat spacetime, in inertial coordinates, any normalized timelike constant vector field satisfies the requirements (although the optical geometry is flat for all choices). There are however also other, not quite so trivial, timelike hypersurface forming Killing fields for a flat spacetime. In particular there is a Killing field parallel to the four-velocities of the points of a rigidly accelerating system (a so called Rindler system[12]). The associated optical geometry is here curved. Note that this Killing field is not a global field however. In summary, there may exist no standard optical geometry, and there may exist more than one (non-trivially related) standard optical geometry, depending on the spacetime in question. 8.2. The generalized optical geometry for the projected curvature In the generalized optical geometry of section 3 (using the projected curvature), the fundamental equation (requirement) is not really an equation for a vector field, but for a congruence of worldlines. There has to exist a non-rotating, non-shearing congruence for this type of generalized optical geometry to work. This is however equivalent to the existence of a rotationfree and shearfree vector field ζ µ (not necessarily normalized) that obeys Pµ ρ Pν σ (∇ρ ζσ + ∇σ ζρ ) = f Pµν ρ
σ
Pµ Pν (∇σ ζρ − ∇ρ ζσ ) = 0 25 26
Assuming four dimensions, it follows from (45) that f = 12 ∇α ξ α . Substitute η µ by ξ µ , and modify the form of Pµ ρ as was just shown.
(47) (48)
Generalizing Optical Geometry
16
Here Pµ ρ = δµ ρ + −ζ1σ ζσ ζµ ζ ρ and again f is some scalar function27 . The first equation corresponds to setting σµν as defined in (4) to zero28 . The second equation corresponds to setting ωµν as defined in (5) to zero29 . Expressed in this form we note that the only difference from the standard optical geometry equations (substituting ζ µ with ξ µ ) lies in the projection operators in (47). Indeed (47) is the projected version of (45). This means that the standard optical geometry equations are more restrictive than the latter two equations. Any field that satisfies (45) and (46), will also satisfy (47) and (48), but the converse is not generally true. So, given a field ζ µ that satisfies (47) and (48) the generalized optical geometry exists. We cannot however in general find the rescaling parameter (modulo a single constant) from the norm of ζ µ30 . To find the rescaling we would instead consider the slices orthogonal to the field and assign a continuous parameter t to these slices. We would then make a rescaling so that g˜αβ ∇α t∇β t = −1. We may understand that the freedom in the labeling t of the time slices, gives a freedom of a scale factor (as a function of t) for the whole (spatial) optical geometry. For any standard optical geometry there are thus a multitude of generalized optical geometries, for the same reference congruence. As we will demonstrate in a forthcoming paper [13] one can also have a standard optical geometry connected to a certain reference congruence, and in the same region of spacetime have a generalized optical geometry for a different congruence. More importantly however, there are cases where there exists no standard optical geometry, but where the generalized optical geometry exists. In the companion paper [13] we illustrate that there exists a generalized optical geometry (within a finite region of any spacetime point) for any spherically symmetric spacetime. Indeed there are infinitely many (in-falling) non-shearing reference congruences for this case. In particular we show that there exists a generalized optical geometry across the horizon of a Schwarzschild black hole (unlike in the standard optical geometry). The equations (47) and (48) are here mainly included for comparison with the standard optical geometry equations. In the companion paper [13], we in fact find it more convenient to work in coordinates, starting from (40). 9. Summary and conclusion A generalization of the optical geometry has been proposed prior to this paper [4] using a different philosophy, but see [14] for criticism. The optical geometry as presented in this article can be done at different levels.
In four dimensions it follows that f = 32 P αβ ∇α ζβ . Substitute ζ µ by ξ µ , and modify the form of Pµ ρ as was just shown. Note that when applying (4) to a non-normalized vector field, the appropriate θ (corresponding to 32 f ) is given by θ = P αβ ∇α ζβ . 29 Substitute ζ µ by ξ µ , and modify the form of Pµ ρ as was just shown. 30 In fact (47) and (48) are independent of the norm of ζ µ in the sense that if ζ µ solves (47) and (48) so will hζ µ where h is some arbitrary function. This is contrary to (45) of the preceeding section, where given a field ξ µ that solves (45), a field hξ µ would solve (45) if and only if h was constant. 27
28
Generalizing Optical Geometry
17
• In any spacetime we can produce an optical geometry where photons move with unit speed. Using the new sense of curvature [8], a geodesic photon follows a spatially straight line and the sideways force on a massive particle following a straight line is independent of the velocity. • In any spacetime that admits a hypersurface orthogonal shearfree congruence of worldlines, we can produce an optical geometry where photons move with constant speed. Here the projected curvature and the new-straight curvature coincide and therefore either one can be used to define what is spatially straight. A geodesic photon follows a spatially straight line and the sideways force on a massive particle following a straight line is independent of the velocity. Furthermore, a gyroscope initially pointing in the direction of motion, following a spatially straight line, will not precess (independently of the velocity) relative to the forward direction. If the gyroscope follows a spatially curved line, it obeys a very simple law of threedimensional precession (23). • In a conformally static spacetime (choosing the preferred congruence) we have the same features of the optical geometry as outlined in the preceding point. Here the optical geometry is static. Furthermore, as demonstrated in [15], Maxwell’s equations take a simple form written in terms of the optical metric. In any generalization of a theory, there are in general several possibilities. Indeed if we consider the standard optical geometry, in conformally static spacetimes, we may use either the projected or the new-straight curvature (they are identical here). Generalizing to more complicated spacetimes we however get two different theories (at least algebraically) depending on what curvature measure we are adopting. One may certainly consider other ways of defining an optical geometry. For instance, one might consider relaxing the spatial geodesic requirement of photons and replace it with some preferably simple law (indeed that is what we get if we consider the projected curvature when there is shear, see (11)). Apart from extending the set of spacetimes where one can introduce optical geometry, this article also aims to present a solid inertial force formalism both for the generalized and the standard optical geometry (again see [14] for criticism of previous works). As regards applications of the generalized optical geometry we refer to a companion paper [13] where a non-static (but shearfree) congruence is employed to do optical geometry across the horizon of a static black hole. Appendix A. The Fermi-Walker equation for the gyroscope spin vector in the rescaled spacetime Consider a rescaled spacetime g˜µν = e−2Φ gµν . Let k µ be a general vector defined along a worldline of four-velocity v µ . Introducing v˜µ = eΦ v µ and k˜µ = eΦ k µ one may show [8] that the relation between the rescaled and the non-rescaled covariant derivative of the
Generalizing Optical Geometry vector k µ is given by ! µ ˜ k˜µ D 2Φ Dk µ ρ α µρ =e − (v k − v kα g )∇ρ Φ . ˜τ Dτ D˜ In particular, for k µ = v µ , we get the transformation of the four-acceleration ˜ vµ Dv µ D˜ = e2Φ − (v µ v ρ + g µρ )∇ρ Φ . ˜τ Dτ D˜
18
(A.1)
(A.2)
The Fermi-Walker equation for a gyroscope of spin vector S µ is given by Dv α DS µ = v µ Sα . (A.3) Dτ Dτ Using (A.1), substituting k µ → S µ , and (A.2) together with the Sα v α = 0 we readily find ˜ S˜µ ˜ vα D D˜ = v˜µ S˜α . (A.4) ˜τ ˜τ D˜ D˜ Thus a spin vector that is Fermi-Walker transported relative to the non-rescaled spacetime is Fermi-Walker transported also relative to the rescaled spacetime if we just rescale the spin vector itself. The converse obviously holds also. Appendix B. Shearfree congruences Assuming that [θ˜µ β t˜β ]⊥ = 0, for all directions t˜µ we have θ˜µ β t˜β ∝ t˜µ .
(B.1)
˜ Knowing that θ˜µ β = σ ˜ µ β + θ3 P˜ µ β we see that (B.1) is equivalent to σ ˜ µ β t˜β ∝ t˜µ . We know that σ ˜ µ β η˜β = 0. Since σ ˜µν is a symmetric tensor it follows that in coordinates adapted to the congruence, only the spatial part of σ˜µν is nonzero. Also, for (B.1) to hold for arbitrary spatial directions t˜i , we must have σ ˜ i j ∝ δ i j . Knowing also that the trace σ ˜αα always vanishes, it follows that σ˜ µ ν must vanish entirely. This in turn is true if and only if the non-rescaled shear tensor vanishes since σ ˜µν = e−Φ σµν .
References [1] [2] [3] [4] [5] [6] [7] [8]
Abramowicz M A (march) 1993 Sci. Am. 266 (no 3) 74-81 Abramowicz M A, Carter B and Lasota J-P 1988 Gen. Rel. Grav. 20 1173-1183 Abramowicz M A and Lasota J-P 1997 Class. Quantum Grav. 14 A23-A30 Abramowicz M A, Nurowski P and Wex N 1995 Class. Quantum Grav. 12 1467-1472 Abramowicz M A 1992 Mon. Not. R. Astr. Soc 256 710-718 Abramowicz M A, Nurowski P and Wex N 1995 Class. Quantum Grav 10 L183-L186 Sonego S, Almergren J and Abramowicz M A 2000 Phys. Rev. D 62 064010 Jonsson R 2006 Inertial forces and the foundations of optical geometry Class. Quantum Grav. 23 1-36 [9] Jonsson R 2006 A covariant formalism of spin precession with respect to a reference congruence Class. Quantum Grav. 23 37-59 [10] Misner C W, Thorne K S and Wheeler J A (1973) Gravitation, (New York: Freeman) 566
Generalizing Optical Geometry
19
[11] Jonsson R 2006 An intuitive approach to inertial forces and the centrifugal force paradox in general relativity Am. Journ. Phys. 74 905-916 [12] Rindler W 2001 Relativity: Special, General and Cosmological, (Oxford: Oxford University Press) pp 267-272 [13] Jonsson R 2006 Optical geometry across the horizon 23 77-89 [14] Bini D, Carini P and Jantzen RT 1997 Int. Journ. Mod. Phys. D 6 14 [15] Sonego S and Abramowicz M A 1998 J. Math. Phys 39 3158-3166 [16] Jonsson R 2007 Gyroscope precession in special and general relativity from basic principles Am. Journ. Phys. 75 463-471
Optical geometry across the horizon Rickard Jonsson Department of Theoretical Physics, Chalmers University of Technology, 41296 G¨ oteborg, Sweden. E-mail:
[email protected] Submitted 2004-12-10, Published 2005-12-08 Journal Reference: Class. Quantum Grav. 23 77 Abstract. In a companion paper (Jonsson and Westman 2006 Class. Quantum Grav. 23 61), a generalization of optical geometry, assuming a non-shearing reference congruence, is discussed. Here we illustrate that this formalism can be applied to (a finite four-volume) of any spherically symmetric spacetime. In particular we apply the formalism, using a non-static reference congruence, to do optical geometry across the horizon of a static black hole. While the resulting geometry in principle is time dependent, we can choose the reference congruence in such a manner that an embedding of the geometry always looks the same. Relative to the embedded geometry the reference points are then moving. We discuss the motion of photons, inertial forces and gyroscope precession in this framework. PACS numbers: 04.20.-q, 95.30.Sf, 04.70.Bw
1. Introduction In [1] it is illustrated how we can generalize the optical geometry (see e.g [2] for an introduction) to a wider class of spacetimes than the conformally static ones. In particular, employing the standard projected curvature (see [1] for alternative curvature measures), the new class of spacetimes consists of those spacetimes that admit a hypersurface forming shearfree congruence of timelike worldlines. We are now curious as to whether any of the standard solutions to Einstein’s equations, that are not conformally static, falls into the new category. The task is then to look for a congruence such that, in the corresponding coordinates, the metric after rescaling takes the form [1]
g˜µν =
1 0
0 2Ω(t,x) ¯
−e
hij (x)
.
(1)
Indeed, as will be shown in the coming section, such a congruence can be found (in a finite four-volume of the spacetime) whenever we have spherical symmetry in the
2
Optical geometry across the horizon
original metric. This includes the inside of a Schwarzschild black hole and the horizon as well. Throughout the article, we will use the timelike (+, −, −, −) convention for the sign of the metric. Also the optical line element will be denoted by d˜ s. 2. A spherical line element A general, time dependent, spherically symmetric line element can be written on the form dτ 2 = a(r, t)dt2 − 2b(r, t)drdt − c(r, t)dr 2 − r 2 dΩ2 .
(2)
This we may rewrite as
τ 2 − dΩ2 . dτ 2 = r 2 d¯
(3)
Here we have introduced a two-dimensional line element a(r, t) 2 2b(r, t) c(r, t) 2 d¯ τ2 = dt − drdt − dr . (4) 2 2 r r r2 In this reduced spacetime we may introduce an arbitrary timelike initial congruence line. From this line we go a proper orthogonal distance ds to create a new line. From the new line we create yet another line in the same manner. Next we introduce a new spatial coordinate x′ that is constant for every congruence line, and where dx′ = ds. Also introduce time slices of constant t′ orthogonal to the congruence lines. The reduced line element then takes the form d¯ τ 2 = f (t′ , x′ )dt′2 − dx′2 .
(5)
The full line element with respect to these coordinates is then on the form !
1 ′2 2 . dx + dΩ dτ = r f (t , x ) dt − f (x′ , t′ ) 2
2
′
′
′2
(6)
Here r is in principle known in terms of x′ and t′ . After rescaling away the factor r 2 f (t′ , x′ ), this line element clearly has the form of (1) needed for the generalized optical geometry. The optical geometry is then 1 ′2 2 . (7) dx + dΩ d˜ s2 = f (x′ , t′ ) So, in whatever spherically symmetric spacetime we consider we can thus do the generalized optical geometry. This includes collapsing stars, the spacetime around the horizon of a Schwarzschild black hole and so forth. Notice however that there is no guarantee for the generalized geometry to work globally in these spacetimes. The way we are constructing our congruence, it may for instance go null before we have come very far from our original congruence line. Also, the geometry which will be determined by how we choose our initial congruence line, may be more or less complicated, time dependent and so forth.
3
Optical geometry across the horizon 2.1. A small note on intuition
We are here considering a congruence that is fixed in the spherical angles. From a dynamical point of view, we have found a radial velocity of infinitesimally separated congruence points such that the proper shape that is spanned by the points is preserved, see figure 1.
Figure 1. To have vanishing shear (which is necessary for a congruence generating the optical geometry), the congruence points (the black dots), must be shifted in such a way that the shape of a little box of congruence points (as seen when comoving with the box) is the same at all times.
If we start with a cube it must remain a cube, but not necessarily of the same size, in a system comoving with the points. For instance considering a flat space and low velocities inwards (towards the origin of the spherical coordinates) we may understand that the velocity of the inner part of the cube must be smaller than that of the outer part to insure that the cube is not elongated in the radial direction. 3. Optical geometry for a black hole including the horizon Let us now study a black hole explicitly, with focus on the horizon. The ordinary Schwarzschild coordinates are ill suited for congruences passing through the horizon. There is however another coordinate system (called Painlev´e coordinates) in which the Schwarzschild line element is given by s
2M 2M dTP2 − 2 dTP dr − dr 2 − r 2 dΩ2 . (8) dτ 2 = 1 − r r This line element is connected to the standard line element through a resetting of the ordinary Schwarzschild clocks1 . In these coordinates there are no problems in passing through the horizon. We also find it practical to introduce dimensionless coordinates r/2M → x, TP /2M → T . Then the line element takes the form
dτ 2 1 1 dT 2 − 2 √ dT dx − dx2 − x2 dΩ2 . = 1− 2 (2M) x x
1
(9)
The clocks are reset in such a way that the coordinate time passed for a freely falling observer initially at rest at infinity corresponds to the proper time experienced by this observer. Inside the horizon one cannot have any material clocks at a fixed r but that doesn’t matter.
4
Optical geometry across the horizon The reduced line element (compare with (4)) is then given by d¯ τ 2 = AdT 2 − 2BdT dx − Cdx2 .
(10)
Here the reduced metrical components are given by2 1 1 1 1 1 B = 2√ A= 2 1− C = 2. x x x x x
(11)
Now we may introduce an arbitrary timelike trajectory that passes the horizon. From this we go a proper distance ds orthogonal to the trajectory, to create a new congruence line and so forth. In general this scheme would completely hide the manifest time symmetry of the black hole. There is however a way to circumvent this as will be shown in the following sections. 4. Keeping the time symmetry, covariant approach Suppose that we can find an initial trajectory such that the second trajectory (go ds orthogonal to the initial trajectory) is related to the first by a simple translation straight in the T -direction (along the Killing field connected to T ). This way we would maintain a certain time symmetry in the optical metric. In figure 2 we see schematically how this would work. T ∆T ∆s
∆s
∆T x Figure 2. Two congruence lines separated along the Killing field with constant proper distance between them.
Zooming in on the two lines around some specific point, they will to first order be two parallel straight lines. Given the tilt of the lines (i.e. the four-velocity) we can find a relation between the displacement along the Killing field and the orthogonal distance between the lines. How these are related is sketched in Fig 3. Here uµ is the four-velocity of the trajectories and v µ is a spacelike vector normed to −1 and orthogonal to uµ . Just adding vectors we find Kdsξ µ = dsv µ + σdsuµ . 2
(12)
One may alternatively use the Eddington Finkelstein coordinates where after corresponding rescalings A = (1 − 1/x)/x2 , B = 1/x2 , C = 0.
5
Optical geometry across the horizon
K ds ξ µ
σ ds uµ
ds v µ Figure 3. The relation between Killing field ξ µ , four-velocity uµ and the orthogonal vector v µ , assuming an in-going congruence.
Notice that K is a assumed to be a constant unlike σ. Multiplying both sides by uµ we get σ = Kξ µ uµ .
(13)
Inserting this back into (12) we get Kξ µ = v µ + Kξ α uα uµ .
(14)
Taking the absolute value of both sides yields shortly 1 . K2 = α 2 (ξ uα ) − ξ α ξα So K is known given uµ . Solving for ξ α uα yields
(15)
s
1 . (16) K2 Outside of the horizon ξ µ is always timelike and thus the sign in front of the root will be positive3 . Inside of the horizon, where ξ µ is spacelike, we can have both signs depending on uµ . On the horizon we however have ξ µ ξµ = 0. Thus for finite K we realize that we must have the positive sign on the inside as well to get a continuous four-velocity across the horizon. Using (16) together with uα uα = 1, we can in principle solve for uµ given K. The equations are however second order, and there will be four different solutions at every point (see section 5.1 for intuition). There is however a simple way to get first order equations. Defining v µ to be the orthonormal vector to uµ that lies less than 180◦ clockwise4 of uµ , we may write ξ α u α = ± ξ α ξα +
1 v = √ ǫµν gνρ uρ g µ
where
µν
ǫ
=
0 -1 1 0
!
.
(17)
Here g = −Det(gµν ). Then we may rewrite (14) into s
1 1 Kξ µ = √ ǫµν gνρ uρ ± K ξ α ξα + 2 uµ . g K 3 4
Assuming uµ and ξ µ to both be future directed. In accordance to figure 3, assuming positive values of K, σ and ds.
(18)
6
Optical geometry across the horizon
As noted before, considering an in-falling congruence at the horizon we should choose the positive sign. We see that (18) is a linear equation system which we, given K, should be able to solve to find uµ . The scheme thus appears successful and there exists an optical (shearfree) congruence that will preserve manifest time symmetry. 5. The congruence for a Schwarzschild black hole Choosing the positive sign of (18) and assuming a positive K in accordance with the discussion above, (18) takes the form s
1 1 Kξ µ = √ ǫµν gνρ uρ + K ξ αξα + 2 uµ . g K
(19)
Assuming the reduced line element to be of the form of (10), and ξ µ = (1, 0) this can be written √ C B K = √ u0 + √ u1 + K 2 A + 1u0 (20) g g √ 1 (21) 0 = √ (Au0 − Bu1 )+ K 2 A + 1u1 . g Recognizing that dx/dT = u1 /u0 we find from the second equation alone that dx A . = √ √ 2 dT B− g K A+1
(22)
Inserting the metrical components of (11) into (22) we find dT = dx
√1 x
−
q
K2 (1 − x1 ) x2 1 − x1
+1
.
(23)
So here we have the tilt of the reference congruence lines in the Painlev´e coordinates. For an infinite value of the free parameter K, outside the horizon, this corresponds to a congruence at rest (i.e. the classical optical congruence). Inside the horizon the root takes a negative value for infinite K and we have no solution. For any finite values of K we notice that at infinity (23) will correspond to an in= −1). In the particular case of K = 0 the congruence will correspond going photon ( dT dx to an in-going photon all the way through the horizon and into the singularity. Photons are however at first sight not particularly well suited for a congruence. For finite K and x < 1 we see that the root goes imaginary unless x3 . (24) 1−x So the conclusion is that we can do optical geometry, while keeping manifest time symmetry, from a point arbitrarily close to the singularity, across the horizon and all the way towards infinity. Notice in particular that with this scheme we not only get the time symmetry, we insure that we can span the full spacetime all the way towards infinite Schwarzschild times. K2 <
7
Optical geometry across the horizon 5.1. Comments
A short comment may be in order regarding the congruence as we approach infinity, where the spacetime approaches Minkowski. Here one would expect that any congruence with fixed coordinate velocity dx/dt would work as an optical congruence, not just leftmoving photons. Indeed from (23) we see that for any large, but finite, x there exists a K such that we can have any in-going coordinate velocity of the congruence. As we go outwards towards infinity from this point the congruence will however start approaching a left moving photon. Another comment may be in order. The existence of an optical congruence, keeping manifest time symmetry, is independent of what coordinates we are using. In the standard Schwarzschild coordinates it is easy to realize that, both on the inside and the outside, the existence of one congruence immediately implies the existence of another5 , as depicted in figure 4. t Right−moving congruence
Right−moving congruence
Left−moving congruence
Left−moving congruence
0
1
x
Figure 4. Illustrating in standard (dimensionless) Schwarzschild coordinates that for any left-moving congruence that fulfills the requirements there is also a right-moving congruence that fulfills the requirements.
Inside the horizon this is manifesting itself in the ± sign of (18). On the outside, where we must have a plus in the ± sign, it manifests itself in the possibility to have negative K. In the latter case we need to consider a slightly different image than that of figure 3, but the mathematics will be identical if we let K assume negative values. In general we may show that dx A . (25) = √ √ 2 dT B∓ g K A+1
The minus in this case corresponds to a left-moving congruence, and the plus a rightmoving.
5
Except if the congruence on the outside would be parallel to the Killing field, or equivalently if the congruence on the inside would be perpendicular to the Killing field.
8
Optical geometry across the horizon 6. The optical metric
Let the spatial coordinate difference dx′ , separating two nearby congruence lines, equal the proper orthogonal distance ds between the lines6 . Let the new coordinate time difference dt′ , separating two time slices, equal the original coordinate difference dT , as measured along the Killing field. The reduced metric takes a new form according to d¯ τ 2 = AdT 2 − 2BdT dx − Cdx2
d¯ τ 2 = f (x′ , t′ )dt′2 − dx′2 .(26)
→
Here f is a function yet to be determined. Recall the relation between the various vectors, as depicted in figure 5. t′ + dt′ dT
σ ds uµ
K ds ξ µ
t′ ds v µ Figure 5. The relation between Killing field, four-velocity and the orthogonal vector. The dotted lines are the new local time slices
Like before we assume the Killing field to be (1, 0) so that dT = Kds = dt′ . The proper distance squared, as measured along a congruence line, separating two time slices can be expressed as d¯ τ 2 = (σds)2 uµ uµ
d¯ τ 2 = f dt′2 .
(27)
From (13) and (16) respectively we have s
1 . (28) K2 Like before we have chosen the positive sign of the root. Putting the pieces together we find 1 (29) f = A + 2. K The total, original, line element in the new coordinates is now given by α
α
σ = Kξ uα
ξ uα =
A+
dτ 2 = (2M)2 x2 f (x)dt′2 − dx′2 − dΩ2 . Using (11) and (29), the optical metric is thus given by: 1 ′2 2 d˜ s2 = 1 . dx + dΩ 1 1 + 1 − x2 x K2 6
Recall that the distance between the lines is by definition constant along the lines.
(30)
(31)
9
Optical geometry across the horizon
Notice however that it is not in explicit form since we do not have x in terms of x′ and t′ . We may however recall figure 5 where we for constant x have dt′ = Kds = Kdx′ (since dx′ per definition equals ds). Thus we know that constant x means dt′ /dx′ = K. In the new coordinates we have therefore a schematic picture as depicted in figure 6. T
t’
The singularity The limit where the optical congruence breaks down The horizon Constant x
x
x’
Figure 6. To the left the congruence in the Painlev´e coordinates. To the right the horizon etc relative to coordinates adapted to the congruence.
Notice that the Killing field is still a constant vector (tilted up and to the right) in the new coordinates. While we still do not have x analytically in terms of x′ and t′ , we know the qualitative relation well enough to understand some basic features. 6.1. The rubber sheet model We see from (31) that at spatial infinity the geometry becomes that of a three-cylinder, except if K is infinite. Also we see that on the inner boundary, where the optical congruence breaks down, the stretching (of d˜ s relative to ds) is infinite. It appears very difficult to do any calculations in our new coordinates, considering that we don’t have any explicit relation for x in terms of x′ and t′ . At every fixed t′ we may however express the optical geometry as a function of x, given that we have a relation between dx and dx′ (as will be derived in section 7). This background geometry is time independent. The scenario (in 2D) can then be exactly described by a rubber sheet sliding snuggly over the fixed background geometry. Photons move on geodesics with unit velocity at every point if we comove with the rubber sheet. The velocity of the rubber sheet will correspond to the velocity of a constant x line relative to the reference congruence. Then we can use our knowledge of geodesics on rotational surfaces, and relative velocities, to find the paths of photons relative this pseudo-optical background geometry. 7. On the relation between x and x′ Given a displacement dx′ along the x′ -axis we want to find dx. From figure 3 we see that dx = v x ds or equivalently dx = v x dx′ .
(32)
10
Optical geometry across the horizon From (17) and (20) we readily find v x = −αux
where α =
√
K 2 A + 1.
(33)
To find ux , we solve the linear equation system of (20) and (21) letting u0 → uT and u1 → ux . The result is K √ . (34) ux = g B2 C √ √ 2 − (K 2 A + 1) + A g g Inserting the explicit metrical functions, this miraculously is reduced to x2 . K Using (32) and (33), the general relation between dx and dx′ is given by 1 dx′ = − x dx. αu In explicit form this is then reduced to K dx′ = r dx. 2 x2 Kx2 1 − x1 + 1 ux = −
(35)
(36)
(37)
Incidentally, using the Eddington-Finkelstein original coordinates yields the same expression, as it must. The expression however turns out not to be particularly easy to integrate analytically except in the limits where K is either infinite or zero. In the limit where K is infinite it however cannot be inverted to find x in terms of x′ . In any case (37) is sufficient to express the background geometry explicitly. 8. The background optical geometry Inserting (37) into (31) we may at a fix time t′ write the optical line element as 1 1 2 dΩ2 . d˜ s2 = 2 dx + 1 1 1 1 1 1 4 + 1 − x x2 1 − x + K 2 x2 x K2
(38)
In the limit of K → ∞ this takes the familiar form of the standard optical geometry d˜ s2 =
1
1−
2
1 x
2 dx
+
x2 dΩ2 . 1 − x1
(39)
At the other end, where K goes to zero, it to lowest non-zero order approaches K4 2 dx + K 2 dΩ2 . (40) x4 Here the x-dependence can be taken away by another coordinate transformation. It is then obvious that in this limit we have a flat space. In a symmetry plane this would correspond to a cylinder, infinitely extended in the direction of the singularity but with finite distance from horizon to infinity. Unfortunately in the same limit the optical velocity of constant x position goes to the velocity of light, to lowest order. This means that, if we just concern ourselves with the lowest order influence of K on metrical d˜ s2 =
Optical geometry across the horizon
11
components and velocities, we will not get any usable dynamics7 . We can for instance not find the photon radius. If we still would like to use the K = 0 limit, we must take higher order terms into account. Perhaps expressions when expanded to the second non-vanishing order in K will be easier to deal with than in the general case. If this would work out it would be no approximation but give the exactly correct dynamics. The point of considering this limit is of course that in this limit the full spacetime is spanned by the optical geometry. We will however not pursue this point further here. In any case we may, for arbitrary K, Taylor expand d˜ s/dx in the limit where we approach the innermost point of the optical geometry. Doing this we readily find that regardless of K the momentaneous distance to the innermost point is infinite. Remember however that the line element of (38) is not strictly the optical geometry. It is not with respect to this element (except in the K → ∞ limit) that photons move on geodesics, as discussed earlier. The speed d˜ s/dt′ of the constant x lines relative to the optical space is easy to derive since we have K = dt′ /dx′ and from (31) we see that d˜ s = Kdx′ /α2 . Then we find 1 d˜ s . (41) =r ′ dt 1 K2 1− x +1 x2
We see that in the limit where K → ∞ this goes to zero as it should. At K = 0 it goes to the velocity of light. Incidentally the velocity is at a minimum at x = 3/2, the photon radius. So now we have everything that we need to make explicit calculations in the generalized optical geometry, using the rubber sheet analogy. In fact we may also embed the background optical geometry and visualize the photon radius. 9. Embedding the background geometry In figure 7 we see a schematic picture of how an embedding of the background geometry would look. Using this qualitative image we will see that the photon radius lies exactly at the neck of the background geometry, just like in standard optical geometry. We will also understand that the way gyroscopes in circular motion precesses, is different inside and outside of the neck. 9.1. Photon geodesics
Study now a photon moving on the surface. At any given radius we can find an angle of the velocity vector of the photon, relative to the rubber sheet, such that instantaneously the photon has no radial velocity relative to the background geometry8 . 7
Think of the rubber sheet model discussed earlier. If we direct the photon directly outwards it will move slowly out towards spatial infinity. On the other hand if we direct it purely azimuthally relative to the rubber sheet it will be dragged inwards with the rubber sheet. Somewhere in between there is obviously an angle such that it has purely azimuthal 8
12
Optical geometry across the horizon Infinity The photon radius
The horizon
Figure 7. The background optical geometry. The author took the liberty of enhancing the radial variations to get more shape without affecting the qualitative behavior.
A free photon, with an initial position and velocity such that it has no radial velocity, will follow a local geodesic on the surface. However, its position relative to the background will be shifted continously according to the sheet velocity. To time evolve the the position and velocity of the photons, we may however consider the following two-step process. First we move a distance corresponding to the time step dt, along a geodesic on the surface. Then we take the resulting forward direction and parallel transport it downwards corresponding to the shift of the congruence points. In the second step, the angle of the forward direction with a purely radial line will be maintained. This follows from that the congruence is non-shearing (see figure 9 for intuition). We may iterate the two-step process to time evolve the position to arbitrary times. For a geodesic on a rotational surface it is easy to show that the angle the geodesic makes with a local line of fixed azimuthal angle (a radial line in this case) is decreasing with increasing radius, and vice versa. In figure 8 we illustrate the effects of this shift of angles for photon geodesics with no momentary radial velocity. Looking at figure 8 we may understand that if we are on the outside of the neck, the photon velocity vector will be directed more and more outwards as time goes. Thus it will leave the radius it started at and move to infinity. We also realize that for the corresponding initial velocity vector inside the neck, the velocity vector will be rotated to be directed less and less outwards and thus will start to move inwards. If we start exactly at the neck however, where the embedding radius doesn’t change to first order, the photon will remain on the same radius. So, just like in standard optical geometry (see e.g [2]), a geodesic photon in circular motion will stay at the neck of the embedding. To further clarify the two-step scenario we may plot the evolution as seen from the normal of the surface. This is depicted in figure 9. Notice that unlike the standard optical geometry (which is a subset of this discussion) we need the velocity of the rubber sheet apart from the background geometry to determine the paths of free photons. We know however that for a given background velocity relative to the background.
13
Optical geometry across the horizon Infinity Instantaneous Geodesic
Background trajectory
Instantaneous Geodesic
Fixed radial position Instantaneous Geodesic
The horizon Towards singularity Figure 8. Understanding the photon radius in the generalized optical geometry. We can time evolve the position and velocity of the photon by a two-step process. First we transport the forward direction along a geodesic on the surface and then we parallel transport it downwards according to the shift of the congruence during the time step.
Radial lines A
B
C
D Local geodesic
Constant radius
A C
B D
Figure 9. The shift of congruence points A,B,C and D as seen from a local background geodesic coordinate system. Since the congruence is non-shearing, the angle the forward direction makes with a pure radial line is unaltered by the shifting of the congruence points. Notice that the net effect is that the forward direction is parallel transported relative to the background geometry.
geometry the sheet velocity depends only on the embedding radius. The bigger the radius the bigger the velocity. Also, the velocity of the sheet at infinity and at the horizon is that of light. This is sufficient to understand the qualitative behavior of geodesic photons.
14
Optical geometry across the horizon 9.2. Gyroscope precession for circular motion
Let us now consider circular motion (fixed x) with constant velocity. Consider first motion, outside the photon radius, directed to the left as seen from outside the embedding as illustrated in figure 10.
II II
I
I
Figure 10. The two step process of moving the forward direction. To the left in 2+1 dimensions, to the right in 2 dimensions.
Moving along a circle of fixed radius with fixed speed, we know that after the twostep process of transporting the forward direction, we must get a forward direction that has the same angle relative to the radial line as we had before the two-step process. As is illustrated in figure 10 this means that the optical curvature has to be directed to the left (looking at the surface from the outside). We know from [1] that a gyroscope undergoes pure Thomas precession relative to the optical geometry. For the case at hand where the trajectory turns left this means that a gyroscope will precess clockwise relative to a corresponding parallel transported vector. Since the forward direction is precessing counterclockwise relative to a parallel transported vector, the gyroscope will precess clockwise relative to the forward direction. It follows that it will precess clockwise relative to the local radial line also. Looking at the embedding from the inside (and from the top), we may say that clockwise circular motion results in counterclockwise precession relative to the forward direction. This is in fact what one expects of gyroscope precession in Newtonian mechanics. Completely analogously, we may understand that inside the photon radius, looking at the embedding from the top, clockwise circular motion results in clockwise precession relative to the forward direction (as seen from the inside looking at the surface). This is not analogous to the Newtonian precession. Indeed gyroscope precession is easier to deal with in the standard optical geometry (where the congruence is static), but as we have seen it can be done also considering an infalling congruence, that allows us to include the horizon.
Optical geometry across the horizon
15
9.3. Inertial forces considering circular motion Orbiting a black hole at a fixed radius outside the photon radius requires a smaller outward comoving force the faster one orbits the black hole (like in Newtonian gravity). Inside the photon radius however the required outward force increases the faster one orbits the black hole. This can be readily understood in the standard optical geometry (see e.g [3]) corresponding to infinite K, but when we have a congruence moving relative to the background it is much more complicated to see. The point is that as we increase the orbital speed, we change the direction of motion relative to the in-falling congruence (tilt the velocity arrow down). This brings about all sorts of changes, for instance the optical curvature (as opposed to the background curvature) changes. This particular feature of black holes apparently cannot be so easily explained, using simple qualitative arguments, when the reference congruence is in-falling. 10. Conclusion We conclude that the generalized optical geometry (assuming shearfree congruences) can be applied to (a finite sized region of) any spherically symmetric spacetime. In particular we can define an optical geometry from spatial infinity across the horizon and arbitrarily close to the singularity of a static black hole. In 2 spatial dimensions we can display the optical geometry as a curved surface, relative to which the reference congruence points are moving. This motion of the reference points certainly makes any argumentation more complicated than in standard optical geometry. We can however do essentially the same type of qualitative arguments concerning photons, and gyroscopes as in the standard optical geometry, and include the horizon. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Jonsson R and Westman H 2006 Generalizing optical geometry Class. Quantum Grav. 23 61-76 Abramowicz M A and Lasota J P 1997 Class. Quantum Grav. 14 A23-A30 Abramowicz M A (march) 1993 Sci. Am. 266 (no 3) pp 26-31 Jonsson R 2006 Inertial forces and the foundations of optical geometry Class. Quantum Grav. 23 1-36 Jonsson R 2006 A covariant formalism of spin precession with respect to a reference congruence Class. Quantum Grav. 23 37-59 Abramowicz M A, Nurowski P and Wex N 1995 Class. Quantum Grav. 12 1467-1472 Abramowicz M A 1992 Mon. Not. R. astr. Soc. 256 710-718 Abramowicz MA, Nurowski P and Wex N 1995 Class. Quantum Grav. 10 L183-L186 Bini D, Carini P and Jantzen R T 1997 Int. Journ. Mod. Phys. D 6 14
