Visualizing curved spacetime (Licentiate thesis)

LICENTIATE THESIS Visualizing curved spacetime RICKARD M. JONSSON Department of Astronomy and Astrophysics Centre for...

Author: Rickard Jonsson

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LICENTIATE THESIS

Visualizing curved spacetime

RICKARD M. JONSSON

Department of Astronomy and Astrophysics Centre for Astrophysics and Space Science Chalmers University of Technology and G¨oteborg University G¨oteborg, Sweden 2001 [email protected]

Visualizing curved spacetime Rickard M. Jonsson Department of Astronomy and Astrophysics Center for Astrophysics and Space Science Chalmers University of Technology and G¨oteborg University G¨oteborg, Sweden Abstract In this thesis I derive two fundamentally different methods of visualizing curved spacetimes, the dual and the absolute. The images are nice pedagogical tools for teaching General Relativity. I also review the concept of space-eigentimes. In the dual scheme, I start from the equations of motion 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. On the embedded surface freely falling particles take the shortest paths. Thus, one can visualize how acceleration in a gravitational field is explained by particles moving on straight lines in a curved spacetime. In the absolute scheme, I start from an arbitrary Lorentzian spacetime with a given field of timelike four-velocities uµ . I then do 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. The new absolute metric can be covariantly related to the original one. This method is particularly well suited for visualizing gravitational time dilation and the horizon of Black Holes.

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Acknowledgements First of all I would like to thank my supervisor Marek Abramowicz for his support of myself and my ideas. I also thank Ulf Torkelsson and Martin Cederwall for their comments on my Paper. I would like to salute all of the guys at the 6:th and 7:th floor, as well as the Onsalapeople, for creating such a nice working environment. In particular I would like to thank Pontus, Martin and Alex for always being available objects for me to pester :) In Astrophysics I would like to mention Rim, Alessandro, Gustav, Tommy and Duilia for nice discussions on all sorts of topics. Thank you Achim for being a good co-builder of longbows as well as a fellow rider. A special thanks also for giving a particular section of this thesis the most thorough read-through I ever saw. Thank you Hans, as always, for your open mind and endless ongoing search for the Ultimate understanding. Thank you Daniel for your contributions to my understanding of space-eigentimes. Off the record I would also like to thank the girls at Salsa & S¨ondag for insuring that at least one evening per week I am not stuck in front of a computer. I am also grateful to the guys at the experimental workshop for their advice in the delicate business of making your own aluminum spacetime! At last a very special thanks to all my students at the F-education for making not only teaching but also life outside the corridors so enjoyable.

April, n˚ adens ˚ ar 2001 Rickard Jonsson

P.S While I am not sure that they will get the message, I am very grateful to a certain family of laughing bottlenose dolphins in the Red Sea. They made a dream come true.

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Contents

1 Introduction 1.1 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A geometrical introduction to gravitation 2.1 Spacetime, what is that? . . . . . . . . . . 2.2 Gravity and curved spacetime . . . . . . . 2.3 Forces and the acceleration paradox . . . . 2.4 The full spacetime . . . . . . . . . . . . . 2.5 Comments and conclusions . . . . . . . . .

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3 The absolute metric 3.1 The absolute line element . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Black hole embedding . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Comment regarding geodesics . . . . . . . . . . . . . . . . . . . 3.2 Generalization to arbitrary spacetimes . . . . . . . . . . . . . . . . . . 3.2.1 A covariant approach . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Freely falling observers as generators . . . . . . . . . . . . . . . . . . . 3.4 On geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Proof regarding geodesic generators . . . . . . . . . . . . . . . . 3.5 Covariant approach to photon geodesics . . . . . . . . . . . . . . . . . . 3.5.1 Photons in 1+1 dimensions . . . . . . . . . . . . . . . . . . . . 3.6 Photon geodesics in static spacetime . . . . . . . . . . . . . . . . . . . 3.6.1 The reference freefaller coordinates . . . . . . . . . . . . . . . . 3.6.2 The absolute metric . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Equations of motion in a general 1+1, time independent metric 3.6.4 Outmoving photons . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Flat embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Spacelike generators? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 A mathematical remark . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Metrics, geodesics and affine connections 4.1 Finding the metric from the geodesics . . . . . . . . . . . . . . . . 4.1.1 Coordinate curvature . . . . . . . . . . . . . . . . . . . . . 4.1.2 The geodesic equation using a coordinate affine parameter 4.1.3 Equivalent affine connections . . . . . . . . . . . . . . . . 4.2 On the construction of the dual metric . . . . . . . . . . . . . . . 4.2.1 Point-dual metrics . . . . . . . . . . . . . . . . . . . . . . 4.3 On the dual metric in freely falling coordinates . . . . . . . . . . . 4.3.1 Finding the coordinate transformation to the freely falling dinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The dual metric in the freely falling coordinates . . . . . . 5 On 5.1 5.2 5.3 5.4

whether the interior dual metric is a Conditions for spheres . . . . . . . . . . The dual interior metric . . . . . . . . . Approximative internal sphere . . . . . . Spheres in the Newtonian limit . . . . .

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6 The Epstein-Berg way 6.1 The main philosophy . . . . . . . . . . . . 6.1.1 Particle trajectories and mappings 6.1.2 Intuition about geodesics . . . . . . 6.1.3 Mathematics about geodesics . . . 6.2 Embeddings . . . . . . . . . . . . . . . . . 6.2.1 The Berg dynamical view . . . . . 6.3 The Epstein internal space . . . . . . . . . 6.3.1 Verification using the dual scheme . 6.3.2 Comments . . . . . . . . . . . . . . 6.4 Comments . . . . . . . . . . . . . . . . . .

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Appended Papers Paper I Embedding spacetime via a geodesically equivalent metric of Euclidean signature Jonsson, R. (2001) - Accepted for publication in GRG, June 2001, Volume 33

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1 Introduction Ever since it was presented in 1916, Einstein’s General theory of Relativity has been extremely successful in explaining all sorts gravitational phenomena. Two examples from the beginning of the century are the gravitational deflection of light from distant stars passing our sun and the precession of the perihelia of Mercury. More recent experiments involve gravitational red shifts, slowing down of atomic clocks and indirect measurements of gravitational waves, the latter matching the predictions of Einstein’s theory to a fantastic precision. Ever since the theory was launched it has also puzzled the minds of physicists and people in general. In fact it is rumored that in the beginning of the century a reporter asked Sir Arthur Eddington to comment on there only being three people in the world who understood Einstein’s theory. The reply from Eddington was “Who’s the third?”. Today the number of physicists understanding General Relativity is quite large. For the more general audience the theory is however still clouded in mystique. In particular, people find the legendary Black Holes fascinating, and incomprehensible. I know, having explained the concepts so many times at the local pizza-place and other places. Einstein’s theory, which is a geometrical theory, is in many ways well suited to be explained by images. For instance the way a star is affecting the space around and inside it can easily be displayed by a curved surface. These kinds of images are invaluable in teaching General Relativity, but also for the seasoned relativist. For a proper understanding of General Relativity, one certainly needs the mathematical background, but it will always be accompanied by examples, mental images and even movies. The very heart of Einstein’s theory, curved spacetime, is however fundamentally difficult to display using curved surfaces. The reason is that in General Relativity there are negative (squared) distances, something that we do not have on ordinary curved surfaces. 1

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1. Introduction

1.1

This thesis

In this thesis I show how one can circumvent the problem of negative distances, and visualize curved spacetimes after all. I do this using two completely different methods. I also re-derive an already existing method. Using the images of spacetime, one can explain how acceleration of particles here on Earth is caused by a curvature of spacetime rather than by a force. One can also, for instance, explain the gravitational slowing-down of clocks as a pure geometrical effect. The basic structure of the thesis is outlined below: • Chapter 2 is 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. In General Relativity there is no such thing as a gravitational force. The underlying mathematics of this chapter is dealt with in Paper I, at the end of this thesis. • Chapter 3 deals with another idea for visualizing curved spacetime. This idea is more suited for understanding the slowing down of clocks close to gravitational sources. It is also well suited for understanding Black Holes and the event horizon. This idea is ideal for people who know Special but not General Relativity. It has however many virtues also for people who know nothing about General Relativity, as well as for seasoned General Relativists. This idea is covariant and applicable to any spacetime. I have yet to put this in a form more suited for a general audience. • Chapter 4 deals with some mathematical aspects of geodesics; it is in principle a comment to Paper I. • Chapter 5 is also a rather mathematical comment to Paper I, regarding the shape of the spacetime inside a star. I show that in the Newtonian limit it corresponds to an exact sphere. • Chapter 6 is a re-derivation of the basic ideas underlying the book by Epstein [2], which has a different way of displaying effects in curved spacetimes. • Paper I Given the geodesic trajectories in a 1+1 static Lorentzian spacetime, such as the Schwarzschild line element, I find a new positive definite metric that has the same geodesics as the original metric. This new dual metric can be embedded as a rotational surface. Apart from the three ways of displaying curved spacetimes dealt with in this thesis there is also another fundamentally different way invented by Marolf [1]. There are thus at least four, and probably more, fundamentally different methods of visualizing curved spacetimes by curved surfaces. They each have different virtues and explaining power. I believe Feynman once said that one doesn’t have a proper understanding of a phenomena until one can explain it in at least five different ways. I think he had a good point.

2 A geometrical introduction to gravitation In this section I would like to give some intuition, without using mathematics, about the General Theory of Relativity, Einstein’s theory of gravity. I will show how one can explain why things to fall to the ground when we drop them, not because a force is acting on them, but because the particles are moving straight in a curved spacetime. To understand curved spacetimes (don’t worry, it’s simple) we need first to understand what a spacetime is.

2.1

Spacetime, what is that?

Imagine a straight line drawn on the floor with meter marks on it: Tid

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Rum (position)

We can describe all events that occur on the line, like an alarm bell ringing, in a spacetime diagram. See Fig 2.1 Time

Event at the 7 m⌧point, 3 o’ clock

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Event at the 4 m⌧point, 1 o’ clock

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Figure 2.1: A diagram over all the events on the line

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2. A geometrical introduction to gravitation

The later the event occurs, the higher up in the diagram it is marked. The more to the right the event occurs on the line, the more to the right it is marked. This graphical connection between space and time is called the spacetime. To be more strict one can say that spacetime is the collection of all events that occur on the line and the spacetime diagram is a map over all these events. In a spacetime diagram it is easy to visualize how particles move on the line. See Fig 2.2. Notice that the more the trajectory of a particle is tilted, the faster it moves. Time

Particle moving with constant velocity to the right

Particle at rest

Particle first moving rapidly to the right, then turning and moving slowly to the left

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Figure 2.2: Particles in motion represented by lines in the spacetime diagram Now imagine the line not drawn on the floor, but placed somewhere between the the stars, in empty space, where one doesn’t feel any effects of gravity. In such a place, if you throw an apple to the right along the line, it will keep following the line moving at a constant speed without accelerating since there are no forces acting on it. In the same manner, if we release an apple on the line such that it was at rest when we let it go, it will remain at rest. In both cases the apple is moving with constant speed. This means that it follows a straight line in the spacetime. Thus we can explain the motion of particles that are not acted on by any forces, by saying that they move on straight lines in the spacetime, instead of saying that they do not accelerate. Particles that are affected by forces however do accelerate. In the spacetime picture this corresponds to a curvature of the trajectory. In other words, the trajectory of a particle that is acted on by a force, is curving away from a straight line. Correspondingly, if the trajectory is curved, we know that there has to be a force acting on the particle. See Fig 2.3 On a flat surface there are many ways to define what we mean by a straight line. One way, which is also working on curved surfaces, is to say that a straight line is the path that a little toy car takes when we push it forward on the surface. We are then assuming that the little toy car is well manufactured and brand new so that it really goes straight forward :) Hopefully we have now developed a feeling for how velocities and accelerations correspond to tilted and curved trajectories respectively in the spacetime picture. It will

2.2. Gravity and curved spacetime

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Time 3 2 1

No force; again straight line Here the trajectory curvs away from a straight line

From t=0 to t=2 there is a force to the right acting on the particle

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Particle at rest, no force; straight line

Figure 2.3: How a force gives a curvature of a spacetime trajectory

then be easy to understand how this is working also in a curved spacetime.

2.2

Gravity and curved spacetime

Suppose that we put our line outside the Earth.

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Figure 2.4: A line placed outside the Earth. The Earth is not in scale but put there to show in what direction the Earth lies Now particles that are only affected by gravity will not move at a constant speed but will be accelerated towards the Earth as depicted in Fig 2.5. Time

Particle thrown directly away from the earth and returning due to gravity

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Figure 2.5: Particle accelerating towards the Earth due to gravity Newton would explain the acceleration by saying that there is a gravitational force acting on the particle, causing it to accelerate towards the Earth.

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2. A geometrical introduction to gravitation

Imagine however if the the mass of the Earth has curved the spacetime, so it really should look like that depicted in Fig 2.6.

Time Space

Figure 2.6: The curved spacetime

To create such a spacetime from the ordinary flat version, we can imagine that we roll it in the time direction, while also stretching it at the left end. Really the spacetime constructed this way should be like a toilet roll with infinitely many layers that are infinitely thin. When we walk one lap around the trumpet-like spacetime we thus come to a new point. From now on I will however suppress this in the images and let time close in on itself. On the trumpet we can mark events, and motion, just like we did in the flat image. Still events that happen far to the right on the line, we mark far to the right on the trumpet, towards the narrow end, and vice versa for events that happened more to the left. However moving in time now means moving around the trumpet. As a warm-up mental exercise, I have in the Fig 2.6 drawn the trajectories of three different particles. The one closest to the Earth (the leftmost one) corresponds to a particle traveling to the left, moving closer to the Earth. The middle trajectory corresponds to a particle at rest, moving only in time. The rightmost trajectory corresponds to a particle moving out from the Earth at high speed. Imagine that we have a real curved surface as the one depicted in Fig 2.6, made of metal for instance. Suppose that we take a little toy car and put it on the outside of the trumpet. We direct it so that it is mostly pointing around the trumpet, but also a little towards the narrow end of the trumpet and push the car forward. We will find that the car first starts to move around, and to the right on the trumpet. Then it will reach a point where it is no longer moving to the right, but instead starts moving to the left. All this is due to the geometry of the trumpet. The path taken by the car will be as that depicted in Fig 2.7.

2.2. Gravity and curved spacetime

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Figure 2.7: The trajectory of a particle in free fall as a straight line on a curved spacetime. Compare with Fig 2.5

The image of the Earth is included to remind the viewer in what direction the Earth lies. Strictly one shouldn’t draw the spacetime in the same picture as the Earth however. The trajectory in Fig 2.7 corresponds to a particle that is first moving away from the Earth, reaching a maximum and then falling back. Just like we we would expect an apple, thrown straight out from the Earth, to first move away from the Earth and then return due to gravity. We see then the possibility to explain the acceleration caused by gravity as an effect of a spacetime geometry (or shape), rather than as an effect of a force. A particle in free fall corresponds to a straight line in the spacetime By ’free fall’ we mean a particle that is not affected by anything except gravity. So it is just like in the preceding section where we had no gravity, we explain the motion of particles that are not affected by anything but gravity by saying that they move on a straight line (a line taken by a little toy car) in the spacetime. The difference is that the spacetime now is curved. Thus in geometrical gravity, there is no such thing as a gravitational force.

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2.3

2. A geometrical introduction to gravitation

Forces and the acceleration paradox

Although gravity is not a force in the geometrical theory of gravity, there are still other forces that can affect particles. For instance there is the force with which we push our old car (Volvo Amazon) when we are trying to jump-start it. Just like in the case where we had no gravity, particles that are affected by a force will not follow a straight line, but will curve away from a straight line. Consider for instance a particle that is held at rest at a certain position in space. Being at rest in space means moving around the trumpet on a circle. If we direct a toy car along such a circle and push it forward it will soon leave the circle and start moving to the left. We can however make the car stay on the circle by turning the wheels to the right a little bit. This means that the circular trajectory is bending away from what is straight, and there must therefore be a force acting on the particle corresponding to the circular trajectory. A particle at rest at fixed position outside the Earth must therefor constantly be pushed away from the Earth.

Figure 2.8: We see how the trajectory of a particle at rest is curving away from what is straight

Consider yourself, sitting on a chair. In Newtonian theory the gravitational force downwards on your body is exactly cancelled by an equally big force upwards from the chair. Therefore we do not accelerate. In Einstein’s theory however there is only one force acting on us, namely the force from the chair. This means that we are accelerating upwards all the time. It may seem paradoxical how we can be accelerating upwards all the time, without going anywhere. It is however just like the car driving around on the spacetime. Although it is turning right all the time, it is never going to the right!

2.4. The full spacetime

2.4

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The full spacetime

So far we have described the spacetime of a line outside the Earth. Now let us study the spacetime of a line going through a hole in the Earth. See Fig 2.9.

Space

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Figure 2.9: A line in a hole through the Earth

If we drop a particle outside the Earth on the line it will accelerate relative the earth towards the center, pass the center and start accelerating back from the other side. It will in this way oscillate back and forth. In the ordinary flat spacetime image this will look something like that depicted in Fig 2.10. Time

To the left of the earth

Particle oscillating around the center of the earth

The in⌧ terior of the earth

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Figure 2.10: Particle oscillating around the center of the Earth

Newton would explain the motion by a gravitational force directed towards the center of the Earth. Now we will instead explain it by a spacetime geometry as that depicted in Fig 2.11. Just like before, moving in time means moving around the spacetime whereas moving in space means moving to the right or to the left. In particular study a trajectory which is following the equator around the central bulge (the dark trajectory in Fig 2.11). This corresponds to a particle that is not moving in space, only in time, and is at rest at the center of the Earth. The trajectory is a also a straight line on the curved surface, since a little toy car pushed in the direction of the equator will remain on the equator. Thus the trajectory corresponds to a particle in free fall (a particle that is only affected by gravity). From Newtonian theory we would expect a particle that is placed at rest at

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2. A geometrical introduction to gravitation

Figure 2.11: The curved spacetime of a central line through the Earth

the center of the Earth to remain at rest since there would be no gravitational forces in this point. Next consider as a trajectory a tilted equator on the bulge (the light trajectory in Fig 2.11). Assuming the bulge to be exactly spherical this would also be a straight line and thus correspond to a particle in free fall. Here as we move in time the particle is first moving to the left, then reaching a maximum before moving back to the right, passing the ordinary equator, reaching a a maximum before moving back to to the left. This corresponds to a particle oscillating around the center of the Earth, just like we would expect from Newtonian theory. It completes one oscillation per lap around the spacetime. In the case of the Earth, one lap around the spacetime is roughly 84 minutes. Notice that it does not matter how much the trajectory is tilted (i.e what amplitude the oscillation has), as long as it remains within the central bulge. Whatever the tilt, it always takes one lap around the spacetime for the trajectory to move away, pass and return again to the center of the Earth. The period time for an oscillation is thus independent of the amplitude. This is also expected from Newtonian theory.

2.5

Comments and conclusions

We have now seen a geometrical explanation of why things fall when we drop them, and why particles dropped in a hole through the Earth will oscillate around the center of the Earth. The explanation was not that the Earth was affecting the thing with a force but that the thing was traveling straight forward in a curved spacetime. In a sense we have however only touched half of Einstein’s theory of gravitation. Not only does the theory explain the motion of particles, as a geometrical effect, but it also predicts how matter generates this spacetime geometry. Suppose for instance that the

2.5. Comments and conclusions

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core of the Earth had consisted of lead. Then there would have been a different matter distribution and we would have gotten a different shape of the spacetime. Matter curves the spacetime and the spacetime determines how matter moves. We have seen how one can explain gravity in two ways (at least), and one may ask which one is the correct one. There is however no such thing as a correct theory. Every theory that we come up with to explain the nature around us, is a model. We can never know if our model is a fundamentally correct theory for the Universe (if even such a model exists). We can only, by measurements, prove that a theory is wrong, never that it is right. It may however be that some day we come up with a model that is so simple and beautiful, that explains everything we have so far tested and seen, that we believe it is a fundamental theory of the Universe. Einstein’s and Newton’s theories for Gravity are both models, describing the world around us. There are however many effects that cannot be explained by Newtonian gravity, but that can be explained with Einstein’s theory. The full theory of Einstein, which we have just tasted a sip of, is much richer than just explaining how particles move. It is also explaining why clocks slow down near gravitational sources, what gravitational waves are and how the Universe can be expanding. These are effects that we do see in nature directly or indirectly. I would like to finish this introduction with an analogy between a curved spacetime and our own Earth. Think back to a time when man had not yet been convinced that the Earth was round (some still debate this) and we merely had ordinary flat maps of the world. Suppose that we did the experiment to send away the most straight-going vessel in the navy, perpendicular to the equator (towards the north). The cartographers would perhaps have noticed that relative their map the ship would turn to the right. Maybe they would have invented a theory of a special force guiding the vessel to the right. But then, they might also have gotten the idea that perhaps it was just the map that had the wrong shape and it should really have been round as depicted in Fig 2.12.

Figure 2.12: A flat map of an Earth that is really round

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2. A geometrical introduction to gravitation

3 The absolute metric The dual metric (see Paper I) is good for explaining acceleration of particles in free fall. To explain time dilation of for instance a star, with a rotational surface, we should however rather have a cylinder with a dip in the middle than a bulge as in the dual scheme. The Schwarzschild time should still be the azimuthal angle, and the proper time elapsed for an observer at rest should be the Euclidean distance traveled on the local circle (corresponding to a fixed position in space). We thus want a rotational body √ where the radius is proportional to gtt . We can also make the surface so that the proper distance (the distance to walk) between two adjacent spatial positions (i.e circles in the spacetime) is the correct one.

3.1

The absolute line element

We realize that the scheme outlined above corresponds to embedding the geometry we get if we simply flip the sign of the spatial part of the metric. Suppose that we have the line element of a central line through a star: dτ 2 = gtt dt2 + gxx dx2

(3.1)

Here gxx will be negative. Now we consider the metric: ds2 = gtt dt2 + |gxx |dx2

(3.2)

An embedding of this metric for a constant proper density star is depicted in Fig 3.1. In this picture the proper time traveled for an observer at rest is exactly the Euclidean distance on the surface. Also spatial distances as experienced by an observer at rest are exactly correct, except for a minus sign. This means that photons everywhere move at exactly 45◦ to the local spatial line. 13

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3. The absolute metric

Figure 3.1: An hourglass embedding of the absolute spacetime of a central line through a star

We can also find the proper distance separating two infinitesimally displaced events on the surface. Given the direction of an arrow pointing from one point to the other, we know the gamma factor connected to the two events with respect to an observer at rest. Thus, knowing a little Special Relativity, we can evaluate any proper distance on the surface, just from the shape of the surface 1 . The key point is however that we see gravitational time dilation. The timelike distance traveled per lap around the spacetime is shorter in the center of the star than outside. To see what this means let an observer, far outside the star, send two photons separated by one lap, to the center of the star. They will arrive at the center of the star still separated by one lap. An observer in the center of the star, will however experience a shorter time separating the two photons than the observer at infinity. We understand that time inside the star runs slow relative time at infinity. Alternatively, we see from Fig 3.1 that the lines of constant Schwarzschild time are lying closer to each other inside the star. An observer inside the star will therefor experience a local Schwarzschild clock to tic a lot faster than his own. In other words he will see the Universe outside the star evolving at a faster rate than that experienced outside the star. I have this vision of standing at some point with big time dilation and looking at galaxies flying by in the sky, like clouds in a storm2 .

3.1.1

Black hole embedding

Consider now a black hole rather than a star. On the outside we would expect the radius of the the rotational surface to go to zero at the horizon, corresponding to infinite time 1 It is sufficient that we have the surface and that we know that moving in the azimuthal direction is timelike motion with proper time equaling the Euclidean distance and that moving along the surface is orthogonal to moving azimuthally and that proper spatial distance along the surface are equaling the Euclidean distance 2 And then everything goes black - and you didn’t get any action except on TV ...

3.1. The absolute line element

15

dilation. We can however also consider the inside of the black hole. Suppose that we create a new line element by taking the absolute value of both gtt and gxx : ds2 = |gtt | · dt2 + |gxx | · dx2

(3.3)

This line element is time independent and positive definite. We should therefor be able to embed it as a rotational body where distances exactly along the surface or purely azimuthal distances are exactly right (up to a minus sign). Using the Schwarzschild black hole line element we easily get the embedding depicted in Fig 3.2.

Figure 3.2: An embedding of the absolute spacetime of a central line through a black hole Both on the inside and the outside, azimuthal angle means Schwarzschild time. Also photons move on 45 degrees both inside and outside. On the inside however moving along the surface is timelike motion contrary to the situation on the outside. We notice that the singularity is not in the picture. While the distance to walk along the internal trumpet from the horizon to the singularity is finite 3 , the embedding radius is infinite at the singularity. Thus we stand no chance of embedding the singularity in this scheme. Let us zoom in on the internal geometry as depicted in Fig 3.3. From symmetry we know that (or see that) following a Schwarzschild time line inside the black hole is timelike geodesic motion. Study then the trajectories of two constant Schwarzschild time lines, starting close to each other near the horizon and extending towards the singularity. The corresponding two observers will to lowest order in the separation between them be at rest with respect to each other in the beginning. As they approach the singularity however they will drift further and further apart. At the singularity, where the embedding radius is infinite, they will be infinitely separated. It is then easy to imagine that the tidal forces between particles approaching the singularity will be infinite. 3

Since we know that it takes a finite eigentime to reach the singularity once inside the horizon

16

3. The absolute metric

Figure 3.3: The absolute internal spacetime of a central line through a black hole

3.1.2

Comment regarding geodesics

Freely falling particles will in this scheme not follow the shortest Euclidean path on the surface. However, we know the true proper distance between points from the shape of the surface. We also know that freely falling particles will choose a way such that the eigentime experienced maximized. Consider then two events separated by some finite azimuthal angle only. It is easy to imagine that a particle traveling between the two events, will gain eigentime by moving out towards a bigger embedding radius, before moving back to the second event. On the other hand it cannot move out too fast since then special relativistic time dilation will slow down the internal clock. Since relativistic time dilation depends to second order on the velocity and the gravitational time dilation is proportional to the radius, it seems very plausible that the path that maximizes the eigentime will indeed be a trajectory that first moves out and then back, like we indeed see in physics.

3.2

Generalization to arbitrary spacetimes

The scheme used in the preceding section was quite specific for that particular metric written in those particular coordinates. There is however a way to generalize the absolute scheme. Given an arbitrary Lorentzian spacetime, we specify a field of four-velocities uµ (x). We then make a coordinate transformation to a local Minkowski system comoving with the given four-velocity 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. Together with the absolute metric, we still need the vector field uµ (x) to keep track of what is timelike and what is not. Doing so, we maintain all information about the

3.3. Freely falling observers as generators

17

original spacetime. We can always do the backwards transform and flip Pythagoras into Minkowski. Since uµ is normalized we have N − 1 degrees of freedom, where N is the total number of dimensions, in generating the absolute metric. In particular studying a 1+1 dimensional spacetime this means that the absolute metric may be specified by a single function of x and t. In the preceding section the generators were simply the Schwarzschild observers at rest 4 . Notice that the observers standing just outside the horizon have infinite proper acceleration. It is then perhaps not surprising that the resulting embedding is singular at the horizon. In fact, by instead considering freely falling observers as generators, we could hope to better resolve the horizon.

3.2.1

A covariant approach

Suppose that with respect to a to a metric and a field of four-velocities, we want to find the absolute metric, from now on denoted g¯µν . We may then go to a system comoving with uµ , insert a Euclidean metric and then transform the metric back to the original coordinates. This is how I first did it. There is however a faster, more elegant way, using tensors. We know that the absolute metric is a tensor and in a comoving frame we have: 

g¯µν =

       

1 0 0 0

0 1 0 0

0 0 1 0

We realize that we must have:

0 0 0 1

        



=

    −   

1 0 0 0

0 -1 0 0

0 0 -1 0

0 0 0 -1





         + 2      

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

        

(3.4)

dxµ (3.5) dτ Notice that both sides of the equality are tensors that equal each other in a certain system, thus they are equal in every system. g¯µν = −gµν + 2 · uµ uν

3.3

uµ = gµν

Freely falling observers as generators

As an example, consider freely falling observers initially at rest at infinity as generators. We also assume a Schwarzschild black hole 1+1 original metric. Using the squared Lagrangian formalism for the equations of motion we readily find the four-velocity of the generating freefallers: ' ( ' ( x 1 1 µ u = ,− ⇒ uµ = 1, (3.6) x−1 x x−1 4

Inside the horizon it was the t = const-observers that were generators

18

3. The absolute metric

The absolute metric is then according to Eq (3.5):    

g¯µν = 

1+

√ 2 x x−1

1 x

√ 2 x x−1

x(x+1) (x−1)2

    

(3.7)

To make an embedding of this metric we are wise to first diagonalize the metric by the coordinate transformation: T = t + φ(x)

g¯tx dφ = dx g¯tt

where

(3.8)

This gives us the line element in the new coordinates as: )

g¯2 ds = g¯tt · dT + g¯xx − tx g¯tt 2

2

*

· dx2

(3.9)

Inserting the explicit absolute metrical components we find: '

ds2 = 1 +

(

'

1 1 · dT 2 + 1 + x x

(−1

· dx2

(3.10)

This metric is easy to remember since it happens to be exactly the Schwarzschild metric but with the minus signs exchanged to plus signs 5 . Notice in particular that nothing special happens with the metrical components at the horizon (x = 1). At the singularity (x = 0) however the absolute metric goes 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. This we do by taking the trajectories as calculated in the Schwarzschild metric, and coordinate transforming the trajectories using Eq (3.8): dx = dT 1+

dx dt g¯tx g¯tt

·

dx dt

(3.11)

We may then numerically solve for x(T ) or T (x). In a similar manner we solve for the lines orthogonal to the freefallers, the local simultaneity lines of the generators (see Section3.6). In Fig 3.4 the freely falling absolute geometry is depicted. Notice how the local Minkowski systems are twisted on the surface. The horizon lies exactly where the generating worldlines are at 45◦ to a purely azimuthal line. Time dilation of observers at rest is now not solely determined by the local embedding radius, but also by the gamma factor of the observer at rest relative the generating observer. This can be calculated directly from what angle the world-line of the observer at rest makes with the generating world-line. For instance an observer at rest 5

Except for the inversion minus sign on the dx2 -term

3.4. On geodesics

19

Figure 3.4: The absolute freefaller geometry. The darker area lies within the horizon.

at the horizon will be at 45◦ to the generating observer, corresponding to an infinite gamma factor, and his clock will therefor not tick at all during a Schwarzschild lap (one circumference), thus being infinitely time-dilated. It turns out that the freely falling generators are geodesics also in the absolute spacetime. This is generally true for freely falling generators as is proven in the following section.

3.4

On geodesics

Studying geodesics we may want an expression for the new absolute affine connection, i.e the affine connection for the absolute metric. For this we need the inverse absolute metric which is given by : g¯µν = −g µν + 2uµuν

(3.12)

Raised indexes on unbarred objects means that they are raised with the original metric. That this is indeed the inverse is easy to verify. It can also be derived straight-forwardly, using the contravariant analogue of the method used in Section 3.2.1. It is a little surprising however that we get the inverse of the new metric by raising the indices with the original metric. This is a unique feature of this particular metrical construction. If we had instead considered a metric of the form −g µν + α · uµ uν , were α is some general number, the inverse would have been −g µν + α/(α − 1) · uµ uν . It is only in the case α = 2 that we can simply raise with the original metric to get the absolute metric. Incidentally we may notice that also in the raising of uµ the new and the old metrics make an equally good job: g¯µν uν = (−g µν + 2uµ uν ) uν = −uµ + 2 uµ = uµ

(3.13)

20

3. The absolute metric

Knowing the inverse absolute metric, the absolute affine connection is given by: ¯µ = Γ αβ

1 2

(-g µρ + 2uµ uρ) ·

(3.14)

(∂α (-gρβ + 2uµ uν ) + ∂β (-gρα + 2uµ uν ) − ∂ρ (-gαβ + 2uα uβ )) From now on objects connected to the absolute metric will be barred. We may rewrite Eq (3.14): ¯ µ = Γµ − 2uµ uρ · Γραβ Γ αβ αβ

(3.15)

+ (-g µρ + 2uµ uρ) (∂α (uρ uβ ) + ∂β (uρ uα ) − ∂ρ (uα uβ ))

3.4.1

Proof regarding geodesic generators

In the new metric, proper distances will in general be very much different from those in the original metric. Distances measured along a generating congruence line will however be the same (these are unaffected by the sign-flip). In particular this means that: dxµ dxµ u = = = u¯µ dτ d¯ τ µ

(3.16)

Then the covariant four-acceleration of the generating congruence lines in the absolute metric can be expressed as: ¯ uµ D¯ duµ ¯ µ α β ¯ τ = dτ + Γαβ u u D¯

(3.17)

Assume now that the generating observers are geodesics in the original metric. We suspect that they are then geodesics in the absolute metric as well. To prove this it is sufficient to show that the absolute covariant four-acceleration is vanishing in one system (it then vanishes in all systems). Let us therefore evaluate Eq (3.17) in that particular originally freely falling system that is exactly 6 comoving with the reference observer uµ . In this system the original affine connection vanishes. Also any derivative of the form uα ∂α has to vanish since it equals dτd in the direction of the preferred observer, in which metrics and four velocities are constant, Minkowski and (1,0,0,0) respectively. 7 ¯ µ uα uβ in this particular coordinate system we are left simply with: Thus forming Γ αβ ∗ µρ α β ¯ µ uα uβ = Γ g u u ∂ρ (uα uβ ) αβ

(3.18)

6 Meaning that we fasten the spatial origin of the freely falling system exactly on the freely falling observer 7 If the generating observers had not been geodesics, and we had chosen that particular freely falling coordinate system that was locally comoving with the freely falling observer, uα ∂α um , would in general be non-zero

3.5. Covariant approach to photon geodesics

21

∗

Here = means that the equality holds in an originally freely falling system comoving with the generator 8 . Notice however that we have, in general coordinates: uκ uκ = 1

⇒

∂ρ (uk uκ ) = 0

⇒

uk ∂ρ uκ + uk ∂ρ uκ = 0

(3.19)

Then we may write: ∗

∗

uκ ∂ρ uκ = −uκ ∂ρ uk = −uκ ∂ρ (gκλ uλ ) = −uκ gκλ∂ρ uλ = −uλ ∂ρ uλ

(3.20)

Again using Eq (3.19) we have then: ∗

uκ ∂ρ uκ = 0

(3.21)

Then we see trivially that the right hand side of Eq (3.18) vanishes in any originally µ = 0 we find from Eq freely falling system. Since also in these particular coordinates du dτ (3.17) that: ¯ uµ ∗ D¯ ¯τ = 0 D¯

(3.22)

The left hand side being covariant we know that the equality hold also without the star over the equality sign. So the covariant four-acceleration of a freely falling generating congruence line, vanishes also with respect to the absolute metric. It is thus a geodesic also in the absolute metric. Incidentally, from Eq (3.21) we realize that: uκ $ρ uκ = 0

(3.23)

uκ ∂ρ uκ = +uκ Γλ ρκ uλ

(3.24)

This implies that:

3.5

Covariant approach to photon geodesics

Out of curiosity I would like to see to what extent we can make photons move on geodesics in the absolute space. Of course we cannot simply make the congruence of observers (uµ ) follow the photon geodesics. Since photons do not experience eigentime, the resulting absolute metric would loose one dimension in a sense. All points along a certain photon congruence would be separated by a zero distance. However, maybe we can choose, an in general non-geodesic uµ , that makes some photons retain their geodesic property. In particular I am interested in to what extent we can do this in 1+1 dimensions. Let us however start in a general dimensionality. 8

I will later on use a star also when the equality holds in any originally freely falling system

22

3. The absolute metric In a freely falling system we may write, using Eq (3.15): ∗ α β ¯ µ f αf β = Γ f f (-g µρ + 2uµ uρ ) (∂α (uρuβ ) + ∂β (uρ uα ) − ∂ρ (uα uβ )) αβ ∗

= f α f β (-g µρ + 2uµ uρ ) · 2 (uβ ∂α uρ + uρ ∂α uβ − uβ ∂ρ uα )

(3.25) (3.26)

I have deliberately violated manifest α, β-symmetry within the second parenthesis for compactness. We could expand this expression further, but we may as well leave it as it is since after all the raising metric in the above expression is trivial in a system comoving ¯ µ is numerically identical to Γ ¯ µαβ in these coordinates. Thus with uµ . This means that Γ αβ let us write: +

∗ ¯ µαβ f α f β = Γ 2 · f β uβ f α ∂α uµ + uµ f β f α ∂α uβ − f β uβ f α ∂µ uα

,

(3.27)

In a comoving system photons will have an absolute four-velocity, in the point in question, given by: 1 f α = √ (1, n ˆ) 2

(3.28)

Also, from the normalization of uµ follows that: ∗

0 = ∂α (gµν uµ uν ) = ∂α ((ut )2 − (ux )2 ) ∗ = 2ut ∂α ut − 2ux ∂α ux ∗ = 2∂α ut

(3.29) (3.30) (3.31)

Thus in a locally comoving freely falling system, all derivatives of ut are zero (not so for ux in general however). Remembering to henceforth treat f α as a constant vector we may move it in over derivatives and rewrite Eq (3.27): +

,

∗ ¯ 0αβ f α f β = Γ 2 · f α ∂α (f i ui) − f 0 ∂0 f i ui = f j ∂j (f i ui)

+

∗ ¯ kαβ f α f β = 2 · f 0 f α ∂α uk − f 0 ∂k f i ui Γ

Here Latin indices take the values 1, 2, 3.

3.5.1

,

(3.32) (3.33)

Photons in 1+1 dimensions

√ Specializing to 1+1 dimensions and f α = 1/ 2 · (1, 1) we find: ∗ ¯ t f αf β = Γ ∂x ux αβ ∗ x α β ¯ f f = ∂t ux Γ αβ

(3.34) (3.35)

We know that in a freely falling system the photon will to second order move on a straight line. This means that the absolute acceleration perpendicular to the photon trajectory vanishes in this system i.e. d2 ∗ (t − x) = 0 2 ds

(3.36)

3.6. Photon geodesics in static spacetime

23

I will from now on sometimes use ds rather than d¯ τ for absolute distance. For the photon to be following a local geodesic in the absolute spacetime, we must then have: ¯ t f αf β + Γ ¯ x f αf β = 0 −Γ αβ αβ

⇒

∗

∂x ux − ∂t ux =0

(3.37)

So this has to hold in the comoving originally freely falling system. Now, can we make this into a covariant statement, that applies in any coordinate system? Indeed we can. Defining v µ to be a vector orthogonal (both originally and absolutely) to uµ and absolutely normalized (¯ gµν v µ v ν = 1), we see that the covariant expression that reduces to Eq (3.37) is given by 9 : v ν v µ $µ uν − v ν uµ $µ uν = 0

(3.38)

In two dimensions v µ is given up to a sign by the original metric and the generating fourvelocities. The sign determines whether we are studying ingoing or outgoing photons. Notice that if we want also infalling photons to be moving on geodesics we would get the constraint: ∂x ux + ∂t ux = 0

(3.39)

This together with Eq (3.37) and ∂α ut = 0 means that all the derivatives of the fourvelocity of the generating observers relative the freely falling system would have to vanish. This would mean that all the derivatives of the absolute metric would vanish as well, making all original geodesics to absolute geodesics. My guess is that this can only happen at every spacetime point, if the spacetime is flat. Thus I draw the conclusion that in a non-trivial spacetime we cannot get all originally geodesic photons to move on absolute geodesics.

3.6

Photon geodesics in static spacetime

We found in Section (3.5) a covariant differential relation, Eq (3.38), usable to find generators such that outgoing photons move on geodesics in a general 1+1 dimensional spacetime. Assuming the original metric to be time independent, and letting the generators be time independent as well, it is however more clever to start over from the beginning. We instead use the power of the Lagrangian formalism, and constants of the motion. The goal is to find an embedding were outgoing photons move on geodesics, and were the horizon is included. In general in 1 + 1 dimensions we have only one freedom in uµ since it must be normalized. A clever choice of parameter for this freedom may be to choose the velocity v of the generating observer relative a freely falling observer originally at rest at infinity, from now on denoted reference observer. That way we have a parameter that is ranging 9 I have incidentally generalized this, in 1+1 dimensions, to apply to an arbitrary direction, not necessarily a photon: (f α vα )3 · v ν v µ $µ uν − (f α uα )3 · v ν uµ $µ uν

24

3. The absolute metric

from -1 to 1, that hopefully has an absolute value smaller than 1 at the horizon 10 . For simplicity let us restrict ourselves to the case where the relative velocity v only depends on x.

3.6.1

The reference freefaller coordinates

For the task at hand it may be clever to express the Schwarzschild line element in coordinates in which it is possible to pass the horizon. There is a certain coordinate system which is ideally suited for reference observers. The line element in these coordinates is given by: '

1 dτ = 1 − x 2

(

2 dT 2 − √ dxdT − dx2 x

(3.40)

These coordinates are related to the standard coordinates via: T = t + φ(x) x=x

(3.41) (3.42)

In principle it is just a resetting of the Schwarzschild clocks. The function φ(x) is such that we have the following nice features for reference observers: • A constant T -line is a local simultaneity line for a reference observer • The eigentime experienced by the reference observer between events on his worldline, is exactly the difference in T between the events • The spatial distance experienced by the reference observer between infinitesimally displaced events separated only in x is exactly dx • For the freefaller we have

dx dT

= − √1x

• An observer moving with velocity v relative the reference observer has v

dx dT

= − √1x +

In particular this means that the local gamma factor of a particle moving relative the local reference observer is simply dT /dτ which will be used below.

3.6.2

The absolute metric

Using the nice properties listed above we readily find the four-velocity of an observer moving with speed v relative a reference observer: uµ = ( 10

dT dx , ) dτ dτ

As opposed to if we had chosen, as a parameter, the velocity relative an observer at rest

(3.43)

3.6. Photon geodesics in static spacetime

25

dx dT (1, ) dτ dT 1 1 = √ (1, v − √ ) 2 x 1−v

(3.44)

=

(3.45)

Lowering the four-velocities with the metric we find: 1 v (1 − √ ) 2 x 1−v −v ux = gxt ut + gxx ux = ... = √ 1 − v2 ut = gtt ut + gtx ux = ... = √

(3.46) (3.47)

From this we may immediately verify that uµ uµ = 1. We have the general expression for the absolute metric: g¯µν = −gµν + 2uµuν

(3.48)

Then we can form the absolute metric as a function of the parameter v: '

(

)

*2

1 1 v g¯tt = − 1 − +2· 1− √ 2 x 1−v x ) * 1 v 1 1− √ · (−v) g¯tx = √ + 2 · 2 x 1−v x 1 · v2 g¯xx = 1 + 2 · 1 − v2

(3.49) (3.50) (3.51)

Now remains to find v, and thus the absolute metric, such that the outmoving photon, whose trajectory is known, is a geodesic in the new absolute metric.

3.6.3

Equations of motion in a general 1+1, time independent metric

Suppose that we have a metric of the form: ds2 = a(x) · dT 2 + 2b(x) · dT dx + c(x) · dx2

(3.52)

Using the squared Lagrangian formalism we find that geodesics obey: a

dx dT +b =K ds ds

(3.53)

Here K is a constant of the motion. Introducing σ = 1/K 2 , and using Eq (3.52), we readily find: )

dx dT

*2

(σb2 − c) +

dx · 2b(σa − 1) + a(σa − 1) = 0 dT

(3.54)

26

3.6.4

3. The absolute metric

Outmoving photons

For an outmoving photon we have: 1 dx = −√ + 1 dT x

(3.55)

Inserting this into Eq (3.54) together with the explicit expressions for the metrical components as functions of v (Eq (3.49),Eq (3.50) and Eq (3.51)), we get a quite nasty expression. At first sight it appears to be a fourth order equation in v. The expression may however be drastically simplified, using Mathematica, and we can easily solve for v. The result is: 4x √ 2 σ(1 + x) + 2x

v =1−

(3.56)

So, demanding outgoing photons to be geodesics yields v uniquely as a function of x up to a single parameter σ. This parameter determines the velocity at infinity through: v∞ = 1 −

4 σ+2

(3.57)

We see from Eq (3.56) that v lies in the range [-1,1] for all values of x and σ (sigma is positive since it is the square of a real number). Choosing to have v = 0 when x goes to infinity immediately yields σ = 2. Then we have v explicitly as a function of x which we may insert into the expressions for the absolute metric.

3.6.5

Embeddings

To easily embed the new metric we are wise to first diagonalize it. A general metric of the form Eq (3.52) is easily diagonalized using the coordinate transformation: T & = T + f (x)

where

b df = dx a

(3.58)

*

(3.59)

The line element in the new coordinates is given by: )

b2 (x) ds = a(x) · dT + c(x) − · dx2 a(x) 2

&2

With the metric in this form it is an easy task to embed it as a rotational surface. To plot the lines of the generators and photons moving on the surface we could use the same technique as that used for the freefallers Eq (3.11). From a numerical point of view however it is better to parameterize both x and t with s, the proper absolute distance

3.6. Photon geodesics in static spacetime

27

along the curve 11 . The four-velocity transforms as a vector under the diagonalization. Thus we readily find: b t u& = ut + ux a x u& = ux

(3.60) (3.61)

Knowing ut (x), ux (x), b(x) and a(x) we can easily numerically integrate these expressions to plot the various world lines. To get a nice grid on the surface we need also to calculate the lines orthogonal to the generators, the lines that are local simultaneity lines of the generators. Letting v µ be a vector determined by gµν v µ v ν = −1 and gµν uµ v µ = 0 we find: )

1 v · v, 1 − √ v = (±) √ 2 x 1−v µ

*

(3.62)

This is a vector that is normalized to unity in the absolute metric (minus unity relative the original metric), and is orthogonal to the generator also in the absolute metric. It is thus the absolute four-velocity of the lines orthogonal to the generators and we may solve for these lines just like we solved for the generator lines above. In Fig 3.5 we show the embedding with the lines of the generators together with outmoving photons.

Figure 3.5: The absolute space using generators uµ (x) such that outmoving photons move on geodesics. The embedding covers the range x'[0.55, 4]. The thick lines with arrows correspond to photons moving out. The left one, being inside the horizon is however forced to follow the geometry into the singularity. Notice 11 That way there is no problem in dealing with trajectories that are first increasing in t! and then decreasing in t! for instance

28

3. The absolute metric

how the photons stay at exactly 45◦ to the generating grid. Notice also that the horizon is exactly where there is a minimum in the embedding radius. This is of course necessary if we want the photon to remain on a certain spatial position and at the same time move on a geodesic on the surface. Also we knew in advance that it would have to be a be a minimum lest a photon could oscillate back and forth around the horizon. It was however not obvious that the scheme would at all be successful. Like before we can embed the internal black hole, but not all the way into the singularity. It is however a little bit unfortunate that the absolute spacetime for these particular generators has so little shape. Demanding photons to move at 45◦ at infinity we know from the equations of√ motion on a general rotational body (see Paper I) that the radius at the neck must be 1/ 2 times the radius at infinity. This seems to imply quite a lot of shape. The radius however turns out to increase quite slowly with increasing r, giving the embedding this, close to cylinder-like, appearance. It may be that we can create more shape by considering other generator-velocities at infinity. In fact we may realize that a positive v at infinity would decrease the radius of the neck. Remember that the spacetime in these pictures is like a toilet roll. When you walk once around the roll you are on a new layer. The outgoing photon should therefore really just be seen in one layer, unless we rip the toilet paper in a clever way. In any case, we should not draw the conclusion that we have three photons moving outwards outside the horizon in the embedding.

3.7

Flat embeddings

Having seen one set of generators produce a spacetime where the radius increases towards the horizon and another where it decreases, it is natural to ask whether we can find generators such that the radius is constant. This is very easy to check, since it corresponds to having g¯tt (v) = C, where C is some constant. Using g¯tt (v) from Eq (3.49) we may readily solve the second order equation in v to find: v=

√ 2 x ± −1 + 2x + x2 (C 2 − 1)

1 + x · (1 + C)

(3.63)

So there are two possibilities for every given C. Similar to when we demanded outgoing photons to be geodesics we have one free parameter determining the generator velocity at infinity. In this case we see that: √ v∞ = ± C − 1 (3.64) We see from Eq (3.63) that if we choose C < 1 the root will go imaginary for sufficiently large x. Solving in x for when the root is zero yields: x=

1 1−C

or

x=

1 1+C

(3.65)

3.8. Spacelike generators?

29

I thus appears that choosing C sufficiently large the cylinder embedding will exist from infinity to arbitrarily close to the horizon. In particular however, choosing g¯tt = 1 at infinity (C = 1) the cylinder will exist into half the Schwarzschild radius. We should however also verify that v stays in the interval [-1,1]. I haven’t analyzed it properly but it appears to stay in this interval whichever x and C we have, given that the root is real.

3.7.1

Comments

Notice that for this kind of visualization the curvature of spacetime is manifested solely as a twist of the local Minkowski systems relative each other. In this picture we can of course also explain why particles in free fall move the way they do. In particular study a particle moving between two events separated only in the azimuthal direction on the cylinder. The task is to find a path that maximizes the eigentime. Moving out is of course good, because then you move to a region where the twist of the local Minkowski systems is smaller, and thus pure azimuthal motion means smaller gamma factor. On the other hand moving out means moving closer to the light-cone which increases the gamma factor. Which one of the effects that would win is perhaps not obvious in this case since the gamma factor is not quadratic in the change of velocity now (as opposed to the case in the hourglass embedding), but a first order effect. In any case, assuming the events to be separated by a vast distance in the azimuthal direction, we immediately understand that it’s preferable to move out to a lower gamma factor before moving back, corresponding to the real physical effect. Notice that the cylinder we are here considering is intrinsically flat. It is somewhat amusing that we are in a sense back to where we started, with a flat surface. Now the geometry is instead represented by a velocity. Unfortunately, just thinking in terms of degrees of freedom, this will not in general work in higher dimensions.

3.8

Spacelike generators?

Out of curiosity we may ask ourselves what would happen to the absolute metric if uµ was spacelike? Assuming |uµ uµ | = 1 we can evaluate the absolute metric in that particular local Minkowski system that has uµ directed along its x-axis. In this system we have: uµ = (0,

dx dx , 0, 0) = (0, , 0, 0) dτ i · ds

→

1 uµ = (0, − , 0, 0) i

(3.66)

30

3. The absolute metric

Then we find:



g¯µν =

1   0 −  0 

0 -1 0 0 0

0 0 -1 0





0 0     0 0 +2   0 0   -1 0

0 -1 0 0

0 0 0 0





 -1 0     0  0 =  0   0   0 0

0 -1 0 0

0 0 1 0



0   0  0   1

(3.67)

In this case we thus get a metric with two time directions. In fact it is those two spatial directions that is orthogonal to the one prescribed by the spacelike generator, that becomes the new timelike directions. This is not particularly exiting but included for completeness.

3.9

A mathematical remark

Having seen the neat covariant expression of Eq (3.38), regarding photon geodesics, I start wondering whether there is a more general covariant relation that reduces to this one in the particular case of photons in two dimensions. For a given particle trajectory of four-velocity q µ , and a given generating field, we µ ¯ q¯µ have a well defined absolute four-velocity q¯µ . We know that DDs and Dq are both ¯ Dτ covariant objects, describing more or less the same thing. It seems plausible that we could relate the two quantities in some manifestly covariant way, probably involving covariant derivatives of the generating field. A good starting point for the search for such a relation is of course to first relate q¯µ to q µ in a manifestly covariant way. This is straightforward as can be seen below. . /

/ dxµ dτ dxµ gµν dxµ dxν = · = qµ · 0 q¯µ = ds dτ ds −gµν dxµ dxν + 2uµ uν dxµ dxν . / = qµ · / / 0

1

−1 + 2uµ uν

(3.68)

dxµ dxν dτ dτ

1 = qµ · 2(uµ q µ )2 − 1

So now we have a manifestly covariant expression relating the four-velocities. Notice that choosing q µ = uµ yields u¯µ = uµ as we realized earlier. Also if we choose q µ to be orthogonal to uµ the four-velocities are related simply via an imaginary i, as expected. ¯ q¯µ Evaluating DD¯ ¯ τ in an originally freely falling system where the original affine connection vanishes, and then setting all unbarred derivatives to their covariant analogue we get (see Eq (3.26)): 



¯ q¯µ D  D qµ 1  = ¯τ µ 2 µ 2 D¯ Dτ 2(uµ q ) − 1 2(uµq ) − 1

(3.69)

3.10. Comments

+2

31 qαqβ (−g µρ + 2uµ uρ ) (uβ $α uρ + uρ $α uβ − uβ $ρ uα ) 2(uµ q µ )2 − 1

So here we have a manifestly covariant relation. The result is however not particularly illuminating, though it is possible that it can be simplified. A small remark regarding the backwards transform from the absolute metric to the original metric. Notice that, since u ¯µ = uµ and u¯µ = uµ , we have a perfect symmetry in going from the Lorentzian system to the absolute system and vice versa: g¯µν = −gµν + 2uµ un

⇒

gµν = −¯ gµν + 2¯ uµu¯n

(3.70)

Completely symmetric.

3.10

Comments

A nice application of the absolute metric scheme would be to apply it to a metric that is time dependent. For instance one may study a radial line through a spherical shell of matter that is collapsing. In fact the desire to see how this would look was the main reason for the generalization of the absolute scheme. I haven’t done it yet. It would however be nice for then one would use more fully the freedoms of two-dimensional surfaces 12 . After all, the rotational bodies that I have produced so far may be described by a single one-dimensional function, namely the embedding radius. Another nice application would be the interior Kerr hole. There are some pretty interesting regions in there that may perhaps be nicely visualized using the absolute scheme. In fact one of the major advantages of the absolute scheme, relative the dual scheme, is that we may visualize the structure of any two-dimensional spacetime slice taken in the full spacetime. With the dual metric scheme this was only meaningful if the slice could “contain” a set of geodesics. For a general slice, particles in free fall will in general not stay in the slice but will pass it, making the dual metric of the slice more or less meaningless. Another advantage, as well as a disadvantage, of the absolute scheme relative the dual scheme is the big freedom in choosing the generators. Depending on what generators we choose, the embedded surface will be very much different. The physics of the surface however, will of course not depend the choice . Remember that the shape is not all, we need also the lines telling us the directions of the local Minkowski systems. In the dual metric scheme we had three parameters determining the embedding while in the absolute scheme we have a whole function. This freedom is good since it increases our chances of producing esthetical pictures with explaining power, while it is also bad since it lacks absoluteness (pardon the small joke). What I like the most with the absolute scheme is the scale invariance. Just like in the dual metric scheme we have local Minkowski systems sitting on the surface. In the 12

I wonder what generators that would be clever to use in this case?

32

3. The absolute metric

dual scheme however the scale of these systems was varying depending on position, and was also not “isotropic” (the opening angle of the light cone was not in general 90◦ ). In the absolute scheme however all the local Minkowski systems have the same scale, we just need a direction to get all the local proper distances right. The absolute scheme is also well suited to display cosmological concepts such as open and closed Universes, Big Bang and Big Crunch, preferred velocities and how it is possible that there was no time “before” the Big Bang. A more detailed discussion of this will however have to wait until I put the absolute scheme in a form more suited for a general audience. Really the dual and the absolute schemes have different virtues depending who is looking at the respective embeddings. For the non-physicist, who doesn’t know about negative distances and Minkowski systems, the dual metric is certainly the more powerful one when it comes to giving an understanding of gravitation as a geometrical feature rather than as a force. For the general relativist however the absolute scheme may be more interesting. If for nothing else, because it is covariant and is applicable to any original Lorentzian spacetime. Lastly some pure speculation, mainly inspired from the dual scheme. Imagine if various particles would correspond to various geometrical signatures, so maybe fermions are in fact just dual particles! Typically the border of the particle would be defined by where the signature changes to Lorentzian (like it does in the artificial singularity in the dual scheme). In fact imagine if all physics is geometry, but not of a fixed signature necessarily?

4 Metrics, geodesics and affine connections In this thesis and in particular in the dual scheme, geodesics are at the center of the discussion. In the dual metric scheme I am, given the geodesics, looking for possible metrics to create these. This I managed in the case of a 1+1 dimensional time independent metric, using constants of the motion. In this chapter I make a more general analysis, for arbitrary dimensions and metrics.

4.1

Finding the metric from the geodesics

The geodesic equations of motion on a manifold bestowed with a metric can be written (see D’Inverno [5] p. 100-101): d2 xµ d2 τ /dλ2 µ µ α β x˙ + Γ x ˙ x ˙ = αβ dλ2 dτ /dλ

here

x˙ µ ≡

dxµ dλ

(4.1)

Here λ is the parameter used to parameterize the worldline in question and τ is proper distance. We have Γµαβ : Γµαβ

)

1 ∂gαρ ∂gβρ ∂gαβ = g µρ + − 2 ∂xβ ∂xα ∂xρ

*

(4.2)

Suppose now that we have chosen some coordinates (imagine a Cartesian coordinate space). In these coordinates we have through every point and in every direction a geodesic. Suppose that there is a metric that generates these geodesics. Can we from the geodesic curves find out what metrics 1 that could generate them ? 1 For any metric that produces the correct geodesics, a constant times this metric will also generate the right geodesics. The metric will thus not be unique.

33

34

4. Metrics, geodesics and affine connections

4.1.1

Coordinate curvature

We thus want to find a metric that gives the right curvature for every geodesic at every point. How then do we specify the curvature of a geodesic line? Normally we would 2 µ use DDτx2 , the four-acceleration, but we do not know the metric so do not know the proper distance τ . Therefor this measure of the curvature is not applicable. A good way to specify the curvature of a particular geodesic in a point, would instead be to give a vector, coordinate-perpendicular (in the most ordinary sense, using a Pythagorean metric to do the dot product) to the geodesic, pointing in the direction towards which the geodesic curves, and with a length that is proportional to how fast it curves 2 Notice that this measure of the curvature is highly dependent on what coordinates that we use. In the particular coordinates that we have chosen we introduce a Euclidean coordinate metric denoted δµν , where δµν ≡ Diag(1, 1, 1, 1). Suppose that we parameterize the geodesic in question with, as I denote it, a coordinate-affine parameter λ: -

-

dλ = K dx2 + dy 2 + dz 2 + dt2 ≡ K δµν dxµ dxν

(4.3)

Here K is any constant. Then we have the metric-independent (but coordinate dependent) measure of the curvature of a single geodesic , or any worldline, simply through: d2 xµ dλ2

(4.4)

This has exactly the properties described above, when λ is given by Eq (4.4). I will call this the coordinate-curvature. Notice that this is not a tensor.

4.1.2

The geodesic equation using a coordinate affine parameter 2 µ

If we use a coordinate-affine parameter, we know that there will be no part of ddλx2 that is parallel to x˙ µ since the (Euclidean) length of x˙ µ is fixed along the curve for such a parameterization. Then we know that the right hand side of the equation of motion, which is manifestly parallel to x˙ µ , must be exactly canceled by the part of Γµαβ x˙ α x˙ β that is parallel to x˙ µ . Thus we may write: δρκ x˙ ρ Γκαβ x˙ α x˙ β µ d2 xµ µ α β + Γ x ˙ x ˙ − x˙ = 0 αβ dλ2 δηξ x˙ η x˙ ξ

(4.5)

This we may rewrite into: + , 1 d2 xµ µ ρ α β κ µ κ = δ x ˙ x ˙ x ˙ Γ x ˙ − Γ x ˙ ρκ αβ αβ dλ2 δηξ x˙ η x˙ ξ 2

This is most easily visualized in three dimensions

(4.6)

4.1. Finding the metric from the geodesics

35

Choosing a parameterization with K = 1 in Eq (4.3), giving δηξ x˙ η x˙ ξ = 1, would compactify the expression a little. It may appear that Eq (4.6) is only a necessary condition for the affine connection to produce the right geodesics. We also have the restriction from the parallel part: x˙ ρ δρκ Γµαβ x˙ α x˙ β µ d2 τ /dλ2 µ x˙ = x˙ δηξ x˙ η x˙ ξ dτ /dλ

(4.7)

This is however automatically satisfied for any affine connection obeying Eq (4.6), and every metric giving rice to this connection, assuming a coordinate affine parameterization. We should thus see this, not as a constraint, but as a piece of information and a reminder that we must still find a metric that produces the correct affine connection.

4.1.3

Equivalent affine connections

It is obvious from Eq (4.6) that all metrics that give rise to the same affine connection, will do equally well, or bad, in producing the right geodesics. It is however not obvious 2 µ that we must have the same affine connection to still get exactly the same ddλx2 as function of x˙ µ . In fact necessary and sufficient conditions for two different affine connections, Γ ˜ to yield the same geodesics are: and Γ, +

,

+

˜ κ yµ − Γ ˜ µ yκ δρκ y ρ y αy β Γκαβ y µ − Γµαβ y κ = δρκ y ρy α y β Γ αβ αβ

,

(4.8)

This must hold for all values of the parameter y µ, at every position xµ for the two affine connections to be geodesically equivalent. As an example of how different affine connections can give rise to the same geodesics, let us study a two-dimensional manifold. We also assume that the coordinate curvature, d2 xµ , vanishes everywhere and in every direction. From Eq (4.6) we find (after some dλ2 calculus) that this will be satisfied for any affine connection of the form: Γ011 = Γ100

(4.9) µ

φ0 (x ) 2 φ1 (xµ ) = 2

Γ000 = φ0 (xµ )

Γ101 = Γ110 =

Γ111 = φ1 (xµ )

Γ001 = Γ010

(4.10) (4.11)

Here φ0 and φ1 are arbitrary functions of spacetime position. So, at least in two dimensions, it is not necessary to have a vanishing affine connection to produce straight lines, and nothing but straight lines, as geodesics. In section 4.3, we will see an example of a non-constant metric that yields zero coordinate-curvature for every direction of the geodesics at a certain point.

36

4.2

4. Metrics, geodesics and affine connections

On the construction of the dual metric

While I were constructing the dual scheme I used to think that what I did was equivalent to jumping to a freely falling system and doing a sign flip of the spatial part of the metric in this system. In this section we will understand that this is not the case 3 .

4.2.1

Point-dual metrics

Suppose that we are in a freely falling coordinate system, where in one point, the metric reduces to Minkowski with vanishing derivatives 4 . Suppose that we just take this metric and flip the sign of every component except gtt , to create a new metric. The new metric is Pythagorean with all derivatives still vanishing, in the point in question. When the derivatives of the metric vanishes, the affine connection also vanishes, see Eq (4.2). This means that both the ordinary and the new metric (if we construct it in this way) have all components of the affine connection equal to zero. This means that they produce the same geodesics, in the point in question. We may say that the new metric is point-dual to the original metric. Notice that under coordinate transformations (Weinberg [3] p. 100), the affine connection transforms according to: Γ&λ µν =

∂x&λ ∂xτ ∂xσ ρ ∂x&λ ∂ 2 xρ Γ + ∂xρ ∂x&µ ∂x&ν τ σ ∂xρ ∂x&µ x&ν

(4.12)

Notice that the first term on the right hand side is zero when the affine connection is zero. The second part depends only on the coordinate transformation itself. This means that if we construct a point-dual metric according to the reasoning above, the pointdual affine connection would equal the ordinary affine connection in any coordinate representation. In the dual metric scheme of Paper I, the dual affine connection does not equal the ordinary affine connection. The explanation is, that we do not get the dual metric by simply flipping the sign of the spatial part in the freely falling coordinates. To get a dual affine connection which is different from the ordinary affine connection, the dual metric in the originally freely falling system must have non-vanishing derivatives. That this is actually the case is shown in Section 4.3.

4.3

On the dual metric in freely falling coordinates

We understood in the preceding section that the dual metric in freely falling coordinates does not equal Pythagoras with vanishing derivatives. Out of curiosity I would like to 3 Having since then constructed also the absolute scheme, I know the limitations of this technique when it comes to geodesics. 4 Compare with Chapter 3 regarding the absolute metric.

4.3. On the dual metric in freely falling coordinates

37

know what the dual metric, given by Eq (10) and Eq (11) in Paper I, does look like in a originally freely falling coordinate system. This will also serve as check of the correctness of the dual scheme. In a freely falling system the trajectories of freely falling particles will be straight lines (to second order) so the dual affine connection has to be of the form given by Eq (4.9), Eq (4.10) and Eq (4.11).

4.3.1

Finding the coordinate transformation to the freely falling coordinates

Around a point, t = 0 and x = x0 , we search for freely falling coordinates u, v where u is timelike 5 and v is spacelike. We assume that x and t depends at most to second ∂x ∂t order on u and v. Also we demand ∂u = 0 and ∂v = 0, that is we want the freely falling system which is at rest. Further on, we assume a diagonal, time independent, two-dimensional original metric (with positive a0 and negative c0 ). We have: ∂xα ∂xβ (4.13) ∂x&µ ∂x&ν Demanding that the coordinate transformed metric must reduce to Minkowski at (u, v) = (0, 0) with vanishing u and v-derivatives yields (in two pages of calculation): & gµν = gαβ

t = k1 · u + k2 · uv x = x0 + k3 · v + k4 · v 2 + k5 · u2

(4.14) (4.15)

Here the constants are uniquely given by: a0 & k2 = − √ 2 −c0 · a0 3/2 c0 & k4 = 4(−c0 )2

1 k1 = √ a0 1 k3 = √ −c0

(4.16) k5 = −

a0 & 4(−c0 )a0

(4.17)

Here prime denotes derivative with respect to x. The index & 0& means as usual the original Lorentzian metric. All the quantities are evaluated at the point x0 .

4.3.2

The dual metric in the freely falling coordinates

Knowing the coordinate transformation, we may calculate the transformation of the dual metric. The result is (after 4 pages of calculations), to first order in u and v:      5

guu guv



guv   gvv

 



α   a − β − k0 · v  0 

=    

−

k0 ·u 2

k0 − ·u 2 αβ 2k0 β ·v − 2 (a0 − β) a0 − β

Increasing u means moving timelike. Increasing v means moving spacelike

         

(4.18)

38

4. Metrics, geodesics and affine connections

The constant k0 is given by: α a0 & √ k0 = (a0 − β)2 −c0

(4.19)

Again all the quantities are defined at the point x0 . So a0 is really a0 (x0 ) here. We may immediately check that setting α = −β = ∞, corresponding to having the dual metric equal to the original metric, yields to first order exactly the Minkowski metric - which we demanded from the start. The dual metric in the freely falling coordinates is of the form:  

gαβ =   

a+b·v

c·u

c·u

d+e·v

    

(4.20)

Here a, b, c, d, e are constants. It is easy to show that the affine connection of such a metric obeys the necessary and sufficient conditions, Eq (4.9), Eq (4.10) and Eq (4.11) for coordinate straight geodesics at the point (u, v) = (0, 0) if and only if: c=

b 2

e=

2bd a

(4.21)

We see immediately that the freely falling dual metric, Eq (4.18), obeys these restrictions. We have thus verified that the dual metric gives the right equations of motion in the freely falling system.

5 On whether the interior dual metric is a sphere 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 also 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 logical to ask whether we really can find parameters such that the internal case becomes spherical. In the following subsections we will see that it is not in general exactly spherical but one can choose parameters such that it is exactly spherical in the weak field limit.

5.1

Conditions for spheres

For a sphere of radius R, introduce definitions according to Fig 5.1:

R

r

z Figure 5.1: Definitions of variables for a sphere

39

40

5. On whether the interior dual metric is a sphere

It is easy to derive, from the Pythagorean theorem, that: 5 '

dr =− dz

R r

(2

−1

(5.1)

Assuming z = z(x), where (see Paper I, Eq (23)): dz = c(x) − r & 2 dx

here

r& =

dr dx

c = gxx

(5.2)

We readily find: R2 =

r2 1 − r & 2 /c

(5.3)

& Also if we have a rolled up sphere such that r = γrsphere and r & = γrsphere the formula is immediately modified to:

R2 =

5.2

r2 γ 2 − r & 2 /c

(5.4)

The dual interior metric

For the Schwarzschild interior dual metric we may show that 2

r & /c =

βk 2 x2 4x60 (a0 − β)

and

r 2 = αk 2

1

a0 a0 − β

: (5.5)

Inserting this into Eq (5.4) 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

(5.6)

Notice that dividing the expression by αk 2 we see that any change in R2 and γ can be cancelled by a corresponding change in α and k respectively. For this expression to be true for all x it must be true for every power in x. a0 has a lot of powers in x which would mean that the factors multiplying a0 would have to vanish. 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. 1 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.

5.3. Approximative internal sphere

5.3

41

Approximative internal sphere

We can however produce something that is very similar to sphere. Set γ = 1, corresponding to a non-rolled sphere, and α = (1 − β)/k 2 meaning unit radius at infinity. We can expand the two occurrences of a0 in Eq (5.6) to second order in x2 . Demanding the equation to hold to zeroth, first 2 and second order yields after some simplification (Mathematica does fine simplifications): β

=

k

=

a00 · (R2 − 1) R2 − a00

where

a00

) 5

1 1 ≡ a0 (0) = 3 1− −1 4 x0

*2

(5.7)

7/4

-

2x0

√ √ 3 x0 − 1 − x0

(5.8)

Notice that Eq (5.8), 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 (5.7) and Eq (5.8) into my 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.

5.4

Spheres in the Newtonian limit

From Eq (5.3) we have the necessary relation for a sphere: r&2 r2 =1− 2 c R

(5.9)

Using the specific expressions for β and k given by Eq (5.7) and Eq (5.8) we may evaluate both the left and the right hand side of Eq (5.9) 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 . 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. 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 3

It is always satisfied to first order Terms like x2 /x30 are of course treated as 1/x0 -terms

42

5. On whether the interior dual metric is a sphere

An embedding diagram of the Earth spacetime is displayed in Fig 5.2. 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 (5.7) and Eq (5.8) with x0 = 7.19 · 108 , suitable for the Earth. As can be seen the bulge is quite spherical.

Figure 5.2: An embedding diagram for the dual spacetime of a central line through the Earth. The time per circumference is roughly 84 minutes. Notice the two freefallers in the sphere, one static in the center, the other one oscillating around the Earth with a period time of roughly 84 minutes.

6 The Epstein-Berg way There is yet another fundamentally different approach to embedding the spacetime, apart from the dual and the absolute schemes. In Epstein’s book [2], ’Relativity Visualized’, there are images which look a lot like my own dual metric embeddings. The exterior spacetime is represented by a trumpet-like shape and a star is a bulge on a cylinder. The interpretation of the diagram is however fundamentally different from my own. In the book, which is a popular scientific book, there is no mathematical background to the images. For me it first appeared as a miracle that he could get the geodesics right. A good friend of mine, Daniel Berg, had however independently come up with an idea to create a positive definite metric. We realized that this idea exactly matched the Epstein view, and must have been how Epstein did it. The basic ideas are outlined below.

6.1

The main philosophy

Consider a time independent metric with no cross terms in the time component: dτ 2 = a · dt2 − hij · dxi dxj

(6.1)

Here we assume that a is positive and hij is positive definite. The expression for the line element can be rewritten as: dt2 =

1 hij · dτ 2 + · dxi dxj a a

(6.2)

This looks like a positive definite line element where τ is now a coordinate and dt is proper distance. Notice that if we would have considered an original line element that was time dependent we couldn’t have done this little trick. Also if we had had off43

44

6. The Epstein-Berg way

diagonal terms we would have had to solve a second order equation to find dt. This would then not have been a quadratic form, and thus not a metric 1 .

6.1.1

Particle trajectories and mappings

To better understand the new line element we may consider a particle trajectory in the ordinary spacetime and in the new space-eigentime. See Fig 6.1. t

!

x

x

Figure 6.1: A trajectory in the original spacetime mapped to a corresponding trajectory in the space-eigentime

For a single particle there is a one-to-one mapping between points along the trajectory in the ordinary space and the trajectory in the space-eigentime. Notice however that where along the τ -axis in the space-eigentime we put the trajectory is completely arbitrary. There is in fact no general one-to-one mapping of events between the two spacetimes. To make this clear, consider two trajectories that have the same starting and ending point in the original spacetime. The eigentime elapsed along the two trajectories will in general not be the same. See Fig 6.2. We understand that in the space-eigentime we cannot then have both same starting and ending point for the two trajectories. Thus, unless we are specifying some special set of trajectories to use for the mapping, there is no unique mapping from the ordinary spacetime to the spaceeigentime. There is however still a unique metric in both spaces. Notice however that the Epstein space-eigentime is not a spacetime in the ordinary sense since a single point in the space-eigentime can correspond to several physical events.

6.1.2

Intuition about geodesics

A trajectory that is a geodesic in the ordinary spacetime will also be a geodesic trajectory in the space-eigentime. This we may understand most easily using the variational principle. Consider first any trajectory in the ordinary space and a variation of this (with fixed end points), and how this is mapped to the space-eigentime. See Fig 6.2 1 Of course, both off-diagonality and time independence depends on what coordinates we are using. I however refrain from a more general analysis for now.

6.1. The main philosophy

45

t

!

x

x

Figure 6.2: Varying the trajectories

Given that we start the trajectories at the same point in the space-eigentime, they will not in general end up in the same point, as discussed earlier. If we however assume the original trajectory in the ordinary space to be a geodesic, we know that for a small variation of the trajectory we do not change the integrated eigentime (to first order). This means that the varied path has the same endpoints in the space-eigentime also. What more is that the coordinate time t which is proper distance in the space-eigentime is exactly (to any order) the same for both the original and the variational trajectory, since it was so in the ordinary space. This would imply that the original trajectory in the space-eigentime must be a geodesic as well. Actually, for strictness, we should also argue that we can get any first order variation in the space-eigentime from first order variations in the original space. This we can probably do, but maybe we should prove the whole geodesic preservation mathematically in any case. This is done below.

6.1.3

Mathematics about geodesics

A geodesic in the original space obeys the Euler Lagrange equations, which for a time independent metric of the form Eq (6.1), becomes: K 0

= =

a

dt dτ

∂k a − ∂k hij

(6.3) dxi dxj d + dτ dτ dτ

)

2hkj

dxj dτ

*

here

∂k ≡

∂ ∂xk

(6.4)

The index k takes the values 1, 2 or 3. Using dτd = Ka · dtd , we may rewrite the spatial equations to a form which is more suitable for comparison with the equations of motion in the space-eigentime: )

)

K2 1 dxi dxj d dxj 1 0= · ∂k a − ∂k hij + 2hkj a a a dt dt dt dt

**

(6.5)

The geodesic equations in the space-eigentime are equivalently: 1 K

=

1 ∂τ a ∂t

(6.6)

46

6. The Epstein-Berg way

0

=

)

∂k a dτ − 2 · a dt

*2

)

a∂k hij − hij ∂k a dxi dxj d dxj + 2hkj − a2 dt dt dt dt

*

(6.7)

Given that the equations of motion hold in the original spacetime, the first equation above is trivially satisfied and the spatial ones can be reduced to: )

dxi dxj a2 1 ∂ a · − + a − h k ij a2 K2 dt dt

*

=0

(6.8)

The expression within the parenthesis is however trivially zero, using the expression for the line element and the definition for the constant of the motion. Thus we find that a geodesic trajectory in the original spacetime is a geodesic also in the space-eigentime. Equivalently we may understand that a geodesic in the space-eigentime is a geodesic also in the original spacetime.

6.2

Embeddings

The Epstein line element Eq (6.2) in the case of a 1+1 Schwarzschild original spacetime can easily be embedded as a rotational surface. See Fig 6.3. Particles in free fall move on geodesics on the surface, just like in the dual metric scheme. Now however it is eigentime that is the azimuthal angle. This means for instance that photons move completely along the surface with no spiral what so ever in the azimuthal direction.

Figure 6.3: The Epstein embedding where x goes from 1.04 to 4. with embedding parameter k = 0.5. Notice that eigentime is the azimuthal angle.

The Schwarzschild time elapsed along a given trajectory on the surface is simply the length of the curve, as measured on the surface. Notice however again that this is not an image of the spacetime in the sense that every point on the surface corresponds to a unique event in the spacetime. For instance if two trajectories cross each other on the

6.3. The Epstein internal space

47

Epstein surface that does not mean that they ever crossed in the ordinary spacetime. To determine whether two trajectories on the Epstein surface really meet in the ordinary spacetime, we need to specify the Schwarzschild time at an arbitrary point along each trajectory. Then we can take a ruler and measure the Schwarzschild time elapsed along each curve, to find the Schwarzschild time at any point along either curve. If ever we find that the Schwarzschild time along the two curves are the same, as the trajectories pass a certain spatial position - then they crossed each other in the real spacetime.

6.2.1

The Berg dynamical view

The procedure above is a little awkward. There is however another, more dynamical view on the scenario. Actually the idea was introduced to me by my friend, Daniel Berg. Take for instance two moving particles, corresponding to two trajectories in the original spacetime. Suppose that at a given slice t = const in the original space, we have mapped the particles to two specific points on the Epstein trumpet, for instance to the same azimuthal angle. Then evolving things in Schwarzschild time, the two points will start moving on the surface. Notice that whichever directions the two particles take on the surface, which is in principle arbitrary so long as it is not backwards 2 in the azimuthal angle, the speed of the points will be the same and constant all the time. This is of course because the Schwarzschild time is measured as the distance on the surface. Looking at the two points moving on the surface, we can see how the eigentime of the two particles evolve since it equals the azimuthal angle for each particle. In this dynamical view, whether the two particles collide or not is determined by if they are ever at the same spatial point (circle on the trumpet) at the same time! This is of course very close to the ordinary non-spacetime view of things. Notice that the azimuthal angle of the particles have nothing to do with whether they collide or not. I find this latter view a quite intriguing. It is like a mixture of spacetime physics and dynamical physics. We are evolving things in 4 dimensions (in general) as opposed to the way one is doing things in numerical relativity, where one is evolving things in 3 dimensions. Of course in the Epstein-Berg dynamical we are not evolving the spacetime itself, we are merely studying particles, and also the spacetime has to be time independent. Still, I find it interesting.

6.3

The Epstein internal space

Just like in the dual scheme we may also consider the spacetime of a radial line through a spherically symmetric object such as a star. Like before we assume the object to be of constant proper density, so we can apply the standard Schwarzschild interior and exterior metrics to generate the Epstein metric. Looking at embeddings of the spaceeigentime, the internal bulge looks quite spherical. In this section I investigate to what 2

For antiparticles one might consider this after all!

48

6. The Epstein-Berg way

extent it is spherical. We saw in Chapter 5 that an exact sphere has to obey Eq (5.3): R2 =

r2 1 − r & 2 /c

(6.9)

In the Epstein case we have: r c

=

1 k·√ a0

=

−c0 a0

where where

) 5

1 1 a0 = 3 1− − 4 x0 )

x2 c0 = − 1 − 2 x0

*−1

5

x2 1− 3 x0

*2

(6.10) (6.11)

Inserting this into the right hand side of Eq (6.9) we find quite easily that the equality cannot hold, to an arbitrary order in x. We may however, just like in the dual scheme, demand that the internal bulge shall be spherical to second order around the central star. This corresponds to a vanishing second order derivative of the right hand side of Eq (6.9), when x = 0. Demanding this, Mathematica yields that the embedding parameter k must obey: k=

. ) 5 / / 0x3 3 1 − 0

1 −1 x0

*

(6.12)

Incidentally, the third order derivative vanishes automatically (this is not in general true for the higher order derivatives). So, the interior bulge becomes quite spherical, as can be seen in Fig 6.4.

Figure 6.4: The Epstein embedding for a star with x0 = 2 and with the embedding parameter chosen to make the interior as spherical as possible

6.4. Comments

6.3.1

49

Verification using the dual scheme

We may verify the correctness of Eq (6.12) using earlier derived formulas in the dual scheme. If we (for a given x0 ) choose k according to Eq (5.8) and Eq (6.12) in the dual and Epstein scheme respectively, we know that we have (to second order) a perfect sphere in each case. Thus for small oscillations around the center, the orbital time will be independent of amplitude. In the dual scheme the, Schwarzschild time 3 per oscillation, will then equal the Schwarzschild time per circumference which is given by 2π · k. Analogously in the Epstein scheme the eigentime 4 per oscillation, will equal the eigentime per circumference which is given by 2π · k. Also, for very small oscillations, √ and thus very small local velocities we must have dτ = g00 · dt. In other words we must have: . ) 5 / / 0x3 3 1 − 0

kEpstein 1 −1 x0

*

?

= ?

=

-

a(0) · kDual ) 5

(6.13) *

7/4

1 1 2x0 3 1− −1 · - √ √ 2 x0 3 x0 − 1 − x0

(6.14)

Indeed we see that the equality holds, thus the dual and the Epstein scheme verify each other.

6.3.2

Comments

Notice that the interior bulge is not a perfect sphere in the Epstein view. This may also be easily seen from numerical embeddings in the limit x0 → 1.125. Also we lack the opportunity to produce shapefull images in the weak field limit. Whichever embedding constant we choose for the space-eigentime, of for instance a radial line through the Earth, the relative variations in the embedding radius will be of the order 10−9 .

6.4

Comments

One might think that there would be no use in having the Epstein trumpet layered, since we cannot distinguish events anyway. For every single particle it makes sense however, since eigentime is certainly not periodic. Notice that time dilation is visualized in exactly the opposite way from that in the absolute scheme. In the Epstein scheme, time dilation is an increase of the embedding radius. In the absolute scheme, with observers at rest as generators, it is a decrease of the embedding radius. A small note about spacelike distances and geodesics. In the Epstein embedding there is no such thing as spacelike distances. However, the equality Eq (6.2), which 3 4

The rescaled and dimensionless Schwarzschild time The rescaled and dimensionless eigentime

50

6. The Epstein-Berg way

is the basis for the embedding, holds also when dτ 2 is negative. This would however correspond to an imaginary τ , and is thus not included in the embedding (where τ is taken as a real-valued angle).

Bibliography [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., 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

51

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