A field approach to the description of gravitational interaction

We then obtain

Of t j {~ (x.)). Thus, in an arbitrary curvilinear coordinate system of pseudo-Euclidean Killing vector has the form

Bm

Generalization of expressions ordinates presents no difficulty. isolated system we obtain

O/(xH) ox~

-

O/(x~) o~

at q -

tensor M t ~ ---

( IO. 27 )

~lsf s (XH).

( 1 0 . 2 4 ) - ( 1 0 . 2 6 ) to t h e c a s e o f a r b i t r a r y curvilinear coProceeding exactly as above, for the four-momentum of an

1 ~ 3 P~ = I l / ' - y (xH) dxHdxHdx~T

The a n t i s y n ~ e t r y

space--time the

o f a n g u l a r momentum i n t h i s

om

(x~)

O/(x~)

case has the form

I I/ --?(x#)dx#dx#dx#T~ i 2 3 ~(x#)[ fro(x#) O/(xH) Ox--~

/_~.(XH).OYm(XH) ~x~H ]"

Thus, t h e p o s s i b i l i t y o f o b t a i n i n g i n t e g r a l c o n s e r v a t i o n laws d e p e n d s on t h e c h a r a c t e r of the geometry of space--time. In the case of four dimensions (physical space--time) only spaces of constant curvature possess all I0 integral conservation laws, while in other spaces the number of them is less than 10. 11.

A Field Approach to the Description of Gravitational

Interaction

In order that the gravitational field may be considered a physical field in the spirit of Faraday--Maxwell with its usual properties of a carrier of energy--momentum, it suffices for us to select as a natural geometry for the gravitational field the geometry of a space of constant curvature. Since experimental data obtained in studying the strong, weak, and electromagnetic interactions bear witness to the fact that for fields connected with these interactions the natural geometry of space--time is pseudo-Euclidean, at least at the present stage of our knowledge it may be assumed that this geometry is the sole natural geometry for all physical processes including gravitational processes. This assertion constitutes one of the basic propositions of the field approach to the theory of gravitational interaction we developed. It is altogether obvious that it will lead to fulfillment of all conservation laws of energy-momentum and angular momentum, ensuring the existence of all 10 Lntegrals of the motion for a system consisting of the gravitational field and the remaining fields of matter. The gravitational field in the field approach, similar to all other physical fields, is characterized by its energy-momentum tensor which contributes to the total energy--momentum tensor of the system. This is the basic difference of our approach from Einstein's theory. It should be noted that, in addition to general simplicity, in pseudo-Euclidean space--time the integration of tensor quantities has an altogether definite meaning. Another key question arising in the construction of a theory of the gravitational field is the question of the nature of the interaction of the gravitational field with matter. In acting on matter the gravitational field can effectively alter its geometry if it enters the terms for the highest derivatives in the equations of motion of matter. The motion of material bodies and other physical fields in pseudo-Euclidean space--time under the action of the gravitational field will then be indistinguishable from their motion in some effective Riemannian space--time. Universality of the action of the gravitational field on matter follows from experimental data, and hence the effective Riemannian space--time will be the same for all matter. This leads us to an assertion which we call the identity principle (the principle of geometrization) which is defined as follows: The equations of motion of matter under the action of a gravitational field in pseudo-Euclidean space--time with metric tensor Yni can be

1752

identically represented as equations of motion of matter in some effective Riemannian space-time with metric tensor gni depending on the gravitational field and the metric tensor 7ni" We introduced and formulated this principle in [10], although it was essentially already expressed in [11]. It means that the description of the motion of matter under the action of a gravitational field in pseudo-Euclidean space--time is physically identical to the description of the motion of matter in a corresponding effective Riemannian space--time. In this approach the gravitational field (as a physical field) in the description of the motion o f matter is eliminated, as it were, and its energy, figuratively speaking, goes to the formation of the effective Riemannian space--time. Thus, the effective Riemannian space--time is essentially a carrier of energy-~omentumo In correspondence with the identity principle as much energy as is contained in the gravitational field is expended on the creation of the Riemannian space--time, and hence the propagation of curvature waves in the Riemannian space--time reflects the usual transport of energy by gravitational waves in pseudo-Euclidean space--time. This means that in the field approach curvature waves in the Riemannian space--time are a direct consequence of the existence of gravitational waves in the spirit of Faraday--Maxwell which possess an energy-momentum density. It should be emphasized that the identity principle does not follow from any other physical principles. This is an independent principle which determines, on the one hand, equivalence of the description of the motion of matter and, on the other hand, it determines the character of the interaction of the gravitational field with matter and thus corresponds to a particular choice of the Lagrangian density of the interaction between them. In particular, it reflects the physical fact that the inertial mass of a point body is equal to its gravitational mass. We note that when we introduce the principle of geometrization we thus preserve Einstein's great idea on the Riemannian geometry of space--time for matter. However, this does not mean that we must unavoidably return to the general theory of relativity. The general theory of relativity is a special realization of this idea and not conversely. Therefore, the idea of the gravitational field as a physical field transportating energy combined with the identity principle leads us to other equations of the gravitation field distinct from Einstein's equations, and it alters our conception of space--time and gravitation. A new theory of gravitation realizing this idea makes it possible to describe all present gravitational experiments, satisfies the correspondence principle, and leads to a number of fundamental consequences. The field approach to the theory of gravitational interaction does not specify beforehand the nature of the gravitational field. We do not know the nature of the real gravitational field. It is possible, for example, that for an adequate description, it is necessary to use spin tensors or, say, a scalar field. In the present work we consider only one of the possible realizations of the field approach in which a symmetric tensor field of second rank is used to describe the gravitational field. However, before proceeding to an exposition of this version of gravitational theory, we compare various classes of gravitational theories that traditionally use a symmetric tensor field of second rank, and we determine which of them introduces the gravitational field in the most acceptable way from a physical point of view. In the construction of a theory of gravitation a key feature is the choice of a natural geometry for the gravitational field. For linear theories the natural geometry is the geometry of flat space--time, and theories of gravitation with linear equations of the free gravitational field can be formulated in terms of a flat space--time with metric tensor Yni o We shall call theories of gravitation formulated in terms of flat space--time theories of class ( A ) . Theories of class (A) can also be nonlinear, but it is important that this nonlinearity not enter in terms with the leading derivatives in the field equations and thus not alter the geometry of the natural space--time. Thus, in theories of class (A) we have a single flat space--time which guarantees the presence of all 10 consideration laws for a closed system. The Riemannian space--time in terms of which the motion of matter is described is the effective space--time which arises as a result of the action of the gravitational field ~ on matter. Among theories of class (A) we note the subclass of two-metric theories in which the gravitational field ~ in combination with the metric tensor Yni forms in the Lagrangian 1753

density of the gravitational field Lg a new field variable -- the metric tensor of the effective Riemannian space--time gni, in terms of which the equations of motion of matter are formulated, and the natural geometry for this field variable is pseudo-Euclidean geometry:

L----Lg(V.~,g., (Vms, ~ms)) + LM(g.~, ~A). An e x a m p l e o f a n o n l i n e a r density

where y is derivative

theory

of this

class

is

Rosen's

theorem

[23] w i t h a L a g r a n g i a n

t h e d e t e r m i n a n t o f t h e m e t r i c t e n s o r o f f l a t s p a c e - - t i m e and Di i s t h e c o v a r i a n t i n f l a t s p a c e - - t i m e ; h e r e and h e n c e f o r t h i t i s a s s u m e d t h a t G = c = ] .

In two-metric theories the gravitational field ~ni i s a c t u a l l y n o t p r e s e n t , since the field variable is the metric tensor gni; therefore, h e r e t h e r e i s no s u f f i c i e n t l y deep physical justification of the connection between the effective R i e m a n n i a n s p a c e - - t i m e and t h e s i n g l e flat space--time. I n t h e o r i e s o f c l a s s (A) we a c t u a l l y h a v e two p h y s i c a l s p a c e - - t i m e s -- t h e f l a t s p a c e - - t i m e with metric tensor u in terms of which the equations of the gravitational field are form u l a t e d and a n o n - E u c l i d e a n s p a c e - - t i m e w i t h m e t r i c t e n s o r g n i i n t e r m s o f w h i c h t h e m o t i o n of matter is formulated. Both t h e s e s p a c e - - t i m e a r e r e a l o b s e r v a b l e s . The f r o n t o f a g r a v i tational wave moves a l o n g g e o d e s i c s o f f l a t s p a c e - - t i m e , and h e n c e g r a v i t a t i o n a l waves c a n be used to determine the geometry of pseudo-Euclidean space--time. The f r o n t o f e l e c t r o m a g n e t i c waves moves a l o n g g e o d e s i c s o f e f f e c t i v e R i e m a n n i a n s p a c e - - t i m e , and h e n c e e l e c t r o m a g n e t i c waves and m a s s i v e p a r t i c l e s can be used to determine the geometry of this Riemannian space-time. If in a nonlinear theory of the tensor field ~ni the nonlinear terms enter in contraction of derivatives in the Lagrangian density (in terms with the leading derivatives in the field equations), then for such a theory non-Euclidean space--time with some effective metric tensor gn~=g~(?~z, ~z) is natural. We shall call theories of gravitation formulated in terms of an effective Riemannian space--time theories of class (B). The Lagrangian density of theories of this class has the form

L=L~(g~, ~ ) +L~(g,~, ~ ) . Theories of this class merit special consideration. Since flat space--time in theories of this class is not observable, here in the general cases it is obvious that there is no sufficient justification of the connection g,i=g~,(?~z, ~z) between the singleRiemannian space--time and the gravitational field ~n~ . The single Riemannian space--time in theories of this class arises on the basis of the gravitational field ~ and a nonobservable flat space--time. It should be noted also that the equations of the gravitational field in theories of class (B) are necessarily nonlinear. A subclass of geometrized theories of class (B) is the set of theories with complete geometrization in which the Lagrangian density of the gravitational field depends on the metric tensor gni:

L=Lg(g~) +L~(g~, ~). Einstein's theory belongs to this subclass of theories and corresponds to the special choice of the Lagrangian density in the form Lg = --~gR. In theories with total geometrization flat space--time is excluded entirely from the description of the motion both of matter and of the gravitational field. Neither the gravitational field ~ i nor the metric tensor Yni appears anywhere in the theory. The variables gni have a dual meaning: as variables of the physical field and the metric tensor of space--time. This leads to the situation that in theories of this subclass the gravitational field is not a field in the spirit of Faraday-Maxwell possessing an energy--momentum density. It should be emphasized that theories of classes (A) and (B) are basically different theories of gravitation. No transformation of the field variables or transformation of the coordinates can transform a theory of one class into a theory of the other class. Thus, in analyzing the possibilities available, we arrive at the conclusion that only theories of class (A) introduce the gravitational field in the most acceptable way from a 1754

physical point of view. Theories of this class make it possible to consider the gravitational field a physical field in the spirit of Faraday--Maxwell and possess all 10 integrals of the motion for a closed system of interacting fields. The effective Riemannian space--time used to describe the motion of matter in theories of this class reflects in a natural way the existence of a physical gravitational field and a single pseudo-Euclidean space--time. Hence, we again arrive at the necessity of primary study of the possibilities of constructing a gravitational theory realizing the field approach to the description of the gravitational interaction. 12.

Conservation Laws for the Gravitational Field and Matter

We shall study the character of conservation laws for all local theories of class (A) without making a specific choice of the Lagrangian density. Proceeding from the basic principles of the field approach, for theories of this class we write the Lagrangian densities of a system consisting of matter and gravitational field in the form

(I2.1)

L = Lg(yn~, ~n~)+LM(gn,, ~a), where Yni is the metric tensor of pseudo-Euclidean space--time, field, and ~A are the remaining fields of matter.

~

is the gravitational

We shall assume with no loss of generality that the metric tensor of Riemannian space-time gni is a local function depending on the metric tensor of flat space--time, the gravitational field ~ , and their partial derivatives through second order:

gmt = gmt (Yni, Os~ni, ~n~, Osj~ni, OsYni, ~sj~i, ~sY hi, ~sj~ nl, yni),

(12.2)

where we have used the notation

ns T

OxnOxs 9

--

We shall assume that the Lagrangian density of matter LM depends only on the fields ~A , their partial derivatives of first order, and the metric tensor gniIt is easy to see that in this case the Lagrangian density of matter contains partial derivatives of the gravitational field through second order. We shall assume that the Lagrangian density of the gravitational field depends on the metric tensor Yni, the gravitational f~eld ~n~ , and their partial derivatives through third order. To obtain conservation laws we use the covariant method of infinitesimally small displacmenets. Since the action J is a scalar, for an arbitrary small coordinate transformation (2.12) the variations of the action of matter ~JM and of the gravitational field 6Jg will be equal to zero. Since the Lagrangian density of matter contains both covariant and contravariant components of the metric tensor of Riemannian space--time, we vary the Lagrangian density with respect to them as if they were independent and then use the relations between their variations

6g.'~

=

- -

g.~g,~t6g u.

We proceed in an altogether similar way in the variation with respect to the components Yni and u of the metric tensor of flat space--time. We write the variation of the action of matter under transformation (2.12) in the form

8JM = f d4x [--A~ni 8LM 6LCpA+ Div}, [ ~LA~8Lgni -~- 8TA

(12.3)

where Div denotes divergence terms whose consideration is inconsequential for our purpose. Introducing the notation

t ~ = --2 (12o4) m

t M~

mg

=

%tim

1755