Linearized Equations of General Relativity and the Problem of Reduction to the Newton Theory

The paper is concerned with the problem of reduction of the general relativity theory to the Newton gravitation theory for a gravitation field with relatively low intensity. This problem is traditionally solved on the basis of linearized equations of general relativity which, being matched to the Newton theory equations, allow us to link the classical gravitation constant with the constant entering the general relativity equations. Analysis of the linearized general relativity equations shows that it can be done only for empty space in which the energy tensor is zero. In solids, the set of linearized general relativity equations is not consistent and is not reduced to the Newton theory equations. Specific features of the problem are demonstrated with the spherically symmetric static problem of general relativity which has the closed-form solution.


Introduction. General Relativity Equations
The basic equation of general relativity which specifies the Einstein tensor has the following form: are the components of the Ricci curvature tensor (we use mixed components because for the spherically symmetric problem considered further they coincide with the physical components). The Einstein tensor is associated with the energy tensor as where χ is the relativity gravitational constant. The energy tensor expressed with the aid of Equations (1) and (2) identically satisfies the conservation equa- For the static problem, where j i σ is the stress tensor and µ is the density.
For gravitation with relatively low intensity, the general relativity must reduce to the Newton theory in which the gravitation potential ψ satisfies the Poisson in which G is the classical gravitation constant. Traditionally, the linearized version of Equation (1) is obtained and matched to Equation (5). The result is

Linearized Equations and the Reduction Problem
Consider the space referred to Cartesian coordinates  approximation, the last term with 44 f belongs to the second-order approximation. Naturally, this does not mean that the last term can be neglected-in this case, the gravitation disappears. This situation is not unique in mechanics of solids. For example, to construct the two-dimensional theory of thin plates from three-dimensional equations of theory of elasticity by the asymptotic method, we need to retain small terms of the first and of the second orders. Neglecting the second-order terms, we arrive at the plate theory which is not physically consistent [1]. Consider further the spherically symmetric problem for which closed-form solutions can be obtained.

Classical Linear Solution
For comparison with the general relativity solutions that are discussed further, consider the problem of the theory of elasticity for a linear elastic isotropic solid sphere with radius R and constant density µ loaded with gravitation forces following from the Newton theory. The gravitational potential ψ is the solution of the Poisson equation Here, ( ) ( ) The solution of this equation that satisfies the boundary condition ( ) 0 The components of the metric tensor depend on the radial coordinate only.
For the foregoing linear solution, the general relativity interpretation of the obtained results is [2] In case 0 g r = , the space is Euclidean and gravitation vanishes.

General Relativity Equations
For a spherically symmetric problem, the field equations following from Equation (1) reduce to [3] in which in accordance with Equations (2) and (4) The only one conservation equation, Equation (3) The solution of the external ( r R ≥ ) problem must satisfy the asymptotic conditions and to reduce to Equations (25) for r → ∞ . The solution for the internal ( 0 r R ≤ ≤ ) problem must satisfy the symmetry condition at the sphere center according to which , , g g g and two stresses , r θ σ σ with one coordinate condition for the metric tensor [4] and one equation for the stresses [5]. For the case of perfect fluid which is considered further, r p θ σ σ = = − and we have four unknown functions and need only one coordinate condition for the metric tensor.

Linearized Solution
Decompose the components of the metric tensor in Equation (24) as 11 and assume that the absolute values of functions ( ) f r are much less than unity. Undertaking linearization of Equations (26)-(28), we arrive at ( ) ( ) . Using Equations (29), we can present these equations as The conservation equation, Equation (30), transformed with aid of Equation Neglecting r σ in comparison with 2 c µ in the last term of this equation, we can conclude that it is analogous to the equilibrium equation, Equation (21).
Proceeding, express f from the first equation of Equation (33), f ′ from the second of these equations, i.e., at the consistent theory, we need to construct the second-order approximation.

Second-Order Asymptotic Approximation
Present the components of the metric tensor entering Equation (24) as in which ε is a small parameter. Substitution in Equation (1) Substituting expressions (36) for the metric tensor, we arrive at the following two equations corresponding to ε and 2 ε : E . However, these two equations include three unknown functions with indices 1, 2 and 4. To solve the problem, we need to supplement the aforementioned two equations with a coordinate condition. According to the condition used further, the so-called space density R E d g g = in which R g and E g are the determinants of the metric tensor in Riemannian and Euclidean three-dimensional spaces is minimized [5]. For the spherically symmetric problem, the introduced coordinate condition has the form [5] So, the sphere mass is the same that in Euclidean space [6].
Using Equations (36), we can write Equation (41) as and get two coordinate conditions for the first and the second approximations, i.e., Consider the external space ( R r ≤ < ∞ ) for which ( ) ( )      . Thus, the classical expression for the gravitational constant in Equation (6) is proved, but under the following conditions: • The problem is spherically symmetric and static.
• The sphere consists of a perfect fluid with constant density.
• The asymptotic equations are of the second order, not of the first.
• The coordinate condition in Equation (41)  1 f Consider the external problem for which 0 Integration of Equations (71) and (73)   g r = . So, Equation (28), being initially of the second order, reduces to the equation of the first order, and its solution contains only one integration constant which is found from the asymptotic condition. The second constant that could be used to satisfy the boundary condition for 11 g is missing and this condition is not sa-ing sphere with radius R, this angle can be determined from the following equa- , we can conclude that the second term is negligible in comparison with the first one, and that the shift angle for the obtained metric coefficients in Equations (69) is specified by the traditional Equation (84) which is in good agreement with experiment.

Conclusion
As follows from the foregoing analysis, the general relativity theory reduces for low intensity gravitation to the Newton theory only for the empty space. For solids, the linearized equations of the general relativity do not describe gravitation and the second-order asymptotic equations should be applied. For this approximation, the traditional expression for the gravitational constant is found for spherically symmetric static problem under special coordinate condition. The linearized Schwarzchild solution is not reduced to the Newton solution. The obtained solution for the metric tensor of the spherically symmetric empty space yields the traditional result for the angle of light ray deviation in the vicinity of Sun.