To the Complete Set of Equations for a Static Problem of General Relativity

The paper is concerned with the formulation of the static problem of general relativity. As known, this problem is reduced to ten equations for the components of the Einstein tensor and the solution of these equations is associated with two principal problems. First, since the components of the Einstein tensor identically satisfy four conservation equations, only six of these equations are mutually independent. So, the set of the Einstein equations ac-tually contains six independent equations for ten components of the metric tensor and should be supplemented with four additional equations which are missing in the original theory. Second, for a deformable solid the Einstein tensor is associated with the energy tensor which is expressed in terms of six stresses induced by gravitation. These stresses are not known and the relativity theory does not propose any equations for them. Thus, the static problem of general relativity cannot be properly formulated because the set of governing equations is not complete. In the paper, the problem of completeness of the general relativity governing set of equations is analyzed in application to the spherically symmetric static problem and the proposed approach is further described for the general case. As an example, linearized axisymmetric problem is considered.


Introduction. General Relativity Equations
The Einstein equation which specifies the Einstein tensor has the following form: where 4 8 G c is the relativity gravitational constant expressed in terms of the classical constant G and the velocity of light c. The energy tensor expressed with the aid of Equation (1) and Equation (2)  T c µ = (5) where j i σ is the stress tensor and µ is the density.
Consider two problems associated with the formulation of the general relativity static problem. First, substituting Equations (2) in Equation (1), we arrive at ten equations for ten components of the metric tensor in four-dimensional Riemannian space. However, the right parts of these equations identically satisfy Equations (4) which means that only six of ten Equation (1) are mutually independent. Thus, we have six equations for ten unknown functions. The additional equations which are usually referred to as coordinate conditions should be imposed on the metric tensor. As known, the metric tensor of the Euclidean space must satisfy the Lame equations. Such equations do not exist for the Riemannian space. However, we can suppose that the Riemannian space induced by gravitation cannot be arbitrary and must be somehow restricted, e.g., by coordinate conditions. The necessity of such conditions was first mentioned by D. Hilbert [1]. By now the widely recognized general form of these conditions has not been proposed. Existing particular conditions are discussed further in application to the spherically symmetric problem.
Second, changing E to T and then to σ in Equations (1), we arrive at the set of equations containing stresses on the left sides. These stresses are not known.

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 loaded with gravitation forces following from the Newton theory. For a sphere with constant density µ , the gravitational potential ϕ is the solution of the Poisson equation Here, ( ) ( ) is the sphere mass. Introduce the so-called gravitational radius The compatibility equation follows from these equations and has the form Express the strains in terms of stresses with the aid of Hooke's law ( ) (13) in which E is the elastic modulus and ν is the Poisson's ratio. Substituting the strains in Equation (12), we finally get Equation (12) means that the geometry of the deformed sphere is Euclidean. In general relativity, the geometry is Riemannian, the displacement u and Equations (11) do not exist. However, there is the second way to obtain Equation (14) not attracting Equations (11). This approach is based on the principle of minimum of the complementary energy ( ) under the condition that the stresses satisfy the equilibrium equation, Equation (10). Introducing this equation with the aid of the Lagrange multiplier λ , construct the augmented functional The Euler equations providing Expressing λ from the second equation and substituting in the first equation, we arrive at the compatibility Equation (14). Thus, we get two equations, Equation (10) and Equation (14) for two stresses. The final solution which satisfies the boundary condition For a sphere of perfect fluid, r p θ σ σ = = − and the pressure p can be found from the equilibrium Equation (10) not attracting the compatibility Equation (14). The result is ( ) In general relativity, the space geometry is Riemannian and the line element in spherical coordinates , , The components of the metric tensor depend on the radial coordinate only. For the foregoing linear solution, these components are [2] In case 0 g r = , the space is Euclidean and gravitation vanishes. For real objects, the ratio g r , as a rule, is extremely small. For example, for Earth

General Relativity Solution
For a spherically symmetric problem, the field equations following from Equation (1), Equation (2) and Equation (5) The only one conservation equation, Equation (4) The solution of the external ( r R ≥ ) problem must satisfy the asymptotic conditions and to reduce to Equation (22) (26) is used. The obtained solution identically satisfies Equation (24).
To solve the problem, we should supplement Equation (23), Equation (25) and Equation (26) which include three components of the metric tensor and two stresses with one coordinate condition for the metric tensor and one equation for the stresses. The first coordinate condition was proposed by K. Schwarzchild [5] who changed the spherical coordinates to x t = and applied the condition 1 g = , where g is the determinant of the metric tensor components in coordinates i x . This condition is equivalent to 2 22 g r = [6] and reduces the order of Equation (25). As a result, the solution does not contain the proper number of integration constants and the first boundary condition in Equation (27) cannot be satisfied [6]. The internal problem was solved for a sphere of perfect fluid [7] and did not require the additional equation. The other way involves the application of the so-called harmonic coordinate conditions which in the general case have the following form [8] External spherically symmetric problem was solved with the harmonic coordinate condition by V. Fock [9]. Internal problem and boundary conditions were not considered.
To obtain the general solution of the spherically symmetric static problem, apply the set of Equation (23), Equation (25) and Equation (26). To simplify these equations, introduce new notations for the components of the metric tensor, i.e., put 2 11 g q = , Equation (31), written for a four-dimensional space, is known in general relativity as a possible way to derive the field equations [12]. However, if the variations   ρ is g r . Assume that this minimum value correspond to the sphere radius g R . Then, substituting 1 The solution of this equation is 1.115 . Thus, the obtained solution gives the critical radius which is larger than the gravitational radius. In contrast to the Schwarzchild solution, for the sphere with the critical radius g R the solution is not singular and gives finite values for the metric coefficients. Particularly, for For g R R < , the solution becomes imaginary which means that the general relativity is not valid for such high levels of gravitation. Dependences of the space metric coefficients on the radial coordinate for the sphere with the critical radius g R is shown in Figure 1.
Consider the propagation of light from the sphere surface. The trajectory of light in the equatorial ( 2 θ = π ) plane is specified by the following equations

The General Theory
Return to Section 1 and consider the general case. Ten Einstein's equations  Minimization with respect to the stresses and λ -multipliers yields 10 equations for six stresses and four multipliers [14]. Thus, we have arrived at the complete set of 20 equations for 10 metric coefficients 6 stresses and 4 multipliers.

Linearized Axisymmetric Problem
Spherically symmetric problem discussed above requires only one coordinate condition. To demonstrate a more complicated case, consider an axisymmetric problem for which we need two conditions. Since the general problem can hardly be solved because the equations are too complicated, obtain the linearized so-