Solution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere

The paper is concerned with the spherically symmetric static problem of the General Relativity Theory (GRT). The classical interior solution of this problem found in 1916 by K. Schwarzschild for a fluid sphere is generalized for a linear elastic isotropic solid sphere. The GRT equations are supplemented with the equation for the stresses which is similar to the compatibility equation of the theory of elasticity and is derived using the principle of minimum complementary energy for an elastic solid. Numerical analysis of the obtained solution is undertaken.

To introduce the proposed approach to GRT problem for elastic solid, consider the problem of the classical theory of elasticity for a sphere whose gravitational field is described by the Newton theory.For a solid sphere with constant density μ and radius , R the Newton gravitational potential φ is the solution of the Poisson equation ( ) Here, r is the radial coordinate (0 ) r R ≤ ≤ , ( ) ( ) / d dr ′ ⋅⋅⋅ = ⋅⋅⋅ and G is the classical gravitational constant.For the external ( ) r R ≥ space, 0 μ = and Equation (1) has the following well known solution: e Gm r φ = − (2) Where index "e" corresponds to the external space and is the mass of a homogeneous solid sphere whose internal space is Euclidean.For the internal (0 ) r R ≤ ≤ space the solution of Equation (1) which satisfies the regularity condition at the sphere center is Here, index "i" corresponds to the internal space.The integration constant 1 C is determined from the boundary condition on the sphere surface according to which ( ) ( ).
, we can present Equation (4) in the following final form: The gravitational body forces which act inside the sphere are Here, r σ and θ σ are the radial and the circumferential stresses which are accompanied by the corresponding elastic strains following from Hooke's law, i.e.

( 2 )
r r 10) Thus, we have two equations, Equations ( 7) and ( 10), for two unknown stresses.To reduce these equations to one equation with respect to the radial stress, express θ σ using Equation ( 7), i.e.

(
) 2 r r r kr and substitute it in Equation ( 10) to get The solution of this equation must satisfy the following boundary conditions:  ) (3 ) 10(1 ) Having in mind to obtain the solution of the problem under study within the framework of GTR, we should take into account the GRT equations are formulated in the Riemannian space in which the displacement u , as well as the strain-displacement equations, Equations (9), do not exist.Thus, we cannot derive the compatibility equation, Equation (10) using the traditional approach.However, theory of elasticity provides another way to obtain this equation not attracting Equations (9).As known, the compatibility equation formulated in stresses follows from the principle of minimum of the complementary energy under the condition that the stresses satisfy the equilibrium equations.The elastic energy of the solid sphere is is the elastic potential.Substituting the strains from Hooke's law, Equations ( 8), in Equation ( 15), expressing θ σ in terms of r σ with the aid of Equation ( 11) and thus satisfying the equilibrium equation, we can reduce the complementary energy to the following functional: where The Euler equation which provides the minimum value of the complementary energy is Substituting F from Equation ( 17), we arrive at the compatibility equation, Equation ( 12).
In conclusion, transform the obtained results introducing the following dimensionless parameters: Here, c is the velocity of light and is the so-called gravitational radius.Using Equations ( 3) and ( 6) for m and k and applying Equations ( 19), we can present the compatibility equation, Equation ( 12) in the following form: The normalized stresses become

Spherically Symmetric Static Problem of General Relativity for an Elastic Sphere
For a spherically symmetric static problem, the line element is traditionally taken in the following form corresponding to the classical Schwarzchild solution: Here , , r θ φ and t are space spherical and time coordinates, ij g are the metric coefficients that depend on the radial coordinate r only.For the spherically symmetric static problem and the line element in Equation ( 23), the conservation equation which is analogous to the equilibrium equation, Equation (7) of the theory of elasticity, is (Synge, 1960) Here, the stresses and the density are expressed in terms of the metric coefficients with aid of Einstein's equations which can be presented as (Synge, 1960) where is the GRT gravitational constant.As known, substitution of Equations ( 25)-( 27) in Equation ( 24) identically satisfies this equation.
Perform some transformations.First, consider Equation ( 27 Finally, substituting the second of Equations ( 29) and Equations (30) in Equations ( 25) and ( 26), we arrive at Following the approach described in Section 1, express f in terms of r σ using Equation ( 31) and substitute this result in Equation ( 32), i.e.
According to the basic idea of GRT, the stresses specified by the Einstein equations, Equations ( 31) and ( 32), identically satisfy the equilibrium equation, Equation ( 24).Transforming this equation with the aid of Equations ( 3), ( 20) and ( 28) for , , g m r χ and substituting θ σ from Equation (33), we can readily prove that the equilibrium equation is satisfied.Thus, to determine r σ , we can apply the principle of minimum of the complementary energy discussed in Section 1.In the Riemannian space, Equation (15) for the complementary energy is generalized as Using Equation ( 32), we can reduce it to the functional in Equation ( 16).Omitting the explicit expression for the function F which is rather cumbersome, present the Euler equation, Equation ( 18), which takes the following final form: To simplify this equation, we use dimensionless parameters in Equations ( 19) and Equations( 3), ( 21) and ( 28) for , g m r and χ .Neglecting the terms with g r in comparison with unity and omitting nonlinear terms, we arrive at equation ( 21) of the theory of elasticity.

Numerical Analysis
For the numerical analysis, we take 0 ν = and reduce Equation (34) to It is interesting to compare the results that follow from Equations ( 35) and ( 36) with the theory of elasticity solution specified by Equations ( 22) and with the interior Schwarzchild solution obtained for a sphere of perfect incompressible fluid.This solution has the following form (Synge, 1960): and demonstrates a specific behavior.Taking 0 r = in Equation ( 37), determine the pressure at the sphere center, i.e.The denominator of this expression is zero for the sphere with radius 8 / 9 0.8888, g r = = and the pressure becomes infinitely high at the sphere center.This result is sometimes used to support the existence of the objects referred to as the Black Holes (Thorne, 1994).The results of the numerical analysis are presented in Figures 1, 2. Figure 1 demonstrates the distributions of the normalized radial (solid lines) and circumferential (dashed lines) stresses over the radial coordinate for 0.2, 0.4, 0.6, 0.8, 0.99.For 1, g r = the numerical solution does not converge.However, it does not look like the stresses are singular in this case.It seems that the tolerance of the applied numerical procedure is not as high as should be.The last result 0.5886 is obtained for 0.99.

Conclusion
The solution of the Schwarzchild spherically symmetric static problem for a fluid sphere is generalized for a linear elastic sphere.The equation for the stresses missing in GRT is derived using the minimum complementary energy principle of the theory of elasticity.In contrast to the singular Schwarzchild solution for the pressure in the fluid, the stresses do not demonstrate singular behavior for the elastic sphere whose radius is equal to the gravitational radius.

(
"e " and "i" correspond to external and internal spaces, whereas g r is the gravitational radius specified by Equation (20).Second, introduce function ( ) numerically integrated under the boundary conditions that follow from Equations (13), i.e

Figure 1 .Figure 2
Figure 1.Distributions of the normalized radial (solid lines) and circumferential (dashed lines) stresses over the radial coordinate for (a) 0.2, 0.4, 0.6 g r = and (b) 0.8, 0.9, 0.99 g r Dashed line in Figure 2 corresponds to the theory of elasticity solution specified by Equations (22).