Spherically Symmetric Problem of General Relativity for a Fluid Sphere

The paper is devoted to a spherically symmetric problem of General Relativity (GR) for a fluid sphere. The problem is solved within the framework of a special geometry of the Riemannian space induced by gravitation. According to this geometry, the four-dimensional Riemannian space is assumed to be Euclidean with respect to the space coordinates and Riemannian with respect to the time coordinate. Such interpretation of the Riemannian space allows us to obtain complete set of GR equations for the external empty space and the internal spaces for incompressible and compressible perfect fluids. The obtained analytical solution for an incompressible fluid is compared with the Schwarzchild solution. For a sphere consisting of compressible fluid or gas, a numerical solution is presented and discussed.

Spherically Symmetric Problem of General Relativity for an Elastic Solid Sphere

The paper is devoted to a spherically symmetric problem of General Relativity (GR) for an elastic solid sphere. Originally developed to describe gravitation in continuum (vacuum, gas, fluid and solid) GR does not provide the complete set of equations for solids and, in contrast to the Newton gravitation theory, does not allow us to study the stresses induced by gravitation in solids, because the compatibility equations which are attracted in the Euclidean space for this purpose do not exist in the Riemannian space. To solve the problem within the framework of GR, a special geometry of the Riemannian space induced by gravitation is proposed. According to this geometry, the four-dimensional Riemannian space is assumed to be Euclidean with respect to the space coordinates and Riemannian with respect to the time coordinate. Such interpretation of the Riemannian space in GR allows us to supplement the conservation equations for the energy-momentum tensor with compatibility equations of the theory of elasticity and to arrive to the complete set of equations for stresses. The analytical solution of the Einstein equations for the empty space surrounding the sphere and the numerical solution for the internal space inside the sphere with the proposed geometry are presented and discussed.

To the Solution of a Spherically Symmetric Problem of General Relativity

The paper is devoted to the spherically symmetric problem of General Relativity. Existing solutions obtained by K. Schwarzschild and V. Fock are presented and discussed. A special geometry of the Riemannian space induced by gravitation is proposed. According to this geometry the four-dimensional Riemannian space is assumed to be Euclidean with respect to the space coordinates and Riemannian with respect to the time coordinate. The solution of the Einstein equations for the empty space with this geometry coincides with the solution in Gullstand-Painlever coordinates. In application to the found solution, the problem of the light trajectory deviation in the vicinity of Sun and the problem of escape velocity are discussed.

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 compo-nents 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 actually 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.

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.

To the Schwarzschild Solution in General Relativity

The paper is concerned with the history of the spherically symmetric static problem solution of General Relativity found in 1916 by K. Schwarzschild [1] [2] which is interpreted in modern physics as the background of the objects referred to as Black Holes. First, the modern interpretation this solution which does not exactly coincide with original solution obtained by K. Schwarzschild is discussed. Second, the basic equations of the original Schwarzschild solution are presented in modern notations allowing us to compare existing and original solutions. Finally, a modification of the Schwarzschild approach is proposed allowing us to arrive at the exact solution of the Schwarzschild problem.

Black Holes or Dark Stars—What Follows from the General Relativity Theory

The paper is concerned with spherically symmetric static problem of the Classical Gravitation Theory (CGT) and the General Relativity Theory (GRT). First, the Dark Stars, i.e. the objects that are invisible because of high gravitation preventing the propagation of light discovered in the 18th century by J. Michel and P. Laplace are discussed. Second, the Schwarzchild solution which was obtained in the beginning of the 20th century for the internal and external spaces of the perfect fluid sphere is analyzed. This solution results in singular metric coefficients and provides the basis of the Black Holes. Third, the general metric form in spherical coordinates is introduced and the solution of GRT problem is obtained under the assumption that gravitation does not affect the sphere mass. The critical sphere radius similar to the Black Hole horizon of events is found. In contrast to the Schwarzchild solution, the radial metric coefficient for the sphere with the critical radius referred to as the Dark Star is not singular. For the sphere with radius which is less than the critical value, the GRT solution becomes imaginary. The problem is discussed within the framework of the phenomenological theory which does not take into account the actual microstructure of the gravitating objects and, though the term “star” is used, the analysis is concerned with a model fluid sphere rather than with a real astrophysical object.