Putting p_e = r in Equations (11), we arrive at

The solutions of these equations are

The integration constants C₁ and C₂ can be found from the asymptotic conditions according to which for r ® ¥ the obtained solution must reduce to the CGT solution that is expressed in terms of the Newton gravitation potential j [7] , i.e., to

As a result, Equations (16) take the well known form [3]

As follows from the derivation of Equations (18), they include the radius r_g which is specified by Equation (2) and corresponds to the classical gravitation theory. Equations (18) show that the Schwarzchild solution determines the metric coefficients of the external space in terms of the sphere gravitational radius which depends on the sphere mass only and does not take into account the internal structure of the sphere, i.e., the structure of the source of gravitation. Such situation is typical for CGR, i.e., for the solution in Equations (17), which is unique. However, for GRT the situation can be different, because the function g₂₂ = r²(r) in Equation (5) can be chosen in different forms. Consider, for example, De-Donder-Fock solution which satisfies the so-called harmonic coordinate condition. Omitting mathematical formulation and physical interpretation of this condition which can be found elsewhere [8] [9] , presume that the function r(r) has the following form: r(r) = r + r_g/2. Then, the solutions of Equations (11) become [8] [9]

For r ® ¥, these solutions asymptotically reduce to Equations (17). Thus, Equations (19) satisfy all GRT equations and asymptotic conditions, but do not coincide with the Schwarzchild solution in Equations (18).

For the internal space of the fluid sphere and r(r) = r, Equations (12) and (13) reduce to

In conjunction with Equation (15) for the pressure we have three equations for three unknown functions g_i(r), h_i(r) and p(r) For the sphere with constant density m, the solution of Equation (21) which satisfies the regularity condition at the sphere center r = 0 is [3]

This solution must satisfy the boundary condition on the sphere surface according to which g_i(R) = g_e(R). Using Equations (18), we have

and finally get [3]

Consider Equation (20) for h_i. Substituting Equation (24), we arrive at

Substituting Equation (25) in Equation (15) and using Equation (10) for c, we get the following equation for the pressure in the fluid:

The general solution of this equation is [3]

The integration constant C₃ can be found from the boundary condition on the sphere surface according to which p(R) = 0. The resulting expression for the pressure is [3]

Thus, the Schwarzchild solution is specified by Equations (18) for the external space and Equations (24), (25), (28) for the internal space of the fluid sphere with constant density.

To demonstrate the specific features of the Schwarzchild solution, consider it for the sphere surface r = R. For this surface, we have

As can be seen, for the sphere with radius R = r_g, g_R becomes singular and h_R is zero. This result has not been accepted by A. Einstein [10] who expected that GRT could not be valid for the sphere with radius close to r_g. Later, the singularity of the Schwarzchild solution gave rise to the idea of Black Holes and r_g was referred to as the radius of the horizon of events of the Black Hole [1] [2] . However, recent results in Mechanics of Solids [11] allow us to conclude that singular solutions do not have physical nature and are the formal consequences of the discrepancy between physical and mathematical models. Particularly, the mathematical model of space specified by Equation (5) is simplified by the condition r(r) = r which is not properly justified. As a result, the Schwarzchild solution suffers from several shortcomings discussed below.

First, the critical radius r_g in GRT solution in Equations (18) and (24) is the same that in Equation (2) which follows from CGT.

Second, the solution for g_i in Equation (24) satisfies the boundary condition on the sphere surface only if Equation (23) is valid. Substituting r_g from Equation (2) and c from Equation (10) in Equation (23), we can derive the equation for the sphere mass which becomes m = 4/3pmR³ and coincides with Equation (3) corresponding to the Euclidean space. However, the space inside the sphere, in accordance with the basic idea of GRT, is Riemannian. The mass of the sphere with constant density corresponding to the metric form for the line element in Equation (5) is

Substituting g_i from Equation (24) and taking r_i = r, we get

The second part of this equation is the power decomposition with respect to r_g/R. As follows from Equations (3) and (30), the sphere mass corresponds to the Euclidean space only if r_g = 0. In the general case, r_g ³ 0 and the sphere mass, corresponding to the Schwarzchild solution must be specified by Equation (30). However, in this case g_e(R) ¹ g_i(R) and the boundary condition for the external and internal spaces cannot be satisfied on the sphere surface.

Third, Equation (28) for the pressure demonstrates rather specific behavior of the function p(r, R) [12] . Consider the pressure at the sphere center. Taking r = 0 in Equation (28), we get

The denominator of this expression becomes zero if R = R_s = 9/8r_g = 1.125r_g [13] . Thus, the pressure becomes infinitely high at the sphere center if R = R_s. This result is traditionally used to justify the existence of Black Holes [1] . However, R_s does not coincide with r_g. Then, Equation (31) specifies the pressure if R < R_s but this pressure is negative (or the density is negative) which does not have physical meaning. Finally, if we use the general solution in Equation (27) and take r = r_g, we get

On the sphere surface r = R, we have p(R) = ?mc²/3 and the boundary condition p(R) = 0 cannot be satisfied.

The original Schwarzchild solution for the internal space discussed in Section 3.2 has been obtained for the sphere with constant density m. Within the framework of the Schwarzchild assumption, i.e. r(r) = r, consider the sphere for which m(r) is an arbitrary function of the radial coordinate. In this case, the solution of Equation (21) that generalizes the solution in Equation (22) for the constant density is

For the sphere surface, we get

where

is the sphere mass corresponding to the Euclidean space. If we use Equations (18), (2) and (10) for g_e, r_g and c, we can conclude that Equation (32) satisfies the boundary condition g_i(R) = g_e(R). However actually, the sphere mass with the metric coefficient specified by Equation (32) does not correspond to the Euclidean space.

Thus, in the general case of the sphere whose density is an arbitrary function of the radial coordinate and r(r) = r, the boundary condition for g(r) can be satisfied only if the geometry of the internal space is Euclidean which is not the case. The reason of this situation follows from Equation (13) which has, in general, the second order. Under the condition r(r) = r, it reduces to the equation of the first order and the corresponding solution does not contain the proper number of integration constants that are required to satisfy the boundary conditions.

Consider the general metric form of the line element specified by Equation (5) in which is some unknown function. For the external space, the second equation in Equations (11) can be reduced to

and has the following general solution:

Substituting this result in the first equation in Equations (11) and integrating, we get

Assume (and further prove) that the following asymptotic conditions are valid for r_e® ¥:

i.e., that at a distance from the sphere r_e degenerates into r. Then, Equations (33) and (34) asymptotically reduce to Equations (17) corresponding to CGT if C₄ = -r_g and C₅ = 1. Finally,

For the internal space, Equation (13) can be transformed to

Assume (and further prove) that at the sphere center

For the sphere with constant density m, the solution of Equation (37) that satisfies the regularity condition at the sphere center is

For r(r) = r Equations (36) and (39) coincide with the Schwarzchild solutions in Equations (18) and (22). However, in contrast to Equations (18) and (22), the obtained equations include the derivative r¢(r).

Now, we need to specify the function r(r). Recall that the main shortcoming of the Schwarzchild solution is the discrepancy between the Euclidean form of the sphere mass which allows us to satisfy the boundary condition g_e(R) = g_i(R) and the actual mass (30) which corresponds to the Riemannian internal space of the sphere.

Note that the mass in Equation (30) depends on gravitation. Being discussed by, e.g. Zeldovich and Novikov [14] , the dependence of mass on gravitation, does not seem to correspond to the principle of equivalence of gravitational and inertial masses. Indeed, the inertial mass shows itself under the action of forces whose nature is different from the nature of gravitation. Anyway, as known [3] , any assumption may exist in GRT if the resulting solution is feasible. Thus, suppose that gravitation does not affect the sphere mass which remains the same that in the absence of gravitation and corresponds to the Euclidean space. As follows from Equation (29), the sphere mass corresponds to the Euclidean space if

which is the equation for the function r_i(r). Now, since the mass is Euclidean, Equation (23) of the Schwarzchild solution is valid and we can transform Equation (39) to the following final form:

Substituting Equation (41) in Equation (40), we arrive at the following differential equation for the function r_i(r):

The solution of this equation which satisfies the condition in Equation (38) is

where

Equation (42) implicitly specifies the function r_i(r) which changes from r_i = 0 that corresponds to the sphere center to r_i(R) = r_R that corresponds to the sphere surface. Taking r = R in Equation (42), we arrive at the following implicit equation for r_R:

The metric coefficients for the external space must satisfy the asymptotic conditions in Equations (35) and the following boundary conditions on the sphere surface r = R:

To satisfy these conditions, assume that Equation (40) is valid not only for the internal space, but for the external space as well, i.e.,

Matching Equations (40) and (45), we can conclude that if the second boundary condition in Equations (44) is satisfied, the first of these conditions is satisfied automatically because r is continuous. Substituting the first equation in Equations (36) in Equation (45), we get the following differential equation for the function r_e(r):

The solution of equation (46) which satisfies the second boundary condition in Equations (44) is

where

This solution is valid for r_R £ r_e < ¥ where r_R corresponds to the sphere surface. For the external space, we need to check the asymptotic conditions in Equations (35). Dividing Equation (47) by and taking r_e ® ¥, we can readily prove that r_e ® r. Differentiating Equation (47) with respect to r_e and dividing the result by, we can prove that the second condition in Equations (35) is also valid. Thus, for r_e ® ¥, Equations (36) reduce to the Schwarzchild solutions in Equations (18) which, in turn, degenerate into the CGT solutions in Equations (17).

Consider Equation (47) from which it follows that the real solution exists if r_e ³ r_R ³ r_g. Thus, the minimum possible value of the function r_R that corresponds to the sphere surface is r_g specified by Equation (2). Taking r_R = r_g in Equation (43), we arrive at the following equation for the minimum possible value of the sphere radius R_g:

The solution of equation (48) is R_g = 1.115r_g. For the sphere with R < R_g, r_R < r_g and the solution in Equation (48) becomes imaginary.

As follows from the foregoing derivation, the obtained solution gives the critical radius similar to the horizon of events radius of the Black Hole in the Schwarzchild solution. However first, this radius does not coincide with r_g which actually follows from CGT. Second, for the sphere with the critical radius R_g, the solution is not singular. Indeed, Equations (40), (45) and (47) for r = R_g yield g_e(R) = g_i(R) = 1.243 and r_e(R) = r_i(R) = 0.8968R. Third, for sphere with radius R < R_g, the solution becomes imaginary which means (in accordance with A. Einstein prediction) that GRT equations are not valid in this case. The dependences r(r) and g(r) corresponding to the limiting case R = R_g = 1.115r_g are shown in Figure 1 with solid lines. Dashed lines correspond to the limiting case (R = r_g) of the Schwarzchild solution.

As can be seen, the obtained solution coincides with the Schwarzchild solution at r = 0 and r ® ¥. The most pronounced difference is observed in the vicinity of the sphere surface. For R = r_g, the Schwarzchild solution is singular at r = R, whereas the obtained solution is finite.

To proceed, find the pressure p in the fluid. Taking s_r = -p in Equation (12) and substituting g_i from Equation (41), we arrive at the following equation for h_i:

in which, as earlier, (…)¢ = d(…)/dr.

Changing r to a new variable r_i, we can transform Equation (49) to

Integration of Equation (50) allows us to satisfy the boundary condition h_e(R) = h_i(R). The pressure p entering Equation (50) is specified by Equation (15).

Changing r to r_i in this equation, we get

Substituting Equation (50) in Equation (51), we obtain the following equation for the pressure:

As can be seen, Equation (52) is analogous to Equation (26) and its solution which satisfies the boundary condition p(r_R) = 0 is similar to Equation (28), i.e.,

The pressure at the sphere center (r_i = 0) is

The denominator of Equation (54) becomes zero if R = 1.013r_g which is less than the critical radius R_g = 1.115r_g. Thus, in contrast to the Schwarzcnild solution in Equation (31), the pressure is not singular. For the sphere with the critical radius R = R_g, we get p₀ = 0.808mc². The dependences of the normalized pressure on the radial coordinate are presented in Figure 2. Solid line demonstrates the obtained solution, Equation (53), for the sphere with the critical radius R = R_g. Dashed line corresponds to the Schwarzchild solution, Equation (28), for the sphere with the limiting radius R = R_s = 9/8r_g.

Thus, the critical radius is R_g = 1.115r_g. Using Equation (2) for r_g we get

As shown by Vasiliev and Fedorov [15] , for the sphere with radius R_g (as for the Black Hole) the escape velocity is equal to velocity of light and, following P. Laplace, this sphere can be referred to as the Dark Star. Naturally, the parameters of this object are different from those found by Laplace.

Thus, we can conclude that GRT equations are valid for the problem under study if R ³ R_g. The sphere with R < R_g is statically stable. Assume that due to high gravitation the Dark Star with radius R ≤ R_g attracts some additional fluid mass with the same density and the star mass becomes higher. Then, in accordance with Equation (55) the critical radius increases in proportion to the mass. However, the sphere radius R is proportional to m^1/3. So, the inequality R < R_g remains valid and the Dark Star remains invisible.

The obtained result is of a qualitative nature because it corresponds to the fluid sphere with constant density, i.e., to the object which is not quite realistic.