_{1}

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.

The existence of Dark Stars was predicted by J. Michel in 1783 and P. Laplace in 1796 [

in which c is the velocity of light and

is the so-called gravitational radius which depends on the gravitational constant G and the sphere mass. For the sphere with constant density µ in the Euclidean space

Substituting Equations (2) and (3) in Equation (1), we get

According to the reconstructed Laplace calculation, for Earth with density m_{E} = 5520 kg/m^{3} and radius R_{E} = 6371032 m, Equation (4) yields v_{e} = 11,000 m/s which is 27,270 times less than the velocity of light. Because v_{e} in Equation (4) is proportional to R increasing R up to 1.62 × 10^{11} m which is 249.6 times higher than the radius of Sun (6.96 × 10^{8 }m) we arrive at v_{e} = c. This calculation allowed Laplace to conclude that the star with the density of Earth and the radius which is about 250 times larger than the radius of Sun is not visible and can be referred to as the Dark Star. Later, the idea of Dark Stars based on CGT and the corpuscular model of light was abandoned by Laplace and the subsequent authors [

The general form of the line element in spherical coordinates r, q, j can be presented as

For spherically symmetric static problems, the components of the metric tensor g_{ij} depend only on the radial coordinate r. Material properties of space are determined by three components of the energy tensor

where (…)¢ = d(…)/dr.

In accordance with the basic mathematical idea of GRT, Equation (6) is satisfied identically if the tensor

Here,

is the GRT gravitational constant. Because the left-side parts of these equations are linked by Equation (6), only two of three Equations (7)-(9) are mutually independent. Traditionally [_{11} and g_{44} whereas Equation (8) is satisfied identically for any function g_{22}(r) Possible forms of the solution of two equations (7) and (9) with three unknown functions are discussed by Vasiliev and Fedorov [

Because the metric coefficients must be positive, introduce new functions, i.e., take g_{11} = g^{2}(r), g_{22 }= r^{2}(r), g_{44} = h^{2}(r). Consider the external and the internal spaces for the sphere with radius R.

For the external empty space (r ³ R) we have

Index “e” corresponds to the external space. Thus, we have two equations for three functions g_{e}(r), r_{e}(r), h_{e}(r) and to solve the problem, we need one more equation.

For the internal space of the solid sphere (0 £ r £ R),

where s_{r} and s_{q} are the radial and the circumferential stresses acting in a solid sphere. Then, Equations (7) and (9) become

Index “i” corresponds to the internal space. The conservation equation, Equation (6), takes the form

This is actually the equilibrium equation for a solid sphere [_{r} = s_{q} = ?p(r) where p is the pressure in the fluid. Then, Equation (14) reduces to

Thus, for the case of a fluid sphere we have three equations, Equations (12), (13) and (15), which include four unknown functions g_{i}(r), r_{i}(r), h_{i}(r) and p(r). Again, to solve the problem we need one more equation.

The solution for the external space must satisfy the asymptotic condition according to which it must degenerate into CGT solution with a distance from the sphere, the solution for the internal space must satisfy the regularity condition at the sphere center, and both solutions must meet the boundary conditions on the sphere surface.

Both external and internal problems for a fluid sphere were solved by K. Schwarzchiuld in 1916. The equation that forms the complete set with the equations presented above was taken in the form p(r) = r.

Putting p_{e} = r in Equations (11), we arrive at

The solutions of these equations are

The integration constants C_{1} and C_{2} 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 [

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

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 g_{22} = r^{2}(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 [_{g}/2. Then, the solutions of Equations (11) become [

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 [

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 [

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 [

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

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 [_{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 [

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 = 4/3pmR^{3} 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) [_{ }= 0 in Equation (28), we get

The denominator of this expression becomes zero if R = R_{s} = 9/8r_{g} = 1.125r_{g} [_{s}. This result is traditionally used to justify the existence of Black Holes [_{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) = ?mc^{2}/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

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 C_{4} = -r_{g} and C_{5} = 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 [

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 _{e} ® ¥, we can readily prove that r_{e} ® r. Differentiating Equation (47) with respect to r_{e} and dividing the result by_{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 _{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 p_{0} = 0.808mc^{2}. The dependences of the normalized pressure on the radial coordinate are presented in _{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 [_{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.

As follows from the foregoing discussion, the singular Schwarzchild solution of the spherically symmetric GRT problem satisfies the boundary condition for the space component of the metric tensor on the sphere surface only if the sphere mass is Euclidean which is not the case in GRT.

The proposed solution of the problem based on the assumption according to which the gravitation does not affect the sphere mass, in contrast to the Schwarzchild solution, is not singular and specifies the critical radius of the sphere limiting the application of GRT equations to the spheres whose radii are equal or larger than the critical value. The obtained solution is valid for the sphere consisting of a perfect incompressible fluid. Thus, the objects that are similar to Dark Stars in the Classical Gravitation Theory can follow from the General Relativity Theory.

The author thanks L.V. Fedorov for collaboration.

Vasiliev, V.V. (2017) Black Holes or Dark Stars―What Follows from the General Relativity Theory. Journal of Modern Physics, 8, 1087-1100. https://doi.org/10.4236/jmp.2017.87070