Spherically Symmetric Problem of General Relativity for a Fluid Sphere ()
1. Introduction
The paper is concerned with the spherically symmetric GR problem for a sphere consisting of perfect fluid. This problem is of primary importance for the theory, because the model of an incompressible fluid allows us to obtain the exact analytical solution which is hardly reachable for more complicated models. The first solution of the problem was found by K. Schwarzchuld [1] who discovered than for a certain radius of the sphere (8/9 of the gravitation radius) the pressure at the sphere center can become infinitely high. This result was later associated with the existence of Black Holes [2] . However, the model of an incompressible fluid is not compatible with GR [3] , because the velocity of sound in such fluid is infinitely high, whereas in GR it is limited by the velocity of light. The numerical solution for a sphere consisting of a compressible gas was obtained by J.R. Oppenheimer and G.M. Volkoff [4] . A class of inverse solutions was constructed by R.C. Tolman [5] . Since three field equations (for the energy-momentum tensor components
and
) include four unknown functions (two metric coefficients
and
, pressure and density), the metric coefficients were linked by an assumed equation and the obtained solutions were studied with respect to physical consistency. Further results are presented elsewhere [6] [7] [8] [9] .
Though a sphere of perfect fluid is frequently used as an idealized model of stellar objects, particular neutron stars [4] , the problem considered in this paper corresponds to the conventional phenomenological GR theory based on the traditional model of space as a homogeneous isotropic continuum whose actual microstructure is ignored.
The aforementioned results are based on the traditional form of the line element in which space and time metric coefficients correspond to the Riemannian space. This paper demonstrates the alternative approach based on the proposed special model of the Riemannian space which is Euclidean with respect to space coordinates and is Riemannian with respect to time only. Within the framework of this model, analytical solutions of the external and internal problems for a sphere of perfect incompressible fluid are obtained and the numerical solution of the internal problem for a sphere consisting of compressible fluid or gas is constructed.
2. Special Riemannian Space Model
Within the framework of the traditional GR theory, the continuum is characterized with the energy-momentum tensor
introduced in a 4-dimensional Riemannian space with the line element
(1)
The energy-momentum tensor must satisfy four conservation equations
(2)
If
are the space coordinates and
is the time coordinate, the first three Equations (2) are the motion equations, whereas the fourth Equation (2) provides the conservation of mass.
To obtain the expressions for
, consider a spherically symmetric problem in coordinates
for the Newton gravitation theory. The field equations including the motion equation and the conservation equationin the Euler coordinates are [10]
(3)
Here,
and
are the radial and the circumferential stresses,
is the radial velocity,
is the density, t is time, and
is the Newton gravitation potential. Using identical transformations and the second equation in Equations (3), we can reduce the right-hand part of the first equation to the following form [10] :
Using this result, we can present Equations (3) as
(4)
Here, c is the velocity of light. To apply Equations (2) for the energy-momentum tensor to the spherically symmetric problem of the Newton theory, we should take the following form of the line element in Equation (1):
(5)
in which the amplitude value of function f is much smaller than unity. Applying Equations (2) for the line element in Equation (5) and undertaking linearization of these equations with respect to f, we arrive at
(6)
Matching Equations (4) to Equations (6), we can conclude that
(7)
The last of these equations in which f is the component of the metric tensor of the Riemannian space and
is the Newton gravitation potential shows that the analogy between the gravitation and the Riemannian geometry actually follows from the Newton gravitation theory. In GR, Equations (7) are generalized as
(8)
It is assumed that
is the same in the first and the rest equations which follows from the principle of equivalence of the gravitation and the inertia masses. It is important to take into account that in the spherical coordinates mixed tensor components coincide with physical components. So, for a spherically symmetric problem, we have
(9)
To determine the geometry of the Riemannian space corresponding to the Newton gravitation theory, the trajectory of a particle in a Newton gravitation field is compared to the equation that specifies a geodesic line in a Riemannian space with the line element
(10)
The result of comparison yields the following expressions for the components of the metric tensor [11] :
(11)
in which
(12)
is the so-called gravitation radius that depends on the sphere mass m and the classical gravitation constant G. Note that Equations (10) and (11) correspond to the Riemannian space which is Euclidean with respect to the space coordinates
and Riemannian only with respect to the time coordinate tc.
The obtained solution is not confirmed with experiment. The angle of deviation of the light beam from the straight line in the vicinity of Sun calculated for the space with the metric coefficients in Equations (11) turns out to be only one half of the measured value. A more general GR theory is based on equation
(13)
according to which the energy-momentum tensor in Equations (8) is proportional to the Einstein tensor
(14)
in which
are the components of the Ricci curvature tensor depending on the metric tensor of the four-dimensional Riemannian space. The coefficient
(15)
is the relativity gravitation constant.
Thus, in accordance to Equations (13) and (14), gravitation is associated with the curvature of the Riemannian space. However, as follows from the foregoing derivation, only one component of the energy-momentum tensor, namely
, allows for gravitation. The rest components in Equations (8) include mechanical stresses and velocities which induce the curvature of the Riemannian space as well. So, we can conclude that according to GR the space can be Euclidean only in the absence of gravitation, stresses and motion. Since this is not the case for the real continuum, the corresponding space is Riemannian and the Euclidean space does not exist. It should be noted that three-dimensional Riemannian space can exist in the Euclidean space with six dimensions [12] , so the actual space is Euclidean but six-dimensional. This strange result follows directly from the GR theory. Fortunately, there is a more simple and realistic model of space. Recall that in the Newton theory Equations (10) and (11) correspond to the space which is Euclidean with respect to space coordinates and Riemannian with respect to time. Thus, introduce, in general, the special Riemannian space which is Euclidean with respect to space coordinates
and Riemannian with respect to the time coordinate
only. In the absence of gravitation, the space is three-dimensional and Euclidean, whereas the continuum is described by the classical theories of deformable solids, fluids and gasses. Gravitation results in a four-dimensional space with space coordinates supplemented with time whereas the continuum is described by GR theory Equations (2), (13) and (14) in which six components of the metric tensor, namely
, correspond to the Euclidean space and are known. The rest four metric coefficients, namely
, are found from equations, following from Equations (13) and (14). In general, there are ten such equations which describe the gravitation in vacuum. However, since the tensor
is proportional to tensor
and hence satisfies four Equations (2), only six of these ten equations are mutually independent. The traditional set of GR equations is not complete [11] [13] [14] [15] and should be supplemented with the so-called coordinate conditions the general form of which is not known. For the proposed special Riemannian space, the set of independent GR equations consists of six equations and includes four unknown metric coefficients. In the theory of partial differential equations, the case in which the number of equations exceeds the number of unknown functions is less critical than the case in which the number of unknown functions exceeds the number of equations, because some equations can be used to determine the integration functions which enter the solutions of the other equations. Moreover, it can occur that some of equations are satisfied identically. This is the case for a spherically symmetric problem considered further for which the set of GR equations is complete. For this problem, the line element corresponding to the proposed special Riemannian space is [16]
(16)
As can be seen, in contrast to the traditional Equation (19), there are two unknown metric coefficients and the space is not “orthogonal” to time.
3. The Schwarzchild Solution
The first solution of the GR problem for a sphere consisting of a perfect incompressible fluid was obtained by K. Schwarzcchild. The set of the Einstein equations, Equations (13), (14), for a spherically symmetric static problem with the line element in Equation (10) has the following form [17] :
(17)
(18)
(19)
where
. For the sphere consisting of a perfect incompressible fluid,
, where p is the pressure and the density is constant, i.e.
. For a static problem,
and Equations (8) yield
Consider the empty space surrounding the sphere with radius R. For this space,
,
and Equations (17)-(19) are homogeneous. As can be seen, Equations (19) and (17) allow us to express
and
in terms of
. The general solution of these equations is [18]
(20)
in which C1 and C2 are the integration constants. Substituting Equations (20) in Equation (18), we can conclude that this equation is satisfied identically with any function
. This result looks natural because the set of the GR equations is not complete. The Schwarzchild solution was obtained under the additional coordinate condition
. Originally, K. Schwarzchild used a different condition which can reduced to the written above [19] . The integration constants can be found from the asymptotic conditions according to which Equations (20) should reduce to Equations (11) corresponding to the Newton theory for
. Taking
, we finally get
(21)
Here
is specified by Equation (12) and index “e” corresponds to the external space. The solution in Equations (21) formally has two singular points—
and
. The second singularity does not appear because Equations (21) are valid for
in which R is the radius of a fluid sphere. The first singularity takes place at the minimum possible value of r which is
. Thus, for the problem under study, the so-called Schwarzchild singularity appears on the sphere surface. Naturally,
cannot be referred to as the radius of the horizon of events because the penetration through this surface is physically not possible.
Consider the internal problem for
. For
and
, integration of Equation (19) yields
(22)
in which index “i” corresponds to the internal space and C3 is the integration constant. We should take
, otherwise,
becomes singular at the center of the sphere of any radius. Thus,
(23)
As can be seen this expression does not include an integration constant. This is associated with the coordinate condition
. Indeed, substitution of this condition in Equation (19) which has initially the second order reduces it to the equation of the first order whose solution (22) includes only one constant. As a result, we arrive at the problem associated with the boundary condition on the sphere surface according to which
(24)
Substituting Equations (21) and (23) and using Equations (12) for
and (15) for
, we arrive at the following expression for the sphere mass:
(25)
which corresponds to the Euclidean space. However, the space inside the sphere is Riemannian and the mass can be found using Equation (23) as
(26)
This result coincides with Equation (25) if
which means the absence of gravitation. Naturally, the actual mass is specified by Equation (26) and the boundary condition (24) is not satisfied in the Schwarzchild solution.
To proceed, transform Equation (23) Equations (12), (15) and (25) for
,
and m which yield
(27)
Then, Equation (23) becomes
(28)
Consider Equation (2) which for the problem under study has the following form:
Substituting Equations (20) for the energy-momentum tensor, we get
(29)
Taking
in Equation (17) and substituting
from Equation (28), we can find
(30)
where
(31)
Equations (29) and (30) yield the final equation for the normalized pressure
The solution of this equation which satisfies the boundary condition
is [17]
(32)
Using this result and integrating Equation (30) under boundary condition
, we can obtain the following expression for the time metric coefficient [17] :
(33)
Substituting the obtained solution, Equations (28), (32) and (33), in the remaining field equation, Equation (18), we can prove that this equation is satisfied identically. Thus, Equations (21), (28), (32) and (33) specify the Schwarzchild solution of the spherically symmetric problem for a sphere consisting of a perfect incompressible fluid. To analyze this solution, calculate the pressure at the sphere center. Taking
in Equation (32), we get
(34)
The denominator of this expression is zero at
. This means that the pressure
becomes infinitely high for the sphere with radius
. It is natural to suppose that the Schwarzchild solution is not valid for the spheres whose radius is less than
. Indeed, for
(
) Equation (33) gives negative pressure (
) which has no physical sense. Thus, the Schwarchild singularity following from Equations (21) and taking place at
is not reached in the Schwarzchild solution for a fluid sphere.
4. GR Theory Equations for the Special Riemannian Space
Consider spherically symmetric problem in the Riemannian space with the line element in Equation (16). The Einstein equations, Equations (13) and (14), have the following form:
(35)
(36)
(37)
(38)
,
(39)
Here,
and
. The energy-momentum tensor satisfies conservation Equations (2) whose explicit form is
(40)
(41)
The energy-momentum tensor is specified by Equations (9) which need some additional comments. The GR problem cannot be strictly static even if there is no motion, because Equations (9) include the velocity of light. If the radial and the time coordinate axes are orthogonal as in Equation (10) used in the previous Section, the velocity of light directed along the time axis does give the projection on the radial axis and for a static problem
. However, for an oblique coordinate frame the situation is different [20] . The velocity of light gives zero contravariant projection on the radial axis, whereas the covariant projection is not zero. Thus, we must take
and
in Equations (9) which take the following form for a sphere consisting of a perfect fluid:
(42)
Simplify Equations (35)-(39). Since
, Equation (39) yields
where
is an unknown function. It is natural to assume that for a static problem the functions in Equations (35)-(38) and (40)-(42) do not depend on time. Then, taking into account Equations (41), we can present Equations (35)-(38) and (40), (41) as
(43)
(44)
(45)
(46)
(47)
(48)
Since we have two Equations (47) and (48), only two of four Equations (43)-(46) are mutually independent. Thus, we have four equations for four functions
and the system of GR equations is complete.
5. External Solution
Consider the empty space surrounding the sphere with radius R. In this space,
and
, Equations (47) and (48) are satisfied identically, whereas (43)-(46) are homogeneous. Taking
in Equations (45) and (46), we get
For
, the obtained solution must reduce to the solution following from the Newton theory, i.e.,to
,
. Thus,
(49)
The obtained solution is not singular and specifies two equivalent spaces that correspond to the positive and the negative values of
. Recall that solution (49) is found from Equations (45) and (46). Direct substitution in the rest homogeneous Equations (43) and (44) satisfies these equations identically. Equations (49) correspond to the so-called Gullstand-Painlever coordinates [21] [22] found as a result of coordinate transformation of the Schwarzchild solution. Here, the metric coefficients in Equations (49) are not associated with the Schwarzchild solution and follow from the proposed model of the Riemannian space.
6. Internal Solution for a Sphere Consisting of a Perfect Incompressible Fluid
Consider an internal space of a fluid sphere for which
. Integration in Equation (45) yields
(50)
in which
. We can transform this result with aid of Equation (27). Note that in contrast to the Schwarzchild solution considered in Section 2, for the problem under study this equation is exact, because the space in coordinates
is Euclidean and the sphere mass is specified by Equation (25). Thus,
Applying Equation (42) and Equation (46) for g, we get
(51)
Consider Equation (48) and express
(52)
Substituting this result in equation (47), we arrive at the following equation:
(53)
which is the same as Equation (29) in the Schwarzchild solution. Transforming Equation (43) with the aid of Equations (27), (42) and (46), we find
(54)
This equation is also the same as Equation (30) in the Schwarzchild solution. Since Equations (53) and (55) coincide with Equations (29) and (30), and the boundary conditions are the same in the Schwarzchild solution, the pressure is specified by Equation (32) and the time metric coefficient
by Equation (33). As in the Schwarzchild solution, the pressure at the sphere center becomes infinitely high for the sphere with radius
. For spheres whose radii are less than
, the solution does not exist. It should be noted that, in contrast to the Schwarzchild solution, the obtained external solution is not singular for
.
The metric coefficient
can be found from Equations (42) and (51), i.e.
(55)
Taking
, we get
and the boundary condition on the sphere surface is satisfied. To complete the analysis, determine the velocity
. Substituting the obtained results in Equation (52), we arrive at
(56)
The maximum absolute value takes place on the sphere surface, i.e.,
Formally,
if
. The same result was obtained for a solid elastic sphere [20] . However, as follows from Equation (56), any point of a fluid has two equal velocities with different signs and we can suppose that
is not associated with any real motion. The obtained solution identically satisfies the field equations, Equations (43)-(48).
7. Internal Solution for a Sphere consisting of a Perfect Compressible Fluid or Gas
The model of a perfect incompressible fluid considered in Sections 2 and 5 is not compatible with GR, because the velocity of sound in this model is infinitely high, whereas in the GR theory it cannot be higher than c. To avoid this shortcoming, consider the case of a compressible fluid or gas and assume that the density is a linear function of the pressure, i.e.,
(57)
in which k is a given coefficient. Recall that only four of six field Equations (43)-(48) are mutually independent. To study the problem, i.e., to determine
and p, we use Equations (43), (45), (47) and (48). In addition to Equations (31), introduce the following normalized parameters:
and
. Then, Equation (57) can be presented as
(58)
Consider Equations (12), (25) and (20) for
and
. The first of them, i.e.
, is valid for a compressible sphere material, because the GR Equations (2) provide the conservation of mass which has the form
(59)
Introduce a formal parameter
which is the gravitation radius of the homogeneous sphere with radius R, density
and mass
. Then, Equation (27) yields
(60)
in which
.
The first equation of the governing set follows from Equation (43) which can be presented with the aid of Equation (60) as
(61)
Taking into account Equations (58), (61) and using Equation (45), we can obtain the second equation of the governing set
(62)
Equation (48) can be used to determine
, i.e.,
Substituting this result in Equation (47), we arrive at the third equation of the governing set
(63)
Thus, we arrived at three Equations (62), (63) and (64) for
and
. In these equations,
and
. Note that we do not know the sphere radius R which depends on pressure. To obtain the solution, we need three boundary conditions according to which
(64)
Since we do not know
, we apply the following iteration procedure. For the preliminary selected parameter
, we take
and undertake numerical integration of the equations. Substituting the obtained solution for
in Equation (59), we find
(65)
The integration is repeated with this value of
and so on. The procedure is terminated when the fourth boundary condition following from the solution for the external space in Equations, i.e.,
(66)
is satisfied with the given accuracy.
To demonstrate the procedure, take
and
. Then, for the first iteration step, we have initially
. Integration yields
and
. We find
and the boundary condition in Equation (66) is not satisfied. For the second step, we take
in Equations (65), so that
. The result of integration is
and the boundary condition in Equation (66) is practically satisfied. Thus, for
we arrive at
and we need only two iteration steps to obtain the solution with the reasonable accuracy.
For
, we have the analytical solution presented in Section 4. According to this solution, the pressure at the sphere center becomes infinitely high for the sphere with the normalized gravitation radius
. Numerical integration for
allows us to conclude that the numerical procedure does not converge if
. Thus, we can associate the absence of the process convergence with the pressure singularity at the sphere center. Assume a relatively small value, e.g.
. The maximum value of
for which the process converges is 0.884 which corresponds to
. The results of the parametric analysis are presented in Table 1.
As can be seen, the higher is the fluid or gas compressibility, the lower is the maximum gravitation radius. In general, the maximum gravitation radius depends on the model of the sphere material. For example, for a solid elastic sphere, it is
[20] .
The dependence of the normalized pressure on the radial coordinate is shown in Figure 1. Solid lines correspond to the compressible fluid or gas, whereas dotted lines demonstrate the analytical solution for an incompressible fluid (the numbers on curves correspond to Table 1).
Table 1. The maximum values of
and
for various values of parameter
.
Figure 1. Dependences of the normalized pressure on the radial coordinate for various values of parameter
, ─: compressible fluid or gas, ∙∙∙: incompressible fluid.
8. Conclusion
The proposed model of the Riemannian space according to which the space is Euclidean with respect to space coordinates of the continuum and is Riemannian with respect to the time coordinate only is applied to a spherically symmetric GR problem for a perfect fluid or gas sphere. The solution to the external problem results in the so-called Gullstand-Painlever metrics. The solution of the internal problem for an incompressible fluid gives the pressure which coincides with the Schwarzchild solution for the same problem. The solution for the fluid or gas whose density is a linear function of pressure is obtained numerically and compared to the Schwarzchild solution for an incompressible fluid.
Acknowledgements
The study was conducted within the framework of the Russian Science Foundation Project No. 23-1100275.