Chapter 2
Schwartzschild and Isotropic Solutions of the Einstein Theory

              This chapter describes the derivation of two commonly used solutions of the Einstein theory: the Schwartzschild solution and the related isotropic solution. These describe the gravitational effects of a star. The Schwartzschild solution is non-isotropic, and so it predicts that the velocity of light is different in the radial and tangential directions, relative to the center of the star. An even stranger property is that spatial compression is different in the radial and tangential directions. Consequently, the ratio of circumference to diameter of a circle centered about the star is not equal to p; it is greater than p. 

        The isotropic solution does not have this strange property of the Schwartzschild solution. However, the interior isotropic solution does not yield an analytical answer for pressure inside the star, and so the theoretical foundation for the isotropic solution is weak. It should be possible to remedy this with numerical computer analysis, but this apparently has not been done

2.1 Description of Schwartzschild Solution  

            Schwartzschld began his analysis by assuming that the metric equation is diagonal in spherical coordinates and has the following elements:

g00  =  en  ;  g11  =  - el  ;  g22  =  - r2   ;  g33  =  - (r sin q)2        (2-1)

  The general form of the metric equation in spherical coordinates is

           ds2  =  g00(dt)2 + g11(dr)2 + g22(dq)2 + g33(dy)2       (2-2)

The variables n and l are assumed to vary only with radial distance r. 

            Section B.5 of Appendix B gives formulas for computing the elements of the Einstein tensor Gmn from the metric tensor, when the metric tensor is diagonal. These were originally derived by Dingle and were published by Tolman [2]. With these formulas one can calculate from the metric tensor values in Eq. 2-1 the following formulas for the non-zero elements of the corresponding Einstein tensor:

            G00  =  e-l { - (l'/r) + (1/r2) } - (1/r2)                           (2-3)

            G11  =  e-l { (n'/r) + (1/r2) } - (1/r2)                           (2-4)

G22  =  G33  =  e-l { ½ (n'/r) - ½ (l'/r) + ½ n" - ¼ l' n' + ¼ (n')2 }   (2-5)

These formulas are given by Tolman [2] (p. 242, Eq. 95.3). The variables n', l' denote the partial derivatives of n, l relative to radial distance r. The variable n" is the second partial derivative of n relative to radial distance r, which is obtained by performing two partial-derivative computations in succession. The Einstein tensor symbol Gmn was not used by Tolman, and so Tolman employed the expression -8pTmn to represent Gmn in his equations.  

            The Schwartzschild analysis has two solutions: (1) the Schwartzschild interior solution, which applies inside the star, and (2) the Schwartzschild exterior solution, which applies in the vacuum of space outside the star. Let us first consider the interior solution.

            Schwartzschild interior solution. Appendix C presents the analysis performed by Schwartzschild to calculate the energy-momentum tensor for the interior of a star. He assumed that the star is a perfect fluid, with constant mass density and no viscous (or shear) forces. As shown in Eq. C-30, the resultant energy-momentum tensor is diagonal and has the following nonzero elements:

            T00 =  r      ;    T11  =  T22  =  T33  =  - p     (2-6)

The parameter r is the mass density of the star, which Schwartzschild assumed to be constant, and p is pressure within the star, which varies with radius. The density and pressure are expressed in normalized relativistic units. The Einstein gravitational field equation is

            Gmn  =   - 8p Tmn                                      (2-7)

Applying this gravitational field equation to the values in Eq. 2-6, gives the following for the required elements of the Einstein tensor Gmn:

            G00  =   - 8pr    ; G11  =  G22  =  G33  =  8pp            (2-8)

Substituting these values into Eqs. 2-3 to 2-5 gives the following equations for the Schwartzschild interior solution, which holds for r  <= rs (where rs is the radius of the star):

            G00  =  - 8pr  =  e-l { - (l'/r) + (1/r2) } - (1/r2)       (2-9)

            G11  =  8pp  =  e-l { (n'/r) + (1/r2) } - (1/r2)            (2-10)

            G22  =  G33  =  8pp                                                  (2-11)  

                    =  e-l{½ (n'/r) - ½ (l'/r) + ½ n" - ¼ l' n' + ¼ (n')2} 

            Schwartzschild exterior solution. In the vacuum of space outside the star, the energy momentum tensor is identically zero, and so the Einstein gravitational field equation requires that the Einstein tensor must be zero for the Einstein theory. Setting Eqs. 2-3 to 2-5 equal to zero gives the following equations for the Schwartzschild exterior solution, which holds for r >= rs:

            0  =  e-l { - (l'/r) + (1/r2) } - (1/r2)                                   (2-12)

            0  =  e-l { (n'/r) + (1/r2) } - (1/r2)                                    (2-13)

            0  =  e-l { ½ (n'/r) - ½ (l'/r) + ½ n" - ¼ l' n' + ¼ (n')2 }   (2-14)

            Results of Schwartzschild analysis. Tolman [2] (pp. 243-247) shows how Eqs 2-9 to 2-14 are solved to obtain the Schwartzschild exterior and interior solutions. As shown in Tolman's Eqs. 96.4 to 96.6, these equations simplify to the following three simultaneous equations, which hold inside the star 

            8pp  =  e-l {  (n'/r) + (1/r2) } - (1/r2)                               (2-15)

            8pr  =  e-l {  (l'/r) - (1/r2) } + (1/r2)                              (2-16)

            dp/dr  =  - ½ (r + p)n'                                                     (2-17)

On pages 246-247, Tolman derives from Eqs. 2-15 to 2-17 the following formula for pressure inside the star:

            p  =  A/B                                                                 

            A  =  3m{ Ö[1 - 2(m/rs)(r/rs)2] - Ö[1 - 2(m/rs)] }

            B  =  4prs3   { 3Ö[1 - 2(m/rs)] - Ö[1 - 2(m/rs)(r/rs)2] }      (2-18)

At the surface of the star, where r is equal to rs, this gives a pressure of zero, as it should be. However if 2m/rs is greater than unity, the pressure becomes imaginary inside the star at small values of radius. Since this is a physically impossible result, the Schwartzchild analysis does not yield a solution for m/rs greater than ½. Within the limits of the analysis (which are for 2m/rs <=1), the Schwartzschild solution yields the metric tensor values in spherical coordinates that are shown in Table 2-1. These metric tensor elements were obtained from Tolman [2] on p. 204 (Eq. 82.9) and on p. 247 (Eqs. 96.9, 96.13).

Table 2-1: Metric tensor values in spherical coordinates for the Schwartzschild exterior and interior solutions, which apply to a single star  

element    exterior                      interior solution

g00          1 - 2(m/r)        ¼{3Ö[1-2(m/rs)] - Ö[1-2(m/rs) (r/rs)2]}2
g11       -1/{1-2(m/r)}         -1/[1 - 2(m/rs) (r/rs)2 ] 
g22             - r2                                       - r2
g33           - (r sin q)2                           - (r sin q)2  

By Eq. 2-18, the pressure p inside the star is infinite if B is zero, which occurs if 

          3Ö[1 - 2(m/rs) ]  =  Ö[1 -  2(m/rs) (r/rs)2 ]                     (2-19)

Squaring this equation gives

          9 - 18(m/rs)  =  1 -  2(m/rs) (r/rs)2                                 (2-20)

Solving for (r/rs)2  gives

          (r/rs)2  =  9 - [4/(m/rs)]                                                   (2-21)

At the value of r given by Eq. 2-21 the pressure inside the star is infinite. When m/rs is 4/9, the pressure is infinite at the center of the star. When m/rs lies between 4/9 and ½, there is a spherical surface inside the star over which the pressure is infinite. As m/rs increases from 4/9 to ½, this spherical surface of infinite pressure expands from the center of the star to the circumference. Since infinite pressure inside a star is physically impossible, it is generally assumed that a star cannot exist if the m/r ratio at the surface of a star exceeds 4/9.

2.2 The Black Hole and Event Horizon Concepts

            Section 2.4 will show that the apparent relative speed of light in the radial direction is equal to

            cap/c  =  Ö[-g00/g11]            (Schwartzschild radial motion)         (2-22)

As will be shown in Section 2.4, this relation holds only in the radial direction for the Schwartzschild solution, but it holds in all directions for the Einstein isotropic solution and for the Yilmaz theory. For tangential motion (perpendicular to the radius), the cap/c ratio for the Schwartzschild solution is  

            cap/c  =  Ö[g00]            (Schwartzschild tangential motion)         (2-23)

Equations 2-22, 2-23 both show that the speed of light is zero when g00 is zero.  

            If m/rs is equal to ½, Table 2-1 shows that g00 should be zero at the surface of the star, and so the speed of light should be zero over the surface. Light presumably cannot escape from such a star, and so the star is called a black hole. A spherical surface over which the speed of light is zero is called an event horizon.

            When m/rs exceeds ½, the pressure inside the star becomes imaginary in the Schwartzschild solution. It was originally assumed that the Schwartzschild analysis does not apply under this condition. However later studies have led to the conclusion that when m/rs exceeds ½ the star must collapse to form a singularity at its center of infinite mass density. Under his assumption, the interior solution no longer exists, and so the exterior solution can still apply. For such a star there should be an event-horizon spherical surface beyond the original radius of the star, which occurs at a radius r equal to 2m. Over this event-horizon the speed of light should be zero. Light presumably cannot escape from the volume inside this event-horizon sphere.

        This event horizon, which is derived from the exterior Schwartzschild solution, falls at or outside the original radius of the star when the m/rs ratio is equal to or greater than ½. Another event horizon is predicted by the interior Schwartzschild solution, and falls inside the star. This event horizon inside the star occurs if m/rs is greater than 4/9 but less than ½. Table 2-1 shows that g00 for the interior solution is zero inside the star under the same conditions where the quantity B in Eq. 2-18 is zero, and so the pressure p inside the star is infinite. 

        At the value of r given by Eq. 2-21, g00 is zero and so the speed of light should be zero. The equation shows that when m/rs is equal to 4/9, the speed of light should be zero at the center of the star and the pressure should be infinite. When m/rs lies between 4/9 and ½, there should be an event-horizon spherical surface inside the star where the speed of light is zero and the pressure is infinite. As m/rs increases from 4/9 to ½, this event-horizon surface should expand from the center of the star to the circumference.

            A star cannot exist in a normal state if it has an event horizon surface inside the star, because electromagnetic fields cannot penetrate this boundary, and the pressure inside the star should be infinite over this surface. Therefore it is generally assumed that a star must collapse into a black hole if m/rs exceeds 4/9. This writer takes the position that the black hole and event horizon concepts do not represent physical reality, because they are not displayed in the Yilmaz theory. They are merely symptoms of a mathematical weakness in the Einstein theory.  

2.3 The Isotropic Solution

            As shown by Tolman [2] (p. 245), a second solution for a single star was derived by assuming that the metric equation is isotropic, which requires that the values of g11, g22, g33 be equal in rectangular coordinates. The resultant solution is called the isotropic solution. The corresponding metric tensor elements in spherical coordinates have the form 

g00  =  en ;  g11  =  - em  ;  g22  =  - r2 g11  ;  g33  =  - (r sin q)2 g11  (2-24)

The variables m and n are assumed to vary only with radial distance r. 

            As shown by Tolman [2] (p. 242, Eq. 95.6), the following formulas for the non-zero elements of the corresponding Einstein tensor are calculated from these metric tensor values:

            G00  =  e-m { m" + (m'2/4) + (2m'/r) }                                  (2-25)

            G11  =  e-m { ¼ m'2 + ½ m'n' + (m'/r) + (n'/r) }                     (2-26)

G22  =  G33  =  e-l { ½ m" + ½ n" + ¼ n'2 + (m'/2r) + (n'/2r) }        (2-27)

            Just as for the Schwartzschild solution, these Einstein tensor values are set equal to zero to obtain equations for the exterior isotropic solution, and are set equal to the energy-momentum tensor values of Eq. 2-6 to obtain equations for the interior isotropic solution. Tolman (p. 244, Eq. 95.15) shows that this analysis reduces to the following three simultaneous equations:

            8pp  =  e-m { ¼ m'2 + ½ m'n' + (m'/r) + (n'/r) }                (2-28)

            8pr  =  - e-m {  m" + ¼ m'2 + 2(m'/r) }                            (2-29)

            dp/dr  =  - ½ (r + p)n'                                                    (2-30)

Equation 2-30 is the same as Eq. 2-17 for the Schwartzschild solution. As explained by Tolman (p. 246), Eq. 2-30 has the following solution, where C is an unspecified constant:

            r + p  =  C e-n/2                                                               (2-31)  

However Eqs. 2-28, 2-29 cannot be reduced to an analytical expression for the pressure p, as was done with the Schwartzschild solution, and so this analysis for the isotropic solution has never been completed. Nevertheless one should be able to obtain a solution with a computer using numerical analysis.  

            As explained by Tolman [2] (p. 205, Eqs. 82.11, 82.12), formulas for the exterior isotropic solution have been derived from the formulas for the exterior Schwartzschild solution as follows. The metric equation for the exterior Schwartzschild solution in spherical coordinates is

   ds2  =  [1 - 2(m/r)](dt)2 - [1 - 2(m/r)]-1(dr)2 - r2 (dq)2 - (r sin q)2 (dq)2                                                                                                  (2-32)
The radial distance r in Eq. 2-32 is replaced by the following expression

            r  =  { 1 + (m/2r) }2 r                                                     (2-33)

This yields the following equation, which is isotropic relative to the variable r:

ds2  =  g00(dt)2 + g11 { (dr)2 + r2(dq)2 + (r sin q)2 (dy)2 }          (2-34)

where g00, g11 are equal to

g00  =  {1 - (m/2r)}2/{1 + (m/2r)}2                                        (2-35)

g11  =  - {1 + (m/2r)}4                                                           (2-36)

When the isotropic solution is used, the variable r is usually treated as the radial distance r. However, it is not clear what the radial distance variable means for this solution.

2.4 Implications of Isotropy

The isotropic solution of the Einstein theory and all solutions of the static Yilmaz theory have isotropic metric tensors. In rectangular coordinates, an isotropic metric tensor is diagonal and the values of g11, g22, g33 are equal. Hence the metric equation for an isotropic metric tensor has the following form in rectangular coordinates:

ds2  =  g00(dt)2 + g11 { (dx)2 + (dy)2 + (dz)2 }              (2-37)

It can be shown that the differential coordinates for rectangular and spherical coordinates are related by.

(dx)2 + (dy)2 + (dz)2  =  (dr)2 + r2 (dq)2 + (r sin q)2 (dq)2       (2-38).

Substituting Eq. 2-38 into Eq. 2-37 gives the isotropic metric equation in spherical coordinates:

 ds2  =  g00(dt)2 + g11 {  (dr)2 + r2 (dq)2 + (r sin q)2 (dq)2 }.

      =  g00(dt)2 + g11 (dr)2 + r2 g11 (dq)2 + (r sin q)2 g11 (dq)2      (2-39)

Thus for an isotropic metric equation, the metric tensor elements g00, g11 in spherical coordinates are the same as in rectangular coordinates, and the elements g22, g33 in spherical coordinates are equal to:

    g22  =  r2 g11   ;    g33  =  (r sin q)2 g11                                (2-40)

Let us replace the differential d by the incremental quantity D. Equation 2-37 shows that an isotropic metric equation in rectangular coordinates can be expressed as

    (Ds)2  =  g00(Dt)2 + g11 (Dxd)2                                       (2-41)

where Dxd is the total distance between two points, which is equal to

   (Dxd)2  =  (Dx)2 + (Dy)2 + (Dz)2                                      (2-42)  

Section 1.4 of the Summary (page 1) of this website shows (in Table 1.4-2) that when the metric equation is isotropic, the apparent speed of light and the apparent compression of distance are given by

         cap/c  =  Ö[-g00/g11]                                                 (2-43)

         Dxap/Dx  =  1/Ö[g11]                                                  (2-44)

For an isotropic metric tensor, these equations apply to motion in any direction. The metric equation for the Schwartzschild solution is not isotropic, and it is diagonal only in spherical coordinates. In spherical coordinates this metric equation is

Ds2  =  g00(Dt)2 + g11(Dr)2 - r2(Dq)2 - (r sin q)2(Dy)2            (2-45)

Consider incremental linear displacements in the q and y directions, and denote these as Dxq and Dxy. These linear displacements are related as follows to the angular displacements of q and y:

     Dxq  =  r Dq                                                                 (2-46)

     Dxy  =  r sin q Dy                                                       (2-47)

Substituting Eqs. 2-46, 2-47 into Eq. 2-45 gives

Ds2  =  g00(Dt)2 + g11(Dr)2 - (Dxq)2 - (Dxy)2                     (2-48)

The displacements Dxq, Dxy are tangential linear displacements that are normal to the radial vector. Let us denote the total tangential displacement as Dxt. This is related as follows to the displacements in the q and y directions:

    (Dxt)2  =  (Dxq)2 + (Dxy)2                                                (2-49)

Substituting Eq. 2-49 into Eq. 2-48 gives:

Ds2  =  g00(Dt)2 + g11(Dr)2 - (Dxt)2                                    (2-50)

This is a convenient form of the metric equation for the Schwartzschild solution. From this we can obtain formulas for the speed of light and for spatial compression for motion in the radial direction by setting Dxt equal to zero. We can obtain the corresponding formulas for motion in the tangential direction by setting Dr equal to zero

    By applying these principles, and using the concepts given in Section 1.4 of the Summary page of this website, the formulas for speed of light and spatial compression for the Schwartzschild solution can be calculated as follows for motion in radial and tangential directions.

Radial:             cap/c  =  Ö[-g00/g11]  =  1 - 2(m/r)             (2-51)

Tangential:            cap/c  =  Ö[g00]  =  Ö[1 - 2(m/r)]            (2-52) 

Radial:       Drap/Dr  =  1/Ö[-g11] =