Appendix C: Schwartzschild Energy-Momentum Tensor

            This appendix describes the general principles for preparing the energy-momentum tensor and it gives the detailed analytical steps that Schwartzschild developed to form the energy-momentum tensor that he used in his solution of the Einstein theory.

C.1 Definition of Energy Momentum Tensor T^mn

            Tolman [2] defines the energy-momentum tensor by considering its elements in proper coordinates, which are coordinates that move with the material at each point of interest. Hence within a small region of proper coordinates, all velocities are zero. Tolman (p. 215, Eq. 85.1) gives the following matrix for the energy-momentum tensor in proper coordinates:

.                        |  r      0         0          0       |
.                       |  0      p_xx         p_xy      p_xz      |      (C-1)
    T_p^mn  =        |  0       p_yx      p_yy      p_yz     |
.                       |  0       p_zx      p_zy      p_zz       |

              In this matrix the variable r is the mass density, which is mass per unit volume. The variable p_xy is the stress, which is force per unit area. As was shown in Fig. 5-7 of Chapter 5 of Universe [1], p_xy is the force per unit area applied in the x-direction to a surface that is perpendicular to the y-axis. The mass density r and the pressure p are expressed in normalized mass units. Remember that the conventional mass M is multiplied by G/c² to obtain the normalized mass m. Therefore the elements in the above matrix should be multiplied by G/c² if r and p are expressed in conventional units.

C.2 Formula for Converting Coordinates of a Tensor

            The matrix of Eq. C-1 is a general expression that can be applied at any point in any physical medium. However each point of a general medium may be moving at a different velocity, and so each point may require its own set of proper coordinates. In order to utilize the information for the whole body, one must convert the tensor values for each point to a single coordinate system. This is achieved by applying the general formula for translating a tensor from one coordinate system to another

            The following formula allows one to translate a contravariant tensor from proper coordinates to any other coordinate system:

            T^mn  =  S_a S_b ( ¶x^m/ ¶x_p^a)( ¶xⁿ/ ¶x_p^b)T_p^ab     (C-2)

Variables with the subscript p apply to proper coordinates. Variables without subscripts apply to the fixed coordinate system into which the data are being translated. (Equivalent formulas apply to covariant and mixed tensors.)

            This formula of Eq. C-2 is the key to tensor analysis. All tensors can be transformed from one coordinate system to another in accordance with this formula (or its equivalent). The subscript p can apply to any coordinate system in which a tensor T_p^ab is specified, and the variables without subscripts can apply to any other coordinate system. By means of this formula Einstein achieved covariance in his physical law of general relativity by specifying his law in terms of tensors.

            Since Eq. C-2 has two summations, performed over the four values of the a and b indices, the equation for the T^mn tensor yields 16 terms for each element of the tensor. Hence 160 terms are required to specify the ten independent elements of the T^mn tensor. To illustrate the application of the tensor formula of Eq. C-2, the following are 8 terms of the 16-term expansion, for a = 0, 1, and for b = 0, 1, 2, 3:

T^mn  =  ( ¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰)T_p⁰⁰ +  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p¹)T_p⁰¹

       + (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p²)T_p⁰² + (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p³)T_p⁰³             

         + (¶x^m/¶x_p¹)(¶xⁿ/¶x_p⁰)T_p¹⁰ +  (¶x^m/¶x_p¹)(¶xⁿ/¶x_p¹)T_p¹¹

         + (¶x^m/¶x_p¹)(¶xⁿ/¶x_p²)T_p¹² + (¶x^m/¶x_p¹)(¶xⁿ/¶x_p³)T_p¹³ + etc.

                                                    (C-3)

        Both of the indices m, n are set equal to 0, 1, 2, 3 to obtain the specific formulas  for the individual elements of the energy-momentum tensor T^mn. A total of 256 terms are needed to calculate all 16 elements of this tensor. However this tensor is symmetric, and so only 10 elements need be computed. This reduces the number of terms to 160.

            The expressions of the form ¶xⁿ/¶x_p² in Eq. C-3 are calculated by examining the equations that specify the two sets of coordinates (proper and fixed). By applying calculus to those equations, partial derivatives are computed that relate the variables that are specified in the two sets of coordinates.

            The task of computing the 160 terms required to translate a general energy-momentum tensor from proper to fixed coordinates is extremely tedious.  It may be possible to implement this with hand computations for one point of the medium. However for a general medium, where the velocity changes from point to point, many different sets of proper coordinates must be used to characterize the medium with reasonable accuracy. A different set of transformation equations is required for each set of proper coordinates.

            For such general applications, computer calculations are essential to specify the energy-momentum tensor. Since computers were not available to Einstein, he and other physicists of his day had to restrict themselves to simple physical models when calculating the energy-momentum tensor.

            The Yilmaz theory of gravitation does not require that the energy-momentum tensor be calculated when the theory is applied. Therefore the practical problems of computing the energy-momentum tensor are irrelevant when the Yilmaz theory is used.

C.3 Calculating the Schwartzschild Energy-Momentum Tensor

            In Eq. C-1 the three stresses on the diagonal (p_xx, p_yy, p_zz) are the compressive stresses, which represent pressure in a fluid. The six nondiagonal stresses (p_xy, p_xy, p_xy, etc.) are the shear stresses. Schwartzschild modeled the star as a perfect fluid, which has no shear (or viscous) forces. The three pressure stresses (p_xx, p_yy, p_zz) are equal, and each is represented by the hydrostatic pressure p. Hence, The Schwartzschild energy-momentum tensor in proper coordinates simplifies from Eq. C-1 to

.                    |   r      0       0       0    |
.                   |   0       p         0       0     |           (C-4)
T_p^mn  =       |   0       0       p       0     |
.                   |   0       0       0       p     |

            Converting from proper to fixed coordinates. Since T_p^mn is diagonal, it has nonzero values only when n = m. Hence in the general formula of Eq. C-2 for converting the coordinates of a tensor, we can set b = a in the expression for T_p^ab. This eliminates the summation over the index b and Eq. C-2 simplifies to

            T^mn  =  S_a ( ¶x^m/ ¶x_p^a)(¶xⁿ/ ¶x_p^a)T_p^aa                              (C-5)

       The summation over the index a can be separated into two parts: the first part is the value for a = 0 and the second is the summation over a = 1, 2, 3.

  T^mn  =  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰)T_p⁰⁰ + S_k(¶x^m/¶x_p^k)(¶xⁿ/¶x_p^k)T_p^kk  (C-6)

  The index in the summation is changed from a to k, where k takes the three values (1, 2, 3). The matrix of Eq. C-4 shows that the elements of the Schwartzschild energy-momentum tensor in proper coordinates are

             T_p⁰⁰ =  r          ;    T_p¹¹  =  T_p²²  =  T_p³³  =  p       (C-7)

  Apply these values to Eq. C-6, noting that T_p^kk is equal to p for all three values of k. This gives

T^mn  =  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰)r + S_k (¶x^m/¶x_p^k)(¶xⁿ/¶x_p^k)p      (C-8)

            In proper coordinates, special relativity applies within a small region, and so the line element in proper coordinates is

             ds²  =  dt² - dx² - dy² - dz² 

                   =  g₀₀(dx⁰)² + g₁₁(dx¹)² + g₂₂(dx²)² + g₃₃(dx³)²             (C-9)

Hence the metric tensor values for proper coordinates, which are the same as for special relativity, are

            g₀₀  =  1  ;    g₁₁  =  g₂₂  =  g₃₃  =  - 1                                  (C-10)

When the covariant metric tensor g_mn is diagonal, the contravariant tensor g^mn is also diagonal and its diagonal elements are the reciprocals of the covariant elements. Hence the contravariant metric tensor values in proper coordinates are

            g_p⁰⁰  =  1  ;    g_p¹¹  =  g_p²²  =  g_p³³  =  - 1               (C-11)

The subscript p has been included here to emphasize the fact that these are the elements of the contravariant metric tensor in proper coordinates. Since g_p^mn is diagonal, we can use a formula equivalent to Eq. C-6 to translate the elements of g_p^mn from proper coordinates to the fixed coordinates of the T^mn tensor. This gives

g^mn  =  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰)g_p⁰⁰ + S_k (¶x^m/¶x_p^k)(¶xⁿ/¶x_p^k)g_p^kk    (C-12) 

Apply to this the values for g_p^mn in Eq. C-11 

g^mn  =  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰) - S_k (¶x^m/¶x_p^k)(¶xⁿ/¶x_p^k)        (C-13)

Multiply each term by pressure p to obtain

g^mnp  =  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰)p - S_k (¶x^m/¶x_p^k)(¶xⁿ/¶x_p^k)p     (C-14) 

Add this to T^mn in Eq. C-8. The S_k summation terms cancel to give

  T^mn + g^mn p  =  (¶x^m/¶x_p⁰)(¶xⁿ/¶x_p⁰) (r + p)             (C-15) 

            Tolman [2] explains (in p. 217, Eq. 85.6) that the partial derivative

¶x^m/¶x_p⁰ is equal to the ordinary derivative dx^m/ds taken along the ds line-element path. Hence Eq. C-15 can be expressed as

            T^mn  + g^mn p  =  (dx^m/ds)(dxⁿ/ds) (r + p)       (C-16) 

Solving for T^mn gives

            T^mn  =  (dx^m/ds)(dxⁿ/ds) (r + p) - g^mn p                           (C-17) 

Tolman gives this formula on page 217 (Eq. 87.5) and on page 243 (Eq. 97.5).

            Mixed form of energy-momentum tensor. We want to convert this formula for the energy-momentum tensor T^mn to the mixed form T_mⁿ. As shown in Appendix A of Universe [1], for a diagonal metric tensor this is calculated from

            T_mⁿ