Appendix B
  Einstein Tensor and Christoffel Symbols for Diagonal Metric Tensor

             This appendix derives the formulas for the Christoffel symbols that apply when the metric tensor is diagonal. The corresponding formulas for the Einstein tensor are obtained from Tolman [2], and were first prepared by Prof. Herbert Dingle. These formulas are translated into our symbolism.

            Section B.1 derives simplified formulas for the Christoffel symbols that hold when the metric tensor g_mn is diagonal. Further simplification is achieved in Section B.2 by considering a static, spherically symmetric, gravitational field. From these general formulas, the specific formulas are derived in Section B.3 for the following two applications of the Yilmaz theory: the single-star solution and the Yilmaz cosmology model. The Christoffel symbol formulas in Section B.2 also apply to the two Einstein theory solutions considered in Chapter 2. Section B.4 derives from the formulas in Section B.1 the Christoffel symbol formulas for the general static Yilmaz theory, expressed in rectangular coordinates. 

            Section B.5 gives the formulas for the values of the Einstein tensor when the metric tensor is diagonal.

B.1 General Christoffel Symbols for Diagonal Metric Tensor

            This section derives simplified expressions for the Christoffel symbols that hold when the metric tensor g_mn is diagonal. As shown in Tolman [2], page 494, Eq. 18, the Christoffel symbols are defined as follows:

            G_mn^a  =  ½ S_b g^ab ( ¶_ng_mb + ¶_mg_nb - ¶_bg_mn )              (B-1)

The summation is performed over the index b. The expression ¶_ng_mb represents ¶g_mb/¶xⁿ. The Christoffel symbols are greatly simplified when the metric tensor g_mn is diagonal. If the metric tensor g_mn is diagonal, g^mn is also diagonal and so is zero except when b = a. Hence the index b is set equal to a, and the summation over b is eliminated. For a diagonal metric tensor, Eq. B-1 simplifies to

            G_mn^a  =  ½  g^aa ( ¶_ng_ma + ¶_mg_na - ¶_ag_mn )                 (B-2)

No summation is implied over the index a. Since g_ab is zero when a is not equal to b, all three components g_ma, g_na, g_mn are zero if the indices (m, n, a) are all different, and so the Christoffel symbol G_mn^a is zero. Of the 64 components of G_mn^a, 24 satisfy this condition. The 64 components can be separated into the following five categories:

            (A)  m ¹ n, m ¹ a , n ¹ a
            (B)  m = n = a                    
            (C)  n = a; m ¹ a             
            (D)  m = a; n ¹ a                         
            (E)  n = m; a ¹ m                         

By applying these separate conditions to Eq. B-2, it can readily be shown that the 64 components of the metric tensor are given by

(A) m, n, a, all ¹    G_mn^a  =  0                         (24 components)   (B-3)
(B)                       G_aa^a  =  ½ g^aa ¶_ag_aa    (4 components)    (B-4)
(C) m ¹ a             G_ma^a  =  ½ g^aa ¶_mg_aa     (12 components)   (B-5)
(D) n ¹ a             G_an^a  =  ½ g^aa ¶_ng_aa     (12 components)   (B-6)
(E) a ¹ m             G_mm^a  =  - ½ g^aa ¶_ag_mm    (12 components)   (B-7)

In these formulas, g^aa is equal to 1/g_aa. The computations of case D can be avoided by recognizing that the Christoffel symbols are symmetric relative to the subscript indices:

            G_ma^a  =  G_am^a                                                        (B-8)

Hence the values for case D can be derived from those of case C.              

B.2 Christoffel Symbols for Static, Spherically Symmetric Gravitational Field

            This section derives the Christoffel symbols for a static, spherically symmetric gravitational field. For such a gravitational field, the metric tensor elements g₀₀, g₁₁, g₂₂ in spherical coordinates are only functions of radial distance r; and the element g₃₃ is equal to g₂₂ sin²q. Thus:

 g₀₀, g₁₁, g₂₂  =  functions of r  ;    g₃₃  =  g₂₂ sin²q              (B-9)

For these conditions, the metric tensor in spherical coordinates has five nonzero partial derivatives, which are:

Nonzero derivatives:

         ¶g₀₀/¶r, ¶g₁₁/¶r, ¶g₂₂/¶r, ¶g₃₃/¶r, ¶g₃₃/¶q   (B-10)

  Since r is x¹ and q is x², these derivatives can be expressed as

Nonzero derivatives: 

             ¶₁g₀₀, ¶₁g₁₁, ¶₁g₂₂, ¶₁g₃₃, ¶₂g₃₃             (B-11)

Comparing these partial derivatives with the Christoffel symbol formulas in Section B.1, shows that only 13 of the Christoffel symbols are nonzero. The formulas for these 13 nonzero components, which were obtained from Eqs. B-3 to B-8, are shown in Table B-1. Note that g^mm is equal to 1/g_mm .

Table B-1: Christoffel Symbols for Static, Spherically Symmetric Gravitational Field                                                     

Symbol                  Formula                                         

G₀₁⁰  =  G₁₀⁰        (1/2 g₀₀) ¶₁g₀₀     
G₀₀¹                  - (1/2 g₁₁) ¶₁g₀₀       
G₁₁¹                    (1/2 g₁₁) ¶₁g₁₁ 
G₂₂¹                  - (1/2 g₁₁) ¶₁g₂₂ 
G₃₃¹                  - (1/2 g₁₁) ¶₁g₃₃ 
 G₂₁²  =  G₁₂²      (1/2 g₂₂) ¶₁g₂₂  
G₃₃²                  - (1/2 g₂₂) ¶₂g₃₃ 
G₃₁³  =  G₁₃³       (1/2 g₃₃) ¶₁g₃₃ 
G₃₂³  =  G₂₃³        (1/2 g₃₃) ¶₂g₃₃     

B.3 Christoffel Symbols for Spherically Symmetric Applications of Yilmaz Theory

            This section applies the Christoffel symbol formulas in Table B-1 to two examples of the Yilmaz theory: the single-star solution and the Yilmaz cosmology model. For the static Yilmaz theory, the metric tensor elements are

g₀₀  =  e^-2f  ;  g₁₁  =  - e^2f  ;  g₂₂  =  - r² e^2f  ;

               g₃₃  =  - r² sin²q e^2f  ;             (B-12) 

The partial derivatives of these are

  ¶₁g₀₀  =  - 2 e^-2f ¶₁f  =  - 2g₀₀ ¶₁f   (B-13)

  ¶₁g₁₁  =  - 2 e^2f ¶₁f  =  2 g₁₁ ¶₁f    (B-14)

¶₁g₂₂  =  - 2 r² e^2f ¶₁f - 2r e^2f   

                        =  2 g₂₂ (¶₁f + 1/r)     (B-15)

¶₁g₃₃  =  - 2 r² sin²q e^2f ¶₁f - 2r sin²q e^2f 

                =  2 g₃₃ (¶₁f + 1/r]                      (B-16)

  ¶₂g₃₃  =  - 2r² sin q  cos q  e^2f 

                            =  2 g₃₃ cot q                  (B-17)

Apply the values in Eqs B-13 to B-17 to the formulas of Table B-1. This gives the formulas in column 2 of Table B-2.

       We are considering two applications of the Yilmaz theory with spherically symmetric gravitational fields, which are: (1) the single star, and (2) the Yilmaz cosmology model. The values for f and ¶₁f for these two cases are 

   Single star:               f =  m/r  ;   ¶₁f  =  - m/r²            (B-18)

   Cosmology model:  f =  r²/2r₀²  ;   ¶₁f  =   r/r₀²           (B-19)

Applying the values of Eqs. B-18, B-19 to the formulas in the second column of Table B-2 gives the values in the last two columns for the two cases.  

Table B-2: Christoffel Symbols of the Yilmaz theory, for a single star and for the Yilmaz cosmology model

Symbol            Formula                 Single Star         Cosmology Mode                       

G₀₁⁰ = G₁₀⁰     - ¶₁f                      m/r²                 - (r/ro²)
G₀₀¹               - e^-4f ¶₁f              (m/r²) e^-4M/R    - (r/ro²) exp[-2(r/ro) ²]
G₁₁¹                  ¶₁f                       - m/r²                 r/ro²  
G₂₂¹               - r²(¶₁f + 1/r)          (m - r)              - r(1 + r²/ro²)
G₃₃¹              - r² sin²q(¶₁f + 1/r]   (m - r) sin²q     - r(1 + r²/ro²) sin²q
G₂₁²  =  G₁₂²   (¶₁f + 1/r)             (1/r) - (m/r²)      (1/r) + (r/ro²)
G₃₃²                - sinq cosq            - sinq cosq           sinq cosq
G₃₁³  =  G₁₃³   (¶₁f + 1/r]            (1/r) - (m/r²)       (1/r) + (r/ro²)
G₃₂³  =  G₂₃³     cot q                  cot q                   cot q

B.4 Christoffel Symbols for Static Yilmaz Theory in Rectangular Coordinates

            This section derives formulas for the Christoffel symbols in rectangular coordinates for the general static Yilmaz theory. For the static Yilmaz theory, derivatives relative to time (coordinate x⁰) are zero. The covariant and contravariant metric tensor elements in rectangular coordinates are

g₀₀  =  e^-2f   ;   g₁₁  =  g₂₂  =  g₃₃  =  - e^2f        (B-20)

g⁰⁰  =  1/g₀₀  =  e^2f    ; 

                g¹¹  =  g²²  =  g³³  =  1/g₁₁ =  - e^-2f    (B-21)

 The partial derivatives of the g₀₀ and g₁₁ values given in Eq. B-20 are  

¶_mg₀₀  =  ¶_me^-2f  =  e^-2f ¶_m(-2f)  =  - 2e^-2f ¶_mf 

        =  - 2g₀₀ ¶_mf                        (B-22)

¶_mg₁₁  =  ¶_m(-e^2f)  =  - e^2f ¶_m(2f)  =  - 2e^2f+ ¶_mf 

        =  2g₁₁ ¶_mf                        (B-23)

            Section B.1 gives formulas for the Christoffel symbols when the metric tensor is diagonal. Applying these to the above metric tensor values and derivatives yields the following Christoffel symbols. In these formulas, the indices i, k represent 1, 2, or 3. The following 34 symbols are zero:

G_mn^a =  0  if m ¹ n, m ¹ a, n ¹ a  (24 components)       (B-24)

G₀₀⁰  =  0                                     (1 component)            (B-25)

G_0k^k  =  G_k0^k  =  G_kk⁰  =  0            (9 components)        (B-26)

The 30 nonzero Christoffel symbols are given by

  G_kk^k  =    ¶_kf                            (3 components)     (B-27)

  G_k0⁰  =  G_0k⁰  =  - ¶_kf             (6 components)     (B-28)

  G_kiⁱ  =  G_ikⁱ  =  ¶_kf   (i ¹ k)     (12 components)   (B-29)

  G_ii^k  =  - ¶_kf           (i ¹ k)      (6 components)    (B-30)

   G₀₀^k  =  - e^-4f ¶ _kf               (3 components)    (B-31)

B.5 Formulas for Einstein Tensor Given by Tolman

       Tolman [2] (pp. 253-257) gives formulas that were prepared by Prof. Herbert Dingle for the Einstein tensor when the metric tensor is diagonal. Dingle defined variables denoted A, B, C, D, which are related as follows to the elements of the metric tensor:

 D  =  g₀₀  ;  A  =  - g₁₁  ;  B  =  - g₂₂  ;  C  =  - g₃₃             (B-32)

A diagonal metric equation of the following form is assumed:

 (ds)²  =   g₀₀(dx⁰)² + g₁₁(dx¹)² + g₂₂(dx²)² + g₃₃(dx³)²        (B-33)

To represent the time coordinate, Tolman uses the index (4) in place of our index (0).

            Tolman did not use the symbol G_mⁿ to represent the Einstein tensor, which is equal to (R_mⁿ - ½ d_mⁿ R). Instead he denoted his values of the Einstein tensor as (-8pT_mⁿ). Thus he assumed that the Einstein gravitational field equation is employed, so that the following relation holds:

            G_mⁿ =  R_mⁿ - ½ d_mⁿ R   =  - 8pT_mⁿ                        (B-34)

We should interpret the Tolman expressions for (-8pT_mⁿ) as representing G_mⁿ. Tolman gives formulas for all 16 elements of the Einstein tensor