Chapter 5: General Time-Varying Yilmaz Theory  

            This chapter studies the general time-varying Yilmaz theory. The static solution is described in Chapter 10 of Universe [1]. The static solution theoretically applies only when the gravitational potential f does not vary with time, yet it gives a very accurate approximation if the velocities are small relative to the speed of light. When the gravitational potential f varies with time, the full gravitational potential tensor f_mⁿ must be considered to achieve an exact solution. The trace of the tensor f_mⁿ (the sum of its diagonal elements) is the gravitational potential f.

            When an index in a tensor equation is repeated, with one index a subscript and the other a superscript, the corresponding term is to be summed over the four values (0, 1, 2, 3) of the index. This Addendum normally indicates the implied summations with S summation signs, but this chapter omits the implied summations in some of the equations to simplify the formulas. Comments are added to indicate that summation over specific indices is assumed. 

5.1 Basic Equations for General Yilmaz Theory

            The general time-varying Yilmaz theory is characterized by two basic formulas. One formula specifies the gravitational potential tensor f_mⁿ in terms of the elements of mass and their velocity components. The second formula is a differential relation that specifies the metric tensor g_mn in terms of the gravitational potential tensor f_mⁿ.

            Formula for gravitational potential tensor. Yilmaz [Y3] (eq. 1.2) has shown that, in the general time-varying form of the Yilmaz theory, the gravitational potential tensor f_mⁿ is calculated from the following integral:

            f_mⁿ  =  ¦ { u_m uⁿ)(dm/r) }_retarded                             (5-1)

where dm is an element of mass and r is the absolute value of the distance from that mass element to the point at which the tensor is calculated. Yilmaz expressed (dm) as the product (r dv) where r is the mass density of the medium, and dv is an element of volume. The subscript "retarded" indicates that a time delay equal to r/c is included in the calculation between a change at the mass element (dm) and the resultant effect at the point (p) where the gravitational potential is calculated. This time delay is the time for a light pulse to propagate from the mass element to the point (p).

            The variables u_m, uⁿ are the relativistic velocities of the mass element. The relativistic velocity uⁿ is equal to dxⁿ/ds. The relativistic velocity u_m is equal to g_ma u^a. Hence u_m is equal to g_ma (dx^a/ds), which should be summed over the four values of the a index. 

            Formula for metric tensor. Yilmaz [Y3] (eq. 1.1) shows that the metric tensor g_mn for the general time-varying Yilmaz theory is calculated from the gravitational potential tensor f_mⁿ with the following differential formula:

            dg_mn  =  2g_mn df - 2 S_a{g_ma df_n^a + g_an df_m^a }               (5-2)    

The variable f is the trace of the gravitational potential f_mⁿ, which is the sum of its diagonal elements, as shown by

            f  =  f₀⁰ + f₁¹ + f₂² + f₃³                                          (5-3)

Appendix F shows in Eqs. F.3-4, F.3-5 that Eq. 5-2 can also be expressed as

            dg_mn  =  2g_mn df - 4 S_a g_ma df_n^a                                (5-4)    

            dg_mn  =  2g_mn df - 4 S_a g_an df_m^a                                (5-5)    

From the covariant metric tensor g_mn one can calculate the contravariant metric tensor g^mn by recognizing that the product of the matrices of the two tensors g_mn, g^mn is a unit matrix.

5.2 Elements of the Yilmaz Gravitational Field Equation

            Like the Einstein theory, the Yilmaz theory is based on its gravitational field equation. However, in the Yilmaz theory the gravitational field equation is not solved when the theory is applied, because the Yilmaz gravitational field equation is automatically satisfied when the Yilmaz theory is appropriately implemented. This chapter (supplemented by Appendix F) will prove that the gravitational field equation is exactly satisfied for the general time-varying Yilmaz theory.

            The gravitational field equation for the Yilmaz theory is

            G_mⁿ  =  R_mⁿ - ½ d_mⁿ R  =  - 2(t_mⁿ + t_mⁿ)                    (5-6)

The Yilmaz theory calls t_mⁿ the “stress-energy tensor for matter”, and t_mⁿ is called the “stress-energy tensor for the gravitational field”.

Formula for Ricci tensor. From the metric tensors g_mn, g^mn computed from Eq. 5-2, one can calculate the Ricci tensor R_mⁿ with the same formulas used with the Einstein theory. One first calculates the 64 Christoffel symbols from the following formula, given by Tolman (1934) (p. 494, eq. 18): 

            G_mn^a  =  ½ g_ab{¶_ng_mb + ¶_mg_nb - ¶_bg_mn }                  (5-7)

The symbol  ¶_m denotes the partial derivative relative to x_m:

              ¶_mf  =    ¶f/¶x^m                                                              (5-8)

Tolman (1934) (p. 495, eq. 25) shows that the Ricci tensor is calculated from the Christoffel symbols by:

            R_mn  =  G_ms^a G_an^s - G_mn^a G_as^s  +   ¶_nG_ms^s -   ¶_sG_mn^s            (5-9)

In accordance with Tolman (1934) (p. 495, eq. 21) the mixed form of the Ricci tensor R_mⁿ is calculated from the covariant form Rmn by:

            R_mⁿ  =  g_na R_ma                                                      (5-10)

In Eqs. 5-7 to 5-10, the terms with the repeated indices a, b, s should be summed over the four values of the repeated indices.

            In his documents, Yilmaz uses the negative of Eq. 5-9 to specify the Ricci tensor. This choice reverses the sign of the Ricci tensor R_mⁿ and thereby eliminates the negative sign in the right hand side of the gravitational field equation in Eq. 5-6. This book uses the Tolman definition for the Ricci tensor in Eq. 5-9, and includes a negative sign in the Yilmaz gravitational field equation.

            Formulas for stress-energy tensors. Based on the formulas for the general Yilmaz theory given in Eqs. 5-1, 5-2, Appendix F derives in Eqs. F.1-34, F.1-35 the following formulas for the stress-energy tensors for the general time-varying Yilmaz theory:

t_mⁿ  = - 2¶_mf_b^a ¶ⁿf_a^b + ¶_mf ¶ⁿf + d_mⁿ{¶_lf_b^a¶^lf_a^b - ½ ¶_lf¶^lf}      (5-11)

t_mⁿ =  ¶_a{¶^af_mⁿ - ¶ⁿf_m^a} + 2 ¶_af{¶^af_mⁿ  - ¶ⁿf_m^a}                         (5-12)

In Eqs. 5-11, 5-12, the terms with the repeated indices a, b, l should be summed over the four values of the repeated indices.

            Appendix F proves that when these stress-energy formulas are applied to the gravitational field equation of Eq. 5-6, that the Yilmaz gravitational field equation is exactly satisfied. Thus we have proven that the Yilmaz gravitational field equation is automatically satisfied for the general time-varying Yilmaz theory, and so never has to be solved when the Yilmaz theory is applied. The general Yilmaz theory is implemented by solving the formulas for f_mⁿ and g_mn given in Eqs. 5-1, 5-2.

5.3 Determinant g of the Metric Tensor

              The determinant of the covariant metric tensor g_mn is denoted g. Section F.3.3 of Appendix F proves that the determinant of g_mn for the general time-varying Yilmaz theory has the following very simple formula:

            g  =  det|g_mn|  =  - e^4f                                          (5-13)

The expression Ö[-g], which is important in tensor analysis, is equal to

            Ö[-g]  =  e^2f                                                           (5-14)

            This formula for g greatly simplifies the calculation of the contravariant metric tensor g^mn for general applications where the metric tensor is not diagonal. The product of the matrices of the metric tensors g_mn, g^mn is a unit matrix. Hence it can be shown that the elements of the contravariant metric tensor are computed from

            g^mn  =  cof[gmn]/det|gmn|  = cof[gmn]/g                    (5-15)

The expression det|g_mn| is the determinant of the g_mn matrix, which is the variable g. The expression cof[g_mn] is the cofactor of the element g_mn, which can be expressed as

cof[g_mn]  =  (-1)^m+1+n+1 minor[g_mn]  =  (-1)^m+n minor[g_mn]     (5-16)    

The expression minor[g_mn] denotes the minor of g_mn, which is obtained by deleting from the g_mn matrix the row and column of the particular g_mn element, and taking the determinant of the result. Combining Eqs. 5-13, 5-15, 5-16 gives the following formula for the elements of the contravariant metric tensor for the general Yilmaz theory:

            g^mn  =  - (-1)^m+n e^-4f minor[g_mn]                                   (5-17)

            When the covariant and contravariant metric tensors are known, one can readily convert any other tensor from one of its forms to another by means of the following formulas:

            A^mn  =  S_a  g^ma A_aⁿ                                                (5-18)

            A_mⁿ  =  S_a  g^na A_ma                                                (5-19)

            A_mn  =  S_a  g_na A_m^a                                                 (5-20)

            A_mⁿ  =  S_a  g_ma A^an                                                  (5-21)

            The factor Ö[-g] is used to form the tensor density, which is a powerful concept in tensor analysis. A tensor density is traditionally represented by Old German script, but we use Old English script instead. A tensor density is formed by multiplying a tensor by Ö[-g]. The tensor density corresponding to the tensor A_mⁿ is denoted A_mⁿ and is defined by:

            A_mⁿ  =  Ö[-g]A_mⁿ            (tensor density)                           (5-22)

In the Yilmaz theory this becomes:

            A_mⁿ  =  e^2f A_mⁿ (tensor density for Yilmaz theory)            (5-23)

5.4 Conservation of Energy and Momentum

            Landau and Lifshitz [10] (p. 280) explain that the conservation of energy and momentum for matter (including electromagnetic energy) requires that the following integral be conserved: ¦ Ö[-g]T_mⁿ dS_n, where Ö[-g]T_mⁿ is the density of the energy-momentum tensor T_mⁿ. They state that to achieve this requires that the following condition be satisfied:

            ¶_n{Ö[-g]T_mⁿ} =  0                                             (5-24)

In the Yilmaz theory, T_mⁿ is replaced by t_mⁿ, which is equal to 4pT_mⁿ, and the requirement of Eq. 5-24 becomes

            ¶_n{Ö[-g]t_mⁿ} =  0                                             (5-25)

Equations 5-24, 5-25 should be summed over the four values of the repeated n index. Equation F.2-35 of Appendix F shows that Eq. 5-25 is always satisfied in the Yilmaz theory because of the Freud identity. Therefore, the Yilmaz theory always achieves conservation of energy and momentum of matter, including the effect of electromagnetic fields.

5.5 Simplification of Formula for Gravitational Potential Tensor

            The discussion that followed Eq. 5-1 shows that this equation can be expressed as follows 

            f_mⁿ  =  ¦ g_ma (dx^a/ds) (dxⁿ/ds) (dm/r)        (5-26)                      

This formula assumes summation over the four values of the repeated a index. The "retarded" condition is still required, but the notation is dropped for simplicity. This expression should be summed over the four values of the a index. Let us indicate the summation over the index a, and replace the integral over the mass elements with a summation. This gives

           f_mⁿ  =  S_Dm ( S_a{g_ma(dx^a/ds) } (dxⁿ/ds) (Dm/r) )        (5-27)

Equation 5-27 can be expressed in a more convenient form by factoring the expression (dt/ds) from each derivative, to obtain

f_mⁿ  =  S_Dm [S_a{g_ma (dx^a/dt)}(dxⁿ/dt) (dt/ds)² (Dm/r) ]         (5-28)

The derivatives are now expressed directly in terms of normalized time t. Let us implement the second summation by setting a equal to 0, 1, 2, 3:

f_mⁿ  = S_Dm{g_m0(dx⁰/dt) + g_m1(dx¹/dt) + g_m2(dx²/dt) + g_m3(dx³/dt)}(dxⁿ/dt)(dt/ds)²(Dm/r)              

     =  S_Dm{g_m0 + g_m1(V_x/c) + g_m2(V_y/c) + g_m3(V_z/c)}(dxⁿ/dt)(dt/ds)²(Dm/r)           (5-29)

Since x⁰ = t, the derivative dx⁰/dt is unity. Since t = ct, dx¹/dt is equal to (1/c)(dx/dt) which represents V_x/c, where V_x is the velocity in the x (or x¹) direction. Similarly V_y, V_z are velocities in the y (or x²) direction, and in the z (or x³) direction. Let us separate Eq. 5-29 into the two cases for n = 0, and for n = k, where k = 1, 2, 3. This gives

f_m⁰  =  S_Dm{g_m0 + g_m1(V_x/c) + g_m2(V_y/c)  + g_m3(V_z/c)}(dt/ds)²(Dm/r)       (5-30)

f_m^k  = S_Dm{g_m0 + g_m1(V_x/c) + g_m2(V_y/c)