Appendix F
  Stress-Energy Tensors of General Yilmaz Theory

          This Appendix derives formulas for the stress-energy tensors t_mⁿ, t_mⁿ of the general time-varying Yilmaz theory. The Primary analysis is presented in Section F.1, and Sections F.2, F.3 derive formulas that are used in Section F.1

Based on formulas given by Pauli [3], Section F.2 derives formulas for the pseudo-tensors denoted u_mⁿ and U_mⁿ by Yilmaz. In Section F.1, these formulas are expressed in harmonic coordinates, and gauge conditions are applied. The application of harmonic coordinates converts the pseudo-tensors to true tensors, which are directly related to the stress-energy tensors -t_mⁿ, t_mⁿ of the Yilmaz theory. In harmonic coordinates, the pseudo tensors u_mⁿ and U_mⁿ are equal, respectively, to -t_mⁿ and t_mⁿ, which are true tensors. Tensor t_mⁿ is the stress-energy tensor for the gravitational field and t_mⁿ is the stress-energy tensor for matter.

              When indices in a tensor equation are repeated, with one index a subscript and the other a superscript, the expression is to be summed over the four values (0, 1, 2, 3) of the index. This book normally represents implied summations by S signs. However in this appendix the implied summations are omitted.

  F.1 Formulas for Stress-Energy Tensors

  F.1.1 Application of Gauge Conditions and Harmonic Coordinates

The formulas for U_mⁿ and u_mⁿ can be simplified by applying gauge conditions and harmonic coordinates. The following gauge conditions apply:

            ¶ _nf_mⁿ =  0                                                          (F.1-1)  

            ¶^mf_mⁿ =  0                                                          (F.1-2)

Yilmaz [Y9] (App. B, p. 959), states that the following identities hold with harmonic coordinates:

            ¶_ng^mn  =  0                                                          (F.1-3) 

            ¶ⁿg_mn  =  0                                                          (F.1-4) 

           The harmonic coordinate condition, defined by these equations, greatly simplifies the analysis. It permits a wave to propagate in a single direction, without having a compensating wave in the reverse direction. Hence it allows energy to be radiated. These conditions can apply to electromagnetic waves and to gravitational waves.

            The following partial derivative definitions are assumed in this appendix:

            ¶_af  =  ¶f/¶x^a                                                   (F.1-5)

            ¶ ^af  =   g^ab ¶_bf  =  S_b  g^ab ¶_bf                          (F.1-6)

F.1.2 Simplification of Formula for Pseudo-Tensor U_mⁿ  

            Equation F.2-13 gives the following formula for the pseudo-tensor U_mⁿ:

U_mⁿ  =  (1/4Ö[-g]) ¶_a{g^al g^nr(¶_rg_ml - ¶_lg_mr) + d_mⁿ ¶_bg^ba - d_m^a ¶_bg^bn}      (F.1-7)

The Old English symbols denote the following functions, which are called tensor densities:

            g^mn  =  Ö[-g] g^mn   ,   g _mn =  g_mn/Ö[-g]                                            (F.1-8)

When the harmonic coordinate condition of Eq. F.1-3 is applied to the last two terms of this expression for U_mⁿ, the terms become zero, and the equation simplifies to

    4Ö[-g]U_mⁿ  =  ¶_a[g^nr{g^al ¶_rg_ml}] - ¶_a[g^al{g^nr ¶_lg_mr}]                           (F.1-9)

By Eq. F.3-38 the expressions within the braces { } are equal to

         { g^al ¶_r g _ml }  =  - 4 ¶_r f_m^a                                                           (F.1-10)

          { g ^nr ¶_l g _mr }  =  - 4 ¶_l f_mⁿ                                                      (F.1-11)

Substituting Eqs F.1-10, F.1-11 into Eq. F.1-9 gives

         Ö[-g]U_mⁿ  = - ¶_a [g ^nr ¶_r f_m^a] + ¶_a [g^al ¶_l f_mⁿ ]                              (F.1-12)

Reverse terms, and divide by Ö[-g]. This gives

         U_mⁿ  =  (1/Ö[-g])¶_a [ g^al¶_l f_mⁿ ] - (1/Ö[-g]) ¶_a [g^nr ¶_r f_m^a]          (F.1-13)

The first term of Eq. F.1-13 is the d'Alembertian of f_mⁿ, which is denoted �²f_mⁿ, and is defined by  

�²f_mⁿ  =  (1/Ö[-g]) ¶_a[ g^al ¶_l f_mⁿ ]                                            (F.1-14)

            The second term of Eq. F.1-13 can be modified by applying the definition of g^mn in Eq. F.1-8 and the following definition derived from Eq. F.1-6:

            ¶ⁿf_m^a  =  g^nr ¶_rf_m^a                                                                         (F.1-15)

The second term of Eq. F.1-13 becomes

         - (1/ Ö[-g]) ¶_a[ Ö[-g]) g^nr ¶_rf_m^a ]  =  - (1/Ö[-g]) ¶_a[Ö[-g]) ¶ ⁿf_m^a ]    (F.1-16)

  Applying Eq. F.1-14, F.1-15 to Eq. F.1-13 gives

             U_mⁿ  =  �² f_mⁿ  - (1/Ö[-g]) ¶_a[Ö[-g] ¶ⁿf_m^a ]                                   (F.1-17)

It can be shown that this is a tensor. As explained in Section F.2, U_mⁿ is equal to the sum (t_mⁿ + z_mⁿ), where z_mⁿ is a non-tensor. By expressing U_mⁿ in harmonic coordinates, the non-tenor component becomes zero, and so U_mⁿ becomes the stress-energy tensor for matter t_mⁿ. Setting U_mⁿ equal to t_mⁿ in Eq. F.1-17 gives

        t_mⁿ  =  �²f_mⁿ  - (1/Ö[-g]) ¶_a[Ö[-g] ¶ⁿf_m^a ]                                  (F.1-18) 

  This agrees with the expression for t_mⁿ given by Yilmaz [Y11] (p. 498).

            The expression for �²f_mⁿ in Eq. F.1-14 can be simplified as follows by replacing g^al by the definition of Eq. F.1-8, and then by replacing Ö[-g] by e^2f in accordance with Eq. F.3-29:

  �²f_mⁿ  =  (1/Ö[-g])¶_a[g^al ¶_lf_mⁿ] =  e^-2f¶_a[Ö[-g]g^al ¶_lf_mⁿ]

               =  e^-2f¶_a[e^2f g^al¶_lf_mⁿ] =  e^-2f¶_a[e^2f ¶^af_mⁿ]                            (F.1-19)

The definition for ¶^af_mⁿ given in Eq. F.1-15 was applied in the last term. Substitute this into Eq. F.1-18, and replace Ö[-g] by e^2f in the second term. Then expand the derivatives. This gives

t_mⁿ  =  e^-2f ¶_a[e^2f ¶^af_mⁿ] - e^-2f ¶_a[e^2f ¶ⁿf_m^a ]                                              

       =  e^-2f{e^2f¶_a¶^af_mⁿ + ¶^af_mⁿ(e^2f 2¶_af)} -  e^-2f{e^2f ¶_a¶ⁿf_m^a + ¶ⁿf_m^a(e^2f 2¶_af)}                             

=  ¶_a¶^af_mⁿ - ¶_a¶ⁿf_m^a + 2¶_af {¶^af_mⁿ  - ¶ⁿf_m^a}                        (F.1-20)

This simplifies to

         t_mⁿ =  ¶_a{¶^af_mⁿ - ¶ⁿf_m^a} + 2¶_af