Chapter 3: Aspects of Einstein and Yilmaz Relativity Theories

            This chapter discusses various aspects of the Einstein and Yilmaz theories. Section 3.1 analyzes the gravitational field equation for the static solution of the Yilmaz theory, and shows that it is exactly satisfied for the general static case. Section 3.2 discusses the Poisson equation and shows that the Einstein gravitational equation is closely related to the Poisson equation. The Poisson equation is used to calculate the gravitational potential for the Yilmaz theory when the mass density is spherically symmetric.

 Section 3.3 discusses the Einstein pseudo-tensor for the gravitational field, which Einstein considered as representing the "energy components of the gravitational field". This pseudo tensor is related to the Yilmaz stress-energy tensor for the gravitational field, which is a true tensor. Section 3.4 discusses the concept of the covariant derivative and shows that the Einstein and Yilmaz gravitational field equations are based on the Einstein tensor Gmn, rather than the Ricci tensor Rmn, because the covariant derivative of the Einstein tensor is zero. 

Section 3.5 presents the basic analysis that Huseyin Yilmaz implemented that led to the Yilmaz theory of gravity. This was an extension of an approximate analysis that Albert Einstein had performed to determine the wavelength shift that is produced by a gravitational field. Yilmaz derived an exact analysis of this effect. 

3.1 Validating Yilmaz Gravitational Field Equation for the Static Solution

            A very important aspect of the Yilmaz theory of gravitation is that its gravitational field equation is automatically satisfied when the gravitational potential tensor (or the gravitational potential for the static solution), is appropriately designed. Chapter 5 gives an elaborate analysis to prove that the gravitational field equation is always satisfied for the general time-varying Yilmaz theory. This section addresses the simpler problem of proving that the gravitational field equation is always satisfied for the static Yilmaz solution

    The gravitational field equation for the Yilmaz theory is

Gmn  =  Rmn - ½ dmn R  =  - 2(tmn + tmn)                                      (3-1)

The tensor tmn is the stress-energy tensor for matter, and tmn is the stress-energy tensor for the gravitational field. The static Yilmaz theory is implemented by calculating the gravitational potential f. The gravitational potential f at the point xp is computed from

            f(xp)  =  S Dmk/|xp - xk|                                          (3-2)

The vector xk denotes the location of the mass element Dmk, and |xp - xk| is the absolute value of the distance from the mass element to the point xp where the gravitational potential is calculated. In rectangular coordinates the metric tensor is diagonal and has the following elements

            g00  =  e-2f  ;   g11  =  g22  =  g33  =  - e2f                  (3-3)

Chapter 5 derives the following formulas for the stress-energy tensors in the static Yilmaz theory:               

            tmn  =  - mf nf + ½ dmn Sl lf lf                         (3-4)    

t00  =  - e-2f Ñ2f  =  - e-2f{2f/x2 + 2f/y2 + 2f/z2}    (3-5)

        In the static Yilmaz theory, the stress-energy tensor for matter tmn reduces to the single element t00 given by Eq. 3-3. All other elements of this tensor are zero.

Appendix B gives formulas derived by Prof.. Herbert Dingle for calculating the elements of the Einstein tensor Gmn when the metric tensor is diagonal. These Dingle formulas were applied to the general  metric tensor values of the static Yilmaz solution given in Eq. 3-3 (with derivatives relative to time set to zero). This yielded  the following general formulas for the Einstein tensor of the static Yilmaz solution in rectangular coordinates:

 Gmn  =  0              for m ¹ n, m or n = 0                              (3-6)

Gjk  =  - 2 e-2f jf kf           for j ¹ k                          (3-7)

G00  =  e-2f Sj {(jf)2 + 2 j2f }                                (3-8)

G11  = - e-2f { (1f)2 - (2f)2 - (3f)2 }                     (3-9)

G22  =  - e-2f { (2f)2 - (1f)2 - (3f)2 }                    (3-10)

G33  =  - e-2f { (3f)2 - (1f)2 - (2f)2 }                 (3-11)

The calculus computations for deriving these elements of the Einstein tensor are very difficult and tedious. A skilled and patient mathematician may require two weeks to calculate these formulas without error, even with the great assistance of the Dingle formulas. 

           From Eqs. 3-4, 3-5, one can derive the following equations for the stress-energy tensor elements in rectangular coordinates:

t00  =  - e-2f Sk k2f                                               (3-12)

t00  =  - ½ e-2f  Sk (kf)2                                       (3-13)

tjj  = e-2f { ( jf)2 - ½ Sk ( kf)2 }   (j = 1, 2, 3)         (3-14)  

tjk = e-2f jf kf             (j ¹ k;  j, k = 1, 2, 3)            (3-15)

tmn  =  0                        (n ¹ m; n or m = 0)                   (3-16)  

The summations are implemented over the three values (1, 2, 3) of the index k.

            Equation 3-12 shows that the stress-energy tensor for matter tmn has only one nonzero element t00, which is equal to

t00  =  - e-2f {12f + 22f + 32f}                (3-17)

Equations 3-13 to 3-16 yield the following formulas for the elements of the stress-energy tensor for the gravitational field tmn  

t00  =  - ½ e-2f { (1f)2 + (2f)2 + (3f)2 }                (3-18)

 t11  =  ½ e-2f { (1f)2 - (2f)2 - (3f)2 }              (3-19)

t22  =  ½ e-2f { (2f)2 - (1f)2 - (3f)2 }                 (3-20)

t33  =  ½ e-2f { (3f)2 - (1f)2 - (2f)2 }        (3-21)

 tjk  =  e-2f jf kf       ( j, k = 1, 2, 3; k ¹ j)             (3-22)   

  tmn  =  0                         (n or m = 0; n ¹ m)            (3-23)  

            Based on the above values of the stress-energy tensors, the expressions for -2(tmn + tmn) are as follows:

            - 2(tmn + tmn)  =  0                    (n ¹ m; n or m = 0)        (3-24)

            - 2(tjk + tjk)  =  - 2e-2f jf kf    (j ¹ k;  j, k = 1, 2, 3)   (3-25)

            - 2(t00 + t00)  =  e-2f Sk { 2k2f + ( kf)2  }             (3-26)

            - 2(t11 + t11)   =  - e-2f { (1f)2 - (2f)2 - ( 3f)2 }     (3-27)

            - 2(t22 + t22)   =  - e-2f { (2f)2 - (1f)2 - (3f)2 }      (3-28)

              - 2(t33 + t33)   =  - e-2f { (3f)2 - (1f)2 - ( 2f)2 }    (3-29)

  In Eqs. 3-24 to 3-29, the values for tmn are all zero, except for the value for t00 in Eq. 3-26, which is obtained from Eq. 3-17.

          Comparing Eqs. 3-24 to 3-29 with Eqs. 3-6 to 3-11 shows that Gmn is exactly equal to -2(tmn + tmn) for all elements of these tensors. This proves that the Yilmaz gravitational field equation is exactly satisfied for all static solutions of the Yilmaz theory.  

             Conclusion. An examination of the formulas required to implement this calculation will show that even for this static case, where the metric tensor is diagonal, the equations for calculating the elements of the Einstein tensor Gmn from the metric tensor gmn are very complicated. These calculations are radically different from those for calculating the elements of the stress-energy tensors tmn, tmn. One would never dream when performing these difficult computations that both paths lead to the same result; that Gmn is exactly equal to the expression - 2(tmn + tmn) for all elements of the tensors.

             This result is remarkable! The analysis shows that the gravitational field equation of the Yilmaz theory is exactly satisfied for all cases of the static theory. This proves that the Yilmaz theory has a very solid and profound mathematical foundation.

3.2 The Poisson Equation

  3.2.1 Relation between Gravitational Field Equation and Poisson Equation

            Newton's laws of mechanics are characterized by equations that describe the forces exerted on bodies, and the resultant velocities and accelerations of the bodies. About 1800 the French mathematician Poisson developed a field theory approach to Newton's laws by expressing them in terms of the gravitational potential. Einstein extended this gravitational field theory concept in developing his gravitational field equation.

             The Poisson equation deals with a gravitational potential f' that is related as follows to the relativistic gravitational potential f of the Yilmaz theory:

            f'  =  (c2/G) f                                                     (3-30)

This shows that the equations for the Poisson gravitational potential f' are obtained from those for the Yilmaz gravitational potential f by replacing the relativistic mass m by the true mass M. Hence, the discussion of the Yilmaz theory in Appendix B of Universe [1] shows that the Poisson gravitational potential f' generated at a point (p) by a collection of masses Mk is equal to 

            f'  =  S Mk/|rpk|                                                      (3-31) 

This represents a summation over the masses. The quantity |rpk| is the absolute value of the distance between the point (p) and the center of gravity of mass Mk. If a test mass Mt that is free to move is placed in this gravitational field, it can be shown (by applying Newton's laws of mechanics) that the acceleration of the test mass in the x-direction is equal to

            Ax  =  - f'/x                                                  (3-32)

Thus, the acceleration in the x-direction is equal to the negative of the partial derivative of the gravitational potential y in the x-direction. Equivalent equations hold for all three spatial directions, x, y, z. The gravitational force applied to the test mass in the x-direction is obtained by multiplying this acceleration by the mass Mt of the test particle to obtain:

           fx  =  Mt Ax =  - Mt (f'/ x)                                  (3-33)

            By applying calculus to Eq. 3-31, it can be shown that the following formula holds at all points that are exterior to the bodies:

            (2f'/x2) + (2f'/y2) + (2f'/