Appendix
D: Analysis
of Geodesic Equations
This appendix supplements Chapter 4, which discusses the application of
the geodesic equations to the Yilmaz theory. The following summarizes the
material in this appendix.
Section D.1 gives the general formula for the geodesic equations. It derives from this the general formulas for the second derivatives in spherical coordinates that correspond to a static, spherically symmetric gravitational field. Section D.2 applies the results of Section D.1 to the Yilmaz theory. It calculates the four specific formulas for the second derivatives in polar coordinates for the Yilmaz solution of a single star and the Yilmaz cosmology model. Section D.3 shows that the geodesic equation for time can be solved directly. Hence, the geodesic equations relative to time are expressed in terms of the first derivative of time.
Section D.4 derives formulas for the geodesic equations of the general static Yilmaz theory expressed in rectangular coordinates. Since these equations do not require a gravitational field with spherical symmetry, they can be applied to multi-body applications. For example, they can be used to calculate the accurate orbits of planets and other bodies in our solar system.
This section considers the general formula that specifies the geodesic
equations. It derives from this the four geodesic equations in spherical
coordinates that apply to a static, spherically symmetric gravitational field.
These four equations specify the second derivatives of the four space-time
variables in spherical coordinates (t,
R, q, y), where these derivatives are with
respect to differential motion ds along the path of the body.
The general formula for the geodesic equations is given as follows by
Tolman [2] (p. 495, Eq. 20):
d2xa/ds2
= -
Sm
Sn Gmna
um un (D-1)
um
= dxm/ds
(D-2)
The
two summations shown in Eq. D-1 are not stated implicitly in Tolman's equation.
However, they are implied, because the indices m
and n are repeated. We will solve Eq. D-1
for values of the index a equal to 0, 1, 2, 3.
Table D-1: Non-zero Christoffel
Symbols for Static, Spherically Symmetric Gravitational Fiel
G010
= G100
G221
G332
G001
G331
G313
= G133
G111
G212
= G122
G323
= G233
Table D-1 lists the non-zero Christoffel symbols when the gravitational
field is static and spherically symmetric. The following discussion applies this
information to Eq. D-1, and derives the four geodesic equations for this case.
When one considers only the Christoffel symbols given in Table D-1, the
Geodesic formula of Eq. D-1 yields the following equations when the index a is
set equal to 0, 1, 2, 3:
d2x0/ds2
= - {
G100
u1 u0
+ G010 u0
u1 }
(D-3)
d2x1/ds2
= - {G001u0u0
+ G111u1u1 + G221u2u2
+ G331u3u3} (D-4)
d2x2/ds2
= - {
G122
u1u2
+ G212 u2
u1 + G332
u3 u3
}
(D-5)
d2x3/ds2
= - (
G133u1u3
+ G313u3u1 + G233u2u3
+ G323u3u2}
(D-6)
In
Eqs. D-1 to D-6, apply the equality relations for the Christoffel symbols given
in Table D-1 and combine terms. This yiel
d2x0/ds2
= - 2 G010
u0u1
(D-7)
d2x2/ds2
= - 2 G122 u1u2
- G332 (u3)2
(D-9)
d2x3/ds2
= - 2 G133 u1
u3 - 2 G233
u2 u3
(D-10)
x0 = t, x1 = r, x2 = q, x3 = y (D-11)
In
accordance with Eq. D-2, replace the um
variables by the following
u0 =
dx0/ds =
dt/ds
(D-12)
u1 =
dx1/ds =
dr/ds
(D-13)
u2 =
dx2/ds =
dq/ds
(D-14)
u3 =
dx3/ds =
dy/ds
(D-15)
Equations
D-7 to D-10 become
d2t/ds2
= - 2 G010 (dt/ds)
(dr/ds)
(D-16)
d2r/ds2
= - G001
(dt/ds)2
- G111(dr/ds)2
- G221(dq/ds)2
- G331(dy/ds)2
(D-17)
d2y/ds2
= - 2 G133 (dr/ds)(dy/ds)
- 2 G233 (dq/ds)(dy/ds)
(D-19)
These
are the general expressions for the geodesic equations in spherical coordinates
for a static spherically symmetric gravitational field.
D.2
Geodesic Equations for Spherically Symmetric Applications of Yilmaz Theory
Let us apply these geodesic equations to the Yilmaz theory. Table D-2
gives the formulas for the Christoffel symbols of the Yilmaz theory that apply
to a static, spherically symmetric gravitational field, which were obtained from
Table B-2 of Appendix B. Applying these values to Eqs. D-16 to D-19 gives the
following geodesic equations:
d2t/ds2
= 2 ¶1f
(dt/ds) (dr/ds)
(D-20)
d2r/ds2
= e-4f
¶1f
(dt/ds)2
- ¶1f
(dr/ds)2 + r2(¶1f
+ 1/r)(dq/ds)2
+ r2 sin2q(¶1f
+ 1/r)(dy/ds)2
(D-21
d2q/ds2
= - 2 (¶1f
+ 1/r)(dr/ds) (dq/ds) + sinq cosq
(dy/ds)2
(D-22)
d2y/ds2
= - 2 (¶1f
+ 1/r] (dr/ds) (dy/ds)