Chapter
4: Application
of Geodesic Equations
(Revised November, 2002)
4.1
Use of the Geodesic Equations
When the orbit of a planet is calculated using Newtonian theory, one
applies Newton's law of gravitational attraction and Newton's law of motion,
which states that force is equal to mass times acceleration. We can combine
these into a single equation, which states that the acceleration of a body is
equal to the gravitational force applied to the body divided by its mass.
Acceleration is the derivative of velocity, and so is the second derivative of
position. Thus, with Newtonian theory, the second derivative of the position of
a body is equal to the gravitational force applied to the body divided by the
mass of the body.
The calculations are entirely different with relativity, because this
theory does not allow the concept of a gravitational force. With general
relativity, the orbit of a planet is calculated from a set of four equations,
which are called the geodesic equations.
These four equations specify the second derivatives, relative to the line
element ds along the planetary path, of the four space-time coordinate
variables. In rectangular coordinates, these four space-time coordinates are t,
x, y, z. In the weak gravitational
fields of our solar system, an increment Ds
along a space-time planetary trajectory is closely approximated by Dt.
In Appendix D general geodesic equations in spherical coordinates are
calculated for a static, spherically symmetric gravitational field. From these,
specific geodesic equations are derived for two applications of the Yilmaz
theory: (a) the single-star model and (b) the Yilmaz cosmology model. These
specific geodesic equations are given in Section 4.2. From the geodesic
equations of the Yilmaz cosmology model, Section 4.3 calculates the radial
velocity of a galaxy.
Appendix D also derives general geodesic equations for the static Yilmaz
theory, expressed in rectangular coordinates. These are simplified to give
practical formulas for calculating the orbits of planets and other bodies in our
solar system. These equations are given in Section 4.4.
4.2
Yilmaz Theory Geodesic Equations for a Static, Spherically Symmetric
Gravitational Field
The geodesic equations of the Yilmaz theory for a static, spherically
symmetric gravitational field are derived in Appendix D. These are expressed in
terms of motions in the radial and tangential directions. The general geodesic
equations for the static Yilmaz theory are derived in Eqs. D-56 to D-58, and are
as follows:
General Static, Spherically
Symmetric Gravitational Field
dt/ds
= e2f
(4-1)
d2r/ds2
= (¶f/¶r){ 1
- (dr/ds)2 } + { (¶f/¶r) +
(1/r)](dxt/ds)2
(4-2)
d2xt/ds2
= - { 2(¶f/¶r) +
1/r } (dr/ds)(dxt/ds)
(4-3)
The
variable dr represents incremental motion in the radial direction, and dxt
represents incremental motion in the tangential direction, which is
perpendicular to the radius.
These general formulas are applied to the Yilmaz theory solutions for a
single star and to the Yilmaz cosmology model. In the Yilmaz theory solution for
a single star, the gravitational potential f is equal to m/r. The resultant
geodesic equations for the Yilmaz single-star model are given in Eqs. D-61 to
D-63, and are as follows:
Yilmaz Single Star Model
dt/ds
= e2m/r
(4-4)
d2r/ds2
= (1/r)(m/r){ (dr/ds)2
- 1} + (1/r)[1 - (m/r)} (dxt/ds)2
(4-5)
d2xt/ds2
= (1/r){ (2m/r) - 1 }
(dr/ds)(dxt/ds)
(4-6)
In
the Yilmaz cosmology model, the gravitational potential variable 2f
is equal to (r/r0)2. The resultant geodesic equations for
the Yilmaz cosmology model are given in Eqs. D-64 to D-66, and are as follows:
Yilmaz Cosmology Model
dt/ds
= exp[(r/r0)2]
(4-7)
d2r/ds2
= (r/r02){
1 - (dr/ds)2 } + (1/r)[1 + (r/r0)2](dxt/ds)2
(4-8)
d2xt/ds2
= - (1/r)[1 + 2(r/r0)2
](dr/ds)(dxt/ds)
(4-9)
When these geodesic equations are applied, one
often needs the metric equation. The general metric equation for the static
Yilmaz theory is expressed as follows in terms of motions in the radial and
tangential directions:
(ds)2 =
e-2f
(dt)2 + e2f
[ (dr)2 + (dxt)2 ]
(4-10)
As shown in Eqs. D-107, D-108 of Appendix D, the geodesic equations for the Yilmaz Single-Star Model given in Eqs. 4-4 to 4-6 can be expressed directly in terms of normalized time as the following two equations:
r(d2r/dt2)
= 3(m/r)(dr/dt)2 - (m/r)e-4m/r + [1 - (m/r)](dxt/dt)2
(4-11)
r(d2xt/dt2)
= {(4m/r)
- 1 }(dr/dt)(dxt/dt)
(4-12)
Equations 4-11, 4-12 can be expressed as
rAr
= 3(m/r)Vr2 - (m/r)c2e-4m/r + [1 - (m/r)]Vt2
(4-13)
rAt = {(4m/r) - 1 }Vr Vt (4-14)
where Vr and Vt are the radial and tangential velocities dr/dt and dxt/dt; and Ar and At are the radial and tangential accelerations dr2/dt2 and dxt2/dt2.
4.3 Solution of Yilmaz Cosmology
Model for Radial Motion
This section derives the formula that specifies the radial velocity of a
galaxy for the Yilmaz cosmology model. We can simplify the equations for the
Yilmaz cosmology model by setting tangential incremental motion dxt
equal to zero, and thereby limiting our attention to radial motion. The metric
equation formula of Eq. 4-10 becomes
(ds)2 = e-2f (dt)2 + e2f (dr)2 = exp[-(r/r0)2](dt)2 - exp[(r/r0)2](dr)2
(4-15)
dt/ds
= exp[(r/r0)2]
(4-16)
d2r/ds2
= (r/r02)
- (r/r02)(dr/ds)2
(4-17)
Solve
the metric equation of Eq. 4-15 for (dr)2:
(dr)2 =
exp[-2(r/r0)2](dt)2
- exp[-(r/r0)2](ds)2
(4-18)
Divide
Eq. 4-18 by (ds)2 to obtain the ratio (dr/ds)2:
(dr/ds)2 =
exp[-2(r/r0)2](dt/ds)2
- exp[-(r/r0)2]
(4-19)
Square
the expression for dt/ds in Eq. 4-16, and substitute
it for (dt/ds)2
in Eq. 4-19. The first term is unity and so Eq. 4-19 becomes
(dr/ds)2 =
1 - exp[-(r/r0)2]
(4-20)
Substituting
Eq. 4-20 into Eq. 4-17 gives