9.1 The Schwarzschild Solution
129
symmetric body. For the sun it is valid for radii greater that the solar radius, which
is about 10
6 km. For smaller radii we must solve a different problem. See Exercises
9.10 and 9.11 and Chap. 10.
Finally it is worth noting that the approximate classical limit for g 00 in (9.3) is, in
this case, exact. This is an accident, due to the choice of coordinates, and has no deep
meaning. In Appendix 1 we will obtain the metric in other coordinates, for which
(9.3) is only approximate.
Digression 9.1 The Birkhoff Theorem states that the Schwarzschild solution is
the unique solution to the vacuum field equations for the exterior of a spherically
symmetric body, given the Minkowski space boundary condition (Birkhoff
1923; Misner 1973). This means that the assumption of time independence of
the metric that we made above is in fact not necessary.
The coordinates used in the metric (9.19) are naturally called Schwarzschild coordinates. Another coordinate system, called spatially isotropic coordinates, is often
used in the linearized theory and in discussions of the observational tests of general
relativity. See Appendix 1 for this form. We will return to it in Chap. 11 (Will 1993,
2014).
9.2 Orbit of a Planet
Schwarzschild’s solution is the key to studying the motion of the planets in the solar
system. Since this is such an important problem we will work out in detail the orbit
of a planet around the sun in this section, and will see that it is very nearly an ellipse,
as in classical theory, but with a small change peculiar to relativity. Our solution will
follow very closely the classical Kepler problem (Goldstein 1980). The reader need
not know the classical theory to follow our solution, but it will be easier and more
transparent if he does. The various transformations and tricks that we will use in this
section are almost the same as those used in the classical problem.
We know from Chap. 7 that the equations of motion for a particle in spacetime are
the Euler-Lagrange equations for a Lagrangian function constructed from the line
element s,
L =
1 −
2m
r
c
2 ˙
t
2
−
1 −
2m
r
−1
˙
r
2
− r
2 ˙
θ
2
− r
2 sin
2
θ ˙
ϕ
2
.
(9.20)
The dot denotes differentiation with respect to the line element s. Recall also that
the numerical value of this Lagrangian function is 1. It is easy to work out the
Euler-Lagrange equations for the coordinates t, θ, ϕ, and we find
129
symmetric body. For the sun it is valid for radii greater that the solar radius, which
is about 10
6 km. For smaller radii we must solve a different problem. See Exercises
9.10 and 9.11 and Chap. 10.
Finally it is worth noting that the approximate classical limit for g 00 in (9.3) is, in
this case, exact. This is an accident, due to the choice of coordinates, and has no deep
meaning. In Appendix 1 we will obtain the metric in other coordinates, for which
(9.3) is only approximate.
Digression 9.1 The Birkhoff Theorem states that the Schwarzschild solution is
the unique solution to the vacuum field equations for the exterior of a spherically
symmetric body, given the Minkowski space boundary condition (Birkhoff
1923; Misner 1973). This means that the assumption of time independence of
the metric that we made above is in fact not necessary.
The coordinates used in the metric (9.19) are naturally called Schwarzschild coordinates. Another coordinate system, called spatially isotropic coordinates, is often
used in the linearized theory and in discussions of the observational tests of general
relativity. See Appendix 1 for this form. We will return to it in Chap. 11 (Will 1993,
2014).
9.2 Orbit of a Planet
Schwarzschild’s solution is the key to studying the motion of the planets in the solar
system. Since this is such an important problem we will work out in detail the orbit
of a planet around the sun in this section, and will see that it is very nearly an ellipse,
as in classical theory, but with a small change peculiar to relativity. Our solution will
follow very closely the classical Kepler problem (Goldstein 1980). The reader need
not know the classical theory to follow our solution, but it will be easier and more
transparent if he does. The various transformations and tricks that we will use in this
section are almost the same as those used in the classical problem.
We know from Chap. 7 that the equations of motion for a particle in spacetime are
the Euler-Lagrange equations for a Lagrangian function constructed from the line
element s,
L =
1 −
2m
r
c
2 ˙
t
2
−
1 −
2m
r
−1
˙
r
2
− r
2 ˙
θ
2
− r
2 sin
2
θ ˙
ϕ
2
.
(9.20)
The dot denotes differentiation with respect to the line element s. Recall also that
the numerical value of this Lagrangian function is 1. It is easy to work out the
Euler-Lagrange equations for the coordinates t, θ, ϕ, and we find