164
11 Linearized General Relativity and Gravitational Waves
This is the same as Poisson’s equation for the classical potential φ, so we identify a
relation between the 0,0 metric component and the classical potential
¯
h 00 =
4
c 2 φ.
(11.21)
The other components of ¯
h αβ we may take to be zero, which is consistent with the
energy-momentum tensor (11.19). To get the physical metric field h μν we use its
relation to ¯
h μν in (11.12) and obtain the metric field and line element in terms of φ
h μν =
2
c 2 φδ μν , ds
2
=
1 +
2φ
c 2
c
2 dt
2
−
1 −
2φ
c 2
d x
2
.
(11.22)
This constitutes a rather general solution for the field created by low density matter
moving slowly, giving the metric in terms of the classical potential. In particular for
a stationary point mass M we have a line element
ds
2
=
1 −
2G M
c 2 r
c
2 dt
2
−
1 −
2G M
c 2 r
d x
2
, r =
√
x 2 .
(11.23)
Equations (11.22) and (11.23) are useful in many practical applications in celestial
mechanics.
Note that (11.23) is spatially isotropic and thus does not have the same form as the
Schwarzschild solution in Schwarzschild coordinates (9.19); instead it has the same
form as the Schwarzschild solution in isotropic coordinates (9.59) and the Eddington
form (9.60), which is useful in discussions of the experimental tests of relativity (Will
2014).
11.3 Gravitational Plane Waves
One of the most interesting properties of the linearized field equations is that they
admit wave solutions, much as Maxwell’s electrodynamics admits electromagnetic
wave solutions. Most of what we do in this section is very similar to the solutions
of Maxwell’s equations for electromagnetic waves in terms of the 4-vector potential
and the Maxwell tensor, so the reader who is familiar with that topic will find it
especially easy. For those not familiar with electromagnetic waves or who desire a
brief review see Appendices 2 and 3.
In vacuum the field equations (11.18) are
¯
h
,λ
μν ,λ =
2 ¯
h μν = 0, ¯
h
μν
,ν = 0.
(11.24)
11 Linearized General Relativity and Gravitational Waves
This is the same as Poisson’s equation for the classical potential φ, so we identify a
relation between the 0,0 metric component and the classical potential
¯
h 00 =
4
c 2 φ.
(11.21)
The other components of ¯
h αβ we may take to be zero, which is consistent with the
energy-momentum tensor (11.19). To get the physical metric field h μν we use its
relation to ¯
h μν in (11.12) and obtain the metric field and line element in terms of φ
h μν =
2
c 2 φδ μν , ds
2
=
1 +
2φ
c 2
c
2 dt
2
−
1 −
2φ
c 2
d x
2
.
(11.22)
This constitutes a rather general solution for the field created by low density matter
moving slowly, giving the metric in terms of the classical potential. In particular for
a stationary point mass M we have a line element
ds
2
=
1 −
2G M
c 2 r
c
2 dt
2
−
1 −
2G M
c 2 r
d x
2
, r =
√
x 2 .
(11.23)
Equations (11.22) and (11.23) are useful in many practical applications in celestial
mechanics.
Note that (11.23) is spatially isotropic and thus does not have the same form as the
Schwarzschild solution in Schwarzschild coordinates (9.19); instead it has the same
form as the Schwarzschild solution in isotropic coordinates (9.59) and the Eddington
form (9.60), which is useful in discussions of the experimental tests of relativity (Will
2014).
11.3 Gravitational Plane Waves
One of the most interesting properties of the linearized field equations is that they
admit wave solutions, much as Maxwell’s electrodynamics admits electromagnetic
wave solutions. Most of what we do in this section is very similar to the solutions
of Maxwell’s equations for electromagnetic waves in terms of the 4-vector potential
and the Maxwell tensor, so the reader who is familiar with that topic will find it
especially easy. For those not familiar with electromagnetic waves or who desire a
brief review see Appendices 2 and 3.
In vacuum the field equations (11.18) are
¯
h
,λ
μν ,λ =
2 ¯
h μν = 0, ¯
h
μν
,ν = 0.
(11.24)