11.1 The Field Equations of the Linearized Theory
163
This is a somewhat nonstandard notation for the d’Alembertian, using a square, in
analogy with the Laplacian ∇
2 in three dimensions. As we noted above, in the last
two lines we have dropped the prime notation, assuming that we always work in a
system where the divergence is zero. Notice that the equations in (11.17) and (11.18)
are consistent since the energy-momentum tensor has a zero divergence as we noted
previously.
The gauge choice we used in the above paragraph is often called the Lorentz
gauge since it is the exact analog of the Lorentz gauge of electromagnetism, but it
is also often called the de Donder or harmonic gauge. Its utility is obvious because
of the way it simplifies the field equations to be a set of Poisson equations with a
divergence constraint. They could hardly be more simple. Unless noted otherwise we
will always work in the Lorentz gauge. In the final form (11.18) the field equations
are very similar to many in classical physics, in particular electromagnetic radiation
theory, so many problems in gravity may be solved using well-known techniques, as
we discuss in the Appendices.
There is one more important fact to be obtained from (11.15) for the transformation
of the metric field divergence. If the original gauge is Lorentz and we transform with
functions f
α to a new gauge then the new gauge is also Lorentzian if and only if the
f
α obey the wave equation
2 f
α
= 0. We will use this sort of gauge transformation
to great advantage in what follows.
11.2 The Classical Limit
By the classical limit we mean that the gravitational fields are weak and independent
of time, and the source is the matter density, independent of velocity as in Poisson’s
equation. We have already treated this situation in Chaps. 7 and 8 where we developed
basic ideas and field equations, but in this section we will go a little deeper and be
more general and systematic.
The source for the classical limit case must be well-described by the energymomentum tensor of slowly moving dust, as we discussed in Chap. 8, with only a
0,0 metric component since the motion is to be neglected. That is
T μν =
⎛
⎜
⎜
⎝
ρ 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
⎞
⎟
⎟
⎠ .
(11.19)
For a time independent system the field equations (11.18) then give
∇
2 ¯
h 00 = −
16π G
c 2
ρ.
(11.20)
163
This is a somewhat nonstandard notation for the d’Alembertian, using a square, in
analogy with the Laplacian ∇
2 in three dimensions. As we noted above, in the last
two lines we have dropped the prime notation, assuming that we always work in a
system where the divergence is zero. Notice that the equations in (11.17) and (11.18)
are consistent since the energy-momentum tensor has a zero divergence as we noted
previously.
The gauge choice we used in the above paragraph is often called the Lorentz
gauge since it is the exact analog of the Lorentz gauge of electromagnetism, but it
is also often called the de Donder or harmonic gauge. Its utility is obvious because
of the way it simplifies the field equations to be a set of Poisson equations with a
divergence constraint. They could hardly be more simple. Unless noted otherwise we
will always work in the Lorentz gauge. In the final form (11.18) the field equations
are very similar to many in classical physics, in particular electromagnetic radiation
theory, so many problems in gravity may be solved using well-known techniques, as
we discuss in the Appendices.
There is one more important fact to be obtained from (11.15) for the transformation
of the metric field divergence. If the original gauge is Lorentz and we transform with
functions f
α to a new gauge then the new gauge is also Lorentzian if and only if the
f
α obey the wave equation
2 f
α
= 0. We will use this sort of gauge transformation
to great advantage in what follows.
11.2 The Classical Limit
By the classical limit we mean that the gravitational fields are weak and independent
of time, and the source is the matter density, independent of velocity as in Poisson’s
equation. We have already treated this situation in Chaps. 7 and 8 where we developed
basic ideas and field equations, but in this section we will go a little deeper and be
more general and systematic.
The source for the classical limit case must be well-described by the energymomentum tensor of slowly moving dust, as we discussed in Chap. 8, with only a
0,0 metric component since the motion is to be neglected. That is
T μν =
⎛
⎜
⎜
⎝
ρ 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
⎞
⎟
⎟
⎠ .
(11.19)
For a time independent system the field equations (11.18) then give
∇
2 ¯
h 00 = −
16π G
c 2
ρ.
(11.20)