**4.7. Physical content of the Einstein equations**

In the last section we showed that the Poisson gauge variables
(,
,
** w**) are given by the

We showed in eq. (4.47) that the Poisson gauge can be
transformed to any other gauge. In the cosmological Lorentz gauge (see
Misner et al. 1973
for the noncosmological version) all metric
perturbation components obey wave equations. Therefore, the solutions
in Poisson gauge *must be causal* despite appearances to the contrary.

There is a precedent for this type of behavior: the Coulomb gauge of
electromagnetism. With
^{.}
** A** = 0, eqs. (4.45) become

(4.57) |

We have separated the current density into longitudinal and transverse
parts. The similarity of the first two (scalar) equations to eqs.
(4.49) and (4.50) is striking. The similarity would be even
more striking if we were to use comoving coordinates rather than treating
** x** and here as
flat spacetime coordinates. As an exercise one
can show that with comoving coordinates,
and

Are we to conclude that electromagnetism *also* violates causality,
because the electric potential
depends only
on the instantaneous
Distribution of charge? No! To understand this let us examine the Coulomb
and Ampère laws in flat spacetime for the *fields* rather than the
potentials:

(4.58) |

The Ampère law has been split into longitudinal and transverse parts.
We see that the *longitudinal* electric field indeed *is* given
instantaneously by the charge density. Because the photon is a massless
vector particle, only the *transverse* part of the electric and
magnetic fields is radiative, and its source is given by the transverse
current density:

(4.59) |

But how does this restore causality? To see how, let us consider the
following example. Suppose that there is only one electric charge in
the universe and initially it is at rest in the lab frame. If the
charge moves - even much more slowly than the speed of light -
*E*_{||} - the solution to the Coulomb equation - is
changed everywhere instantaneously. It must be therefore that
*E*_{}
*also* changes instantaneously in such a way as to
exactly *cancel* the acausal behavior of *E*_{||}.

This indeed happens, as follows. First, note that the motion of the
charge generates a current density
** J** =

Now that we understand how causality is maintained, what is the use of the
longitudinal part of the Ampère law,
- _{}
*E*_{||} =
4
*J*_{||}? The answer is, to ensure charge
conservation, which is
implied by combining the time derivative of the Coulomb law with the
divergence of the Ampère law:

(4.60) |

*Charge conservation is built into the Coulomb and Ampère laws.*
This remarkable behavior occurs because electromagnetism is a *gauge*
theory. Gauge invariance effectively provides a redundant scalar field
equation whose physical role is to enforce charge conservation. From
Noether's theorem (e.g.,
Goldstein 1980),
a continuous symmetry (in this
case, electromagnetic gauge invariance) leads to a conserved current.

General relativity is also a gauge theory. Coordinate invariance - a
continuous symmetry - leads to conservation of energy and momentum.
As a result there are redundant scalar and vector equations [eqs.
(4.50), (4.52), and (4.54)] *whose role is to enforce
the conservation laws* [eqs. (4.24) and (4.25)]. We are free
to use the action-at-a-distance field equations for the scalar and vector
potentials in Poisson gauge because, when they are converted to fields and
combined with the gravitational radiation field, the resulting behavior is
entirely causal.

The analogy with electromagnetism becomes clearer if we replace the gravitational potentials by fields. We define the "gravitoelectric" and "gravitomagnetic" fields (Thorne, Price & Macdonald 1986; Jantzen, Carini & Bini 1992)

(4.61) |

*using the Poisson gauge variables*
and
** w**. In section 4.8 we shall see
how these fields lead to "forces" on particles
similar to the Lorentz forces of electromagnetism. For now, however, we
are interested in the fields themselves.

Note that ** g** and

In the limit of comoving distance scales small compared with the curvature
distance |*K*|^{-1/2} and the Hubble distance
^{-1}, and
nonrelativistic shear stresses, the gravitoelectric and gravitomagnetic
fields obey a gravitational analogue of the Maxwell equations:

(4.62) |

where ** f** =
( +

Obtaining this physical insight into general relativity is *much*
easier in the Poisson gauge than in the synchronous gauge. This fact alone
is a good reason for preferring the former. When combined with the other
advantages (simpler equations, no time evolution required for the scalar
and vector potentials, reduction to the Newtonian limit, no nontrivial
gauge modes, and lack of unphysical coordinate singularities), the
superiority of the Poisson gauge should be clear.

Although the physical picture we have developed for gravity in analogy with electromagnetism is beautiful, it is inexact. Not only have we linearized the metric, we have also neglected cosmological effects in eqs. (4.62). We shall see in section 4.9 how to obtain exact nonlinear equations for (the gradients of) the gravitational fields.