4.6. Poisson gauge
Recall that our general perturbed Robertson-Walker metric (4.11)
contains four extraneous degrees of freedom associated with coordinate
invariance. In the synchronous gauge these degrees of freedom are
eliminated from g00 (one scalar) and
g0i (one scalar and one transverse vector) by requiring
=
wi = 0. There are other ways to
eliminate the same number of fields. As we shall see, a good choice is
to constrain g0i (eliminating one scalar) and
gij (eliminating
one scalar and one transverse vector) by imposing the following gauge
conditions on eq. (4.11):
![]() |
(4.46) |
I call this choice the Poisson gauge by analogy with the Coulomb
gauge of electromagnetism
( .
A = 0).
(2)
More conditions are required here than in electromagnetism because gravity
is a tensor rather than a vector gauge theory. Note that in
the Poisson gauge there are two scalar potentials
(
and
), one
transverse vector potential (w), and one
transverse-traceless tensor potential h.
A restricted version of the Poisson gauge, with
wi = hij = 0, is known
in the literature as the longitudinal or conformal Newtonian gauge
(Mukhanov, Feldman &
Brandenberger 1992).
These conditions can be
applied only if the stress-energy tensor contains no vector or tensor parts
and there are no free gravitational waves, so that only the scalar metric
perturbations are present. While this condition may apply, in principle,
in the linear regime
(|
/
|
<< 1), nonlinear
density fluctuations generally induce vector and tensor modes even if
none were present initially. Setting
w = h = 0 is analogous to
zeroing the electromagnetic vector potential, implying B =
0. In
general, this is not a valid gauge condition - it is rather the
elimination of physical phenomena. The longitudinal/conformal Newtonian
gauge really should be called a "restricted gauge." The Poisson gauge,
by contrast, allows all physical degrees of freedom present in the metric.
To prove the last statement, and to find out how much residual gauge freedom is allowed, we must find a coordinate transformation from an arbitrary gauge to the Poisson gauge. Using eq. (4.40) with hats indicating Poisson gauge variables, we see that a suitable transformation exists with
![]() |
(4.47) |
where w comes from the longitudinal part of w
(w|| =
- w), while
h and hi come from the longitudinal and
solenoidal parts of h in eq. (4.14). Because these
conditions are algebraic in
,
, and
(they are not differentiated, in contrast with the transformation to
synchronous gauge of eq. 4.41), we have found an almost unique
transformation from an arbitrary gauge to the Poisson gauge. One can
still add arbitrary functions of time alone (with no dependence on
xi)
to
and
i. (Adding a function of time alone to
has no effect at all because the transformation, eq. 4.39,
involves only the gradient of
.)
Spatially homogeneous changes in
represent changes in
the units of time and length, while spatially homogeneous changes in
represent shifts in the origin of the spatial coordinate system. These
trivial residual gauge freedoms - akin to electromagnetic gauge
transformations generated by a function of time, the only gauge freedom
remaining in Coulomb gauge - are physically transparent and should cause
no conceptual or practical difficulty.
It is interesting to see the coordinate transformation from a synchronous gauge to the Poisson gauge. As an exercise the reader can show that this is given by
![]() |
(4.48) |
Comparing with eq. (4.43), we see that the two Poisson-gauge
scalar potentials are
=
A and
=
-
H.
(Kodama & Sasaki 1984
call these variables
=
and
=
-
.) The vector
potential
wi in Poisson gauge is related simply to the
solenoidal potential hi of the synchronous gauge
(eq. 4.31).
Thus, the metric perturbations in the Poisson gauge correspond exactly with several of the gauge-invariant variables introduced by Bardeen. By imposing the explicit gauge conditions (4.46), we have simplified the mathematical analysis of these variables.
Now that we have seen that the Poisson gauge solves the gauge-fixing problem, let us give the components of the perturbed Einstein equations. They are no more complicated than those of the synchronous gauge:
![]() |
(4.49)
(4.51)
(4.53) (4.54) (4.55) |
As in the synchronous gauge, the scalar and vector modes satisfy
initial-value (ADM) constraints (eqs. 4.49-4.51) in
addition to evolution equations. However, it is remarkable that in the
Poisson gauge we can obtain the scalar and vector potentials directly
from the instantaneous stress-energy distribution with no time integration
required. This is clear for
-
and
w, both of which obey
elliptic equations with no time derivatives (eqs. 4.53 and
4.51, respectively). By combining the ADM energy and longitudinal
momentum constraint equations we can also get an instantaneous equation
for
:
![]() |
(4.56) |
Bardeen (1980) defined the matter perturbation variable
m
(
+
3
f /
and
noted that it is the natural
measure of the energy density fluctuation in the normal (inertial)
frame at rest with the matter such that
v + w = 0 (recall the
discussion in section 4.3). However, for our
analysis
we will remain in the comoving frame of the Poisson gauge, in which case
/
and not
m is the
density fluctuation.
We can show that for nonrelativistic matter the field equations we have
obtained reduce to the Newtonian forms. First, it is clear that in the
non-cosmological limit ( = K = 0), eq. (4.56) reduces to
the Poisson equation. For
0 the longitudinal momentum
density
f is also a
source for
,
but it is unimportant for perturbations with
|
/
|>>
vHv / c2 where
vH is the Hubble
velocity across the perturbation. Next, consider the implications of
the fact that the shear stress for any physical system is at most
O(
cs2) where cs is the
characteristic thermal speed
of the gas particles. (For a collisional gas the shear stress is much less
than this.) Equation (4.53) then implies that the relative difference
between
and
is no more than
O(cs / c)2. Third, eq.
(4.51) implies that the vector potential
w ~ (vH /
c)2v.
Thus, the deviations from the Newtonian results are all
O(v / c)2.
Poisson gauge gives the relativistic cosmological generalization of
Newtonian gravity.
There are still more remarkable features of the Poisson gauge. First,
the Poisson gauge metric perturbation variables are almost always small
in the nonrelativistic limit
(|| <<
c2, v2 << c2),
in contrast with the synchronous gauge variables hij,
which become large when
|
/
|> 1.
(However,
Bardeen 1980
shows that the relative numerical merits of these two gauges can reverse
for isocurvature perturbations of size larger than the Hubble distance.)
Second, if (
,
,
w, h) are very small, they - but not
necessarily their derivatives! - may be neglected to a good approximation,
in which case the Poisson gauge coordinates reduce precisely to the Eulerian
coordinates used in Newtonian cosmology. Finally, it is amazing that the
scalar and vector potentials depend solely on the instantaneous
distribution of stress-energy - in fact, only the energy and momentum
densities and the shear stress are required. Only the tensor mode
- gravitational radiation - follows unambiguously from a time
evolution equation. In fact, it obeys precisely the same equation as in
the synchronous gauge (with a factor of 2 difference owing to our different
definitions) because tensor perturbations are gauge-invariant -
coordinate transformations involving 3-scalars and a 3-vector cannot change
a 3-tensor (leaving aside the special case of eq. 4.17 for a closed space).
2 The
same gauge has been proposed recently by
Bombelli, Couch &
Torrence (1994),
who call it "cosmological gauge." However, I prefer the
name Poisson gauge because cosmology - i.e., nonzero
-
is irrelevant for the definition and physical interpretation of this
gauge. Although I have seen no earlier discussion of Poisson gauge
in the literature, its time slicing corresponds with the minimal shear
hypersurface condition of
Bardeen (1980).
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