7.1 Perturbations and gauge-invariant variables 295rotational motions. The correspondingcovariantcomponents of the rotational ve-
locityδu⊥i,satisfying the condition(δu⊥i),i= 0 ,are also gauge-invariant.
Tensor perturbationsFor tensor perturbations,
ds^2 =a^2[
dη^2 −(
δij−hij)
dxidxj]
(7.25)
andhijdoes not change under coordinate transformations. It already describes the
gravitational waves in a gauge-invariant manner.
7.1.3 Coordinate systems
Gauge freedom has its most important manifestation in scalar perturbations. We
can use it to impose two conditions on the functionsφ,ψ,B,E,δεand the potential
velocity perturbationsδui=φ,i. This is possible since we are free to choose the two
functionsξ^0 andζ. Imposing the gauge conditions is equivalent to fixing the (class
of) coordinate system(s). In the following we consider several choices of gauge and
show how, knowing the solution for the gauge-invariant variables, one can calculate
the metric and density perturbations in any particular coordinate system in a simple
way.
Longitudinal (conformal-Newtonian) gaugeLongitudinal gauge is defined by the
conditionsBl=El= 0 .From (7.18), it follows that these conditions fix the co-
ordinate system uniquely. In fact, the conditionEl=0 is violated by anyζ= 0 ,
and using this result we see that any time transformation withξ^0 =0 destroys
the conditionBl= 0 .Hence there is no extra coordinate freedom which preserves
Bl=El=0. In the corresponding coordinate system the metric takes the form
ds^2 =a^2
[
( 1 + 2 φl)dη^2 −( 1 − 2 ψl)δijdxidxj]
. (7.26)
If the spatial part of the energy–momentum tensor is diagonal, that is,δTji∝δij,
we haveφl=ψl (see the following section) and there remains only one vari-
able characterizing scalar metric perturbations. The variableφlis a generalization
of the Newtonian potential, which explains the choice of the name “conformal-
Newtonian” for this coordinate system. As can be seen from (7.19)–(7.21), the
gauge-invariant variables have a very simple physical interpretation: they are the
amplitudes of the metric, density and velocity perturbations in the conformal-
Newtonian coordinate system, in particular, =φl,=ψl.
Synchronous gaugeSynchronous coordinates, whereδg 0 α= 0 ,have been used
most widely in the literature. In our notation, they correspond to the gauge choice
φs=0 andBs= 0 .This does not fix the coordinates uniquely; there exists a whole