9. Introduction into perturbation theory in general relativity 2 (Inflationary perturbations 3)

Posted by Dmitry
March 27, 2008

Today I will continue our discussion of the perturbation theory in general relativity (please see the previous post here); in particular, I will focus my attention on the issue of gauge invariance.

1. Gauge transformations

Let us consider how the metric modes we have constructed in the previous post change under coordinate transformations. We introduce an infinitesimal (i.e., \xi^{\alpha}\to0) coordinate transformation

\tilde{x}^{\alpha}=x^{\alpha}+\xi^{\alpha}(x^{\beta}), (1)

under which the metric tensor is transformed as

\tilde{g}_{\alpha\beta}(\tilde{x})=\frac{\partial x^{\gamma}}{\partial\tilde{x}^{\alpha}}\frac{\partial x^{\delta}}{\partial\tilde{x}^{\beta}}g_{\gamma\delta}(x)\approx

\approx g_{\alpha\beta}^{(0)}(x)+\delta g_{\alpha\beta}-g_{\alpha\delta}^{(0)}\xi_{;\beta}^{\delta}-g_{\beta\delta}^{(0)}\xi_{;\alpha}^{\delta}=

=\tilde{g}_{\alpha\beta}^{(0)}(\tilde{x})+\delta\tilde{g}_{\alpha\beta}(\tilde{x}).

Therefore,

\delta\tilde{g}_{\alpha\beta}=\delta g_{\alpha\beta}-g_{\alpha\beta,\gamma}^{(0)}\xi^{\gamma}-g_{\alpha\delta}^{(0)}\xi_{;\beta}^{\delta}-g_{\beta\delta}^{(0)}\xi_{;\alpha}^{\delta}.

Exercise 2.4. What is the origin of the second term in the right hand side?

Let us represent the 3-component of the 4-vector \xi^{\alpha} in the form

\xi^{i}=\xi_{{\rm df}}^{i}+\zeta^{,i},

where \xi_{{\rm df}}^{i} is the divergence free part of \xi^{i}. Than, one has

\delta\tilde{g}_{00}=\delta g_{00}-2a(a\xi^{0})',

\delta\tilde{g}_{0{}i}=[/\delta g_{0i}+a^{2}(\xi_{{\rm df}\,;i}'+(\zeta'-\xi_{0})_{,i}),

\delta\tilde{g}_{ij}=\delta g_{ij}+a^{2}\left(2\frac{a'}{a}\delta_{ij}\xi_{0}+2\zeta_{;ij}+(\xi_{{\rm df}\, i;j}+\xi_{{\rm df}\, j;i})\right),

where prime denotes derivative with respect to conformal time \eta. Now one can understand how various modes present in the metric tensor are transformed.

We start with scalar modes. Their contribution into the overall perturbed metric is given by

ds^{2}=a^{2}\left((1+2\phi)d\eta^{2}+2B_{,i}d\eta dx^{i}-\left((1-2\psi)\delta_{ij}-2E_{;ij}\right)dx^{i}dx^{j}\right),

so one finds find that

\tilde{\phi}=\phi-\frac{1}{a}(a\xi^{0})',

\tilde{\psi}=\psi+\frac{a'}{a}\xi^{0},

\tilde{B}=B+\zeta'-\xi^{0},

\tilde{E}=E+\zeta

Exercise 2.5. Check these transformation rules.

Exercise 2.6. Construct scalars which do not change under coordinate transformations (1).

Similarly, vector modes give the following contributions into the overall metric:

ds^{2}=a^{2}\left(d\eta^{2}+2S_{i}d\eta dx^{i}-\left(\delta_{ij}-F_{i;j}-F_{j;i}\right)dx^{i}dx^{j}\right),

and one has for the vector perturbations

\tilde{S}_{i}=S_{i}+\xi_{{\rm df}\, i}',

\tilde{F}_{i}=F_{i}+\xi_{{\rm df}\, i}.

Exercise 2.7. Construct vector which does not change under coordinate transformations (1).

Finally, one can find that the tensor mode h_{ij} does not change under coordinate transformations (1).

2. Important gauges

Discussing various possible gauges, we will focus our attention on scalar perturbations, because these are the ones the most influenced by the choice of the gauge.

a) Synchronous gauge

Since in 4-dimensional world we are allowed to do four coordinate transformations, by these transformations we can always choose

g_{00}=1, g_{0i}=0, (2)

which can be recast in the form \phi=0, B=0 using scalar modes we introduced above.

The resulting metric will look like

ds^{2}=dt^{2}-\gamma_{ij}(t,{\bf x})dx^{i}dx^{j},

where dt=ad\eta.

The gauge defined by these conditions is called synchronous, since in this gauge clocks everywhere in the universe (or, more accurately, in a given Hubble patch) are synchronized. Conditions (2) do not fix the gauge completely, because the freedom remains to make arbitrary 3-dimensional coordinate transformations.

b) Longitudinal (or newtonian) gauge.

This gauge is fixed by the conditions B=0, E=0, so that the metric looks like

ds^{2}=a^{2}(1+2\phi)d\eta^{2}-a^{2}(1-2\psi)\delta_{ij}dx^{i}dx^{j}.If T_{i}^{j}\propto\delta_{i}^{j}, the scalar modes \phi and \chi coincide. The variable \phi in this gauge reduces to the Newtonian potential \varphi in the non-relativistic limit.

In the next post I will discuss gauge invariant variables and their dynamics.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft

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