Show metric perturbation transformation

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1. Dec 3, 2017

Valeriia Lukashenko

1. The problem statement, all variables and given/known data

Consider following transformation: Transformation: $$X^{\mu}\rightarrow \tilde{X^{\mu}}= X^{\mu}+\xi^{\mu}(\eta, \vec{x})$$
where $\xi^0=T, \xi^i=L_i$
Show transformation of metric perturbation $B_i\rightarrow \tilde{B_i}=B_i+\partial_iT-\partial_{\eta}L_i$

2. Relevant equations

Perturbed metric: $$ds^2=a^2(\eta)[(1+2A)d\eta-2B_idx^id\eta-(\delta_{ij}+h_{ij})dx^idx^j]$$
$$g_{\mu\nu}(X)=\frac{\partial \tilde{X^{\alpha}}}{\partial X^{\mu}}\frac{\partial \tilde{X^{\beta}}}{\partial X^{\nu}}\tilde{g_{\alpha\beta}}(\tilde{X})$$
where $\tilde{g_{\alpha\beta}}(\tilde{X})$ is metric in new coordinates.

3. The attempt at a solution

$$g_{0i}(X)=\frac{\partial \tilde{X^{\alpha}}}{\partial \eta}\frac{\partial \tilde{X^{\beta}}}{\partial X^{i}}\tilde{g_{\alpha\beta}}(\tilde{X})$$

$$\frac{\partial \tilde{X^{\alpha}}}{\partial \eta}\frac{\partial \tilde{X^{\beta}}}{\partial X^{i}}\tilde{g_{\alpha\beta}}(\tilde{X})=\frac{\partial \tilde{X^{0}}}{\partial \eta}\frac{\partial \tilde{X^{0}}}{\partial X^{i}}\tilde{g_{00}}(\tilde{X})+\frac{\partial \tilde{X^{0}}}{\partial \eta}\frac{\partial \tilde{X^{i}}}{\partial X^{i}}\tilde{g_{0i}}(\tilde{X})+\frac{\partial \tilde{X^{i}}}{\partial \eta}\frac{\partial \tilde{X^{0}}}{\partial X^{i}}\tilde{g_{i0}}(\tilde{X})+\frac{\partial \tilde{X^{i}}}{\partial \eta}\frac{\partial \tilde{X^{j}}}{\partial X^{i}}\tilde{g_{ij}}(\tilde{X})$$

$$\frac{\partial \tilde{X^{\alpha}}}{\partial \eta}\frac{\partial \tilde{X^{\beta}}}{\partial X^{i}}\tilde{g_{\alpha\beta}}(\tilde{X})=(1+\partial_{\eta}T)\partial_iT\tilde{g_{00}}(\tilde{X})+(1+\partial_{\eta}T)(1+\partial_iL^i)\tilde{g_{0i}}(\tilde{X})+(\partial_{\eta}X^i+\partial_{\eta}L^i)\partial_iT\tilde{g_{i0}}(\tilde{X})+(\partial_{\eta}X^i+\partial_{\eta}L^i)(\delta^{ij}+\partial_{i}L^j)\tilde{g_{ij}}(\tilde{X})=(1+\partial_{\eta}T)\partial_iTa^2(\eta+T)(1+2\tilde{A})-2(1+\partial_{\eta}T)(1+\partial_iL^i)a^2(\eta+T)\tilde{B_i}-2(\partial_{\eta}X^i+\partial_{\eta}L^i)\partial_iTa^2(\eta+T)\tilde{B_i}-(\partial_{\eta}X^i+\partial_{\eta}L^i)(\delta^{ij}+\partial_{i}L^j)a^2(\eta+T)(\delta_{ij}+\tilde{h_{ij}})$$

I don't see how to get the answer because at first order I get $2B_i$ and don't see how to get $\partial_{\eta}L^i$. Could anyone give me a hint how to get the answer?

Last edited: Dec 3, 2017
2. Dec 8, 2017 at 6:00 PM

PF_Help_Bot

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