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Show metric perturbation transformation

  1. Dec 3, 2017 #1
    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. jcsd
  3. Dec 8, 2017 at 6:00 PM #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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