Show metric perturbation transformation

[Valeriia Lukashenko]

Homework Statement

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##

Homework 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.

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?

 

[mark726]

One possible hint is to consider the transformation of the perturbed metric components individually. Specifically, focus on the transformation of ##B_i## and see if you can identify any terms that will give you ##\partial_{\eta}L^i## when you apply the transformation. Additionally, it may be helpful to keep in mind that ##\partial_{\eta}L^i## is a function of both ##\eta## and ##\vec{x}##.