 Solution

In Questions 3–4, use primes for conformal-time derivatives and $\mathcal{H}=a^{\prime }/a$; proper-time derivatives will be stated separately. Let ${\xi }^{\mu }=(T,L^i)$. The metric is a covariant tensor, so under the passive coordinate change ${\tilde{x}}^{\mu }=x^{\mu }+{\xi }^{\mu }$,The inverse Jacobian is ${\delta }_{\mu }^{\alpha }-{\partial }_{\mu }{\xi }^{\alpha }$ to first order. At the same new coordinate argument, the homogeneous background satisfies ${\bar{g}}_{\alpha \beta }(x)={\bar{g}}_{\alpha \beta }(\tilde{x})-T{\bar{g}}_{\alpha \beta }^{\prime }$. Shifting the argument of $\delta g$ or multiplying it by a Jacobian perturbation produces only second-order terms, which we discard. Subtracting the unperturbed metric therefore givesThis is the printed Jacobian expression expanded to linear order, equivalently $\delta \tilde{g}=\delta g-{\mathcal{L}}_{\xi }\bar{g}$. It is a gauge transformation in cosmological perturbation theory, not a change of the physical spacetime.