Linearized Einstein equations

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Let us consider small perturbations on Minkowski background, such that in some frame the metric can be presented in the form \begin{equation} \label{WFL} g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x), \qquad |h_{\mu\nu}(x)|\ll 1. \end{equation} We can also consider perturbations on the background of other exact solutions of the Einstein equations by replacing $\eta_{\mu\nu}$ with the corresponding $g_{\mu\nu}^{(0)}$. Thus cosmological perturbations are naturally studied in the Friedmanninan background. The linearized Einstein equations are obtained in the first order by $h_{\mu\nu}$, discarding quadratic terms. On Minkowski background the zero-order terms for the curvature tensor and its contractions vanish, so from the Einstein's equation the stress-energy tensor in the considered region must also be small (if non-zero) and $\sim h$. The constraints this places on matter will be considered in more detail in the next section.

Problem 1: Inverse metric

Show that on Minkowski background the inverse metric is \[g^{\mu\nu}(x)=\eta^{\mu\nu}-h^{\mu\nu}(x)+O(h^2), \quad\text{where}\quad h^{\mu\nu}\equiv\eta^{\mu\rho}\eta^{\nu\sigma} h_{\rho\sigma},\] and we agree to use $\eta$ for raising and lowering of the indices.

Problem 2: Raising indices

Show$^*$ that using the background metric $g_{\mu\nu}^{(0)}$ to raise and lower indices instead of the true metric $g_{\mu\nu}$ only makes difference in the next order by $h$.
Consider for definiteness a second rank tensor $A_{\mu\nu}$: \begin{align*}A_{\mu\nu} =g_{\mu\rho}g_{\nu\sigma}A^{\rho\sigma} =g_{\mu\rho}^{(0)}g_{\nu\sigma}^{(0)}A^{\rho\sigma} +O(hA). \end{align*}

$^*$ Is it really a problem at all?

Problem 3: Linearized curvature tensors

Derive the curvature, Ricci and Einstein tensors in the first order by $h_{\mu\nu}$.

Problem 4: Trace-reversed perturbation

Write the Einstein's tensor in terms of the trace-reversed metric perturbation \[\bar{h}_{\mu\nu} =h_{\mu\nu}-\frac{1}{2}h\;\eta_{\mu\nu}.\]