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I am currently reading a paper by Bredberg $et.al$ arXiv:1101.2451 titled "From Navier-Stokes to Einstein". In this paper, the authors have considered a metric of the form \begin{eqnarray}ds^2_{p+2} = -r d\tau^2+2 d\tau dr +dx^idx_i\\-2(1-\frac{r}{r_c})v_idx^id\tau-2\frac{v_i}{r_c}dx^idr\\+(1-\frac{r}{r_c})\Big[(v^2+2p)d\tau^2+\frac{v_iv_j}{r_c}dx^idx^j\Big]+\big(\frac{v^2}{r_c}+\frac{2P}{r_c}\Big)d\tau dr\\ +\mathcal{O(\epsilon^3)} \end{eqnarray}

The first line conains terms of the order $\mathcal{O(\epsilon^0)}$, second line of the order $\mathcal{O}(\epsilon)$ and the third line of the order $\mathcal{O}(\epsilon^2)$. Here $v_i \sim \mathcal{O}(\epsilon)$, $P \sim \mathcal{O}(\epsilon^2)$. The authors have considered timelike hypersurfaces $r = r_c$. I am trying to find out the unit normal to these hypersurfaces $n^a$. The Unit normal to these hypersurfaces are given by $n^{\mu}\partial_{\mu} = \frac{1}{\sqrt{r_c}}\partial_{\tau}+\sqrt{r_c}(1-\frac{P}{r_c})\partial_{r}+\frac{v^i}{\sqrt{r_{c}}}\partial_i+\mathcal{O}(\epsilon^3)$. To achieve this form of the contravariant unit normal $n^a$, i need to invert the metric and find $g^{\mu \nu}$ upto order $\mathcal{O(\epsilon^2)}.$ I am currently stuck in this. Any help regarding the inversion of this metric either via hand or by CAS would be highly regarded. PS: Notation:- the spacetime dimension is $p+2$ with $\tau$ being the time coordinate ,$r$ being the radial coordinate and $i = 1,2,....p$ being the angular coordinates.

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I am assuming that you're working with the background field method formalism, where we define

$$g_{\mu \nu} = \bar{g}_{\mu \nu} + h_{\mu \nu}$$

so that every other quantity is expressed as an infinite series in $h_{\mu \nu}$. Then, the inverse metric up to $\mathcal{O}(h^2)$ is given by

$$g^{\mu \nu } = \bar{g}^{\mu \nu } - h^{\mu \nu } + h_{\alpha }{}^{\nu } h^{\mu \alpha }$$

which you can easily compute once you know $\bar{g}_{\mu \nu }$ and $h_{\mu \nu }$. Note that all contractions, traces, etc. are taken with respect to the background metric $\bar{g}_{\mu \nu}$.

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