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I have to compute the contravariant metric tensor with off-diagonal terms, such as $g_{0i}\neq 0$. I started with the condition $$g_{\mu\nu}g^{\nu\lambda}=\delta^\lambda_\mu$$ but I don't know how to proceed. I can write $$ \left\{ \begin{array}{ll} g_{00}g^{00}+g_{0i}g^{i0}=1\\ g_{i0}g^{0j}+g_{ik}g^{kj}=\delta^j_i\\ \end{array} \right.$$ but I have 3 unknowns with 2 equations. How can I find the contravariant components?

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  • $\begingroup$ What do you mean when you say you want to find them? What information do you already have, and what is your goal to achieve with that? $\endgroup$
    – Avantgarde
    Commented Apr 14, 2023 at 23:34
  • $\begingroup$ I have this metric and I need to find the inverse, but since it is not diagonal I can't invert the term on the diagonal, the paper I'm reading suggests to use the condition i wrote. $\endgroup$
    – Stefano98
    Commented Apr 14, 2023 at 23:36
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    $\begingroup$ Is this really that much different from calculating the inverse of a matrix in linear algebra? It's quite a standard procedure. $\endgroup$
    – rhomaios
    Commented Apr 14, 2023 at 23:36
  • $\begingroup$ But I don't know how to do it since the metric is expanded up to second order and the expression is not so easy $\endgroup$
    – Stefano98
    Commented Apr 14, 2023 at 23:39
  • $\begingroup$ I'm not sure I understand the issue. Can you not use Cramer's rule or Gaussian elimination to find the inverse? Of course things can get quite messy for $4 \times 4$ matrices (or worse), but it's simply a matter of volume of calculations, no? $\endgroup$
    – rhomaios
    Commented Apr 14, 2023 at 23:50

2 Answers 2

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First of all, your counting of unknowns and equations is incorrect. There are $10$ unknowns (inverse metric components) and $10$ equations. The second equation in your question can be split into $6$ independent equations ($3 \times 2$, for $i$ and $j$). The first equation is only the $00$ part, and you forgot to write the $0i$ part of $g_{\mu \nu}g^{\nu 0}=\delta^0_\mu$ by putting $\mu = i$, to give $g_{i \nu}g^{\nu 0}=\delta^0_i = 0$. This ($0 \mu$) equation are in total 4 independent equations. So you have $10$ equations and $10$ unknowns.

Anyway, you don't really need to worry about all these several equations. And it doesn't matter how complicated your metric is. These days you can use a Mathematica package and just invert the metric. I believe xAct (xCoba, in particular is what you would need) should have this feature.

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I solved doing the following calculations up to second order (which is what I needed to do, actually). I started with \begin{equation*} (\bar{g}_{\mu\nu}+\delta g_{\mu\nu}+\frac{1}{2}\delta^2g_{\mu\nu})(\bar{g}^{\nu\lambda}+A_1^{\nu\lambda}+B_1^{\nu\lambda})=\delta^\lambda_\mu, \end{equation*} where $\bar{g}_{\mu\nu}$ is diagonal. Up to second order I got an equation for $A_1^{\lambda\rho}$ and one for $B_1^{\lambda\rho}$, namely I have \begin{equation} A_1^{\lambda\rho}=-\delta g_{\mu\nu}\bar{g}^{\lambda\nu}\bar{g}^{\rho\mu} \end{equation} and \begin{equation} B_2^{\lambda\rho}=-\bar{g}^{\rho\mu}\delta g_{\mu\nu} A_1^{\nu\lambda}-\frac{1}{2}\bar{g}^{\rho\mu}\bar{g}^{\nu\lambda}\delta^2 g_{\mu\nu}. \end{equation} In this way I easily solved my problem and found the correct expression for the contravariant metric up to second order.

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