# Contravariant metric tensor with off-diagonal terms

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?

• 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? Commented Apr 14, 2023 at 23:34
• 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. Commented Apr 14, 2023 at 23:36
• Is this really that much different from calculating the inverse of a matrix in linear algebra? It's quite a standard procedure. Commented Apr 14, 2023 at 23:36
• 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 Commented Apr 14, 2023 at 23:39
• 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? Commented Apr 14, 2023 at 23:50

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.
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 $$$$A_1^{\lambda\rho}=-\delta g_{\mu\nu}\bar{g}^{\lambda\nu}\bar{g}^{\rho\mu}$$$$ and $$$$B_2^{\lambda\rho}=-\bar{g}^{\rho\mu}\delta g_{\mu\nu} A_1^{\nu\lambda}-\frac{1}{2}\bar{g}^{\rho\mu}\bar{g}^{\nu\lambda}\delta^2 g_{\mu\nu}.$$$$ In this way I easily solved my problem and found the correct expression for the contravariant metric up to second order.