I was reading Weinberg's book on Gravitation. In chapter 13 he introduces the metric (eq. (13.3.4)):

$$g_{\mu\nu}=C_{\mu\nu}+\frac{K}{1-K C_{\rho\sigma}x^{\rho}x^{\sigma}}C_{\mu\lambda}x^{\lambda}C_{\nu\kappa}x^{\kappa}, \tag{13.3.4}$$

where $C_{\mu\nu}$ are constants and $K$ is the curvature. So, for flat space $K=0$. Following that, he mentions that, straightforwardly, one can calculate the components of the Riemann tensor. To do that you need to find first the inverse metric. Do I have to suppose that $C_{\mu\nu}C^{\nu\lambda}=\delta^{\lambda}_{\mu}?$

Is it that easy to guess the inverse metric? If so what is the inverse metric in that case?

  • 1
    $\begingroup$ Hint: Try a somewhat similar ansatz (13.3.4) for the inverse metric (rather than trying to use Cramer's formula directly). $\endgroup$ – Qmechanic Mar 26 '18 at 5:43

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