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from the geodesic equation for non-relativistic case where $$v_i\ll c$$

$$\frac{dx^i}{dt}\ll1,{\rm for }\ c =1$$

$$\frac{dx^i}{d\tau}\ll\frac{dt}{d\tau}$$using this the geodesic equation for proper time $\tau$ becomes $$\frac{d^2x^\mu}{d\tau^2}+\tau{^\mu}_{00}(\frac{dt}{d\tau})^2=0.$$ if $g_{ij}\ne f(t)$ then $$\tau^{\mu}_{00}=-g^{\mu s}\frac{d g_{00}}{2d x^s}.$$Now if we assume that the curved part of the metric is a perturbation to the flat part

$$g_{ij}=\eta_{ij}({\rm flat})+h_{ij}({\rm perturbation})$$

Can anyone please help in how to calculate $g^{ij}$ if off diagonal terms are non-zero (for general case)?

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  • $\begingroup$ What do you exactly mean by "calculating" $g^{ij}$? Are you asking how to obtain the raised-index $g^{ij}$ given you have the lowered-index $g_{ij}$? Or, are you asking how to obtain either of them in the first place? $\endgroup$ – Dvij Mankad Feb 23 at 18:13
  • $\begingroup$ He is askin how to obtain either of them, i think. $\endgroup$ – Žarko Tomičić Feb 23 at 18:32
  • $\begingroup$ See my answer here physics.stackexchange.com/a/330277/133418 $\endgroup$ – Avantgarde Feb 23 at 21:38

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