Pg 171 of "Tensors, Relativity and Cosmology"

The non-relativistic limit of the metric in a static gravitational field is defined as $$ds^2=\left(1+\frac{2 \phi}{c^2}\right)(dx^0)^2+g_{\alpha \beta}dx^\alpha dx^\beta \tag{1}$$ where $\alpha, \beta=1,2,3$

In the non-relativistic limit in the static gravitational field, with the approximate metric given by (1), the only non-trivial component of the Ricci Tensor is the one with k=n=0 $$R_{00}=\partial_0 \Gamma^j_{0j}-\partial_j\Gamma^j_{00}+\Gamma^p_{0j}\Gamma^j_{p0}-\Gamma^p_{00}\Gamma^j_{pj} \tag{2}$$

But why? I understand that in a static gravitational field the components of the metric tensor $(g_{kn})$ (k,n=1,2,3,4) are independent of the time coordinate (i.e. $\partial_0 g_{kn}=0)$ (and could this be related to the answer of my question?)

I tried to verify this with other components like $R_{\alpha \beta}$ (spatial) by expressing the Christoffel symbols in terms of the metric tensors but none of them seem to cancel each other out completely.

  • $\begingroup$ The other components probably are higher order in $\phi$ $\endgroup$ – Sounak Sinha Jun 23 at 19:33
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    $\begingroup$ "non-relativistic" as in the static limit or newtonian? In the latter case note that $T_{00}$ is the only nonzero component of the stress-energy tensor. $\endgroup$ – Quantumness Jun 23 at 19:35
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    $\begingroup$ @EXINT $T_{00}$ is the energy density whereas $T_{0i}$ and $T_{ij}$ are the momentum density and stress tensor, assumed negligible in the static and weak field limits. See en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor $\endgroup$ – Quantumness Jun 23 at 20:24
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    $\begingroup$ In the Newtonian limit maybe consider $g_{\alpha\beta}\approx\eta_{\alpha\beta}+h_{\alpha\beta}$? Where $O(h^2)\approx 0$ so you should in theory be able to cancel out a lot of nonlinearity from $\Gamma$'s and $R$'s $\endgroup$ – Jepsilon Jun 23 at 20:26
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    $\begingroup$ I'm a bit rusty in this but after some scribbling I think the Christoffel symbols with that form for $g_{\alpha\beta}$ reduce to $\Gamma^k_{ij}=-\frac12 \partial^k h_{ij}$ continuing off my previous comment $\endgroup$ – Jepsilon Jun 23 at 20:47

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