ONLY non-relativistic limit of general relativity From my study of GR I learnt that to reach the “Newtonian” limit of the Einstein field equation we have to assume:

*

*weak field $g_{\mu\nu} = \eta_{\mu\nu} + \epsilon h _{\mu\nu}$ with $\epsilon << 1$

*non-relativistic limit $c \to \infty$
My question is: what does it happen if we only perform the non-relativistic limit (2)? Which kind of equation comes out?
 A: 
… what does it happen if we only perform the non-relativistic limit … ?

You would get non-relativistic gravity theory, but one that would allow description of effects associated with strong gravity regime. It can be formulated using Newton–Cartan geometry. Effects different from purely Newtonian theory (which also could be formulated using Newton–Cartan geometry but with additional constraints imposed) are:

*

*torsionful connection (there is a specific variant, twistless torsion leading to the usual galilean causality but with position-dependent time dilation);


*purely spatial (Ricci) curvature;


*spatially inhomogeneous Coriolis fields, which allow descriptions of such effects as rotational frame-dragging.
References

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*Van den Bleeken, D. (2017). Torsional Newton–Cartan gravity from the large c expansion of general relativity. Classical and Quantum Gravity, 34(18), 185004, doi:10.1088/1361-6382/aa83d4, arXiv:1703.03459.


*Hartong, J., Obers, N. A., & Oling, G. (2022). Review on Non-Relativistic Gravity. arXiv:2212.11309.
