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Einstein field equations seem very nonlinear of second order derivative. Expressing LHS of Einstein field equations in purely metric tensor and derivatives, it consists somewhat of second-order derivatives of metric tensor. RHS of course contains stress-energy tensor.

This seems to mean that when calculating metric tensor at some spacetime, it depends on future metric tensor (at that spacetime point and surrounding points) as well.

But geodesic equation needs metric tensor to derive the least action path for matter fields/particles.

  1. The first question: So when calculating metric tensor, are Einstein field equations and geodesic equations coupled together for calculation?

  2. The second question: Suppose we wish to compute metric tensor at some given spacetime. Only the form of stress-energy tensor is given. Do we need entire form of stress-energy tensor at every spacetime to compute metric tensor at some given spacetime, or can we localize stress-energy tensor and use only that information?

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EFE & Geodesic Equations

To answer the first question, the only information needed to pose and solve the geodesic equations is the metric tensor, $g_{\mu\nu}$. Thus, if $g_{\mu\nu}$ is unknown, one must solve,

$$R_{\mu\nu}-\frac12 g_{\mu\nu} R = 8\pi T_{\mu\nu}$$

and then solve the geodesic equations. I don't see how you could think they are coupled in some way and need to be solved together, since the solutions to the geodesic equation are geodesics which do not enter into the Einstein field equations.


Localised Stress-Energy Tensors

As for the second question, it is sufficient to know $T_{\mu\nu}$ say in some region $\Sigma \subset M$ of the space-time and one can solve the Einstein field equations to know the form of the metric in this region.

This is done all the time; one may consider some shell with a surface stress-energy, and consider the form of space-time within and outside the shell, two separate regions.

If we know the metric only in some region of space-time, due to the fact we only know the stress-energy for a particular region, then we cannot make any inferences about how it behaves outside. However, there are strong constraints on how the metric behaves at the boundary of two regions described by different stress-energy tensors.

Among those is the Israel junction condition, which relates the jump in extrinsic curvature on both sides to a stress-energy at the boundary between both regions.


Further Resources

For more information on the junction condition and metric tensors localised to a particular region of space-time, see Singular hypersurfaces and thin shells in general relativity.

For a pedagogical resource, see section 32 of Misner, Thorne and Wheeler's Gravitation on the collapse of stars as well as Brane-Localised Gravity by Mannheim.

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