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The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: what about the actual manifold? Are we assuming a certain type of manifold and then looking for a metric on it, or are we looking for the whole (pseudo-)Riemannian manifold? Because the known solutions all seem to have similar manifolds from a topological POV, but why wouldn't there be solutions with crazy topology? On the other side we do change the manifold for some solutions, for example the Schwarzschild solution removes the center, making it not simply connected anymore right?

So what exactly do we looking for when solving the equations? Maybe manifolds with certain characteristics (for example 3d etc)? I'm terribly confused and would like any clarification on this regards

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    $\begingroup$ Einstein's equations do not say anything about the topology of the manifold. That is an extra input. Solutions can certainly have crazy topologies if you want. Whether those are relevant for physics is a different question. $\endgroup$
    – Prahar
    Commented May 30 at 11:57
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    $\begingroup$ See Is it possible to have a spacetime described by a piecewise metric. The POV from PDEs is that you should think of Einstein’s equations as an initial-value problem (whose local well-posedness was established by Yvonne Choquet Bruhat, and later globalized with Geroch); this gives us maximal globally-hyperbolic developments. Just arbitrarily writing down Lorentzian metrics on a certain manifold solving $G_{ab}=8\pi T_{ab}$ isn’t really that instructive/physical (just as we don’t want “all” solutions to Poisson’s/Maxwell’s equations). $\endgroup$
    – peek-a-boo
    Commented May 31 at 4:33

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There are basically two general ways of solving the Einstein field equations (EFE). The EFE gives the relationship between the distribution of matter and the geometry of the spacetime. Specifically the stress energy tensor of the matter and the Einstein tensor of the spacetime (plus the cosmological constant if needed).

So the two general approaches for solving the EFE are either to specify the distribution of matter and then solve for the resulting spacetime or to specify the spacetime and then solve for the required distribution of matter.

There are several manifolds with unusual topology that are studied. They are generally investigated using the second approach. A spacetime geometry is specified, including any strange topology, and then the distribution of matter required is determined using the EFE. Probably the most common are the “useful” manifolds like wormholes or warp bubbles but this approach also includes topological spaces like toruses (tori?). Most of these spacetimes require matter distributions that violate one or more of the typical energy conditions. So they are not considered realistic, but they are studied.

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