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The Schwarzschild metric is relatively easy to visualize even though the metric is singular at r=2M and r=0. Once we make the Kruskal extension, we find whole new regions in the manifold including regions we call White Hole and if we take a t=constant slice we get an Einstein-Rosen Bridge too.

Mathematically, this is how I think about the Kruskal extension. (Please correct me if I am wrong) To get a solution from the Einstein's equations we can think about a metric which has some particular symmetry. If we have a spherically symmetric system, then we argue the form for the metric (the coordinates in which that symmetry is most obvious are called the Schwarzschild coordinates.) and resubstitute it in the Einstein Field equations to find the form of the components of the metric.

Now, we found that if we change the coordinate system from Schwarzschild to Kruskal we get new regions and the reason for that is that the Schwarzschild metric didn't cover the whole manifold and because Kruskal is the maximal extension we know the Kruskal coordinates cover the whole manifold. But, on the other hand, if we had somehow started with the form of the metric which looks like the Schwarzschild metric in Kruskal coordinates, we could have got an analytic solution in a coordinate system which covers the whole manifold in one go.

But, is there a way to gain physical intuition over how the whole blackhole (i.e in Kruskal coordinates) looks in reality? I know I am being vague here but, I am not able to explain it better. If we take a constant time slice we find a geometry which we call the Einstein-Rosen bridge. So, does a Schwarzschild Black Hole which forms in our universe really have that geometry.

Or, is it that once we find an analytical solution to Einstein's Equations and find the maximal extension we have to rule out certain region through physical intuition?

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In GR (general relativity) the Schwarzschild metric describes a spherically symmetric vacuum solution of the Einstein's equations. The Schwarzschild solution presents spherical coordinates $(t, r, \theta, \phi)$ which offer also an intuitive picture of the geometry, however they are applicable only in the exterior region of a black hole, that is outside the event horizon.
Yet the coordinate system is just an artifact of the human mind to represent the physical reality and if a coordinate system proves inadequate we can look for another one. If we change from Schwarzschild to Eddington-Finkelstein coordinates we can cross the horizon and proceed until the physical singularity.
If we then change to Kruskal coordinates we get the so-called maximally extended Schwarzschild solution which shows four regions: I) the exterior region of the black hole, II) the interior region (infinite future), III) the white hole (infinite past) and IV) the other region, which can be thought connected to the exterior region via an Einstein-Rosen bridge.
As for I know, astrophysical measurements confirm regions I and II (indirectly). As for regions III and even more IV there is no evidence till now. However I do not think we can exclude certain solutions as long as they do not clash with experimental evidences or provide physical inconsistencies.

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