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The Schwarzschild metric represents a stationary (and static), spherically-symmetric, spacetime. These characteristics are manifested by the four Killing vector fields: one for time translation and three for rotational symmetries.

Inside the horizon, however, the time coordinate (and its associated Killing vector field) becomes space-like. The interior solution is no longer stationary: all worldlines head to an endpoint, the future singularity at $r=0$.

How should one think about symmetries of the spacetime in the interior of a Schwarzschild black hole? It seems to me that the spherical symmetry remains (the other three Killing fields are unchanged by crossing the horizon). But now there is a (one-dimensional) spatial translation symmetry, in the $t$ direction. What direction is that? Or how should one think of it?

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  • $\begingroup$ [moved this postscript into a comment] Maybe my question is more general. In the exterior, the spatial coordinate, $r$, doesn't represent a single direction, but parameterizes the approach to the 2-sphere on the horizon. Perhaps interior $t$ has some similar meaning? But while $2M < r < \infty$, the interior coordinate time is still $-\infty < t < \infty$. Does the Killing field in Kruskal-Szekeres coordinates help with intuition? $\endgroup$
    – Ben H
    Commented Feb 20 at 3:36
  • $\begingroup$ Related: physics.stackexchange.com/questions/427415 - and: physics.stackexchange.com/questions/418183 $\endgroup$
    – safesphere
    Commented Feb 22 at 5:36
  • $\begingroup$ You are overthinking. On the outside, a black hole is a 3D sphere infinitely stretched along its worldline in the temporal 4th dimension. This shape is a 4D cylinder of the spherical type called spherinder. On the inside the shape is the same along the same coordinates, only the temporal vs. spatial nature of the coordinates is different. The 3D sphere remains, but now it is stretched in space, not in time, while its radius is now temporal. It is the same geometry, but physically instead of a spatial sphere along its timeline we now have a spatial spherinder surface rapidly shrinking in time. $\endgroup$
    – safesphere
    Commented Feb 22 at 5:49
  • $\begingroup$ Also note that with spatial $t$, $r=0$ is no longer a spatial point evolving in time, but a spatial (spacelike) line that exists momentarily when time ends. See this question with an intuitive diagram and the accepted answer for details: math.stackexchange.com/questions/2929400 $\endgroup$
    – safesphere
    Commented Feb 22 at 5:55
  • $\begingroup$ Finally notice that the horizon (the surface of the shrinking spherinder) that is infinitely thin in the distant Schwarzschild coordinates, for a local observer appears thicker than the size of the universe due to the diverging gravitational length contraction. This makes the full geometry a bit harder to visualize. $\endgroup$
    – safesphere
    Commented Feb 22 at 6:13

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