Is the space-time outside an eternal isolated Schwarzschild black hole time invertable in general relativity? I think for an eternal black hole, one coordinate system describes all of space-time as a moving sidewalk moving towards the singularity in a roughly Newtonian way with space moving faster than light beyond the event horizon. Since simultaneity is relative, maybe the region outside the event horizon can be given another coordinate system where the exact time inverse of space getting dragged in is occurring and the time inverse of the event horizon is called the antihorizon. Anything coming out of the black hole from the antihorizon in finite time in the second coordinate system will have always been outside the antihorizon in the first coordinate system.
Since the first coordinate system describes all of space, does that mean the region of space beyond the antihorizon doesn't exist at all so the second coordinate system doesn't describe the space beyond the event horizon which actually exists but does describe the space beyond the antihorizon which doesn't exist?
In other words, space has an edge in the second coordinate system. I think that according to the big bang theory, black holes can exist but white holes can't exist because they can never be created and when a black hole first forms, regions of space closer to its antihorizon in the second coordinate system start existing but after the black hole gets really old, space is so severely warped really close to the antihorizon in the second coordinate system that the region of space beyond it doesn't exist at all.
 A: The coordinate system you refer to in your opening sentence is the Gullstrand-Painlevé coordinates. This is often referred to as the river model by analogy to observers being swept along by a river.
But it is important to understand that a coordinate system is not a physical object. It just a way of labelling points in spacetime. Spacetime is not flowing inwards towards the black hole - it is just the coordinate system that is flowing inwards. For example we could use the Schwarzschild or Krusal-Szekeres coordinates to describe the same black hole and in neither of these is there any flow.
Given the above it is impossible to answer your question because you have confused the coordinates with the spacetime. You are free to make up any coordinates you choose to label points in the spacetime, and you may well choose a coordinate system that cannot label points in some regions of the spacetime, though this is unlikely to be useful. The failure of this coordinate system does not mean that the region of spacetime doesn't exist, but just that your (badly chosen) coordinate system cannot label points in those regions.
