B Do black holes determine time's arrow?

[stevendaryl]

Note that the only possible meaning for "eternal black hole" here is actually the maximally extended Kruskal-Szekeres geometry; no other "black hole" is eternal. And, as I noted in post #118 just now, in that geometry, there are two "hole" regions, not one: there is a black hole region and a white hole region, and they are not the same. So while it is possible to have a fully time symmetric trajectory for a free-falling particle in this geometry, any such trajectory will start on the white hole singularity, emerge from the white hole horizon, rise to some maximum altitude in the exterior region, fall back inside the black hole horizon, and end on the black hole singularity. It will not fall back into the same region of spacetime from which it emerged.

Well, it’s always impossible to fall back to the same region of spacetime that you came from. Time will be different if nothing else.

 

[stevendaryl]

This is impossible inside an actual black hole.

If we take the Schwarzschild geometry as the background geometry, and consider test particles moving in that background, then there are solutions of the equations of motion where the particle is “falling” as a function of proper time and solutions where the particle is “rising” as a function of proper time. In the region with r less than the Schwarzschild radius, the sign of dr/ds can never change.

 

[PeterDonis]

If we take the Schwarzschild geometry as the background geometry, and consider test particles moving in that background, then there are solutions of the equations of motion where the particle is “falling” as a function of proper time and solutions where the particle is “rising” as a function of proper time.

Yes. The former solutions are inside the black hole region, and the latter solutions are inside the white hole region.

In the region with r less than the Schwarzschild radius, the sign of dr/ds can never change.

That's correct, but there are two such regions, one for each possible sign of dr/ds.

 

[PeterDonis]

it’s always impossible to fall back to the same region of spacetime that you came from.

By "region of spacetime" I meant one of the four regions in the maximally extended Schwarzschild geometry: the "right" exterior, the black hole, the "left" exterior, and the white hole.

 

[Dale]

The former solutions are inside the black hole region, and the latter solutions are inside the white hole region.

And in the region outside the horizons. Both types of solutions exist there.