Time reversal in General Relativity, and black holes

[georgir]

Plain old Newtonian mechanics is time-reversal invariant, i.e. if you view a recording of some events played backwards, it would still appear to be following the same physical laws (gravity attraction law in particular). This type of "time reversal" is exactly equivalent to just turning every object's velocity in the opposite direction.

I read that the same is true for General Relativity. But I can not imagine how black holes fit into all of this.

In a simple example, you have an object or photon outside a black hole, and it falls inside the black hole. Then turn all object velocities around - the object or photon just as easily is able to leave the "black hole"... Except now we call it a "white hole", and say that no objects can enter it... wtf?

How are black holes and white holes different? How does "time reversal" or "turning velocities around" convert one to the other? Why do we say that any concentration of matter (or energy) would create a black hole, why not a white hole? What are the conditions for the existence of a white hole then?

Shouldn't any "hole" be simultaneously "black" and "white"? That would be so much simpler to understand...

 

[clamtrox]

For sure, thinking about these things can be confusing. I think what gets you here is that you think time reversal just means changing particle directions -- that's not at all what happens in GR. Remember that in GR, time is a dimension in the manifold. In fact, even that is not completely correct. The correct statement is that there is a dimension, which is timelike, and this is the thing that we are interested in "reversing".

In black holes a strange thing happens at the event horizon. The coordinate measuring (roughly) the distance from the singularity is space-like beyond the horizon, and timelike inside it. Outside the event horizon you are able to move toward and away from the hole. Inside, you can only move towards the singularity, because that is the timelike direction. Imagine now that you do time reversal -- what happens?

 

[mfb]

A time-reversed black hole would be a white hole - a valid solution of Einstein's equations, but something never observed in the universe. Nothing can enter a white hole, but things can leave it.

 

[bcrowell]

One thing to realize here is that when we say GR is time-reversal invariant, that doesn't necessarily mean that for a particular spacetime there is necessarily a notion of globally time-reversing it. Generically, solutions to the Einstein field equations don't come equipped with a global coordinate that we can naturally identify as "the" time coordinate, and aren't even time-orientable.

WP has a brief discussion of the symmetries of the Schwarzschild solution: http://en.wikipedia.org/wiki/Schwarzschild_metric#Symmetries This discussion isn't particularly clear to me, since I'm not sure if they're talking about the minimal version of the solution or its maximal extension http://jila.colorado.edu/~ajsh/insidebh/penrose.html , which does include a black hole, a white hole, and two copies of Minkowski space. The maximal extension is obviously highly symmetric, but if time-reversing it means flipping the Penrose diagram upside-down, that doesn't seem to be consistent with what WP describes as taking "the time axis (trajectory of the star) to itself." The singularity is spacelike, not timelike.

 

[georgir]

I'm not sure I buy that "time is not well defined, so reversing it isn't".
If I can define a "now" (a cauchy surface?), however relative and subjective it may be in the grand scheme of things, all objects on this "now" have a specific velocity. I just want to flip that. What's not well defined about that?

I will be the first to admit, I have no clue about the maths of this all, so I do not even know if a "now" can be defined for both me outside a black hole and something on the inside of it at once. But intuitively, it seems to me that if it were not possible, then the inside of the black hole should not be something I care about at all. I would not consider it part of my universe, and would not expect to ever observe anything entering or interacting with it.

Are we sure that anything can really fall into a black hole? I mean, I know it can in its own reference frame. But what about viewed from the outside? Would it not seem to get more and more time-dilated as it approaches the horizon, so it never reaches it in a finite outside-observer time?

 

[bcrowell]

I will be the first to admit, I have no clue about the maths of this all, so I do not even know if a "now" can be defined for both me outside a black hole and something on the inside of it at once.

It can't.

But intuitively, it seems to me that if it were not possible, then the inside of the black hole should not be something I care about at all. I would not consider it part of my universe,

This is a perfectly valid point of view, in the sense that all that's externally observable is the mass, charge, and angular momentum.

and would not expect to ever observe anything entering or interacting with it.

But this is not viable. We observe stars orbiting Sagittarius A*, so they're interacting with it. We observe quasars, so matter is entering black holes (although there is no meaningful way to say whether it's in there "now").

Are we sure that anything can really fall into a black hole? I mean, I know it can in its own reference frame. But what about viewed from the outside? Would it not seem to get more and more time-dilated as it approaches the horizon, so it never reaches it in a finite outside-observer time?

We just had a thread about this: https://www.physicsforums.com/showthread.php?t=632543

I'm not sure I buy that "time is not well defined, so reversing it isn't".
If I can define a "now" (a cauchy surface?), however relative and subjective it may be in the grand scheme of things, all objects on this "now" have a specific velocity. I just want to flip that. What's not well defined about that?

It's not well defined because it implicitly assumes that we can fix a global frame of reference, which doesn't exist in GR. The notion of reversing velocity vectors (i.e., reversing the spacelike components of 4-velocities) implies that there are some velocity vectors whose spacelike parts are zero, so that they aren't changed by a flip. This amounts to choosing a frame of reference. To be able to do the flip globally, you'd have to have some sensible notion of a global frame of reference, but we don't have that.

You will have additional issues in cases where there are closed timelike curves, because initial conditions on a Cauchy surface are constrained.

None of this means that you can't define global time-reversal in many cases of interest.

 

[TobyC]

I've wondered about this before, and some of the answers here are really helpful. I think something that hasn't been mentioned that I've often thought must have some relevance is the fact that in the coordinates of an outside observer, falling objects never actually fall into a black hole. They get closer and closer to the horizon as time goes to infinity. So at any time an external observer could effectively flip their time coordinate by reversing the coordinate velocity of every particle they see, and what they subsequently observe wouldn't involve anything escaping the black hole.

Or at least this is what seems like should happen in the simple case of a particle falling into a perfect Schwarzschild black hole that has been around for all eternity, but real black holes were formed at a finite time in the past, and I think they're supposed to be able to accumulate new matter as well, and I have no idea how any of that works based on the simplistic knowledge I have of GR and black holes. I've asked a question on these forums before about how black holes actually accumulate new matter when stuff is supposed to get stuck at the horizon, I don't think it was really satisfactorily answered and I'd appreciate it if someone could answer it now.

EDIT: Just realized that I didn't skim read well enough and this has been touched upon, I'll leave the post anyway.