# Do black holes determine time's arrow?

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PeterDonis
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in the reverse process this energy go in into the white hole
No. Nothing can go into a white hole.

PeterDonis
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I am not aware of any actual exploding white hole solutions
I don't know that anyone has actually written a paper on it, but one obvious such solution would be the time reverse of the Oppenheimer-Snyder collapsing solution. Such a solution would be physically unrealistic since the white hole singularity would be in the past, not the future, and the white hole horizon would be a past horizon, and neither of those things have any physically reasonable way of coming into existence.

Dale
PeterDonis
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there really is no distinction between a black hole and a white hole unless you introduce a thermodynamic arrow of time.
It's a little more complicated than that.

In any system of time-symmetric laws, such as GR, there will be two types of solutions. One type is itself time-symmetric; the maximally extended Schwarzschild spacetime is an example of this type. In such a solution, one has to make an arbitrary choice of which time direction is the "future" direction. (Since the solution is globally hyperbolic, one only has to make one such choice, which holds for the entire spacetime.)

The other type of solution comes in pairs, each of which is the time reverse of the other. The Oppenheimer-Snyder collapsing solution, which describes a spherically symmetric star collapsing to a black hole, is one of such a pair; it will have a time reverse that describes a spherically symmetric white hole exploding and becoming a star. For this type of solution, "choosing a time direction" corresponds to choosing which solution of the pair you want to look at, rather than choosing which direction of time is the "future" in a time symmetric solution (as with the first type above).

PeroK
Dale
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I don't know that anyone has actually written a paper on it, but one obvious such solution would be the time reverse of the Oppenheimer-Snyder collapsing solution.
D’oh! Yes that is obvious (once someone points it out). There is no corresponding black hole in that one. Just the white hole that spews dust out of one part of the singularity.

Ibix
2020 Award
I don't know that anyone has actually written a paper on it, but one obvious such solution would be the time reverse of the Oppenheimer-Snyder collapsing solution. Such a solution would be physically unrealistic since the white hole singularity would be in the past, not the future, and the white hole horizon would be a past horizon, and neither of those things have any physically reasonable way of coming into existence.
I was thinking along these lines earlier (which is why @shlosmem thinks I disagree with @Dale - I don't, we're talking about different things), although considering a time reverse of an evaporating black hole. The white hole would come about from a very violent inrushing of blackbody radiation (time reversed Hawking radiation) and grows from a steady infall of blackbody radiation before starting to emit matter and radiation then finally un-collapsing into a star. You have a point about the singularity, though, which remains difficult to explain. Although we don't really expect the singularity to be a real thing, so perhaps this shouldn't worry us too much.

PeterDonis
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The white hole would come about from a very violent inrushing of blackbody radiation (time reversed Hawking radiation)
You can't create a white hole this way. The white hole's singularity (as you appear to realize) would have to magically already be there, as would its horizon.

To see this, look at the Penrose diagram for a black hole that is formed by gravitational collapse of a null shock wave and then emits Hawking radiation and evaporates. For example, see Fig. 10 of this paper:

https://arxiv.org/pdf/hep-th/9501071.pdf

If we time reverse this (flip the diagram over so ##i^+## is at the bottom and ##i^-## is at the top), we can see that the singularity and the region inside the event horizon are not "reachable" from the flat region at the bottom of the flipped diagram (which is the time reverse of the region at the top of Fig. 10 as given). So while you could add a "shock wave" that was the time reverse of the last pulse of Hawking radiation in the evaporating black hole (i.e., the time reverse of extending the "event horizon" line in Fig. 10 all the way up and to the right, to ##I^+##), that shock wave could not create the white hole singularity or the region inside the white hole horizon (since that horizon is now a past horizon, i.e., nothing can get inside it). So those things would have to magically already be there.

PeterDonis
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we don't really expect the singularity to be a real thing, so perhaps this shouldn't worry us too much
Sure it should, because models in which the singularity is replaced by something else, but still otherwise look just like the models we've considered so far at the classical (or semi-classical) level, simply don't exist. That is pretty much the upshot of the last few decades of research on black hole evaporation and the information paradox. In other words, the original view that, since the singularity is hidden deep inside the horizon, it doesn't really affect anything outside, is not really tenable any more; any model that replaces the singularity with something else will have at least some effect on the region outside the horizon (or it will not even have an event horizon at all, only an apparent horizon).

stevendaryl
Staff Emeritus
It's a little more complicated than that.

In any system of time-symmetric laws, such as GR, there will be two types of solutions. One type is itself time-symmetric; the maximally extended Schwarzschild spacetime is an example of this type. In such a solution, one has to make an arbitrary choice of which time direction is the "future" direction. (Since the solution is globally hyperbolic, one only has to make one such choice, which holds for the entire spacetime.)

The other type of solution comes in pairs, each of which is the time reverse of the other. The Oppenheimer-Snyder collapsing solution, which describes a spherically symmetric star collapsing to a black hole, is one of such a pair; it will have a time reverse that describes a spherically symmetric white hole exploding and becoming a star. For this type of solution, "choosing a time direction" corresponds to choosing which solution of the pair you want to look at, rather than choosing which direction of time is the "future" in a time symmetric solution (as with the first type above).
I’m not sure I understand the distinction you are making.

The issue is whether an arrow of time is an intrinsic attribute of the spacetime geometry or is additional structure imposed by us. Given a spacetime geometry corresponding to a star collapsing into a black hole, we can choose how we describe that geometry using coordinates. Flipping the sign of the time coordinate is just changing the description, not the geometry. So the white hole solution isn’t a different solution, it’s the same solution using different coordinates.

If you include an observer complex enough to have a thermodynamic arrow of time, then relative to such an observer, you could define a black hole to be a solution in which there is a future-directed timelike path reaching the central singularity. A white hole would be a solution where there is a past-directed timelike path reaching the singularity. But that distinction is relative to the is observer’s own arrow of time, rather than being something intrinsic in the geometry.

PeterDonis
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The issue is whether an arrow of time is an intrinsic attribute of the spacetime geometry or is additional structure imposed by us.
Yes. And for the first type of solution (time symmetric), the answer is that it's imposed by us, because we have to decide which direction of the solution to call the "future" direction. While for the second type of solution (comes in pairs, each the time reverse of the other), it's an intrinsic attribute of the spacetime geometry, because each individual spacetime geometry is not time symmetric; one "end" of time is different from the other.

Given a spacetime geometry corresponding to a star collapsing into a black hole, we can choose how we describe that geometry using coordinates. Flipping the sign of the time coordinate is just changing the description, not the geometry.
No. The two time-asymmetric solutions (star collapsing into black hole, white hole exploding into star) are not the same solution with the sign of the time coordinate flipped. They are a pair of distinct solutions, each of which is the time reverse of the other. The fact that you can take a description of one in a particular coordinate chart, flip the sign of the time coordinate, and have a description of the other, does not change the fact that they are two distinct solutions. Physically, the two distinct solutions correspond to two distinct physical scenarios.

If you include an observer complex enough to have a thermodynamic arrow of time, then relative to such an observer, you could define a black hole to be a solution in which there is a future-directed timelike path reaching the central singularity. A white hole would be a solution where there is a past-directed timelike path reaching the singularity. But that distinction is relative to the is observer’s own arrow of time, rather than being something intrinsic in the geometry.
No. The spacetime geometry in any time asymmetric solution has an intrinsic direction; as I have already said, such solutions always come in pairs (since the underlying laws in GR are time symmetric), each of which is the time reverse of the other. One solution has the singularity in the future; the other has it in the past. That is an intrinsic property of each spacetime geometry. The fact that we, as observers with a thermodynamic arrow of time, choose the solution that matches our actual physical experience in order to build models of our actual universe, i.e., in which the future direction that is intrinsic to the spacetime geometry is the same as the future direction of our thermodynamic arrow of time, does not mean the spacetime geometry does not have an intrinsic direction of time.

PeterDonis
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Given a spacetime geometry corresponding to a star collapsing into a black hole, we can choose how we describe that geometry using coordinates.
Note, btw, that while you can indeed choose any coordinates you like, that doesn't change invariants, and there are invariants, like the expansion scalar of the matter, that distinguish the two solutions (star collapsing into black hole vs. white hole exploding into star--the former has a negative expansion scalar, the latter has a positive expansion scalar).

stevendaryl
Staff Emeritus
Note, btw, that while you can indeed choose any coordinates you like, that doesn't change invariants, and there are invariants, like the expansion scalar of the matter, that distinguish the two solutions (star collapsing into black hole vs. white hole exploding into star--the former has a negative expansion scalar, the latter has a positive expansion scalar).
How is that defined?

stevendaryl
Staff Emeritus
Yes. And for the first type of solution (time symmetric), the answer is that it's imposed by us, because we have to decide which direction of the solution to call the "future" direction. While for the second type of solution (comes in pairs, each the time reverse of the other), it's an intrinsic attribute of the spacetime geometry, because each individual spacetime geometry is not time symmetric; one "end" of time is different from the other.

The asymmetry doesn’t imply that the two solutions are different. If one is the time reverse of the other, then they are the same manifold, since manifolds are equivalence classes.
No. The two time-asymmetric solutions (star collapsing into black hole, white hole exploding into star) are not the same solution with the sign of the time coordinate flipped. They are a pair of distinct solutions, each of which is the time reverse of the other.

I don’t understand the distinction you are making.

stevendaryl
Staff Emeritus
Let me explain why I am skeptical of what is being said about the distinction between black hole and white hole solutions.

Take a simple black hole solution: You have an otherwise empty universe, with a cloud of gas that collapses into a black hole.

Now, let's add an observer. An observer has an internal arrow of time determined by thermodynamics. This arrow of time is a matter of initial conditions. It's not actually possible for an object consisting of more than a handful of particles, but in principle, there exists the possibility of a time-reversed observer. You're not reversing time for the whole universe, just for the observer. Classically, we can understand this by just reversing the momenta of each particle (and maybe time-reversing internal electromagnetic fields, as well).

My claim is that what is interpreted as a black hole by one observer would be interpreted as a white hole by a time-reversed observer. Not reversing time for the universe as a whole, but just for the observer.

stevendaryl
Staff Emeritus
My claim is that what is interpreted as a black hole by one observer would be interpreted as a white hole by a time-reversed observer. Not reversing time for the universe as a whole, but just for the observer.

Thinking about this a little more, I realize that you have to reverse a lot more than just the observer. You also have to reverse all the photons that interact with the observer, for him to have the full time-reversed experience.

But my point is that giving the large-scale description of spacetime, that you have a star collapsing into a black hole, doesn't determine how observers would experience it, because to have an observer with functioning memories, you have to have an entropy difference between the two ends of time. Whichever end is low entropy would be experienced by observers as "the past". If there is a singularity in that direction, then it will be experienced as a white hole. If instead there is a singularity in the future, then it will be experienced as a black hole.

The gross structure of spacetime doesn't determine which direction is low entropy.

martinbn
The asymmetry doesn’t imply that the two solutions are different. If one is the time reverse of the other, then they are the same manifold, since manifolds are equivalence classes.

I don’t understand the distinction you are making.
I think this is a matter of terminology. For some space-time is not just the manifold with the matric, it also includes a time orientation. If you change it you get a different space-time.

martinbn
To add, in order to talk about black and white holes you need the notion of future and past null infinity. Time orientation is needed.

PeterDonis
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In https://en.m.wikipedia.org/wiki/Congruence_(general_relativity)

expansion is relative to a vector field ##X##.
Yes. In the cases under consideration, the vector field ##X## is the tangent vector field of the appropriate timelike congruence. For the star collapsing to the black hole and the white hole exploding into the star, the appropriate congruence is the congruence that is comoving with the matter.

So what is the basis for choosing ##X## versus ##-X##?
The physical difference between proper time for the worldlines in the congruence increasing towards the singularity (star collapsing to black hole) vs. away from the singularity (white hole exploding into star). These are different solutions describing physically different scenarios.

PeterDonis
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My claim is that what is interpreted as a black hole by one observer would be interpreted as a white hole by a time-reversed observer. Not reversing time for the universe as a whole, but just for the observer.
The universe doesn't contain just one observer. It contains lots of them, all of whom observe that they share a common future direction of time. In the models under consideration, this is captured by the proper time along the congruence of timelike worldlines that represents observers comoving with the matter, increasing in the same direction along all of the worldlines. I'm not sure how one could have a consistent model that didn't have that property, since it would break continuity of the tangent vector field of the congruence; there is no continuous way to go from one future direction of time to the other (the "time reversal" transformation is discrete, not continuous).

PeterDonis
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I'm not sure how one could have a consistent model that didn't have that property, since it would break continuity of the tangent vector field of the congruence; there is no continuous way to go from one future direction of time to the other (the "time reversal" transformation is discrete, not continuous).
I believe this issue is discussed in Hawking & Ellis, but I don't have my copy handy now to check. IIRC continuity of the tangent vector field is related to global hyperbolicity of the spacetime.

PeterDonis
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to have an observer with functioning memories, you have to have an entropy difference between the two ends of time.
The idealized GR models we are considering do not have to have observers with functioning memories; they just have to have worldlines with proper time that increases in a consistent direction, either towards the singularity (black hole) or away from it (white hole).

The gross structure of spacetime doesn't determine which direction is low entropy.
In the idealized GR models we are considering, the entropy is zero everywhere, since the microstate of the universe at every event is exactly known.

You can add entropy and observers with functioning memories to the model, but then we are talking about a more complicated model, where you have to include some more detailed description of the matter that allows entropy to increase in one direction or the other based on some kind of coarse-graining of the matter's state space. Then the question would be whether such a more complicated matter model can be consistently added to both of the idealized models (black hole and white hole) or only to one (presumably the black hole). I don't see why it wouldn't be possible for both (since we already expect that we can construct consistent models in which entropy increases when things fall into black holes, and since the white hole exploding into a star model, at least in its matter portion, is just like an expanding FRW universe, and we already expect that we can also construct consistent models in which entropy increases in the direction of time in which the universe is expanding). But that still doesn't make them the same model; they are still physically different.

PeterDonis
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If one is the time reverse of the other, then they are the same manifold, since manifolds are equivalence classes.
GR is a theory of physics, not math. The fact that the two solutions might belong to some mathematical equivalence class does not make them physically the same. Physically, they are different solutions, and that's the sense that matters for GR.

stevendaryl
Staff Emeritus
GR is a theory of physics, not math. The fact that the two solutions might belong to some mathematical equivalence class does not make them physically the same. Physically, they are different solutions, and that's the sense that matters for GR.

I'm claiming that physically, they are NOT different solutions. They become different solutions when you throw in observers who have a thermodynamic arrow of time.

stevendaryl
Staff Emeritus
The idealized GR models we are considering do not have to have observers with functioning memories; they just have to have worldlines with proper time that increases in a consistent direction, either towards the singularity (black hole) or away from it (white hole).

Proper time is ambiguous up to a sign. We choose the sign of the proper time so that the timelike path is future-pointing, but that assumes that you already have a notion of which direction is future. It's circular reasoning to say that proper time tells us which direction is the future.

stevendaryl
Staff Emeritus