B Do black holes determine time's arrow?

[stevendaryl]

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]

How is that defined?

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

expansion is relative to a vector field $X$. So what is the basis for choosing $X$ versus $-X$?

 

[stevendaryl]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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.

Yes! That is a fact about thermodynamics. That's my point.

For whatever reason, the time immediately after the Big Bang was very low entropy (comparatively). The entropy sets the arrow of time, and defines the difference between a Big Bang and a Big Crunch. If the singularity is in the thermodynamic past, it's a Big Bang, and if it's in the thermodynamic future, it's a Big Crunch.

 

[stevendaryl]

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.

A tangent vector is ambiguous up to a sign! If you have a parametrized path $\mathcal{P}(s)$, it will have a certain tangent vector. If you parametrize it differently, say using $\lambda = -s$, then the resulting tangent vector will point in the opposite direction.

It's circular to use tangent vectors to specify the arrow of time, because the tangent vectors of timelike paths are chosen to be future-pointing, in the first place.

 

[PeterDonis]

I'm claiming that physically, they are NOT different solutions.

And this claim is wrong, because the direction of proper time of observers comoving with the matter, relative to the singularity (toward vs. away) is part of the physical interpretation of the solution.

Proper time is ambiguous up to a sign.

Only for a time symmetric solution. The direction of proper time for a time asymmetric solution is part of the geometric properties of the solution. Reversing that direction means a physically different solution. Again, GR is a theory of physics, not math; proper time increasing towards the singularity is physically different from proper time increasing away from the singularity. I am frankly baffled as to how you can fail to accept this.

Yes! That is a fact about thermodynamics.

No, it isn't, it's a geometric fact about the solution: there is a congruence of timelike worldlines comoving with the matter, and that congruence has a tangent vector field. That tangent vector field's future direction, relative to the singularity, is different for the two solutions (toward for black hole, away for white hole). That is true for the idealized solutions we are discussing, which, as I have already pointed out, have zero entropy everywhere and do not have the thermodynamic properties you refer to.

For whatever reason, the time immediately after the Big Bang was very low entropy

In the idealized solutions we are considering, the entropy is zero everywhere, as I have already pointed out: the microstate of the matter is exactly known everywhere.

If you want to talk about a universe with a low entropy Big Bang and entropy increasing away from the Big Bang, you are talking about a different model, in which there are two arrows of time: the entropy arrow you describe, and the geometric proper time arrow for the congruence of comoving worldlines that I described; and for the solution to be physically valid for our actual universe, those two arrows have to match. But that does not mean that in an idealized solution where there is no thermodynamic arrow, the proper time arrow doesn't exist.

 

[PeterDonis]

A tangent vector is ambiguous up to a sign!

No, it isn't. The two possible signs are two physically different solutions. That is true even in the absence of a thermodynamic arrow, which, as I have repeatedly pointed out now, is not present at all in the idealized solutions we have been discussing.

 

[Dale]

But the point is that, absent the 2nd law of thermo, it is impossible to experimentally determine if you are going forward in time in a collapsing black hole or backwards in time in an exploding white hole. That is the arbitrariness.

 

[stevendaryl]

No, it isn't. The two possible signs are two physically different solutions.

I don’t think what you’re saying is true. The parameter used for parametrizing a path is a choice for how the path is described. It’s not physically meaningful.

 

[stevendaryl]

And this claim is wrong, because the direction of proper time of observers comoving with the matter, relative to the singularity (toward vs. away) is part of the physical interpretation of the solution.

I don’t think what you’re saying is true. There is no physical meaning to the choice of one sign over the other, unless the direction is chosen for a physical reason, such as pointing in the direction of the thermodynamic arrow of time.

 

[PeterDonis]

absent the 2nd law of thermo, it is impossible to experimentally determine if you are going forward in time in a collapsing black hole or backwards in time in an exploding white hole.

Sure it is; if it's a black hole, you can fall into it and hit the singularity, and you can't have come out of it because none of it is in your causal past. If it's a white hole, you came out of it, from the singularity, and it's in your causal past, and you can't fall back into it. These situations are distinguishable experimentally, although in the black hole case you won't be able to communicate your results back to any journals. And the experiments don't require any knowledge of thermodynamic properties; there is no need to even measure any entropy or any other thermodynamic parameter.

The parameter used for parametrizing a path is a choice for how the path is described. It’s not physically meaningful.

Yes, it is. Every GR textbook I've read gives the curve parameter along a timelike curve the physical interpretation I have described. If you dispute it, please give me a reference that says otherwise.

There is no physical meaning to the choice of one sign over the other, unless the direction is chosen for a physical reason

Putting the singularity in your past vs. your future is a physical reason, and can't possibly require any correspondence with thermodynamics in and of itself, since, as I have pointed out several times now, there is no thermodynamics in the idealized models we have been discussing, yet there is still the physical difference of which way the curve parameter increases relative to the singularity.

As I said above, this is a basic matter of physical interpretation of GR, and what I am saying is in accord with what I have seen in every GR textbook that I have read. I have never seen anything in any GR literature that says you can freely reverse the sign of the curve parameter along a timelike curve without changing the physical interpretation; all the GR literature I have read says the opposite.

 

[stevendaryl]

I’m perfectly fine with a convention that a specification of a solution to GR includes an assignment of an arrow of time to each timelike path. However, I think that such an assignment has no physical meaning unless it aligns with the thermodynamic arrow of time.

We can perform measurements to determine whether the universe is expanding from a Big Bang or contracting to a Big Crunch, relative to the thermodynamic arrow of time. But there is no measurement that can distinguish between the two if we allow for the thermodynamic arrow of time to be inconsistent with the cosmological arrow of time. There is no measurement that can determine whether proper time is increasing or decreasing along a path except by comparing it to a process that has a preferred arrow of time by thermodynamics.

What is the empirical evidence that would distinguish between (1) our universe is in a Big Bang cosmology and our thermodynamic arrow is pointing in the direction of the expansion? (2) our universe is in a Big Crunch cosmology and our thermodynamic arrow of time is pointing in the opposite direction of the contraction?

I claim that there isn’t any empirical distinction, so we may as well call both Big Bang cosmologies.

 

[stevendaryl]

Sure it is; if it's a black hole, you can fall into it and hit the singularity, and you can't have come out of it because none of it is in your causal past.
If it's a white hole, you came out of it, from the singularity, and it's in your causal past, and you can't fall back into it. These situations are distinguishable experimentally,

No, they are not! Not without a thermodynamic arrow of time. Thermodynamics is what distinguishes between “causal past” and “causal future”. The laws of physics are symmetrical under time reversal (or at least electromagnetism and gravity are). The fact that we remember the past and not the future is a fact about thermodynamics. If you somehow arranged the universe so that it had lowest entropy in the far future, we would remember the future, rather than the past.

 

[Dale]

If it's a white hole, you came out of it, from the singularity, and it's in your causal past, and you can't fall back into it.

Not if you are going backwards in time. If it is an exploding white hole and you are going backwards in time then it is indistinguishable from a collapsing black hole with you going forwards in time (absent thermodynamics).

It is thermodynamics that allows you to experimentally distinguish the two. If the ice cubes in my drink melt then I know I am going forward in time to a black hole, and if the ice cubes in my drink are forming then I know I am going backwards in time to a white hole.

 

[stevendaryl]

Let me illustrate the problem with empirically determining whether you or rising or falling without assuming a thermodynamic arrow of time.

Suppose you are near a planet with very weak gravity. You want to figure out if you are rising or falling. Here’s a toy model for doing this. Suppose that someone constructs a tall tower, maybe 400 meters tall. On the outside of the tower, there are “sticky notes”, at one-meter intervals, there is a pad of sticky notes on which is written the distance above the ground.

So someone who is in slow freefall next to the tower can measure his progress by once per minute taking one of the nearest sticky notes and pasting it into his notebook, filling the pages in order.

If he is falling, then the numbers in his notebook will be decreasing—maybe 100, 81, 64, 49, …, 1

If he is rising, then the numbers will be increasing—1, 4, 9, 16, …, 100.

But now the trickster god of thermodynamics messes with the initial conditions. The scientist starts at a height of 100 meters, the notebook already filled with the numbers 1, 4, 9, …, 100. And all the processes inside the scientist’s body ard reversed, so that instead of taking a sticky note from the tower and placing it into his notebook, he takes the last sticky note from his notebook and places it onto the tower.

Then even though the scientist is falling, the numbers in his notebook will be in increasing order. So the ordering of the numbers give the exact wrong impression.

As long as all the fundamental laws of physics are reversible, there will always be a way (in principle) to set up initial conditions so that the question of whether the scientist is rising or falling will give the wrong answer.

You can only do normal empirical reasoning except under the assumption that your arrow of time isn’t messed with.

 

[PeterDonis]

If the ice cubes in my drink melt then I know I am going forward in time to a black hole, and if the ice cubes in my drink are forming then I know I am going backwards in time to a white hole.

Here you are actually agreeing with me: you are implicitly making use of two distinct concepts of "direction of time". One direction is the direction in which the ice cubes melt; the other direction is the direction in which your proper time increases. Since your scenario says these are not the same, you agree with what I have been saying all along.

You could observe that both of the concepts you use, ice cubes melting and proper time increasing, are local; but so is the direction in which the singularity is, since if the singularity is in your past you can see it, and if it is in your future you can't, and that is a local test. That particular local test just happens to reflect a global property of the spacetime you're in, whereas the ice cube melting test and the proper time increasing test don't.

 

[PeterDonis]

If he is falling, then the numbers in his notebook will be decreasing—maybe 100, 81, 64, 49, …, 1

If he is rising, then the numbers will be increasing—1, 4, 9, 16, …, 100.

In other words, you are taking the ordering of the pages, a spatial ordering, to reflect the ordering in time; you have set up a physical process that records a time ordering in a spatial ordering. So once the experiment is done, we can look at the ordering of the pages and what sticky notes are on them to determine what happened. Ok so far.

But now the trickster god of thermodynamics messes with the initial conditions. The scientist starts at a height of 100 meters, the notebook already filled with the numbers 1, 4, 9, …, 100. And all the processes inside the scientist’s body ard reversed, so that instead of taking a sticky note from the tower and placing it into his notebook, he takes the last sticky note from his notebook and places it onto the tower.

Then even though the scientist is falling, the numbers in his notebook will be in increasing order.

No, his notebook will be empty at the end of the experiment, so it will convey no information whatever.

Also, when you say "all the processes" inside the scientist are reversed, is that supposed to include his consciousness as well? Is he supposed to experience this experiment backwards somehow? (And what would that even mean?) Or is he still experiencing it forwards, but experiencing himself following a different set of rules (put the notes onto the tower instead of taking them off)?

 

[PeterDonis]

Thermodynamics is what distinguishes between “causal past” and “causal future”.

Not according to every GR textbook I have ever read. Again, if you disagree, please give me a reference that supports your claim.

I don't see the point of continuing to just assert our respective positions. I submit that if you look in any GR textbook, you will find it using the terms "causal past" and "causal future" as geometric properties, without any hint of thermodynamics whatever. The same goes for "proper time" and the other terms I have used in previous posts. I take what I have described to be the standard physical interpretation of GR used by physicists. If you insist on adopting your own idiosyncratic interpretation, I can't stop you, but I'm not going to agree with you either without at least some reference to go by.

 

[PeterDonis]

Thermodynamics is what distinguishes between “causal past” and “causal future”.

Btw, if you were to assert this claim as an empirical claim about our actual universe, rather than as a theoretical claim about properties of models in GR, I would have no problem with it. Or if you were to say that, yes, you can have idealized solutions in GR where there is no meaningful thermodynamics at all (for the reasons I have already described), but there is still proper time along timelike curves and there is still a physical distinction being made between corresponding pairs of solutions like "star collapsing to black hole" and "white hole exploding into star", even though that distinction, in those idealized models, has no other theoretical basis and is just stated as a rule of physical interpretation without any deeper justification (in the context of those models)--but then you were to say that you don't consider those idealized models to be physically realistic, because there is no thermodynamic arrow of time, I would have no problem with that either.

But you are making the much stronger claim that the rule of physical interpretation I have described is simply invalid, and that is what I disagree with. And, as I have said, every GR textbook I have ever read says that such a rule of physical interpretation is valid. So that's what I'm going with unless I see a reference that says otherwise.