Validity of theoretical arguments for Unruh and Hawking radiation

[Elias1960]

So the only way to stop a black hole from radiating in QFT on curved spacetimes is to have no black hole.

Yes. This is not questioned: QFT on a curved background predicts that a BH resulting from a collapse radiates.

But the domain of applicability of QFT on curved background is limited, and the limits can be seen in the theory itself. Namely, it is not a consistent theory, because it has no back-reaction of the quantum fields on the classical background. And the rough estimate for when we need more, full QG, is when we reach effects of order of Planck distance, Planck time or Planck energy. The trans-Plackian problem is that the semiclassical QFT derivation depends on semiclassical QFT remaining valid deep in the trans-Planckian domain. This assumption is nonsensical.

The question what one names Hawking raditation is irrelevant. There is a mass M, if it is of an actually collapsing star or a BH is irrelevant, because what we can measure outside is anyway the same. Hawking radiation is simply thermal radiation with a particular temperature depending on this mass M. If we see it coming from the direction of a BH candidate, and trace back the corresponding classical light ray, it ends (starts) from the collapsing surface before horizon creation, which comes from a trans-Planckian distance from the horizon, has gone through trans-Planckian time dilation (this is when a Planck time on the surface translates into more than the time after BB for the outside observer) an redshifted down from a trans-Planckian energy. This is what makes the whole thing trans-Planckian. But if it is really a BH or not yet observation cannot decide.

 

[QLogic]

There is a mass M, if it is of an actually collapsing star or a BH is irrelevant, because what we can measure outside is anyway the same. Hawking radiation is simply thermal radiation with a particular temperature depending on this mass M. If we see it coming from the direction of a BH candidate, and trace back the corresponding classical light ray, it ends (starts) from the collapsing surface before horizon creation

This isn't my understanding of Hawking radiation. Hawking radiation is usually defined as radiation that results from the state restricted to observables outside the horizon being a mixed state of KMS form.

If you have a star, i.e. no horizon, the mixed state has quite a different form that lacks Hawking radiation's completely thermal profile. Also Hawking radiation does not originate from the surface of the star since it's a fully kinematic effect in the post-horizon formation spacetime.

 

[Elias1960]

This isn't my understanding of Hawking radiation. Hawking radiation is usually defined as radiation that results from the state restricted to observables outside the horizon being a mixed state of KMS form.

Maybe some people consider it that way, but IMHO it makes no sense.

You need an initial state, the star before the collapse, with the usual ~ Minkowski vacuum. Then you have a collapse and the vacuum state changes together with the metric. The result differs from the vacuum at that time, the difference is the radiation. As long as the collapse continues, the change of the vacuum state continues, thus, new radiation appears.

If you have a star, i.e. no horizon, the mixed state has quite a different form that lacks Hawking radiation's completely thermal profile. Also Hawking radiation does not originate from the surface of the star since it's a fully kinematic effect in the post-horizon formation spacetime.

In Hawking's derivation, the Hawking modes are those modes which go through the star during the collapse and, because of the change which happens while they are inside, appear different than they have started, thus, differ from the vacuum modes. To name this a "fully kinematic effect in the post-horizon formation spacetime" is at best misleading (but I would guess simply nonsense). You need the element of change, without it there is no Hawking radiation. Which is what the paper

Paranjape, A., Padmanabhan, T. (2009). Radiation from collapsing shells, semiclassical backreaction and black hole formation, Phys.Rev.D 80:044011, arxiv:0906.1768v2

proves for a Hawking-like situation, but which is a quite general principle which follow from the way Hawking radiation is derived - it is caused by the difference between the time-evolved in-vacuum and the out-vacuum. But without change, the vacuum is stable and therefore there is no difference.

 

[QLogic]

Maybe some people consider it that way, but IMHO it makes no sense.

"fully kinematic effect in the post-horizon formation spacetime" is at best misleading (but I would guess simply nonsense)

I've given two papers, one from an expert on QFT in curved spacetime where the fact that it is a kinematic effect is not only mentioned but described as "well known". I can give several other sources that say this. One of the papers even explains exactly how it is a kinematic effect. What is actually wrong with what Visser is saying?

As long as the collapse continues, the change of the vacuum state continues, thus, new radiation appears.

It's not thermal until a horizon has formed. Prior to the formation of the horizon the vacuum has a statistical profile that differs from the Minkowski profile but it's not yet thermal so it usually isn't called radiation.

proves for a Hawking-like situation, but which is a quite general principle which follow from the way Hawking radiation is derived - it is caused by the difference between the time-evolved in-vacuum and the out-vacuum. But without change, the vacuum is stable and therefore there is no difference

Of course. The radiation is still a property of the state itself though, not of the dynamics. Hawking radiation is a property of the statistics inherent in the vacuum. You need a horizon for states to have those statistics and the horizon itself can only be created by the dynamics, but the radiation is not produced by the collapse dynamics that form the horizon. It's simply a property of the state.

I will say as a side note that I am the one using standard terminology here. It's difficult to have a discussion if you don't stick to standard terminology and in standard terminology Hawking radiation is kinematic. It's a bit strange to have to argue for textbook terminology.

 

[Elias1960]

I've given two papers, one from an expert on QFT in curved spacetime where the fact that it is a kinematic effect is not only mentioned but described as "well known". I can give several other sources that say this. One of the papers even explains exactly how it is a kinematic effect. What is actually wrong with what Visser is saying?

That it ignores the trans-Planckian problem.

It's not thermal until a horizon has formed. Prior to the formation of the horizon the vacuum has a statistical profile that differs from the Minkowski profile but it's not yet thermal so it usually isn't called [Hawking?] radiation.

If you use Schwarzschild time, the horizon never forms. So, the radiation will be never thermal. In fact, the difference is completely negligible already in the Planckian region.

The radiation is still a property of the state itself though, not of the dynamics. Hawking radiation is a property of the statistics inherent in the vacuum. You need a horizon for states to have those statistics ...

A radiation which stops immediately after the dynamics stops and therefore the state remains unchanged, and you need a horizon to have it even if in Schwarzschild coordinates you have no horizon at any finite time. Sorry, I continue to consider such claims as misleading at best.

I will say as a side note that I am the one using standard terminology here. It's difficult to have a discussion if you don't stick to standard terminology and in standard terminology Hawking radiation is kinematic. It's a bit strange to have to argue for textbook terminology.

I have no doubt that this is textbook terminology. There is no doubt that I'm questioning the mainstream approach, given that (better as far as) it ignores the trans-Planckian problem. Any textbook which introduces Hawking radiation but does not discuss the trans-Planckian problem is misleading at best. In democratic physics, you can easily find the 51% votes to prove me wrong.

 

[QLogic]

That it ignores the trans-Planckian problem.

That it's a statistical property of a state restricted to observables outside the horizon has nothing to do with the Trans-Planckian problem.

Let me put it this way. Hawking radiation is due to detection events outside the horizon having a thermal profile. The radiation doesn't emanate from the black hole. It's not driven by the dynamics. It requires a horizon but it is not dynamical. It's just a correlation property of the vacuum not driven by dynamics.

A radiation which stops immediately after the dynamics stops

It doesn't stop. Prior to the hole's formation there is no Hawking radiation. After the horizon forms there is. The radiation emitted during collapse discussed in Padmanabhan's paper is not Hawking radiation.

If the collapse stops it means there never is Hawking radiation.

 

[PeterDonis]

That it's a statistical property of a state restricted to observables outside the horizon has nothing to do with the Trans-Planckian problem.

Mathematically, it doesn't, no. But physics is not mathematics.

I have read the Visser paper you linked to; basically, by "kinematic property" he means "you can derive it from the math of generic classical Lorentzian spacetimes with horizons, without having to make any use of the particular dynamical law that determines the spacetime geometry". That's math, not physics. Math can't emit actual radiation that actual detectors detect. You would need to make a physical argument that the math in Visser's paper is physically relevant for the case under discussion. So far you have not made any such argument in this thread, nor does Visser in his paper.

 

[Elias1960]

That it's a statistical property of a state restricted to observables outside the horizon ...
It doesn't stop. Prior to the hole's formation there is no Hawking radiation. After the horizon forms there is.

This would be a contradiction with relativistic causality or relativistic symmetry.

Once it is a property detectable by observables outside the horizon, it has to be detectable before horizon formation, because it has to be detectable in Schwarzschild time too (assuming the effect does not depend on coordinates). But Schwarzschild time covers only the part before horizon formation, and cannot be causally influenced by anything happening after horizon formation according to Einstein causality. It would follow that there cannot be any Hawking radiation in the whole region of spacetime covered by Schwarzschild coordinates. So, in the form you present it here, it is self-contradictory, and makes no sense.

 

[PeterDonis]

Schwarzschild time covers only the part before horizon formation

This is only true for an "eternal" black hole that never evaporates. If the hole evaporates, it is no longer true that "horizon formation" occurs at $t = + \infty$ in Schwarzschild coordinates. In fact, the usual definition of Schwarzschild coordinate time doesn't even work in such a spacetime.

I think a better way of phrasing your underlying (valid) point would be that @QLogic needs to be a lot more specific about exactly what he means by "before" and "after" horizon formation. Most ways of specifying what those terms mean are highly coordinate-dependent. There are ways of doing it in an invariant manner, but it's not easy, and it's not altogether clear whether such an invariant definition has all of the properties he is implicitly assuming.

 

[PAllen]

A further point is simply that is false that Schwarzschild time only covers events before horizon formation, even in a classical non evaporating collapse. Instead, there is a clear demarcation of exterior events into which horizon formation is the causal future, versus not the causal future, thus possibly present. For the whole region in which horizon formation is not causal future, it is plausible to consider the horizon as currently existing.

 

[Elias1960]

This is only true for an "eternal" black hole that never evaporates. If the hole evaporates, it is no longer true that "horizon formation" occurs at $t = + \infty$ in Schwarzschild coordinates.
In fact, the usual definition of Schwarzschild coordinate time doesn't even work in such a spacetime.

What happens during the evaporation process is already part of a different theory, it is not covered by standard semiclassical QFT, because one has, somehow, to incorporate backreaction. How to do this without specifying preferred coordinates is completely unclear.

I think a better way of phrasing your underlying (valid) point would be that @QLogic needs to be a lot more specific about exactly what he means by "before" and "after" horizon formation. Most ways of specifying what those terms mean are highly coordinate-dependent. There are ways of doing it in an invariant manner, but it's not easy, and it's not altogether clear whether such an invariant definition has all of the properties he is implicitly assuming.

Ok, one can formulate it this way. But I think my remark remains correct, given that I specify the time coordinate I use, so that I can use "before" and "after" horizon formation in their usual meaning as referring to the particual system of coordinates used. Those who don't specify the coordinates should live with the fact that their statements can be rejected as false if applied to some systems of coordinates they are false.

A further point is simply that is false that Schwarzschild time only covers events before horizon formation, even in a classical non evaporating collapse. Instead, there is a clear demarcation of exterior events into which horizon formation is the causal future, versus not the causal future, thus possibly present. For the whole region in which horizon formation is not causal future, it is plausible to consider the horizon as currently existing.

One can, indeed, use the possibility of a system of coordinates with a time so that the horizon already exists as a reasonable way to give "before horizon formation" and "after horizon formation" a coordinate-independent meaning.
But this does not make my claim about the Schwarzschild time coordinate false. .

 

[PAllen]

...

One can, indeed, use the possibility of a system of coordinates with a time so that the horizon already exists as a reasonable way to give "before horizon formation" and "after horizon formation" a coordinate-independent meaning.
But this does not make my claim about the Schwarzschild time coordinate false..

But then who cares about Schwarzschild time coordinate? Physics s about coordinate independent statements.

 

[Elias1960]

But then who cares about Schwarzschild time coordinate? Physics s about coordinate independent statements.

Then one would better avoid talking about "before/after horizon creation" too.

It is, moreover, far from sure that the theory we need to describe back-effects will not have preferred coordinates.

 

[PAllen]

Then one would better avoid talking about "before/after horizon creation" too.

It is, moreover, far from sure that the theory we need to describe back-effects will not have preferred coordinates.

The statement that the horizon is not in the causal future of some exterior event is a coordinate independent statement.

 

[PeterDonis]

I can use "before" and "after" horizon formation in their usual meaning as referring to the particual system of coordinates used.

No, you can't, because standard Schwarzschild coordinates don't cover the horizon, and don't cover the event of horizon formation. So talking about "before" and "after" horizon formation is meaningless in Schwarzschild coordinates.

 

[Elias1960]

No, you can't, because standard Schwarzschild coordinates don't cover the horizon, and don't cover the event of horizon formation. So talking about "before" and "after" horizon formation is meaningless in Schwarzschild coordinates.

Given that we know that the complete solution has such a domain, and we also now that the part not covered is in the causal future of events covered, it makes clearly sense. To claim that it is meaningless is, in my opinion, artificial.

The statement that the horizon is not in the causal future of some exterior event is a coordinate independent statement.

But translating the everyday "before/after" using such a "not in the causal future" seems quite artificial. I think to use "before/after" as in classical physics, assuming some particular time coordinate which should be identified in the context.

 

[PeterDonis]

the part not covered is in the causal future of events covered

This is not correct. The entire region not covered by Schwarzschild exterior coordinates is not in the causal future of the entire region that is covered by Schwarzschild exterior coordinates.

What is true is that the event of horizon formation is not in the causal past of any event in the region covered by Schwarzschild exterior coordinates.

 

[PeterDonis]

I think to use "before/after" as in classical physics, assuming some particular time coordinate which should be identified in the context.

Then you need to pick a time coordinate that actually distinguishes "before" and "after" the event of horizon formation. The Schwarzschild time coordinate does not.

 

[Elias1960]

This is not correct. The entire region not covered by Schwarzschild exterior coordinates is not in the causal future of the entire region that is covered by Schwarzschild exterior coordinates.

??
The part covered contains the star before the collapse. Even if you restrict Schwarzschild time (without necessity) to the part outside the collapsing star, the causal future of this part covers also the whole inner part of the star.

Then you need to pick a time coordinate that actually distinguishes "before" and "after" the event of horizon formation. The Schwarzschild time coordinate does not.

No. Once I know the event is causally after the ones covered by the Schwarzschild time coordinate, I can use the relations "before" and "after" too. This is simply the region where I can assign the Schwarzschild time being $t=\infty$.

I don't understand the point of these remarks. One can easily extend exterior Schwarzschild time by some continuation inside the collapsing star if this seems, for whatever reason, necessary. If I would have to define them precisely, I would use, instead, harmonic coordinates with initial values as defined already by Fock for insular systems. They differ from Schwarzschild coordinates, but qualitatively it gives the same picture.

The relevant physical content is that the consideration of the part covered by these coordinates is sufficient to explain completely all what becomes, whenever, visible to the external observer. As long as the theory used to describe Hawking radiation (even together with backreaction and evaporation) does not depend on preferred coordinates and follows Einstein causality, nothing changes this, thus, nothing can force us to use other coordinates than these extended Schwarzschild coordinates to describe everything visible for the outside observer.

 

[PeterDonis]

Even if you restrict Schwarzschild time (without necessity) to the part outside the collapsing star, the causal future of this part covers also the whole inner part of the star.

No, this is not correct. It is true that there is a portion of the exterior region (outside the horizon) that has the entire interior region (inside the horizon) in its causal future. But this portion is very, very far from being the entire exterior region. And even for the portion of the exterior region that does have the entire interior region in its causal future, the Schwarzschild time coordinate still does not cover that interior region: it goes to infinity as the horizon is approached. An observer in the exterior region must adopt some other time coordinate if he wants to have one that will cover events at and inside the horizon.

 

[PeterDonis]

This is simply the region where I can assign the Schwarzschild time being $t=\infty$.

No, it isn't. If you take the limit $t \rightarrow \infty$ in the Schwarzschild exterior coordinate chart, you get the horizon. You don't get any events inside the horizon. To cover events inside the horizon, you have to switch
charts.

One can easily extend exterior Schwarzschild time by some continuation inside the collapsing star if this seems, for whatever reason, necessary.

We're not talking about inside the collapsing star. We're talking about inside the horizon. Big difference. We all understand that there is a portion of the spacetime region occupied by the collapsing star that is outside the horizon, and that portion can be covered by exterior Schwarzschild coordinates. But the region inside the horizon (which includes both a portion occupied by the collapsing star and a vacuum portion) cannot.

 

[PeterDonis]

The relevant physical content is that the consideration of the part covered by these coordinates is sufficient to explain completely all what becomes, whenever, visible to the external observer.

"The region which happens to be covered by these coordinates" is not the same as "these coordinates". As @PAllen has already pointed out, physics is contained in coordinate-independent invariants, and all the claims that need to be made, including the one of yours quoted above, can be made completely independent of any choice of coordinates.

 

[Elias1960]

"The region which happens to be covered by these coordinates" is not the same as "these coordinates". As @PAllen has already pointed out, physics is contained in coordinate-independent invariants, and all the claims that need to be made, including the one of yours quoted above, can be made completely independent of any choice of coordinates.

You can, but you are not obliged to. If the theory is covariant, you can restrict yourself to one particular choice of coordinates, do all the computations in these coordinates, and present the conclusions in these coordinates too.

No, it isn't. If you take the limit $t \rightarrow \infty$ in the Schwarzschild exterior coordinate chart, you get the horizon. You don't get any events inside the horizon. To cover events inside the horizon, you have to switch charts.

But if all what I want to do is to define (identify correctly) the before-after relationship between pairs of events where at least one is covered by the chart, it is not necessary to switch. For every event A inside the horizon there exists for every value of the time coordinate t an event $A_t \to A$ which is in the chart, has there the time coordinate t, and is in the causal past of A, so that it makes sense to say that $A_t$ is before A independent of any coordinates. Then, for event B inside the chart, with time coordinate $t_B$, we can say, already using the notion of "before/after" defined by the time of the chart, that B happens before $A_{t_B+\varepsilon}$ simply because $t_B < t_B+\varepsilon$. Once t is time-like, both orders are never in conflict with each other. So we can also combine them, and say that B happens before A, given that B happens before $A_{t_B+\varepsilon}$ and $A_{t_B+\varepsilon}$ happens before A.

We're not talking about inside the collapsing star. We're talking about inside the horizon. Big difference. We all understand that there is a portion of the spacetime region occupied by the collapsing star that is outside the horizon, and that portion can be covered by exterior Schwarzschild coordinates. But the region inside the horizon (which includes both a portion occupied by the collapsing star and a vacuum portion) cannot.

Full agreement. If you agree that the region inside the star which is outside the horizon can be covered by some extension of the Schwarzschild time too, and are ready to tolerate that a reference to Schwarzschild time is simply sloppy language for the time coordinate of a system of coordinates on $\mathbb{R}^4$ which covers the inner part of the star outside the horizon as well, and uses Schwarzschild time outside the star, fine.

the causal future of this part covers also the whole inner part of the star.

No, this is not correct. It is true that there is a portion of the exterior region (outside the horizon) that has the entire interior region (inside the horizon) in its causal future. But this portion is very, very far from being the entire exterior region. And even for the portion of the exterior region that does have the entire interior region in its causal future, the Schwarzschild time coordinate still does not cover that interior region: it goes to infinity as the horizon is approached. An observer in the exterior region must adopt some other time coordinate if he wants to have one that will cover events at and inside the horizon.

Of course, except for the "not correct". What would be the causal future of a connected set? The natural definition is that an event outside the set itself is in its causal future if it is in the causal future of at least one event of the set. What else? The other natural definition would be that it has to be in the causal future of them all. But then myself tomorrow would be not in the causal future of the actual (as defined by myself via Einstein synchronization) Milky way, so this is hardly plausible.

 

[PeterDonis]

If the theory is covariant, you can restrict yourself to one particular choice of coordinates, do all the computations in these coordinates, and present the conclusions in these coordinates too.

You can compute invariants in any coordinate chart you want, yes. But your conclusions still need to be stated in terms of invariants, not coordinate-dependent quantities.

if all what I want to do is to define (identify correctly) the before-after relationship between pairs of events where at least one is covered by the chart, it is not necessary to switch.

Having just one of the events covered by the chart is not enough. Both need to be.

I don't understand what you are doing in the rest of this paragraph with trying to define "before" and "after", but in any case, for any event $A$ in the exterior region that is not in the past light cone of the event of horizon formation, there are events inside the horizon that are spacelike separated from $A$ and for which no invariant time ordering relative to $A$ can therefore be defined. This is a coordinate-independent statement.

are ready to tolerate that a reference to Schwarzschild time is simply sloppy language for the time coordinate of a system of coordinates on $\mathbb{R}^4$ which covers the inner part of the star outside the horizon as well

I'm not ready to tolerate this unless you can show me such a coordinate chart and demonstrate that it covers the entire spacetime.

The natural definition is that an event outside the set itself is in its causal future if it is in the causal future of at least one event of the set. What else?

That an event outside the set is in its causal future if it is in the causal future of all the events of the set. See below.

The other natural definition would be that it has to be in the causal future of them all. But then myself tomorrow would be not in the causal future of the actual (as defined by myself via Einstein synchronization) Milky way, so this is hardly plausible.

It's perfectly plausible, because the Milky Way is much, much bigger than one light-day in size. But the Earth isn't, it's much, much smaller than one light-day in size, so you tomorrow will be in the causal future of the actual (Einstein synchronized in the inertial frame in which the center of the Earth is at rest) Earth today. You tomorrow will even be in the causal future of a much larger region than that, a region roughly one light-day in size centered on the Earth, which gets you well out into the Oort cloud.

To have you in the future be in the causal future of the entire Milky Way today, you would have to look at you 100,000 years or so in the future. That's how relativistic causality works. Only a very tiny portion of the Milky Way today can possibly causally affect you tomorrow, so you do not want to say that you tomorrow is in the causal future of the entire Milky Way today--only of that tiny portion that can causally affect you tomorrow.

 

[Elias1960]

Having just one of the events covered by the chart is not enough. Both need to be.
I don't understand what you are doing in the rest of this paragraph with trying to define "before" and "after",

I have shown that this claim is wrong, by extending the notion of "before" and "after" to particular situations where only one of the events is covered by the chart. Means, in the way I have used them, "before" and "after" can be used in a well-defined and meaningful way, which is compatible with the common sense notions of "before" and "after".

If you don't like these notions, ok. I have, as far as I have used them, used in a well-defined way, which is all I can do.

but in any case, for any event $A$ in the exterior region that is not in the past light cone of the event of horizon formation, there are events inside the horizon that are spacelike separated from $A$ and for which no invariant time ordering relative to $A$ can therefore be defined. This is a coordinate-independent statement.

Yes, it is. The point being? Coordinate-dependent statements remain meaningful statements once the coordinates are defined. In the usual coordinate-dependent notion, where "A happens before B" means $t(A) < t(B)$, it may be as well that they are space-like separated. The coordinate-independent statement which follows (but is weaker) is that A is not in the future light cone of B.

I'm not ready to tolerate this unless you can show me such a coordinate chart and demonstrate that it covers the entire spacetime.

Once the claim is that these coordinates cover only the part before horizon formation, not the whole spacetime, this is obviously impossible.

That an event outside the set is in its causal future if it is in the causal future of all the events of the set. See below.
Only a very tiny portion of the Milky Way today can possibly causally affect you tomorrow, so you do not want to say that you tomorrow is in the causal future of the entire Milky Way today--only of that tiny portion that can causally affect you tomorrow.

So, here it remains to say that we obviously completely disagree about what makes sense to define as notions of before and after some set of events. If I would like to refer to your notions, I would formulate them immediately in terms of light cones, any use of "before" and "after" would only, for obvious reasons, cause confusion. "Before" and "after" make sense in a coordinate-dependent way, partially, as I have shown, even outside the chart.

 

[PeterDonis]

I have shown that this claim is wrong, by extending the notion of "before" and "after" to particular situations where only one of the events is covered by the chart.

I don't see how you've done that. Your definition of "before" and "after" uses the time coordinate of the chart:

"A happens before B" means $t(A) < t(B)$

Obviously this is only well-defined if both A and B are covered by the chart.

 

[Elias1960]

I don't see how you've done that. Your definition of "before" and "after" uses the time coordinate of the chart:
Obviously this is only well-defined if both A and B are covered by the chart.

Reread this:

But if all what I want to do is to define (identify correctly) the before-after relationship between pairs of events where at least one is covered by the chart, it is not necessary to switch. For every event A inside the horizon there exists for every value of the time coordinate t an event $A_t \to A$ which is in the chart, has there the time coordinate t, and is in the causal past of A, so that it makes sense to say that $A_t$ is before A independent of any coordinates. Then, for event B inside the chart, with time coordinate $t_B$, we can say, already using the notion of "before/after" defined by the time of the chart, that B happens before $A_{t_B+\varepsilon}$ simply because $t_B < t_B+\varepsilon$. Once t is time-like, both orders are never in conflict with each other. So we can also combine them, and say that B happens before A, given that B happens before $A_{t_B+\varepsilon}$ and $A_{t_B+\varepsilon}$ happens before A.

So, we have two different partial order relations. One, $<_1$, is defined between pairs of events inside the chart, the other one, $<_2$, is the global one restricted to pairs of time-like separated events. Both are compatible with each other. We can combine them. This is a third partial order relation $<_3$ containing both of them. That means, if $A <_1 B$ or $A <_2 B$ then also $A <_3 B$, and $<_3$ is also transitive.

So, we have $B <_3 A_{t_B+\varepsilon}$ because $B <_1 A_{t_B+\varepsilon}$ and $A_{t_B+\varepsilon} <_3 A$ because $A_{t_B+\varepsilon} <_2 A$, and therefore $B <_3 A$ by transitivity of $<_3$.

 

[PeterDonis]

Reread this

By the method you describe, I can construct an argument for any event whatever outside the horizon being before any event whatever inside the horizon. So I don't see how your method is useful.

 

[PAllen]

...So, here it remains to say that we obviously completely disagree about what makes sense to define as notions of before and after some set of events. If I would like to refer to your notions, I would formulate them immediately in terms of light cones, any use of "before" and "after" would only, for obvious reasons, cause confusion. "Before" and "after" make sense in a coordinate-dependent way, partially, as I have shown, even outside the chart.

I have only ever thought of Peter's definition (of causal future of a set of events) as sensible, and I have never seen yours used anywhere. Just wondering if you have any reference defining your notion. Note that above, you have subtly changed 'causal future of a set of events', which was your prior claim, to simply 'before and after'.

Just to recap, here is what @Peter responded to:

"What would be the causal future of a connected set? The natural definition is that an event outside the set itself is in its causal future if it is in the causal future of at least one event of the set. What else? The other natural definition would be that it has to be in the causal future of them all. But then myself tomorrow would be not in the causal future of the actual (as defined by myself via Einstein synchronization) Milky way, so this is hardly plausible. "

Note also, I disagree with using the term "actual" to refer the result of convention, even a very useful convention. This is separate and apart from issues with ambiguity of Einstein synch in the presence of strong gravitational lensing - which certainly exists in the Milky Way.

 

[vis_insita]

I have only ever thought of Peter's definition (of causal future of a set of events) as sensible, and I have never seen yours used anywhere. Just wondering if you have any reference defining your notion.

Wald, General Relativity, defines $J^+(S)=\bigcup_{p\in S}J^+(p)$, which looks exactly like the definition @Elias1960 is using.

The natural definition is that an event outside the set itself is in its causal future if it is in the causal future of at least one event of the set.

I have never seen a different definition of $J^+(S)$ (MTW, and Hawking and Ellis seem to use the same) and as far as I understand the discussion (which is admittedly not very far) this seems to be the relevant one. So, I find the objections to this particular point a little surprising.

 

[Elias1960]

By the method you describe, I can construct an argument for any event whatever outside the horizon being before any event whatever inside the horizon. So I don't see how your method is useful.

Correct, and this is the aim. The argument is useful to reject any claims about something happening only after horizon formation. Such claims have been made, for example, here:

It's not thermal until a horizon has formed. Prior to the formation of the horizon the vacuum has a statistical profile that differs from the Minkowski profile but it's not yet thermal so it usually isn't called radiation.

Note that above, you have subtly changed 'causal future of a set of events', which was your prior claim, to simply 'before and after'.
Note also, I disagree with using the term "actual" to refer the result of convention, even a very useful convention.

Feel free to disagree with my use of of words. All I can/have to do is to explain what I mean using this words, and this I have done. I do not claim, or care, that my use is 100% established. This is, last but not least, a forum containing necessarily a lot of informal talk.

It is clear that if informal talk is used, one may ask what exactly that means, and the one who used the informal talk has, then, to specify what he means, already with details. This I have done. Beyond this, further discussion about such trivialities like the common sense compatible use of "before" and "after" or "actual" seems useless.

Contributions about the trans-Planckian problem would be more interesting.

 

[PeterDonis]

The argument is useful to reject any claims about something happening only after horizon formation.

But it can't reject such claims unless you can show that the claim you're trying to reject is using the same definition of "before" as you are.

Feel free to disagree with my use of of words.

I'm not disagreeing with your use of words in itself. I'm just saying that I don't see how your use of words contributes anything to the discussion. However, to be fair, I don't see how @QLogic's use of words was contributing anything either.

 

[PAllen]

Wald, General Relativity, defines $J^+(S)=\bigcup_{p\in S}J^+(p)$, which looks exactly like the definition @Elias1960 is using.
I have never seen a different definition of $J^+(S)$ (MTW, and Hawking and Ellis seem to use the same) and as far as I understand the discussion (which is admittedly not very far) this seems to be the relevant one. So, I find the objections to this particular point a little surprising.

Thanks. I can see how that is a useful mathematical definition for proofs. However, it seems to have little utility as a definition of physically meaningful or every day sense of 'before or after', especially for unbounded sets.

Consider Minkowski space as a whole. This definition, applied to all of Minkowski space as a set, says all of Minkowski space is in its own causal future and also in its own causal past. I think most people's common sense would be that the future of 'all there ever was or will be' is empty, similarly for the past, which follows from definition @PeterDonis proposed as the physically meaningful one.

Noting that this definition seems to have least meaning for unbounded sets, consider the following fact about a BH spacetime:

For any exterior causal diamond, however large, part of the horizon is in the causal future of the diamond, while part of it is not. [in a BH from collapse, some exterior causal diamonds will have all the horizon in the causal future, while others will not].

Also, of course, in spacetime with BH formed from collapse, for every exterior world line, there will be a first event on it for which the horizon is not all in the causal future. Following this, more and more of the horizon and interior are not in the causal future.

 

[Elias1960]

But it can't reject such claims unless you can show that the claim you're trying to reject is using the same definition of "before" as you are.

Given the context of an informal discussion in a forum, I can presuppose that a common sense compatible notion of "before" is used. I was unable to expect that such a discussion will be around this word (which I continue to consider as essentially unproblematic).

I'm not disagreeing with your use of words in itself. I'm just saying that I don't see how your use of words contributes anything to the discussion. However, to be fair, I don't see how @QLogic's use of words was contributing anything either.

This is something I can accept. To answer arguments of others, I try to use their language, as far as I'm able to interpret it. This may fail.

 

[Elias1960]

Consider Minkowski space as a whole. This definition, applied to all of Minkowski space as a set, says all of Minkowski space is in its own causal future and also in its own causal past. I think most people's common sense would be that the future of 'all there ever was or will be' is empty, similarly for the past, which follows from definition @PeterDonis proposed as the physically meaningful one.

I would consider this example as artificial. If one asks about the future of some set, the set is usually one much less than everything. Then, it may be quite natural to exclude the actual set itself from the future of this set. This can be done explicitly if one likes, replacing $J^+(S)=\bigcup_{p\in S}J^+(p)$ which obviously contains S by $\left(\bigcup_{p\in S}J^+(p)\right)\setminus S$. But to exclude beyond this regions where every trajectory reaching that region has to go through the set some time before from the future of the set is IMHO simply absurd.

Already the fact that for the intersection $\bigcap_{p\in S}J^+(p)$ the causal future of the whole set would be smaller than that of each of its points seems absurd.

About this imho absurd notion we read:

Also, of course, in spacetime with BH formed from collapse, for every exterior world line, there will be a first event on it for which the horizon is not all in the causal future. Following this, more and more of the horizon and interior are not in the causal future.

Fine. But what would be the point of considering this set? It has certainly no relation to Hawking radiation.