A Validity of theoretical arguments for Unruh and Hawking radiation

[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.

 

[vis_insita]

Thanks. I can see how that is a useful mathematical definition for proofs.

Yes, my understanding is that some of these proofs are relevant to Hawking radiation/Black hole thermodynamics. That is why I assumed Wald's definition would be the most relevant one in this discussion too.

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.

It also follows from that physically meaningful definition that the future and past of everything happening "right now," as defined by an inertial observer in Minkowski space, is empty, because every event lies outside the causal future and past of some event at t=0. This observer is in the strange situation that although his next birthday comes after his previous birthday, he is unable to state any temporal relationship between his next birthday, and some events in the universe that he knows happened simultaneously to his last birthday. Even worse, literally nothing happens to that observer anymore after everything that happened at his last birthday.

As a common sense definition of chronological order this doesn't seem to be a very happy effort either.

Also, I think you are arguing against a much stronger claim than was made. No one suggested a definition of chronological order over arbitrary sets of spacetime based on the notion of causal future alone. It was argued that a before/after-relationship defined relative to one chart may be extended to events which are not covered by that chart. In that situation causal (or chronological) future works in agreement with common sense, while your alternative definition doesn't seem to work at all even in simpler cases.

 

[PAllen]

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.

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

I'll dispute your judgment of absurd in a later post, pointing out separate utility for the different notions.

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

Actually, that is exactly the point of the alternative concept.

About this imho absurd notion we read:

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

For the world line, I was not talking about any notion at all of future of a set, but the hopefully unambiguous notion of causal future of an event. This statement is true of all external world lines using just the notion of causal future of an event. I cannot cannot conceive of what you can claim as absurd about a statement of the evolution of causal future along a world line.

 

[PeterDonis]

It was argued that a before/after-relationship defined relative to one chart may be extended to events which are not covered by that chart. In that situation causal (or chronological) future works in agreement with common sense

I don't think it does, for the reasons I explained earlier. As I explained (and @Elias1960 agreed that this was an intended implication), by this definition the entire region inside the horizon is in the causal future of the entire region outside the horizon. That does not seem to be in agreement with common sense.

I think the underlying issue here is that the boundary of these two regions, the horizon, is a null surface. Our intuitions about "before" and "after" do not work well with null surfaces.

 

[PAllen]

What bugs me about calling the union of causal futures the causal future of a set is that it doesn't preserve the primary notion of causality: a point in this notion of future cannot necessarily be influenced by the set as a whole.

Where I see the union definition as useful, as hinted at by @vis_insita is for chronological ordering as opposed to causal ordering. Specifically, it is useful for defining for defining a valid foliation. For example, one may say:

A foliation of region of spacetime is a one parameter family of spacelike surfaces such that each is either in the future or past of every other, and the parameter is chosen consistent with this time ordering. One also requires that every point in the region is in some surface (it can be derived that it is in at most 1). This definition automatically precludes intersections.

To me, I would prefer to call this union definition something different from causal, even though defined in terms of causal relations for points. For example, chronological future of a set, while calling the intersection definition as causal future of a set. Henceforward, in this thread, I will call the union definition simply future, and the intersection definition cfuture (to make up my own term).

I argue that it is not very useful to talk about future of a set that includes timelike curves with unbounded future and past proper time. It makes most sense to me for spacelike surfaces.

As to extending the time ordering of a foliation of a region outside it without defining a more global foliation, I think it is important look at an additional property of a foliation of a region. This applies specifically to an open region like BH exterior. The issue is whether the union of closures of the foliation is equal to the closure of the region. To me, a foliation that doesn't meet this property is pathological for the purposes of extending its time ordering. In particular, for Schwarzschild foliation of exterior, the closure of any slice is the same 2-sphere of events, which would thus be labeled with all time coordinates from minus to plus infinity. This is the cause of property of extension by @Elias1960 definition that @PeterDonis objected to. In contrast, the following is true for a foliation that is not closure degenerate:

For any foliation of the Schwarzschild BH exterior such that the union of foliation closures is the same as the closure of the BH exterior:

a) every slice after a certain one has only part of the (future) horizon and interior in its future.
b) Using @Elias1960 algorithm for extending time ordering outside the region, some interior points are not in the future of some exterior events.

Only a pathological foliation has the property the @Elias1960 seems to think is an essential feature of the relation between interior and exterior.

[edit: just thought I would add one more point about the closure degeneracy concept I introduced. Another way of stating it is that a well behaved foliation of an open set that has a closure is one that such that the closure of the foliations forms a foliation of the closure of the open set. This is false for the Schwarzschild foliation of the BH exterior, but is true for all other commonly used foliations.]

 

[PAllen]

...

This observer is in the strange situation that although his next birthday comes after his previous birthday, he is unable to state any temporal relationship between his next birthday, and some events in the universe that he knows happened simultaneously to his last birthday. Even worse, literally nothing happens to that observer anymore after everything that happened at his last birthday.

As a common sense definition of chronological order this doesn't seem to be a very happy effort either.
...

In principle, you never know what happened simultaneous to your birthday. This is entirely a matter of convention beyond the requirement of spacelike separation.

@PeterDonis definition was never proposed as a definition of chronological ordering, which is coordinate dependent. Instead it was a definition of how to extend the notion of causal future to a set. The key concept of causal future of a point is that the point can influence any future point. To generalize to a set, it seems most meaningful to require the set as a whole can influence can influence any point it its causal future.

 

[PeterDonis]

Instead it was a definition of how to extend the notion of causal future to a set. The key concept of causal future of a point is that the point can influence any future point. To generalize to a set, it seems most meaningful to require the set as a whole can influence can influence any point it it’s causal future.

Since Wald and Hawking & Ellis were previously brought up, it seems appropriate to refer to their term for what is being described in the above quote, and what I was describing earlier. That term is Domain of Dependence. In Wald, Chapter 8, the future domain of dependence of an achronal set $S$, denoted $D^+(S)$, is defined as the set of all points $p$ in the spacetime such that every past inextendible causal curve through $p$ intersects $S$. (Note the "every".)

A key property of the set $S$ in the above is that it must be achronal, i.e., no two points in the set can be connected by a timelike curve. This restriction is not made in the earlier definition of the causal future of a set; however, if you think about it, it makes sense to restrict attention for practical purposes to the causal future of achronal sets, since if we have a set $S$ that is not achronal, we can always find some achronal set $S^\prime$ that has the same causal future as $S$ (heuristically, we just remove any points in $S$ that are in the causal future of other points in $S$, since they add nothing to the causal future of $S$ as a whole).

 

[PAllen]

Since Wald and Hawking & Ellis were previously brought up, it seems appropriate to refer to their term for what is being described in the above quote, and what I was describing earlier. That term is Domain of Dependence. In Wald, Chapter 8, the future domain of dependence of an achronal set $S$, denoted $D^+(S)$, is defined as the set of all points $p$ in the spacetime such that every past inextendible causal curve through $p$ intersects $S$. (Note the "every".)

I think this is actually a third concept. In SR terms, this is talking about the set of events whose past light cone is spanned by S. What you originally proposed (and I called cfuture) would be the set of events for which S is a subset (possibly improper) of the their past light cone (including interior).

A key property of the set $S$ in the above is that it must be achronal, i.e., no two points in the set can be connected by a timelike curve. This restriction is not made in the earlier definition of the causal future of a set; however, if you think about it, it makes sense to restrict attention for practical purposes to the causal future of achronal sets, since if we have a set $S$ that is not achronal, we can always find some achronal set $S^\prime$ that has the same causal future as $S$ (heuristically, we just remove any points in $S$ that are in the causal future of other points in $S$, since they add nothing to the causal future of $S$ as a whole).

This is interesting and shows that the statement "the whole (future) horizon and interior is in the future of the whole exterior" is an absurd triviality. Replacing the whole exterior with the unique equivalent achronal surface, you find this surface is past infinity. Thus the statement I just gave in quotes is really saying nothing more than "the whole (future) horizon and interior is in the future of past infinity". So what ??!