Validity of theoretical arguments for Unruh and 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 ??!