On the nature of the infinite fall toward the EH

[Dale]

I got drawn into this topic of black holes because of what appears to be an issue about physical interpretation of Schwarzschild's coordinate system

That is easy. There is NO physical interpretation of ANY coordinate system (incl. SC and all of the other coordinate systems that we have discussed); what has physical interpretations are the invariants.

The purpose of any coordinate system is simply to make calculations possible or even easy. In some coordinate systems the calculation of specific invariants becomes particularly easy, but even then it is the physical invariants which are easily calculated from the coordinates which have a physical interpretation, not the coordinates themselves.

That sounds reasonable to me, and then no informed person would object if someone else says that a crossing of a black hole horizon never happened; if I'm not mistaken that's simply according to our standard time convention on Earth (perhaps the ECI frame is based on Schwarzschild).

Yes, as long as it is clear that "never" is a coordinate-dependent statement meaning "not at any finite SC coordinate time". Unfortunately, that is rarely clear.

However, it remains paradoxical to me, in the sense that in my experience such things always led to contradictions. Different from SR's relativity of simultaneity it seems to imply a possible disagreement if events will occur or not

This is why I recommend that you read the Carroll's lecture notes. It seems to me that you don't yet understand the basic relationship between coordinate charts and manifolds. Essentially, you seem to not get the fact that a coordinate chart need not cover the entire manifold, and indeed, some simple manifolds are impossible to cover in a single chart (see p. 38). Whether or not a specific chart covers a given point has nothing to do with whether or not that point is in the manifold. The numerous examples of Rindler and Zeno coordinates in flat spacetime should make that abundantly clear.

And in the context of this topic, there may be an issue of consistent physical interpretation: there has been talk of an "internal Schwarzschild solution" according to which an object crosses the horizon, and which is combined with it as a single solution. Is that correct? So far I don't understand how such a combined solution can give a consistent physical interpretation of events.

Please read the Carroll notes again. You cannot have understood that chapter and not understand this concept.

 

[Dale]

This implies that whether or not an object falls across the EH according to a distant observer is simply a matter of convention.

for example, according to Dalespam (post #150) it is a matter of convention if an object falls across the EH while according to you objects can cross the horizon.

I feel like my name is being taken in vain

Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through. So the two statements are not contradictory.

Btw, I had hoped from the context that it was clear that I was talking about whether or not it happened in a finite amount of coordinate time for the different conventions. I do apologize for any lack of clarity. This highlights the importance of the math over the English.

Again, the difficulty lies in the translation of the English word "never" into math. If by "never" you mean "not in any finite SC time" then it is true that it never crosses. If by "never" you mean "not at any event in the manifold" then it is false that it never crosses.

I think you misinterpreted what Dalespam said. I think he was talking about how things look from the point of view of the distant observer. If the distant observer uses one coordinate system, then the crossing takes an infinite amount of time, and if he uses a different coordinate system, the crossing takes a finite amount of time. "Takes an infinite amount of time" doesn't mean "Never happens". Rindler coordinates clearly show that these are not the same. I don't know why you accept the conclusion for Rindler coordinates (or for the Zeno coordinates), but reject it for Schwarzschild coordinates.

Yes, a better way to say it.

 

[harrylin]

I think you misinterpreted what Dalespam said. I think he was talking about how things look from the point of view of the distant observer. If the distant observer uses one coordinate system, then the crossing takes an infinite amount of time, and if he uses a different coordinate system, the crossing takes a finite amount of time. [...]

Sorry for the ambiguity, but that is indeed how I understood it - this whole discussion 'On the nature of the "infinite" fall toward the EH' is obviously meant from the distant Schwarzschild observer's point of view.
[addendum

[..] Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through. So the two statements are not contradictory.

I understood him to mean that it will fall through according to us - he did not write that, but that is what he seemed to be arguing for several threads. If that is not the case, then some of my discussion with him was based on miscommunication.

Btw, I had hoped from the context that it was clear that I was talking about whether or not it happened in a finite amount of coordinate time for the different conventions. I do apologize for any lack of clarity. This highlights the importance of the math over the English.

That was, as I stated earlier, very clear (at least to me!) ]

Again, the difficulty lies in the translation of the English word "never" into math. [..]

Yes, exactly - as I also remarked.

That is easy. There is NO physical interpretation of ANY coordinate system (incl. SC and all of the other coordinate systems that we have discussed); what has physical interpretations are the invariants.

The purpose of any coordinate system is simply to make calculations possible or even easy. In some coordinate systems the calculation of specific invariants becomes particularly easy, but even then it is the physical invariants which are easily calculated from the coordinates which have a physical interpretation, not the coordinates themselves.

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²) and such findings are what Einstein called a "physical interpretations". Similarly the Schwarzschild simulation to which I linked could be called a physical interpretation.

Yes, as long as it is clear that "never" is a coordinate-dependent statement meaning "not at any finite SC coordinate time". Unfortunately, that is rarely clear.

I think that most people who start questions do understand that; but only a poll would tell us.

This is why I recommend that you read the Carroll's lecture notes. It seems to me that you don't yet understand the basic relationship between coordinate charts and manifolds. Essentially, you seem to not get the fact that a coordinate chart need not cover the entire manifold, and indeed, some simple manifolds are impossible to cover in a single chart (see p. 38). [..]

I did read and understand that, which doesn't mean that I agree with his way of thinking*; do you mean that the two parts indeed don't provide a consistent physical description?

*there turn out to be different interpretations of GR just as there are of SR: think of block universe discussions

 

[PAllen]

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²) and such findings are what Einstein called a "physical interpretations". Similarly the Schwarzschild simulation to which I linked could be called a physical interpretation.

Actually, gravitational time dilation is a coordinate dependent convenience for calculating physical observations:

1) It is possible to define at all only for sufficiently static spacetimes such that you can introduce coordinates where the metric components are essentially unchanging with time (thus depend only on positions). This is a very special class of spacetimes. Even for these, just like any other time dilation (strictly coordinate dependent), you simply have relation between a particular class of clock (clock on static world line) and coordinate time.

2) It is never necessary to use gravitational time dilation to compute any observation. To compute clock comparison for two clocks that synchronize, separate, and come together, you just integrate proper time along their paths. This is the universal way to compute this physical observation. It works for non-static gravitational situations (e.g. the vicinity of inspiralling neutron stars, where gravitational time dilation is undefinable), and it works for coordinates for a static situation that don't manifest the static character (e.g. the extension of the local frame of free fall observer).

3) To compute redshift between an emitter and a receiver, the universal method (works for every case in SR, GR, however general) is to parallel transport the emitter 4-velocity along the null path to the receiver, and use pure SR formula for Doppler based on transported emitter velocity and null path propagation vector expressed in local frame of receiver.

So, yes, gravitation time dilation, like all forms of time dilation is a coordinate convention. The observations (2) and (3) above are independent of how different coordinate conventions differently manifest time dilation.

 

[PeterDonis]

Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through.

I agree with you on the physics, but I actually have to admit that I too find this terminology confusing. It seems to imply something that I know you didn't mean, that what is "real" is a matter of convention. IMO it would be better to say something like: the infalling object *does* fall through the horizon, but once it reaches the horizon it can't send light signals back out to the distant observer.

Then, if we wanted to talk about coordinates or simultaneity conventions, we could say that the distant observer's most natural simultaneity convention can't assign finite time coordinates to any event at or below the horizon. But even that is causing confusion; maybe it would be better to leave out coordinates altogether and insist on talking only about invariants, at least in discussions like this where the old Army axiom seems to be in full force: any statement which *can* be misinterpreted, *will* be misinterpreted.

This highlights the importance of the math over the English.

Definitely.

Again, the difficulty lies in the translation of the English word "never" into math.

Yes, and part of the problem is that many people do interpret "never" as being ontological, something that can't be a matter of convention, whereas as you put it here "never" *is* a matter of convention. The math, as you say, is unambiguous.

 

[PeterDonis]

I think that most people who start questions do understand that

I would strongly disagree; if people who start these questions understood that "never" as used in this context is coordinate-dependent, threads like this one would be over in two or three posts. You yourself give an example:

do you mean that the two parts indeed don't provide a consistent physical description?

Of course not; they do. If you understood that "never" is coordinate-dependent, you wouldn't even need to ask this question. The answer would be obvious.

 

[stevendaryl]

I understood him to mean that it will fall through according to us - he did not write that, but that is what he seemed to be arguing for several threads.

There is no "according to us". It's according to this or that coordinate system. As people have pointed out to you many times, a coordinate system only describes what happens in a patch, a region of spacetime. The fact that some event isn't included in a coordinate patch doesn't imply that the event never happens. I think you understand this well enough when it comes to Rindler coordinates or Zeno coordinates. But for some reason, you don't think it relevant when it comes to Schwarzschild coordinates. Why not?

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²) and such findings are what Einstein called a "physical interpretations".

But $t$ has NO direct physical meaning. It's chosen to make the metric look as simple as possible. If you use a different time coordinate, you would get a different "gravitational time dilation" formula relating proper time to coordinate time. In particular, the claim that "an infinite time is required for an infalling observer to reach the event horizon" is NOT a physically meaningful statement. It's true according to some coordinate systems, and false according to other coordinate systems.

 

[harrylin]

[..] Of course not; they do. If you understood that "never" is coordinate-dependent, you wouldn't even need to ask this question. The answer would be obvious.

As before, it seems that you misunderstood my question which doesn't relate to a time label but to what Einstein called physical interpretation, which I here called physical description, and which PAllen called "differently manifest".

 

[harrylin]

There is no "according to us". It's according to this or that coordinate system. [..]

We discussed that our common conventional time is that of the ECi frame, and that we are allowed to use it.

[..] It's true according to some coordinate systems, and false according to other coordinate systems.

Exactly, that is what we elaborated here.

 

[harrylin]

Combining the earlier mentioned opinions:

This is already a statement whose meaning is rejected by relativity. [..] It is absolutely possible for a distant observer to assign remote times in a consistent way such that they consider the object to have crossed the horizon in finite time. They can also consistently assign remote times so that never happens. It will never be possible to verify one assignment over another precisely because event horizon crossing will never be seen.

. [..] it doesn't make SC uniquely represent the viewpoint of a distant observer. This implies that whether or not an object falls across the EH according to a distant observer is simply a matter of convention.

[..] Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through. So the two statements are not contradictory. [..] I was talking about whether or not it happened in a finite amount of coordinate time for the different conventions [from the point of view of the distant observer.]

[..] IMO it would be better to say something like: the infalling object *does* fall through the horizon [..]

I take it that you mean, as implied by the topic of this thread, from the point of view of the distant observer. If so, then I understood you correctly, and Dalespam misunderstood you.

So, once more: doing what you do is making a metaphysical claim that pertains to what we cannot measure - it is your physical interpretation. Others hold that GR does not make such physical claims except for the predictions of events. If an event at (x,y,z,t) has taken place according to us (or if it will take place according to our reckoning) is a matter of convention. So, while I agree with your desire to want to establish a "physical reality", the situation looks, as I said earlier today, very similar to SR in which different interpretations of physical reality are possible - and threads that are not ended voluntarily by the participants are locked when the Mentors have enough of it (of course information exchange about such ideas can be helpful; but that has been done both there and here).
It appears that the best we can do is describe the interpretations/decriptions that relate to different coordinate systems. That is constructive and helpful; philosophy debates of the block universe kind are not.

 

[PeterDonis]

I take it that you mean, as implied by the topic of this thread, from the point of view of the distant observer.

No, I meant "in reality". You can't change reality by changing point of view. So since it seems like all this talk about different points of view is only increasing the confusion, I'm no longer talking about that. I'm talking about invariant properties of reality. It is an invariant property of the reality predicted by the Schwarzschild solution of the Einstein Field Equation that objects do reach and fall through the horizon.

So, once more: doing what you do is making a metaphysical claim that pertains to what we cannot measure - it is your physical interpretation.

No, it's a physical *prediction* made by GR, and more specifically by the Schwarzschild solution of the EFE. (By which I mean not "the solution Schwarzschild published" or "what Schwarzschild believed", or "what Einstein believed about what Schwarzschild wrote", but the best modern understanding of the solution of the EFE for a spherically symmetric vacuum spacetime.) If you're going to contest that prediction, you will be claiming that the prediction doesn't match actual reality; it's not a matter of "interpretation".

How might the prediction not match reality? I can see a couple of possible ways:

(1) The EFE is wrong; or at least, it's wrong sometimes (for example, it's wrong if it predicts an event horizon). This doesn't strike me as a promising approach, and as far as I know nobody has tried to argue for it.

(2) The Schwarzschild solution isn't applicable to actual collapsed objects, because when you get close enough to forming a horizon, either the collapse stops, or the stress-energy tensor changes, in such a way that the horizon never actually forms. In other words, no real spacetime is ever vacuum down to the horizon radius; it always ends up becoming non-vacuum at some larger radius than this. This could be because:

(2a) Some classical effect intervenes: for example, there is always enough pressure present, or enough mass gets radiated away, etc., so that a horizon never actually forms; instead the collapse stops at some other stable state. This doesn't seem promising either: all the stable states of matter that we know of other than black holes have maximum mass limits, and we know of black hole candidates whose observed masses are many orders of magnitude larger than those limits.

(2b) Some quantum effect intervenes: for example, in some thread or other we have discussed possible quantum effects that could change the stress-energy tensor close enough to a horizon, so that a horizon was prevented from forming. This hasn't been ruled out, but does not appear to be promising.

Now: where in any of the above did I talk about coordinates or "interpretations"? Nowhere. I just talked about physical predictions and what sorts of physical phenomena might affect how accurate they are.

(Note: I was only talking about the horizon above, not the singularity at r = 0; that's a different can of worms. I also wasn't talking about Hawking radiation, because whether a horizon eventually evaporates away is a different question from whether it forms in the first place. If a horizon forms, then there will be some possibility of objects falling through it.)

Others claim that GR does not make such claims except for the predictions of events.

The statement that objects fall below the horizon *is* a "prediction of events".

And if an event at (x,y,z,t) has taken place according to us, is a matter of convention.

See my comments to DaleSpam on the use of the word "convention".

 

[harrylin]

No, I meant "in reality". [..]

Yes, exactly - as I specified next. I correctly understood that you were making claims about physical reality according to a distant observer. Such discussions are similar to those started by bob2c and Vandam about SR. And I clarified why I stopped participating in such discussions.

 

[Dale]

I understood him to mean that it will fall through according to us

What do you mean by "according to us"?

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²)

No, the easy solution is not too easy, it is exactly right. What you have just demonstrated is that time dilation is a coordinate effect. Specifically dτ is coordinate independent, but dt is coordinate dependent.

I think that most people who start questions do understand that; but only a poll would tell us.

I did read and understand that, which doesn't mean that I agree with his way of thinking*;

I don't think they do, and despite your claims here I don't think that you understand either. For instance, you refer to an interpretation of GR, but that section of the lecture notes isn't even about GR. It is simply about the math of Riemannian manifolds, tensors, and coordinate charts.

Could you identify what specifically you disagree with or are having trouble with?

do you mean that the two parts indeed don't provide a consistent physical description?

Which two parts of what don't provide a consistent physical description of what?

 

[harrylin]

What do you mean by "according to us"?

We are distant observers of black holes.

[..] time dilation is a coordinate effect. Specifically dτ is coordinate independent, but dt is coordinate dependent.

Probably you meant that time dilation is coordinate dependent; a coordinate effect is no physical effect at all - in which case no prediction of a measurable effect such as gravitational time dilation of clocks could be detected.

[..] that section of the lecture notes isn't even about GR. It is simply about the math of Riemannian manifolds, tensors, and coordinate charts. Could you identify what specifically you disagree with or are having trouble with?

He starts basically the same discussion as I started in the Flowing space thread, with roughly a contrary suggestion: according to him "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system." However he next admits that that isn't necessarily necessary, as follows: "(Although some can, even ones with nontrivial topology." His suggestion that we need different maps that next must be linked together looks contrarian to Einstein's non-rigid measuring bodies of reference.

Which two parts of what don't provide a consistent physical description of what?

The "inner" and "outer" Schwarzschild solutions, do they - as PAllen phrased it - "manifest" such things as time dilation, speed of light, speed of particle, energy and momentum conservation etc. consistently around the Schwarzschild radius from a distant perspective?

 

[stevendaryl]

As before, it seems that you misunderstood my question which doesn't relate to a time label but to what Einstein called physical interpretation, which I here called physical description, and which PAllen called "differently manifest".

Is it possible for you to either ask a precise question, or make a precise statement? When people try to guess what you are talking about, you say that they are wrong, but you don't actually clarify what you really do mean.

So, your reason for believing that an infalling observer never crosses the horizon isn't because the Schwarzschild coordinate t goes to infinity? Then what reason do you have for believing that?

 

[stevendaryl]

Yes, exactly - as I specified next. I correctly understood that you were making claims about physical reality according to a distant observer.

Physical reality doesn't have a modifier "according to a distant observer". I think what you mean is "as described by the coordinate system of the distant observer". But that doesn't uniquely say anything, either, because the distant observer can use different coordinate systems.

 

[PeterDonis]

I correctly understood that you were making claims about physical reality according to a distant observer.

If that's what you understood, you understood incorrectly. I was making claims about physical reality, period. Reality is not "according to" any observer. It's just reality.

More precisely, as I clarified in my last post, I was making claims about a *prediction* of what physical reality is like. That prediction may turn out to be wrong, but if it is, it won't be because somebody's "interpretation" was wrong. It will be because the prediction was based on incomplete knowledge, as all predictions are. I even gave specific ways in which our knowledge might be incomplete.

Such discussions are similar to those started by bob2c and Vandam about SR.

I'm not sure I agree. In those discussions, everybody agreed on the global spacetime model that we were working with, and on all invariant quantities within that model. I don't see a similar agreement in this discussion.

 

[Dale]

I agree with you on the physics, but I actually have to admit that I too find this terminology confusing. It seems to imply something that I know you didn't mean, that what is "real" is a matter of convention. IMO it would be better to say something like: the infalling object *does* fall through the horizon, but once it reaches the horizon it can't send light signals back out to the distant observer.

Sorry about that. If even someone with your background finds the terminology confusing then I should avoid it in the future.

 

[Dale]

We are distant observers of black holes.

Yes, I am aware of that. But it doesn't clarify enough to answer the question. You need to specify which coordinate system you are referring to by "according to us", or if you are referring to invariants instead such as local observations.

according to him "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system." However he next admits that that isn't necessarily necessary, as follows: "(Although some can, even ones with nontrivial topology."

Yes, some manifolds cannot be covered with a single coordinate chart, he even provides an example on p 38 of a simple manifold which cannot be covered with a single chart. That in no way contradicts the fact that some manifolds can be covered in a single coordinate chart, i.e. some manifolds do require multiple charts, some manifolds do not.

It appears to me that you do not understand this point if you think that there is any discrepancy in these statements.

Also, even if a manifold can be covered by a single chart, surely you must admit that we may want to consider multiple charts, e.g. Cartesian and spherical. Do you have any objection to that?

His suggestion that we need different maps that next must be linked together looks contrarian to Einstein's non-rigid measuring bodies of reference.

Why? What seems contrarian to you in that? Again, this has nothing to do with GR at this point, it is just the mathematical framework.

Consider the manifold of a sphere. Suppose that you use non-rigid rulers to lay out a traditional lattitude and longitude grid on the sphere. If you do that, then in order for your resulting system to have the mathematical properties that Carroll identifies then you must exclude one longitude line from pole to pole. So you will require at least two such charts to cover the entire sphere. So the use of non-rigid bodies of reference does not imply that a single chart is possible.

The "inner" and "outer" Schwarzschild solutions, do they - as PAllen phrased it - "manifest" such things as time dilation, speed of light, speed of particle, energy and momentum conservation etc. consistently around the Schwarzschild radius from a distant perspective?

They don't overlap anywhere, so it is hard to see how they could either agree or disagree about anything.

 

[Nugatory]

Such discussions are similar to those started by bob2c and Vandam about SR.

I'm not sure I agree. In those discussions, everybody agreed on the global spacetime model that we were working with, and on all invariant quantities within that model. I don't see a similar agreement in this discussion.

And I am quite sure that I don't agree, for exactly PeterDonis's reason...

But that does suggest yet another way forward. Maybe we could try describing some invariant properties, without regard to ANY coordinate system, see if we agree about them.

We have two worldlines, one corresponding to Lucky who isn't falling into the black hole and one corresponding to Unlucky who is. Lucky and Unlucky both carry wristwatches which are recording their proper time along these worldlines; both Lucky and Unlucky will agree about what Lucky's watch reads at any point on Lucky's worldline, and about what Unlucky's watch reads at any point on Unlucky's worldline (one of them has to calculate it, while the other can just look at his wrist, but they'll get the same result). There is a point on Unlucky's worldline such that at and beyond that point, there is no point on the worldline from which a forward-going null geodesic can be drawn that will intersect Lucky's worldline; and for all points before that point, such a geodesic can be drawn. However, for all points on Lucky's worldline, I can draw a forward-going null geodesic that will intersect Unlucky's worldline. And finally... Unlucky's worldline terminates at the central singularity.

OK, do I have the coordinate-independent description of the physics right? (It's already been pointed out a number of times, correctly, that it's hard to be precise and accurate without the math). And if I do, then Harrylin is this the the physical situation as you understand it?

 

[Dale]

for all points on Lucky's worldline, I can draw a forward-going null geodesic that will intersect Unlucky's worldline. And finally... Unlucky's worldline terminates at the central singularity.

Some points on Lucky's worldline will have inward-going radial null geodesics that reach the singularity after Unlucky's worldline has already intersected it. But other than that minor detail everything else sounds right to me.

 

[JesseM]

There are a few reasons, but the simplest one is that if Bob calculates that Alice "never" crosses the EH, and can witness in a finite time the dissipation of the BH, then I'm having a problem accepting that Alice would ever get an opportunity to cross the EH regardless of what SC analysis shows her experience to be.

But Bob only calculates this in Schwarzschild coordinates, it's just a quirk of how that coordinate system is constructed. Even in flat spacetime it's trivial to construct coordinate systems where it takes an infinite coordinate time for someone to cross a horizon that they do cross in finite proper time, and finite coordinate time in inertial coordinates; for example, look at the Rindler horizon in Rindler coordinates. And we could construct even more trivial examples--for instance, if x and t represent coordinates in some inertial frame, then define a new coordinate system where x'=x but t'=(c * t^2)/x, in this coordinate system any event at x=0 and finite t in the original system will have an infinite t' coordinate, so anything approaching x=0 will take an infinite coordinate time to get there. Presumably you don't think this has any real physical significance, why take Schwarzschild coordinates any more seriously?

 

[bobc2]

Spot on, JesseM. And where have you been all these years since the old Michio Kaku forum days? We need Sunfist here to keep people stirred up. I wonder if Yukki is doing string theory math.

 

[Nugatory]

Some points on Lucky's worldline will have inward-going radial null geodesics that reach the singularity after Unlucky's worldline has already intersected it.

Hmmm... show me one? I'm asking, not arguing here.

[above is an edit... I initially responded "yes of course" or some such, then started thinking more about it]

 

[PeterDonis]

Hmmm... show me one? I'm asking, not arguing here.

[above is an edit... I initially responded "yes of course" or some such, then started thinking more about it]

Your first response was correct.

The easiest way to see it is to look at Lucky's and Unlucky's worldlines on a Kruskal chart. Take the point where Unlucky's worldline hits the singularity, and draw a 45 degree line down and to the right from it; this is the worldline of an ingoing light ray that hits Unlucky's worldline at the instant that worldline hits the singularity. The point where that 45 degree line intersects Lucky's worldline (which is a hyperbola in the right wedge of the diagram) is the last point on Lucky's worldline that can send an ingoing light signal to Unlucky. Any light signal Lucky sends after that will hit the singularity, not Unlucky's worldline.

 

[Dale]

Hmmm... show me one? I'm asking, not arguing here.

[above is an edit... I initially responded "yes of course" or some such, then started thinking more about it]

Sure, no problem. Consider a Kruskal Szkeres diagram. Unlucky's worldline intersects with the r=0 hyperbola at some event. Trace a null (45deg) line outwards and back in time until it intersects with Lucky's worldline. Any light emitted by Lucky after that event will reach the singularity after Unlucky does.

EDIT: Peter got it first!

 

[Nugatory]

Your first response was correct.

The easiest way to see it is to look at Lucky's and Unlucky's worldlines on a Kruskal chart. Take the point where Unlucky's worldline hits the singularity, and draw a 45 degree line down and to the right from it; this is the worldline of an ingoing light ray that hits Unlucky's worldline at the instant that worldline hits the singularity. The point where that 45 degree line intersects Lucky's worldline (which is a hyperbola in the right wedge of the diagram) is the last point on Lucky's worldline that can send an ingoing light signal to Unlucky. Any light signal Lucky sends after that will hit the singularity, not Unlucky's worldline.

D'oh - yes, thanks.

With that added detail, we have a description of the physics without reference to any coordinate system at all (PeterDonis mentioned Kruskal coordinates, but only to make it easier to visualize the null geodesics between the two worldlines).

And my question still stands: Harrylin is this the the physical situation as you understand it?

 

[Nugatory]

EDIT: Peter got it first!

and you both beat me to it...
next time I'll try figuring it out for myself BEFORE I break to cook dinner.

 

[Dale]

and you both beat me to it...
next time I'll try figuring it out for myself BEFORE I break to cook dinner.

No worries, it is always reasonable to ask me to justify a claim I make.

 

[PAllen]

There is a really subtle point about describing the physics of Lucky and Unlucky world lines, and light paths from them, and about the singularity, in a coordinate independent way.

That is, while as Peter and Dalespam pointed out, there is a light path from Lucky's world line that reaches Unlucky at the moment Unlucky reaches the singularity; and there are light paths from later than this on Lucky's world line that reach the singularity; it does not follow, in a coordinate independent way, that these two light paths arrive at the singularity at different times - even though you know their origination events are invariantly time ordered. The reason is that the singularity for SC geometry is spacelike, and thus can validly be considered to exist all at one time. Put another way, there does not exist a timelike world line connecting the event of Unlucky reaching the singularity and the event of the 'later' light ray reaching the singularity. Only a spacelike path connects these events, therefore, there time ordering is not specified in a coordinate independent way. You could even construct valid coordinates that reverse the time ordering of these events as compared to the assignment of Kruskal time. That is, you could validly consider that the light emitted later on Lucky's world line reaches the singularity before light emitted earlier on Lucky's world line. Perverse, but such coordinates would not reverse the time ordering of any causally connected events, thus they would be valid.

 

[PeterDonis]

Only a spacelike path connects these events, therefore, there time ordering is not specified in a coordinate independent way.

I agree that the events are spacelike separated, but there is still an ordering, because the null paths of the light rays ending on those events don't intersect. See below.

You could even construct valid coordinates that reverse the time ordering of these events as compared to the assignment of Kruskal time.

I don't think you can do this without also changing the "direction of time" globally; in other words, such coordinates *would* reverse the time ordering of causally connected events, contrary to your statement at the end of your post. As I noted above, the null worldlines of the ingoing light rays are "time ordered" by the order in which they cross any timelike worldline coming into the singularity. This ordering imposes a similar ordering on the events of their hitting the singularity; changing that ordering would require changing the ordering of the null rays crossing any timelike worldline. So the events on the singularity can be considered to be "time ordered" even though the singularity is spacelike.

I understand the intuition underlying your statement, but curved spacetime sometimes throws curve balls. In ordinary relativity of simultaneity scenarios, you compare arrival times of light rays going in opposite directions, and the relative ordering of those can change when you change timelike worldlines. But the singularity is at r = 0; *all* null rays intersecting it are ingoing, so they are all going in the same direction. That's a key difference.

 

[Nugatory]

There is a really subtle point about describing the physics of Lucky and Unlucky world lines, and light paths from them, and about the singularity, in a coordinate independent way.

Agree, and it would probably be a good habit to avoid using the words "after" and "before" with respect to arrival at the central singularity.

We can reword DaleSpam's #196 to avoid using these words: There's a point on Lucky's worldline such that on one side of the point the inwards null geodesics intersect Unlucky's worldline and terminate at the singularity; on the other side they terminate at the singularity without intersecting Unlucky's worldline.

We've gotten to where I think we're fine-tuning the words we're using to describe a spacetime structure that we understand and agree about.

 

[PAllen]

I agree that the events are spacelike separated, but there is still an ordering, because the null paths of the light rays ending on those events don't intersect. See below.
I don't think you can do this without also changing the "direction of time" globally; in other words, such coordinates *would* reverse the time ordering of causally connected events, contrary to your statement at the end of your post. As I noted above, the null worldlines of the ingoing light rays are "time ordered" by the order in which they cross any timelike worldline coming into the singularity. This ordering imposes a similar ordering on the events of their hitting the singularity; changing that ordering would require changing the ordering of the null rays crossing any timelike worldline. So the events on the singularity can be considered to be "time ordered" even though the singularity is spacelike.

I understand the intuition underlying your statement, but curved spacetime sometimes throws curve balls. In ordinary relativity of simultaneity scenarios, you compare arrival times of light rays going in opposite directions, and the relative ordering of those can change when you change timelike worldlines. But the singularity is at r = 0; *all* null rays intersecting it are ingoing, so they are all going in the same direction. That's a key difference.

I disagree. I focus only on coordinate patch for the BH interior (that is sufficient for my purpose; I don't have to cover all of spacetime with a given coordinate patch). First, consider simply interior SC coordinates, and let T' = - SC r coordinate, and lines of constant T' are lines of constant SC r. Now, I clearly have a coordinates where every event 'on' the singularity has the same time; every event 'just before' hitting the singularity has an earlier time; etc. No crossings or time reversals have occurred.

Now re-render these coordinates so the constant T' lines are horizontal. Now tilt them a little up on the left and down on the right, and use the cartesian x and y on this picture as spatial and time coordinate, respectively. Seems to me that all time orderings on all timelike and lightlike paths in the interior remain unchanged, and I have met my stated goal.

[Edit: For the whole interior region, the final tilt step above will time reverse some null or timelike lines. So for the whole interior, we can achieve the goal that all singularity arrivals are simultaneous. However, for the whole interior we cannot achieve a complete reversal of Kruskal time ordering on the singularity. However, for chosen subsets of the interior including 'parts' of the singularity, it is possible. Thus, it is possible to state that for specific light signals emitted in one order from Lucky's world line, when viewed in specific coordinates covering both of their approaches to the singularity, their arrivals are the reverse of their emissions, while all time orderings on all causal paths (time or light) are the same as in Kruskal), for this specific spacetime region of the interior. ]

 

[PeterDonis]

I disagree. I focus only on coordinate patch for the BH interior (that is sufficient for my purpose; I don't have to cover all of spacetime with a given coordinate patch). First, consider simply interior SC coordinates, and let T' = - SC r coordinate, and lines of constant T' are lines of constant SC r. Now, I clearly have a coordinates where every event 'on' the singularity has the same time; every event 'just before' hitting the singularity has an earlier time; etc. No crossings or time reversals have occurred.

Ok so far, the singularity is at T' = 0 and the horizon is at T' = - 2M.

Now re-render these coordinates so the constant T' lines are horizontal.

And integral curves of T' are vertical? Also, what is the spatial coordinate supposed to be? It can't be r. Is it the Kruskal "X" (or "U")? I'm assuming it is that, or something similar, since you don't appear to have changed the "left to right" ordering of events on the singularity.

Now tilt them a little up on the left and down on the right, and use the cartesian x and y on this picture as spatial and time coordinate, respectively.

If I'm picturing this right, this would make events "to the left" of the spatial origin occur at negative T'' (if we use that label for the new time coordinate in this chart), and events "to the right" of the spatial origin occur at positive T''. So it would give the *same* time ordering as the one implied by the ingoing null curves. To reverse that, you would want to tilt the constant T' surfaces up on the right and down on the left, so events "to the left" of the spatial origin occur at positive T'' and events to the right occur at negative T'', i.e., before those to the left. That would reverse the standard time ordering.

I'll defer further comment until I'm sure we're both considering the same transformation.

 

[PAllen]

Ok so far, the singularity is at T' = 0 and the horizon is at T' = - 2M.

Yes

And integral curves of T' are vertical? Also, what is the spatial coordinate supposed to be? It can't be r. Is it the Kruskal "X" (or "U")? I'm assuming it is that, or something similar, since you don't appear to have changed the "left to right" ordering of events on the singularity.

yes

If I'm picturing this right, this would make events "to the left" of the spatial origin occur at negative T'' (if we use that label for the new time coordinate in this chart), and events "to the right" of the spatial origin occur at positive T''. So it would give the *same* time ordering as the one implied by the ingoing null curves. To reverse that, you would want to tilt the constant T' surfaces up on the right and down on the left, so events "to the left" of the spatial origin occur at positive T'' and events to the right occur at negative T'', i.e., before those to the left. That would reverse the standard time ordering.

I'll defer further comment until I'm sure we're both considering the same transformation.

Actually, what I proposed would reverse the K-S time ordering for arrival events for the 'right half' K-S diagram, while leaving them unchanged for the left half. To fully reverse, you would need to tilt the T' horizontal lines down on the right and down on the left, forming e.g. a shallow upward pointed cone. However, I'm now convinced this can't succeed for the whole interior (that is, the reversal can't; the treatment of all singularity arrival events as simultaneous is fine)[see my edit to #208]. But it can succeed for 'large' subsets of the interior that include a set of singularity arrival events, but do not go all the way to the horizon. Which is still sufficient to justify my claim.

[Edit: A further example is that Lemaitre coordinates time reverse 'half' the singularity arrival events compared to K-S. By choosing the interior portion of these adapted for K-S II + I, or K-S II+III, you can reverse either half of singularity arrivals you want.]