On the nature of the infinite fall toward the EH

[PeterDonis]

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.

Yes, I see that, but the time ordering I was referring to, that imposed by the ingoing null worldlines that hit the singularity, is *not* the same as the K-S time ordering for events on the singularity. In the K-S time ordering, one particular event on the singularity, the one at X = 0 (or U = 0, depending on how you label the horizontal coordinate), has the earliest Kruskal time, and Kruskal time increases going to the left *and* to the right along the singularity. The time ordering I was referring to a is monotonic ordering of events on the singularity: events "to the left" are earlier, and events "to the right" are later. This is the same time ordering as Eddington-Finkelstein time or Painleve time. That is the time ordering that would be reversed on the singularity by starting with surfaces of constant T' horizontal, and then tilting the constant T'' surface up on the right and down on the left.

The reason I'm focusing on this time ordering is that, as should be evident from the fact that it matches E-F time and Painleve time, this is the time ordering of events along any timelike curve in regions I and II of the spacetime. The reason it's different from the K-S time ordering is that the latter time ordering includes curves in all four regions; but regions III and IV are not there in any real case, where a black hole is formed from a collapsing object. So the only curves we need to worry about are those in regions I and II.

Given that, to see the intuition underlying what I said about not being able to reverse the time ordering along the singularity without also reversing it on at least some timelike curve (I said "causal" earlier, but I should have said "timelike"), tilt the K-S diagram 45 degrees counterclockwise, so the "antihorizon" (the past horizon that starts at the lower right and goes up and to the left) is horizontal. Ingoing null rays are now all horizontal lines parallel to the antihorizon, and any timelike curve in regions I and II can be time ordered by the order in which it crosses those null rays, and that time ordering is necessarily the *same* as the natural past to future ordering of proper time along that causal curve. And that time ordering is *also* the same as the "left to right" ordering along the singularity, since that's the order in which the null rays intersect the singularity: the closer the null ray is to the antihorizon, the further to the left it hits the singularity.

Now, look at what surfaces of constant time have to look like in any chart that has the same time ordering as above. These surfaces must be spacelike, as in any chart, and each one must intersect the singularity at the same event as some ingoing null ray. This gives a one-to-one mapping between ingoing null rays and surfaces of constant time, and that mapping relates the ordering of surfaces of constant time to the ordering of ingoing null rays. So any timelike curve in regions I and II must cross the surfaces of constant time in the same order as it crosses the ingoing null rays.

The above seems sufficient to me to show that any chart that reverses the time ordering of events on the singularity, so they go "right to left" instead of "left to right", must also reverse the time ordering along timelike curves in regions I and II.

 

[PAllen]

I think we've ended close to agreement. Since my edit to #208, I have claimed only the following: - you can treat all singularity arrival events as simultaneous in a coordinate patch that covers the entire interior region.

- you can pick an interior subset not including the horizon, but including a portion of singularity arrival events, and reverse singularity arrival order with reversing time flow along any timelike world line within that region.

- Lemaitre coordinates adapted for II+III suggest it might be possible to reverse for the whole interior (not including horizon), without reversing time order on an interior causal curve.

- What you can't do is include the horizon or exterior. You cannot include a region of Lucky's world line, plus events bounded by light paths to the singularity. But you can take an interior subset of this and reverse the singularity arrival times without time reversing any curve within this subset region. Thus, by changing patches, you can consider arrival times reversed.

[Edit: Another way to state the limitation, as I see it, is that Lucky cannot build a coordinates system based on their world line, extending to the singularity, that reverses arrival order relative to emission order for light paths reaching the singularity.However, Lucky can treat all singularity arrivals as simultaneous. And a different observer (e.g. an interior one) can set up coordinates that say Lucky's second signal arrived at the singularity before the first signal. The difference from SR, is that if they do this, they cannot include Lucky's world line the coordinates.]

 

[PAllen]

Hmm, now I wonder if Lucky can treat singularity arrival times as reversible if they narrow their scope of interest:

Consider only the region of spacetime bounded by Lucky's world line from e1 to e2; and light signals from e1 to singularity, and e2 to singularity. That is, all events reachable by ingoing light signals sent by Lucky between e1 and e2 inclusive. I am not seeing any reason a chart covering just this region cannot be smoothly deformed approaching the singularity, to have the singularity arrival events in either order, without changing time order on any causal paths within this region.

 

[Mike Holland]

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.

Sorry, PAllen, but I still have a problem with this.

We observe time dilation at the surface of the Sun by studying the effect on spectral lines and such. But what we measure is affected by the dilation we experience here on Earth, so there is a coordinate factor present. However, we can calculate the effect of Earth's gravity field and also the effect of our orbital movements, etc, and derive an ideal reference frame, like that of the remote observer used in the O-S calculations. Once we have eliminated local effects, which are all calculable, we arrive at an "absolute" time dilation value for the surface of the Sun. Any other obserrver in the galaxy can do the same compensating calculations and arrive at the same value for time dilation at the Sun's surface. So this value is coordinate-independant.

Admittedly this coordinate frame we use in our calculations is an unattainable ideal, just as absolute zero is in temperature measurements. But using this ideal we can ascertain time dilation values which depend only on the mass present and the distance from it, just as O-S did in their calculations.

Edit: OK, so my ideal coordinate frame is still a coordinate frame, but is is one that would be used by astronomers everywhere, and it is a "special" one, much as you might dislike that idea.

 

[harrylin]

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.

With "us" I was referring to people on Earth who typically use the ECI coordinate system. However, it turns out that Peter's answer didn't match what we were talking about; instead he continued to argue about what happens in hidden reality despite the fact that I made clear that I'm not interested.

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?

I'm sure that I understood his point perfectly. And if I wanted to travel to the North pole then I would certainly insist on using a single map - spherical is best but unhandy to carry, second best for me would be a single projection with a clear description of the kind of assumed deformation. So, if you were trying to sell me something else and started to argue that I must buy a map that I'm not interested in then you would have no luck with me: I would simply walk out (I did encounter such salesmen in real shops).

[..]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.

Sorry, but here you actually sound like the salesman of my example...

 

[Mentz114]

Sorry, but here you actually sound like the salesman of my example...

You seem to have become petulant because GR does not live up to some ideal you have. I hope the thousands of words expended in trying to educate you have not been wasted.

What is your problem ?

 

[harrylin]

[..] 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.

Those were informative but never ending discussions about different interpretations of physical reality, related to different interpretations of SR - and happily you don't have to agree with me.

 

[harrylin]

You seem to have become petulant because GR does not live up to some ideal you have. I hope the thousands of words expended in trying to educate you have not been wasted.

What is your problem ?

What is your problem? I just don't need that. A discussion forum is not suited for education, for that we have textbooks and articles and a few links suffice. I'm not sure if I managed to clarify some questions and different views that many people have, but in the process I also asked a few questions myself and - despite some people trying to start arguments - I got a better understanding of both the questions and the answers. Likely there are many onlookers who also benefited from it. However we don't know if the OP of this thread is satisfied.

 

[harrylin]

[..] 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). [..] 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?

Personally I'm not interested in reiterating those things that we all agree on and which I believe do not really answer the issues that seem to be inherent in the title of this thread.
However, we noticed that it's especially things like the meaning of "infinite time" that make black hole discussions difficult; and I got the impression that several issues that came up are not directly concerned with such infinities (see for example the last post by MikeHolland here above). Thus I now think that a thread on the theoretical effects according to a distant observer on an object falling towards a body that is nearly a black hole (without caring about how to sustain such a theoretical object in practice) could be a helpful reference.

 

[Dale]

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.

Point well taken. However, what is coordinate independent is that the later light paths from Lucky do not reach Unlucky before Unlucky hits the singularity.

 

[Dale]

I'm sure that I understood his point perfectly. And if I wanted to travel to the North pole then I would certainly insist on using a single map - spherical is best but unhandy to carry, second best for me would be a single projection with a clear description of the kind of assumed deformation.

A projection is a better map than lattitude and longitude lines, Carroll specifically discusses this on p. 39. As you can see, even with that approach you still cannot cover the entire sphere in one chart since you miss the north pole. So it still requires at least two charts to cover a sphere.

Sorry, but here you actually sound like the salesman of my example...

Except that you are the one who is trying to sell the position that the standard GR math of charts and manifolds is somehow "contrarian to Einstein's non-rigid measuring bodies of reference", an assertion which you have neither explained nor supported in any way, and for which I provided a counter-example as well as a reference with other counter-examples. Can you support that claim now?

A discussion forum is not suited for education, for that we have textbooks and articles and a few links suffice.

Yes, I have provided them and I am still trying to understand what your objection is. You claim that you understand Carroll's notes, but the statements you make and questions you ask seem incompatible with that claim. Particularly your odd resistance to the idea of using multiple coordinate charts to cover a manifold.

Neglecting the connection to GR, how can you justify your continued assertion that all manifolds can be covered with a single chart? Particularly given the clear counter-examples provided.

 

[PeterDonis]

- you can treat all singularity arrival events as simultaneous in a coordinate patch that covers the entire interior region.

Agreed; your T' coordinates are an example.

- you can pick an interior subset not including the horizon, but including a portion of singularity arrival events, and reverse singularity arrival order with reversing time flow along any timelike world line within that region.

Do you mean *without* reversing time flow along any timelike worldline within the region?

- Lemaitre coordinates adapted for II+III suggest it might be possible to reverse for the whole interior (not including horizon), without reversing time order on an interior causal curve.

I agree that these coordinates would order singularity events from "right to left", i.e., the opposite way from Painleve coordinates on regions I + II. As long as we are only looking at timelike curves in region II, I think I agree that their time ordering will be unchanged.

Lucky cannot build a coordinates system based on their world line, extending to the singularity, that reverses arrival order relative to emission order for light paths reaching the singularity.

Yes.

However, Lucky can treat all singularity arrivals as simultaneous.

Yes. Note that if he does, though, he will be in the weird position of maintaining that, if he emits two light rays, one radially ingoing and one radially outgoing, once he is inside the horizon, both of these light rays will hit the singularity simultaneously with him. I'm not saying this position is inconsistent, just weird.

And a different observer (e.g. an interior one) can set up coordinates that say Lucky's second signal arrived at the singularity before the first signal. The difference from SR, is that if they do this, they cannot include Lucky's world line the coordinates.]

Yes.

 

[PeterDonis]

Consider only the region of spacetime bounded by Lucky's world line from e1 to e2; and light signals from e1 to singularity, and e2 to singularity. That is, all events reachable by ingoing light signals sent by Lucky between e1 and e2 inclusive. I am not seeing any reason a chart covering just this region cannot be smoothly deformed approaching the singularity, to have the singularity arrival events in either order, without changing time order on any causal paths within this region.

I'm not sure I agree; I think in this case the same reasoning would apply that I gave before, at least in part: there will at least be a subset of surfaces of constant time in such a chart can be mapped one-to-one to the ingoing null rays that Lucky emits from e1 to e2, and must be ordered the same way. It won't be *all* of the surfaces of constant time now, because the region covered by the chart is bounded by two ingoing null rays, so any spacelike surfaces in this region will "exit" the chart on at least one side. But I think that having even a subset of spacelike surfaces mapped one-to-one to ingoing null rays is sufficient.

 

[stevendaryl]

I'm sure that I understood his point perfectly. And if I wanted to travel to the North pole then I would certainly insist on using a single map - spherical is best but unhandy to carry, second best for me would be a single projection with a clear description of the kind of assumed deformation.

The usual sort of map, with the coordinates described using latitude and longitude, is useless at the North Pole, because all directions are "south". Your suggestion of using a sphere as your map is closer to the mark, except for the fact that if you're navigating at the North Pole, most of the sphere is irrelevant except for a small "cap" representing the North Pole and surrounding areas.

In the case of the universe, the complication is that we don't KNOW what the entire universe looks like. We've only seen a tiny piece of it. We're in the situation of explorers on the planet Earth before the entire planet had been explored. But the point is that to navigate around England, you don't NEED a map of the whole world; you only need a map of England. Every populated region has a map that is good locally, and if you are traveling from one region to another, you have to know how the local maps fit together.

 

[stevendaryl]

Personally I'm not interested in reiterating those things that we all agree on and which I believe do not really answer the issues that seem to be inherent in the title of this thread.

People keep going over the same ground because you don't make it clear what it is that you are claiming. Or if you're not claiming anything, but are just asking questions, you don't make it clear what question it is that you are asking. I've asked for clarification a number of times, and you really haven't said anything very clear. You seem to believe that the use of "patches" to describe the universe is somehow contrary to what Einstein intended. But why do you believe that? You seem to believe that there is a contradiction between the descriptions of black holes as given by Schwarzschild coordinates and KS coordinates. Why do you believe that? You've made many posts, but have managed to say very little.

 

[PeterDonis]

Edit: OK, so my ideal coordinate frame is still a coordinate frame, but is is one that would be used by astronomers everywhere, and it is a "special" one, much as you might dislike that idea.

Mike, I think what PAllen was referring to is that this "ideal coordinate frame" of yours is only valid if all the objects involved are at rest relative to one another, since gravitational time dilation can only be defined in a system that is static. The actual objects are not at rest relative to one another, so your "ideal coordinate frame" has to impose a simultaneity convention that does not match the "natural" one for the objects. You're right that this is a useful convention to adopt for many practical purposes, but it's still a convention.

On the scale of the entire universe, it is true that there is a particular frame that is "special"; it's the one in which the universe is isotropic, i.e., it looks the same in all directions. We are *not* at rest in such a frame; we see a large dipole anisotropy in the CMBR, indicating that we are "moving" relative to this special frame. But the universe is not static either, so even observers who are at rest in the "special" frame can't define a useful notion of gravitational time dilation.

 

[stevendaryl]

However, we noticed that it's especially things like the meaning of "infinite time" that make black hole discussions difficult;

There is nothing difficult about it. It's confusing to the neophyte, but it shouldn't continue to be confusing after you've seen the mathematical details.

and I got the impression that several issues that came up are not directly concerned with such infinities (see for example the last post by MikeHolland here above). Thus I now think that a thread on the theoretical effects according to a distant observer on an object falling towards a body that is nearly a black hole (without caring about how to sustain such a theoretical object in practice) could be a helpful reference.

The notion of "effects according to a distant observer" is not a coherent notion. You can talk about effects as described using this or that coordinate system, but that isn't according to a particular observer; any observer can use any coordinate system.

 

[PeterDonis]

With "us" I was referring to people on Earth who typically use the ECI coordinate system.

Which can't even extend much further than the Moon, let alone to any distant black holes.

However, it turns out that Peter's answer didn't match what we were talking about; instead he continued to argue about what happens in hidden reality despite the fact that I made clear that I'm not interested.

To paraphrase the old saying, you may not be interested in reality, but reality is interested in you. Whether you like it or not, GR's predictions about what you call "hidden reality" are relevant, so they're going to get brought up.

Personally I'm not interested in reiterating those things that we all agree on and which I believe do not really answer the issues that seem to be inherent in the title of this thread.

But if you really agreed on the things you claim "we all agree on", you would see that they *do* answer the title question in this thread. The problem is, you don't.

However, we noticed that it's especially things like the meaning of "infinite time" that make black hole discussions difficult

They make it difficult *for you*. They're not an issue at all for those of us who understand the difference between coordinate quantities and invariants.

and I got the impression that several issues that came up are not directly concerned with such infinities (see for example the last post by MikeHolland here above).

You're right, his question wasn't. See my response to him.

Thus I now think that a thread on the theoretical effects according to a distant observer on an object falling towards a body that is nearly a black hole (without caring about how to sustain such a theoretical object in practice)

I assume you mean a body that is *static* at a radius that is close to, but larger than, the horizon radius for its mass? If such a theoretical object is impossible according to GR, which it is for any radius less than 9/8 of the horizon radius for its mass, what theory are we supposed to use?

 

[Mentz114]

However we don't know if the OP of this thread is satisfied.

You have a way of clouding even the simplest arguments. The original question has been fully answered !

You're in a minority arguing that there's something 'fishy' going on.

 

[stevendaryl]

You have a way of clouding even the simplest arguments. The original question has been fully answered !

You're in a minority arguing that there's something 'fishy' going on.

Well, the original poster went on to talk about Hawking radiation, and the question of reconciling two points of view:

1. From the point of view of Schwarzschild coordinates (modified suitably to allow for a slow time-dependence in the M parameter), the black hole evaporates BEFORE the infalling observer reaches the event horizon.
2. From the point of view of the infalling observer, the infalling observer reaches the singularity in a finite amount of proper time, presumably long before Hawking radiation would be relevant.

There really is no definitive way to resolve this without a quantum theory of gravity, although it seems that there should be a qualitative way of understanding how these are not contradictory. For someone falling into a black hole, it's all over in a short amount of time--you pass through the event horizon and hit the singularity pretty quickly (for small black holes, anyway). It wouldn't seem that Hawking radiation would change this picture very drastically, because Hawking radiation is pretty puny; it shouldn't make a big change to the geometry of the black hole, except after long, long, long periods of time. On the other hand, from the point of view of a distant observer, the black hole evaporates in a finite amount of time. What happens to the infalling observer, then?

This puzzle is really not about classical General Relativity, since it involves quantum corrections. But if there are any real black holes in the universe, then they're going to be quantum black holes, not classical black holes. So it would be nice to have a qualitative understanding of quantum black holes, even if a definitive understanding is years away (if ever). It would be nice to have a feel for which features of the classical description of a black hole are likely to be present (approximately, anyway) in a more realistic black hole, and which features are likely to be completely tossed out in a quantum theory of black holes.

 

[Nugatory]

Personally I'm not interested in reiterating those things that we all agree on

OK, but I'm not asking for a reiteration... I'm asking whether the description of the geometry in my #195 is one of those things that we all agree on. Do you agree with that description, or do you believe that it is incorrect in some way?

and which I believe do not really answer the issues that seem to be inherent in the title of this thread.

But none of these issues can even be clearly stated without a shared understanding of what ISN'T an issue.

 

[stevendaryl]

Well, the original poster went on to talk about Hawking radiation, and the question of reconciling two points of view:

Actually, this was already the topic of a thread in the astrophysics area:

https://www.physicsforums.com/showthread.php?t=609947&page=2

 

[Mentz114]

Well, the original poster went on to talk about Hawking radiation, and the question of reconciling two points of view:

1. From the point of view of Schwarzschild coordinates (modified suitably to allow for a slow time-dependence in the M parameter), the black hole evaporates BEFORE the infalling observer reaches the event horizon.
2. From the point of view of the infalling observer, the infalling observer reaches the singularity in a finite amount of proper time, presumably long before Hawking radiation would be relevant.

There really is no definitive way to resolve this without a quantum theory of gravity, although it seems that there should be a qualitative way of understanding how these are not contradictory. For someone falling into a black hole, it's all over in a short amount of time--you pass through the event horizon and hit the singularity pretty quickly (for small black holes, anyway). It wouldn't seem that Hawking radiation would change this picture very drastically, because Hawking radiation is pretty puny; it shouldn't make a big change to the geometry of the black hole, except after long, long, long periods of time. On the other hand, from the point of view of a distant observer, the black hole evaporates in a finite amount of time. What happens to the infalling observer, then?

This puzzle is really not about classical General Relativity, since it involves quantum corrections. But if there are any real black holes in the universe, then they're going to be quantum black holes, not classical black holes. So it would be nice to have a qualitative understanding of quantum black holes, even if a definitive understanding is years away (if ever). It would be nice to have a feel for which features of the classical description of a black hole are likely to be present (approximately, anyway) in a more realistic black hole, and which features are likely to be completely tossed out in a quantum theory of black holes.

Your trenchant summation just reinforces my assertion that the question has been answered as well as it can be.

I don't agree with your point 1, but there's little purpose in arguing about it until some data is available.

 

[PeterDonis]

[*]From the point of view of Schwarzschild coordinates (modified suitably to allow for a slow time-dependence in the M parameter), the black hole evaporates BEFORE the infalling observer reaches the event horizon.

This isn't correct. Adding Hawking radiation and evaporation of the hole changes the spacetime, so that light rays from the infalling observer as he gets closer and closer to the horizon no longer take a time approaching infinity to get out; instead, they take a time approaching the time it takes for the distant observer to see the hole's final evaporation. In the limit, the distant observer will see the infalling observer reach the horizon at the same instant that he sees the hole's final evaporation; in fact, light from *every* event that took place on the horizon will reach the distant observer at the same time the light from the hole's final evaporation does.

For someone falling into a black hole, it's all over in a short amount of time--you pass through the event horizon and hit the singularity pretty quickly (for small black holes, anyway). It wouldn't seem that Hawking radiation would change this picture very drastically, because Hawking radiation is pretty puny; it shouldn't make a big change to the geometry of the black hole, except after long, long, long periods of time.

All correct, at least to the best of our current knowledge.

On the other hand, from the point of view of a distant observer, the black hole evaporates in a finite amount of time. What happens to the infalling observer, then?

See above. (Note that the distant observer will still never see any light rays from events inside the horizon.)

Edit: If you mean what happens to the infalling observer once he's inside the horizon, that depends on whether there is still a singularity there when quantum effects are taken into account. The problem with the singularity is that it destroys quantum information, violating unitarity. The problem with not having a singularity is figuring out what replaces it.

This entry in the Usenet Physics FAQ is relevant:

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/fall_in.html

 

[Dale]

Adding Hawking radiation and evaporation of the hole changes the spacetime, so that light rays from the infalling observer as he gets closer and closer to the horizon no longer take a time approaching infinity to get out; instead, they take a time approaching the time it takes for the distant observer to see the hole's final evaporation.

This seems intuitively like the right conclusion. Do you have a supporting reference?

 

[PAllen]

Sorry, PAllen, but I still have a problem with this.

We observe time dilation at the surface of the Sun by studying the effect on spectral lines and such. But what we measure is affected by the dilation we experience here on Earth, so there is a coordinate factor present. However, we can calculate the effect of Earth's gravity field and also the effect of our orbital movements, etc, and derive an ideal reference frame, like that of the remote observer used in the O-S calculations. Once we have eliminated local effects, which are all calculable, we arrive at an "absolute" time dilation value for the surface of the Sun. Any other obserrver in the galaxy can do the same compensating calculations and arrive at the same value for time dilation at the Sun's surface. So this value is coordinate-independant.

False. What you are doing is isolating a family emitter world lines that encounter a nearly static metric in the vicinity of the sun, and using these to define coordinates near the sun. This is not even possible in general. See next comment.

Admittedly this coordinate frame we use in our calculations is an unattainable ideal, just as absolute zero is in temperature measurements. But using this ideal we can ascertain time dilation values which depend only on the mass present and the distance from it, just as O-S did in their calculations.

Edit: OK, so my ideal coordinate frame is still a coordinate frame, but is is one that would be used by astronomers everywhere, and it is a "special" one, much as you might dislike that idea.

It is possible to do even remotley as you suggest because there is only one major gravitating body in some large region. Replace the sun with pair tightly co-orbiting neutron stars and you are SOL (hint: time varying metric perturbations not centered on either body would be significant). Meanwhile, the coordinate independent definition of GR Doppler is unaffected, and defines how any emitter, on any world line, anywhere in the vicinity of the binary would be shifted for any given receiver world line further away. Both (dynamic) curvature and different states of emitter and receive motion would have an impact. But you would be unable to define something you call gravitational time dilation. The latter is not a general GR feature; it is something you can define in sufficiently simple spacetimes to simplify calculation. It never necessary. It is not manifested in perfectly good coordinates for simple spacetimes (e.g. the Fermi-Normal coordinates of a free fall observer).

So, I still claim, no exceptions in SR or GR:

- time dilation is a coordinate feature, not an observable.
- Doppler between a chosen emitter and a chosen receiver is an invariant observation.
- Differential aging between different space time paths between given events is an invariant observation.

 

[PAllen]

Point well taken. However, what is coordinate independent is that the later light paths from Lucky do not reach Unlucky before Unlucky hits the singularity.

Correct, indisputable.

 

[PeterDonis]

This seems intuitively like the right conclusion. Do you have a supporting reference?

There's a Penrose diagram of an evaporating black hole on p. 200 of Carroll's lecture notes. Also, the thread in the astrophysics area that was linked to earlier has some good discussion and links; see in particular this post by George Jones:

https://www.physicsforums.com/showpost.php?p=3936159&postcount=23

 

[PAllen]

Well, the original poster went on to talk about Hawking radiation, and the question of reconciling two points of view:

1. From the point of view of Schwarzschild coordinates (modified suitably to allow for a slow time-dependence in the M parameter), the black hole evaporates BEFORE the infalling observer reaches the event horizon.
2. From the point of view of the infalling observer, the infalling observer reaches the singularity in a finite amount of proper time, presumably long before Hawking radiation would be relevant.

There really is no definitive way to resolve this without a quantum theory of gravity, although it seems that there should be a qualitative way of understanding how these are not contradictory. For someone falling into a black hole, it's all over in a short amount of time--you pass through the event horizon and hit the singularity pretty quickly (for small black holes, anyway). It wouldn't seem that Hawking radiation would change this picture very drastically, because Hawking radiation is pretty puny; it shouldn't make a big change to the geometry of the black hole, except after long, long, long periods of time. On the other hand, from the point of view of a distant observer, the black hole evaporates in a finite amount of time. What happens to the infalling observer, then?

This puzzle is really not about classical General Relativity, since it involves quantum corrections. But if there are any real black holes in the universe, then they're going to be quantum black holes, not classical black holes. So it would be nice to have a qualitative understanding of quantum black holes, even if a definitive understanding is years away (if ever). It would be nice to have a feel for which features of the classical description of a black hole are likely to be present (approximately, anyway) in a more realistic black hole, and which features are likely to be completely tossed out in a quantum theory of black holes.

Yes it would. Unfortunately, the correct answer is not known. Somewhere in this thread I posted links to a 2007 paper by Krauss et.al. that argues one position; and a paper by Padmanabhan et.al. from 2009 that claims to refute the former. My belief is that the 2009 paper represents the 'majority view' (and I can't find any response to it from the 2007 authors), but it is far from 'settled physics'. Without responding to the 2009 paper, there are certainly new papers written in the framework of the 2007 paper. It appears to me that both string theory and LQG are more consistent with the framework of the 2009 paper, as is Hawking's proposal for resolving the information paradox.

 

[PAllen]

I'm not sure I agree; I think in this case the same reasoning would apply that I gave before, at least in part: there will at least be a subset of surfaces of constant time in such a chart can be mapped one-to-one to the ingoing null rays that Lucky emits from e1 to e2, and must be ordered the same way. It won't be *all* of the surfaces of constant time now, because the region covered by the chart is bounded by two ingoing null rays, so any spacelike surfaces in this region will "exit" the chart on at least one side. But I think that having even a subset of spacelike surfaces mapped one-to-one to ingoing null rays is sufficient.

I don't agree. I think this will be forced only if e1 and e2 are too far apart (or if you try to include too much outside the region defined by two ingoing null paths, Lucky world line, and the singularity). That is, there is a global prohibition, but not quasi-local problem.

Think of the Kruskal chart, and singularity region in the right half (that matches the GP singularity ordering). Specifically, to make it easy, think of a two singularity arrival events that are nearly horizontal in the chart, and close together, and connect them out with light paths to some static, radial, external world line. Now, within this sliver, we just change all simultaneity surfaces by one degree from horizontal, counterclockwise.

[Edit: Actually, K-S shows a stronger result. It is global, and reverses GP singluarity arrival ordering for 'half' the singularity arrival events. It seems to me, that distortions of K-S can reverse GP ordering in a consistent global chart for all singularity arrival events before (in GP sense) any chosen one. What you can't do is accomplish this for the whole singularity arrival sequence.]

 

[PeterDonis]

think of a two singularity arrival events that are nearly horizontal in the chart, and close together, and connect them out with light paths to some static, radial, external world line. Now, within this sliver, we just change all simultaneity surfaces by one degree from horizontal, counterclockwise.

I see what you mean, but I'm not sure the time ordering on the singularity will be monotonic if you do this. I don't really trust my powers of visualization for this, so I'll have to think about it some more to see if I can come up with a mathematical way to tackle it.

 

[PAllen]

I see what you mean, but I'm not sure the time ordering on the singularity will be monotonic if you do this. I don't really trust my powers of visualization for this, so I'll have to think about it some more to see if I can come up with a mathematical way to tackle it.

Just consider the simple transform (producing ugly metric):

V' = V - k U
U' = U

using the conventions where V is the K-S time coordinate, -1 < k < 1. Lines of constant V' are spacelike everywhere; lines of constant U' are the same as lines of constant U. While the metric gets ugly, it is not hard to see that increasing k towards 1 shifts the inflection in singularity ordering as far to the right as desired; decreasing k towards -1 shifts the inlection to the left. So, for any two events on the singularity, you can get an ordering where the left is first for some k close to -1, and where the right is first for some k close to 1. Each of these charts is global, with the same time ordering for causal curves as the original K-S chart.

This fully justifies (better late than never) my original statement that for any two light signals reaching the singularity from Lucky, Lucky can consider the arrival events to be the reverse of the emission events. The only thing Lucky can't do is achieve such an inversion over the whole history of a static world line. It can be achieved for any segment of interest, but not for the whole past/future eternal history.

Also, note that none of this contradicts Dalespam's improved wording: If light from e1 reaches Unlucky as Unlucky reaches the singularity, light from any event e2, later on Lucky's world line, will not reach Unlucky at all. This wording is coordinate independent. Wording on the order of the two singularity arrival events is coordinate dependent (as expected by the spacelike relation between them).

A final observation is that Lucky can achieve total time order on the singularity consistent with their world line time order using a variety of coordinates (Lemaitre, GP, EF, etc.). A mirror Lucky in region III would use a mirror version of each these coordinate systems to achieve a total singularity ordering consistent with their world line.

 

[PeterDonis]

increasing k towards 1 shifts the inflection in singularity ordering as far to the right as desired; decreasing k towards -1 shifts the inlection to the left.

Yes, but there will always *be* an inflection point; you can never produce a completely monotonic ordering on the singularity this way. That's all I am saying; that a *monotonic* ordering on the singularity can't be reversed without also reversing the time ordering of events on timelike curves (unless you restrict attention only to portions of timelike curves inside the horizon). No K-S style chart gives a monotonic ordering.

This fully justifies (better late than never) my original statement that for any two light signals reaching the singularity from Lucky, Lucky can consider the arrival events to be the reverse of the emission events.

Only if he's willing to accept a non-monotonic ordering of events on the singularity. The emission events are outside the horizon, so there's no way to obtain a reversed monotonic ordering of all events on the singularity that keeps the ordering of emission events the same. If you only want to reverse the arrival events, but allow the complete ordering to be non-monotonic, then yes, you can always do that, as you have shown.

Also, note that none of this contradicts Dalespam's improved wording: If light from e1 reaches Unlucky as Unlucky reaches the singularity, light from any event e2, later on Lucky's world line, will not reach Unlucky at all. This wording is coordinate independent.

Yes, agreed.

A final observation is that Lucky can achieve total time order on the singularity consistent with their world line time order using a variety of coordinates (Lemaitre, GP, EF, etc.).

Yes, and once he's done this, he can't reverse that order while still keeping the ordering the same on his own worldline. (In fact, he can't even reverse it and still *cover* his own worldline; see below.)

A mirror Lucky in region III would use a mirror version of each these coordinate systems to achieve a total singularity ordering consistent with their world line.

Yes, but any such coordinate chart won't cover region I at all. So Lucky and mirror Lucky can never have a common chart that (1) covers both of their worldlines, and (2) agrees on a monotonic ordering of events on the singularity.

 

[PAllen]

Only if he's willing to accept a non-monotonic ordering of events on the singularity. The emission events are outside the horizon, so there's no way to obtain a reversed monotonic ordering of all events on the singularity that keeps the ordering of emission events the same. If you only want to reverse the arrival events, but allow the complete ordering to be non-monotonic, then yes, you can always do that, as you have shown.

Except possibly as a brief, initial speculation, corrected almost immediately, I never claimed monotonic was possible. After backing off from that, everything else I thought turned out to be justified; much more than just a chart bounded e1 to e2 on Lucky's world line, light rays to the singularity, and the singularity arrival events - that reverses arrival order relative to transmission order. Instead, the whole of the K-S manifold can be covered, reversing e1 and e2 arrival; all that can't be done is to reverse the entire singularity arrival ordering.

 

[PAllen]

Yes, but any such coordinate chart won't cover region I at all. So Lucky and mirror Lucky can never have a common chart that (1) covers both of their worldlines, and (2) agrees on a monotonic ordering of events on the singularity.

Yes, I completely understand this. I don't think I suggested otherwise.