The Arrow of Time: The Laws of Physics and the Concept of Time Reversal

[A-wal]

I take it the circle works then?

I know that the formula for finding length contraction/time dilation shows that it becomes more pronounced at higher speeds. If you draw four straight lines joining up the time and space lines to make a diamond shape then the four triangles around the edge would be the difference between a smooth motion (ie .5c = .5 time dilation/length contraction) and the actual equation. In the circle if their proper time stays equal then the curve takes it more up than left at lower speeds then catches up at the end.

This isn't just a graphical representation. This is literally what's happening. All I'm doing is substituting time for another spatial dimension.

 

[A-wal]

How come as soon as you lot think I've made a mistake you can't wait to post "No! You're wrong! You don't understand!" but when I'm asking about something that might work you've all gone away? It's not as even as if it's vague and undefined. Does the circle work or not?

 

[Dale]

Sorry, in our last exchange you basically told me to butt out so I was trying to do so.

It looks like you are doing a spacetime diagram. Traditionally they are drawn with time vertical, but that is just a convention and there is no reason not to do it the way you have done. You are correct that an inertial object follows a straight line and that an accelerating object follows a curved line.

I don't understand why you have a "c" on the space axis (things on the space axis should have units of distance, not speed), and I don't know what significance (if any) you are attaching to the circle.

 

[A-wal]

I wasn't telling you to butt out. It's just that you said twice that I didn't get something to do with simultaneity but wouldn't tell me what it was. I don't think I was missing anything. It just seems that you're much more interested in trying to make others look stupid rather than helping them understand something. I'm going by things I've read in other threads as well. I'm still grateful that you've taken the time to reply. Some of it's been very helpful.

Anythings speed through space-time is always c from any frame. The speed through time and space individually depends which frame you're measuring it from. From you're own perspective you're always moving through space at zero, which makes sense because you can't run away from yourself. So you're always moving through time at c. The circle is just because I originally thought of it as a simple way to compare time dilation and length contraction between two frames and you need to use the same length of proper time to do that, so it always goes to the edge of the circle without acceleration and you then just draw a straight line down to see how much it's shortened by. It also works for gravity/acceleration with the c line being the event horizon.

What about the matter entering a black hole being converted into energy and released at the singularity? It seems to me to be simple and the only way it could work. It answers two questions at once and it's true that no time has passed at the singularity no matter how long the black hole's been there from the outside, and that's where matter that's crossed the horizon has to end up.

I have another question though. I've had a chance to let what I've been told sink in and I've figured out the problem I was having. I was told that if an object is moving in at a faster rate behind another slower object then the faster one will undergo more time dilation from the perspective of a third distant observer, and that there will be a point when it's too late for anything including a signal sent at light speed to reach the closest observer from the closest observers perspective. But what about from the second faster moving observers perspective? It's following the first observer in and it can't possibly enter the horizon after the first observer from its own perspective. But it can't possibly enter the horizon before the first observer from the first observers perspective. We have a contradiction. Which one crosses the horizon first? The only conclusion is that all observers cross at the same time. This makes sense from the third distant observers perspective because they can't see anything cross the horizon until it's gone, which is what I meant before by a black hole being the same as the singularity. Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space. The fact that the second observer is slowing at a faster rate than the first from the third observers perspective just explains how the second observer may never be able to overtake the first observer before reaching the horizon from the third observers perspective.

Let's assume that black holes do loose mass during their lives for whatever reason. The first observer measures the size of the horizon just before it crosses and sees that the second observer hasn't set off yet. The second observer catches up to the first at the event horizon. It should never be too late because it can never witness the first one crossing. Now they both cross together but the second observer measures the size of the horizon as well and says it's smaller than the first one made it, yet they're now in the same place at the same time. If they do all cross at the same time then it makes sense as long as black holes have a life span. The first observer measures the size of the horizon just before it crosses and sees that the second observer hasn't set off yet, and knows that it can't reach the horizon before the black hole's gone. It still works even if the second observer moves in slower than the first because time dilation would slow the first one down so the second one catches up.

This is the only way I can get my head round it without breaking the law that nothing can be destroyed and without contradicting itself. If I've made a mistake somewhere along the line please point out where exactly it went wrong.

 

[Dmitry67]

It's following the first observer in and it can't possibly enter the horizon after the first observer from its own perspective. But it can't possibly enter the horizon before the first observer from the first observers perspective

Begin from here.
There are 2 types of horizons, apparent and absolute. For an observer at infinity, apparent and absolute horizons are at the same position (almost).

But it is not true for a falling observer: when it falls, apparent horizon recedes in front of him. So, if you 'jump' inside the BH to 'resque' a ship 'frozen near the horizon', then you would see how horizons moves deeper and deeper inside the BH, ship 'unfreezes' and falls down too; so you can't catch up with him and can't resque.

And for obvious reasons, no observer can ever pass thru his own apparent horizon, because it would mean that infinite tidal forces had torn his body apart. Fortunaly, in GR freely-falling observer can always assume that his part of space is locally flat, so apparent horizon is always at some distance from him

 

[JesseM]

I have another question though. I've had a chance to let what I've been told sink in and I've figured out the problem I was having. I was told that if an object is moving in at a faster rate behind another slower object then the faster one will undergo more time dilation from the perspective of a third distant observer

I didn't say that, I just said that its coordinate velocity would decrease at a faster rate.

But what about from the second faster moving observers perspective? It's following the first observer in and it can't possibly enter the horizon after the first observer from its own perspective.

Why not? And what do you mean by "its own perspective"? Are you just talking about local coordinate-independent facts like whether the event of the second observer crossing the horizon happens at a later or earlier proper time than the event of the second observer passing the first observer? (if so, the answer is that crossing the horizon happens at an earlier proper time than passing the first observer, if he manages to pass the first observer at all before hitting the singularity) Or are you talking about a "frame" for the second observer, so it can make judgments about things like the velocity of the first observer as a function of time and the time at which the first observer crosses the horizon? If the latter, what coordinate system would this "frame" correspond to?

As always, any statement you can make about observers falling into a black hole has an analogous statement involving observers crossing the Rindler horizon. If you have two inertial observers moving towards the horizon, it may be that the second departed too late to catch up with the first before the first crossed the horizon, even if the second has a higher velocity. In Rindler coordinates, neither would reach the horizon in finite coordinate time, but the second one's velocity would decrease faster than the first so that the second would never pass the horizon at any finite coordinate time. Would you say in this case that the second observer with a higher velocity is "following the first observer in and it can't possibly enter the horizon after the first observer from its own perspective"? If you wouldn't say that in the case of two inertial observers crossing the Rindler horizon, why do you think the case of the black hole is any different?

Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space.

Once and for all, do you agree that "length contraction and time dilation" can only be defined relative to a particular choice of coordinate system, that the fact that length contraction and time dilation approach infinity as you approach the event horizon in Schwarzschild coordinates has no more coordinate-independent "reality" than the fact that length contraction and time dilation approach infinity as you approach the Rindler horizon in Rindler coordinates?

The fact that the second observer is slowing at a faster rate than the first from the third observers perspective just explains how the second observer may never be able to overtake the first observer before reaching the horizon from the third observers perspective.

No, it's a statement about how things work in a parcticular coordinate system, namely Schwarzschild coordinates. What do you mean when you talk about a given observer's "perspective"?? It's not uncommon in SR to talk about an inertial observer's "perspective" as a shorthand for their inertial rest frame, because every inertial observer has a unique inertial rest frame. However, non-inertial observers don't have unique non-inertial rest frames, there are an infinite number of different ways to construct a non-inertial coordinate system where a particular non-inertial observer is at rest. Physicists just don't talk about the "perspectives" of observers in GR, because it would be totally unclear what coordinate system they were referring to! And I suspect that you don't have any well-defined coordinate system in mind when you talk about the "perspectives" of different observers falling into the black hole--if you don't, your comments along these lines would seem to be totally meaningless.

Let's assume that black holes do loose mass during their lives for whatever reason. The first observer measures the size of the horizon just before it crosses and sees that the second observer hasn't set off yet.

"Measures the size" is another meaningless phrase unless you are referring to a particular coordinate system. The only way to define the "size" of an extended object in GR is to have a coordinate system in which you can calculate things like coordinate length or coordinate volume.

The second observer catches up to the first at the event horizon.

Why "at the event horizon"? Depending on how late the second observer departs, it may be impossible for the second observer to catch up with the first until after they are both inside the horizon, or it may even be impossible for the second observer to ever catch up to the first observer before hitting the singularity.

It should never be too late because it can never witness the first one crossing.

If by "witness" you just mean "see the light from the event of the crossing", the second observer will witness the first observer crossing at the moment the second observer crosses the horizon himself (the event of the first crossing the horizon may look like it happened a great distance away at the moment of the second crosses, though). Exactly the same is true if the two observers are crossing the Rindler horizon in flat spacetime, and the second observer isn't able to catch up with the first observer before the first crosses the horizon. Every argument you make I am going to meet with an exact analogy involving the Rindler horizon, so it really is important that you understand the analogy and explain where you think it breaks down (you seemed confused about how the analogy worked before, please reread my clarification in post #148, starting from the paragraph that begins "But I'm not talking about..." right above the second graphic I included in that post)

 

[A-wal]

Begin from here.
There are 2 types of horizons, apparent and absolute. For an observer at infinity, apparent and absolute horizons are at the same position (almost).

But it is not true for a falling observer: when it falls, apparent horizon recedes in front of him. So, if you 'jump' inside the BH to 'resque' a ship 'frozen near the horizon', then you would see how horizons moves deeper and deeper inside the BH, ship 'unfreezes' and falls down too; so you can't catch up with him and can't resque.

And for obvious reasons, no observer can ever pass thru his own apparent horizon, because it would mean that infinite tidal forces had torn his body apart. Fortunaly, in GR freely-falling observer can always assume that his part of space is locally flat, so apparent horizon is always at some distance from him

Yea, that's pretty much exactly what I was describing earlier with the receding horizon but it was shot down.

I have another question though. I've had a chance to let what I've been told sink in and I've figured out the problem I was having. I was told that if an object is moving in at a faster rate behind another slower object then the faster one will undergo more time dilation from the perspective of a third distant observer

I didn't say that, I just said that its coordinate velocity would decrease at a faster rate.

Okay, that was misleading. Loses more velocity through time dilation, not undergo more time dilation.

Originally Posted by A-wal
But what about from the second faster moving observers perspective? It's following the first observer in and it can't possibly enter the horizon after the first observer from its own perspective.

Why not? And what do you mean by "its own perspective"? Are you just talking about local coordinate-independent facts like whether the event of the second observer crossing the horizon happens at a later or earlier proper time than the event of the second observer passing the first observer? (if so, the answer is that crossing the horizon happens at an earlier proper time than passing the first observer, if he manages to pass the first observer at all before hitting the singularity) Or are you talking about a "frame" for the second observer, so it can make judgments about things like the velocity of the first observer as a function of time and the time at which the first observer crosses the horizon? If the latter, what coordinate system would this "frame" correspond to?

I meant the former. How can crossing the horizon happen before the second observer catches the first one? The second observer can't witness the first one crossing the horizon.

As always, any statement you can make about observers falling into a black hole has an analogous statement involving observers crossing the Rindler horizon. If you have two inertial observers moving towards the horizon, it may be that the second departed too late to catch up with the first before the first crossed the horizon, even if the second has a higher velocity. In Rindler coordinates, neither would reach the horizon in finite coordinate time, but the second one's velocity would decrease faster than the first so that the second would never pass the horizon at any finite coordinate time. Would you say in this case that the second observer with a higher velocity is "following the first observer in and it can't possibly enter the horizon after the first observer from its own perspective"? If you wouldn't say that in the case of two inertial observers crossing the Rindler horizon, why do you think the case of the black hole is any different?

Either the second observer can witness the first crossing the horizon before they themselves get there or they can't. You can't have it both ways.

Originally Posted by A-wal
Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space.

Once and for all, do you agree that "length contraction and time dilation" can only be defined relative to a particular choice of coordinate system, that the fact that length contraction and time dilation approach infinity as you approach the event horizon in Schwarzschild coordinates has no more coordinate-independent "reality" than the fact that length contraction and time dilation approach infinity as you approach the Rindler horizon in Rindler coordinates?

Once and for all you say? Does that mean you're not going to ask me this same question ever again? If the first and second ships meet up again then less time would have passed for the first ship, so there is a coordinate-independent reality.

Originally Posted by A-wal
The fact that the second observer is slowing at a faster rate than the first from the third observers perspective just explains how the second observer may never be able to overtake the first observer before reaching the horizon from the third observers perspective.

No, it's a statement about how things work in a parcticular coordinate system, namely Schwarzschild coordinates. What do you mean when you talk about a given observer's "perspective"?? It's not uncommon in SR to talk about an inertial observer's "perspective" as a shorthand for their inertial rest frame, because every inertial observer has a unique inertial rest frame. However, non-inertial observers don't have unique non-inertial rest frames, there are an infinite number of different ways to construct a non-inertial coordinate system where a particular non-inertial observer is at rest. Physicists just don't talk about the "perspectives" of observers in GR, because it would be totally unclear what coordinate system they were referring to! And I suspect that you don't have any well-defined coordinate system in mind when you talk about the "perspectives" of different observers falling into the black hole--if you don't, your comments along these lines would seem to be totally meaningless.

I meant the third observer is sitting there watching the other two so we see what's happing from the perspective of someone who's maintaining a constant distance from the black hole. I spose you'd call it Schwarzschild coordinates. Let me tell you what's meaningless. Those coordinate systems your so found of! I'm not saying they can't be useful, but they're meaningless in themselves. It's what they're describing that has meaning (my one's different because it's not describing it, it's literal), so let's just stick to describing things in real terms. What happens if they do this? What would they see if this happens? Otherwise we're translating it into a graph then translating it back into conceptual reality. It's very limiting to use graphs to form an understanding of something. I think it's meant to be the other way round.

Originally Posted by A-wal
Let's assume that black holes do loose mass during their lives for whatever reason. The first observer measures the size of the horizon just before it crosses and sees that the second observer hasn't set off yet.

"Measures the size" is another meaningless phrase unless you are referring to a particular coordinate system. The only way to define the "size" of an extended object in GR is to have a coordinate system in which you can calculate things like coordinate length or coordinate volume.

I meant that if the first were to measure the size of the black hole just before crossing the horizon and the second one does the same using the same method when reaching the horizon then if they got different results it would mean that they crossed at different times in the black holes life despite crossing the horizon at the same time.

Originally Posted by A-wal
The second observer catches up to the first at the event horizon.

Why "at the event horizon"? Depending on how late the second observer departs, it may be impossible for the second observer to catch up with the first until after they are both inside the horizon, or it may even be impossible for the second observer to ever catch up to the first observer before hitting the singularity.

Because the second observer can't witness the first crossing the horizon until they cross it themselves.

Originally Posted by A-wal
It should never be too late because it can never witness the first one crossing.

If by "witness" you just mean "see the light from the event of the crossing", the second observer will witness the first observer crossing at the moment the second observer crosses the horizon himself (the event of the first crossing the horizon may look like it happened a great distance away at the moment of the second crosses, though).

"See the light from the event of the crossing" is exactly the same as "the event of the crossing". The light's moving slowly because time is. The event of the first one crossing may happen at a great distance away at the moment the second one crosses? The first one can't cross at all from the perspective of the second one at least until the second one reaches the horizon, at which point the first object jumps to a great distance away? WTF?

 

[JesseM]

Okay, that was misleading. Loses more velocity through time dilation, not undergo more time dilation.

I don't see that the faster loss of velocity has anything to do with time dilation, it's just a consequence of the coordinate transformation.

Why not? And what do you mean by "its own perspective"? Are you just talking about local coordinate-independent facts like whether the event of the second observer crossing the horizon happens at a later or earlier proper time than the event of the second observer passing the first observer? (if so, the answer is that crossing the horizon happens at an earlier proper time than passing the first observer, if he manages to pass the first observer at all before hitting the singularity) Or are you talking about a "frame" for the second observer, so it can make judgments about things like the velocity of the first observer as a function of time and the time at which the first observer crosses the horizon? If the latter, what coordinate system would this "frame" correspond to?

I meant the former. How can crossing the horizon happen before the second observer catches the first one? The second observer can't witness the first one crossing the horizon.

Sure he can, the second observer can witness the first crossing the horizon at the moment the moment he himself crosses the horizon. Again just consider the case of two observers that cross the Rindler horizon, where the second one won't see the first crossing it until the moment the second crosses the horizon. Of course that doesn't mean the second observer is right on top of the first observer as they cross, the second observer may see the first crossing the horizon at a great distance away (remember that it is only in Rindler coordinates that the Rindler horizon has an unchanging coordinate position, in any inertial coordinate system the Rindler horizon is expanding outward at the speed of light...similarly, for a black hole it is true in Schwarzschild coordinates that the event horizon has a fixed position, but in Kruskal-Szekeres coordinates it is also expanding outward at the speed of light, and likewise the event horizon is moving outward with a speed of c in the local inertial frame of a freefalling observer at the moment she crosses the horizon). Do you think this is a problem?

Either the second observer can witness the first crossing the horizon before they themselves get there or they can't. You can't have it both ways.

The second observer cannot witness the the first crossing before the second crosses the horizon, the second will only see this at the moment the second crosses the horizon.

Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space.

Once and for all, do you agree that "length contraction and time dilation" can only be defined relative to a particular choice of coordinate system, that the fact that length contraction and time dilation approach infinity as you approach the event horizon in Schwarzschild coordinates has no more coordinate-independent "reality" than the fact that length contraction and time dilation approach infinity as you approach the Rindler horizon in Rindler coordinates?

Once and for all you say? Does that mean you're not going to ask me this same question ever again? If the first and second ships meet up again then less time would have passed for the first ship, so there is a coordinate-independent reality.

Sure, overall elapsed proper time for two observers who cross paths twice is coordinate-independent. But the very fact that it is coordinate-independent means all coordinate systems will agree on this, so you cannot possibly use this fact to support the claim that time dilation and length contraction go to infinity on the horizon, since even a coordinate system (like Kruskal-Szekeres coordinates) where they don't go to infinity at the horizon will make the same prediction about elapsed proper time for any two observers who cross paths twice.

So, do you agree that the statement that time dilation and length contraction go to infinity at the horizon is a purely-coordinate dependent one that's true in Schwarzschild coordinates but has no coordinate-independent reality? If you do agree with that, then I don't understand what you meant by the statement that "Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space".

The fact that the second observer is slowing at a faster rate than the first from the third observers perspective just explains how the second observer may never be able to overtake the first observer before reaching the horizon from the third observers perspective.

No, it's a statement about how things work in a parcticular coordinate system, namely Schwarzschild coordinates. What do you mean when you talk about a given observer's "perspective"?? It's not uncommon in SR to talk about an inertial observer's "perspective" as a shorthand for their inertial rest frame, because every inertial observer has a unique inertial rest frame. However, non-inertial observers don't have unique non-inertial rest frames, there are an infinite number of different ways to construct a non-inertial coordinate system where a particular non-inertial observer is at rest. Physicists just don't talk about the "perspectives" of observers in GR, because it would be totally unclear what coordinate system they were referring to! And I suspect that you don't have any well-defined coordinate system in mind when you talk about the "perspectives" of different observers falling into the black hole--if you don't, your comments along these lines would seem to be totally meaningless.

I meant the third observer is sitting there watching the other two so we see what's happing from the perspective of someone who's maintaining a constant distance from the black hole. I spose you'd call it Schwarzschild coordinates.

No, I most certainly would not. In my view, the only meaningful statements you can make about the "perspectives" of different observers in GR are coordinate-independent statements about when light beams from various faraway events hit their own worldlines, in terms of their own proper time. An observer in GR does not have a "frame" or any other notion of a "perspective" besides this simple visual one, and coordinate systems are just coordinate systems, they don't represent the perspective of any particular observer. From your 'I meant the former' comment in response to my own 'Why not? And what do you mean by "its own perspective"?' above, perhaps you would actually agree with this, and I was misunderstanding you by thinking you were invoking some notion of frames--can you clarify this? Whenever you talk about an observer's "perspective", are you always just talking about coordinate-independent facts about what their clocks read when light rays from various events reach their eyes? If so, then do you agree that such facts will be the same regardless of what coordinate system we use, so it's fine to figure them out using Kruskal-Szekeres coordinates (where it takes only a finite coordinate time to reach the event horizon, just like with observers crossing the Rindler horizon when viewed in inertial coordinates) rather than Schwarzschild coordinates (where it takes an infinite coordinate time, just like with Rindler coordinates)?

Let me tell you what's meaningless. Those coordinate systems your so found of! I'm not saying they can't be useful, but they're meaningless in themselves. It's what they're describing that has meaning (my one's different because it's not describing it, it's literal), so let's just stick to describing things in real terms.

In general relativity, all calculations about coordinate-independent facts depend on having a coordinate system and a metric expressed in terms of those coordinates. Of course once we have figured out the coordinate-independent facts themselves we don't have to talk about the coordinate systems any more, but you seem unsatisfied when I just tell you what the coordinate-independent facts are, like the fact that both observers pass the horizon in finite time according to their own clocks, and that the second observer sees the image of the first observer crossing the horizon at a significant distance at the moment the second observer himself crosses the horizon.

I meant that if the first were to measure the size of the black hole just before crossing the horizon

Again, how do you "measure the size" of anything without using a coordinate system? Are you just talking about the apparent visual size, i.e. what percentage of your field of vision is occupied by the black hole just before crossing the horizon? (if you're interested in visual appearance, see the animations and images on this page)

Because the second observer can't witness the first crossing the horizon until they cross it themselves.

Yes, but they see it happening far away if they waited a significant time before pursuing the first observer.

It should never be too late because it can never witness the first one crossing.

If by "witness" you just mean "see the light from the event of the crossing", the second observer will witness the first observer crossing at the moment the second observer crosses the horizon himself (the event of the first crossing the horizon may look like it happened a great distance away at the moment of the second crosses, though).

"See the light from the event of the crossing" is exactly the same as "the event of the crossing". The light's moving slowly because time is. The event of the first one crossing may happen at a great distance away at the moment the second one crosses? The first one can't cross at all from the perspective of the second one at least until the second one reaches the horizon, at which point the first object jumps to a great distance away? WTF?

No, there is no "jump", the visual position of the horizon (which could be seen as a black region against a background filled with stars) continues to look far-off as you approach it, and even at the moment you cross it yourself. Again see this page, particularly the section labeled "Through the horizon". Also, by the same token if you're following another observer in, the moment you see the event on the first observer's worldline that occurred when they crossed the horizon, their apparent visual distance may be different than the apparent visual distance of the black sphere you see below you, i.e. it may not look like they're crossing the horizon at that moment.

 

[A-wal]

I don't see that the faster loss of velocity has anything to do with time dilation, it's just a consequence of the coordinate transformation.

Um, I was specifically referring to the effect of gravitational time dilation on relative velocity. NOTHING is a consequence of coordinates.

Sure, overall elapsed proper time for two observers who cross paths twice is coordinate-independent. But the very fact that it is coordinate-independent means all coordinate systems will agree on this, so you cannot possibly use this fact to support the claim that time dilation and length contraction go to infinity on the horizon, since even a coordinate system (like Kruskal-Szekeres coordinates) where they don't go to infinity at the horizon will make the same prediction about elapsed proper time for any two observers who cross paths twice.

So, do you agree that the statement that time dilation and length contraction go to infinity at the horizon is a purely-coordinate dependent one that's true in Schwarzschild coordinates but has no coordinate-independent reality? If you do agree with that, then I don't understand what you meant by the statement that "Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space".

I would have thought it was obvious. Any external observer can never witness an object crossing the horizon.

No, I most certainly would not. In my view, the only meaningful statements you can make about the "perspectives" of different observers in GR are coordinate-independent statements about when light beams from various faraway events hit their own worldlines, in terms of their own proper time. An observer in GR does not have a "frame" or any other notion of a "perspective" besides this simple visual one, and coordinate systems are just coordinate systems, they don't represent the perspective of any particular observer. From your 'I meant the former' comment in response to my own 'Why not? And what do you mean by "its own perspective"?' above, perhaps you would actually agree with this, and I was misunderstanding you by thinking you were invoking some notion of frames--can you clarify this? Whenever you talk about an observer's "perspective", are you always just talking about coordinate-independent facts about what their clocks read when light rays from various events reach their eyes? If so, then do you agree that such facts will be the same regardless of what coordinate system we use, so it's fine to figure them out using Kruskal-Szekeres coordinates (where it takes only a finite coordinate time to reach the event horizon, just like with observers crossing the Rindler horizon when viewed in inertial coordinates) rather than Schwarzschild coordinates (where it takes an infinite coordinate time, just like with Rindler coordinates)?

Yes, that is what I meant and I don't want to use any coordinate systems apart from what's actually observed. You seem to desperately want to keep using them though.

Again, how do you "measure the size" of anything without using a coordinate system? Are you just talking about the apparent visual size, i.e. what percentage of your field of vision is occupied by the black hole just before crossing the horizon? (if you're interested in visual appearance, see the animations and images on this page)

Any, that will do. As long as they use the same method so they can compare.

Sure he can, the second observer can witness the first crossing the horizon at the moment the moment he himself crosses the horizon. Again just consider the case of two observers that cross the Rindler horizon, where the second one won't see the first crossing it until the moment the second crosses the horizon. Of course that doesn't mean the second observer is right on top of the first observer as they cross, the second observer may see the first crossing the horizon at a great distance away (remember that it is only in Rindler coordinates that the Rindler horizon has an unchanging coordinate position, in any inertial coordinate system the Rindler horizon is expanding outward at the speed of light...similarly, for a black hole it is true in Schwarzschild coordinates that the event horizon has a fixed position, but in Kruskal-Szekeres coordinates it is also expanding outward at the speed of light, and likewise the event horizon is moving outward with a speed of c in the local inertial frame of a freefalling observer at the moment she crosses the horizon). Do you think this is a problem?

The second observer cannot witness the the first crossing before the second crosses the horizon, the second will only see this at the moment the second crosses the horizon.

In general relativity, all calculations about coordinate-independent facts depend on having a coordinate system and a metric expressed in terms of those coordinates. Of course once we have figured out the coordinate-independent facts themselves we don't have to talk about the coordinate systems any more, but you seem unsatisfied when I just tell you what the coordinate-independent facts are, like the fact that both observers pass the horizon in finite time according to their own clocks, and that the second observer sees the image of the first observer crossing the horizon at a significant distance at the moment the second observer himself crosses the horizon.

Yes, but they see it happening far away if they waited a significant time before pursuing the first observer.

No, there is no "jump", the visual position of the horizon (which could be seen as a black region against a background filled with stars) continues to look far-off as you approach it, and even at the moment you cross it yourself. Again see this page, particularly the section labeled "Through the horizon". Also, by the same token if you're following another observer in, the moment you see the event on the first observer's worldline that occurred when they crossed the horizon, their apparent visual distance may be different than the apparent visual distance of the black sphere you see below you, i.e. it may not look like they're crossing the horizon at that moment.

Not a single mention of the event horizon seemingly being in two places at once and now FIVE! That info would have been very handy before, especially when I was talking about the receding horizon. It could have saved us both a lot of time. Right, how does that happen? If you're saying they're seeing a time delayed image of them crossing where the horizon used to be then the black hole has to have grown in the time between the two objects crossing. The event horizon by definition is the point at which an object can no longer possibly find the energy to escape, so how can it be in two places at the same time? It should be (and appear to be) in the same place for all objects. Also if the event horizon continues to look far off as you approach it then light is escaping from the “actual” horizon.

 

[JesseM]

Um, I was specifically referring to the effect of gravitational time dilation on relative velocity. NOTHING is a consequence of coordinates.

There is no notion of "relative velocity" (or gravitational time dilation) outside of coordinate systems, they are frame-dependent concepts in GR.

So, do you agree that the statement that time dilation and length contraction go to infinity at the horizon is a purely-coordinate dependent one that's true in Schwarzschild coordinates but has no coordinate-independent reality? If you do agree with that, then I don't understand what you meant by the statement that "Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space".

I would have thought it was obvious. Any external observer can never witness an object crossing the horizon.

No, it's not obvious at all, so you need to explain it. "An external observer can never witness an object crossing the horizon" has no clear connection to the statement "Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space". How does not seeing things cross the horizon have anything to do with length contraction or time dilation or the notion that "the black hole exists at a single point in time and space"?

Yes, that is what I meant and I don't want to use any coordinate systems apart from what's actually observed. You seem to desperately want to keep using them though.

I'm fine with not using them as long as you don't try to sneak in inherently coordinate-dependent notions like "relative velocity" or "length contraction" (and we can only talk about 'time dilation' in a coordinate-independent way if we're talking about a twin-paradox-like situation where two observers meet twice and compare the elapsed time on their clocks between meetings, any notion of 'time dilation' other than this is also inherently coordinate-dependent).

Again, how do you "measure the size" of anything without using a coordinate system? Are you just talking about the apparent visual size, i.e. what percentage of your field of vision is occupied by the black hole just before crossing the horizon? (if you're interested in visual appearance, see the animations and images on this page)

Any, that will do. As long as they use the same method so they can compare.

OK, if you just want to talk about apparent visual size, then for a Schwarzschild black hole both will see the BH as being the same apparent size at the moment they each cross the horizon.

Not a single mention of the event horizon seemingly being in two places at once and now FIVE! That info would have been very handy before, especially when I was talking about the receding horizon.

I thought you were talking about what "really" happens in some frame, you didn't specify that you wanted to talk about visual appearances.

It could have saved us both a lot of time. Right, how does that happen?

I don't know of any way to "explain" visual phenomena like this except by calculating which events have light beams that all converge on the observer at the moment he crosses the horizon, which would require using a coordinate system to work out the timing of events and the light travel times (and in any cases the calculations for the apparent size and the horizon would be above my head I think, though the appendix of this paper discusses angular size of spheres in GR, and http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000078000002000204000001&idtype=cvips&gifs=yes&ref=no probably goes into more detail but isn't available for free online).

If you're saying they're seeing a time delayed image of them crossing where the horizon used to be then the black hole has to have grown in the time between the two objects crossing.

"Grown" in what sense? Remember, the only notion of size you wanted to talk about was apparent visual size--do you just mean the apparent visual size grows as the falling observer approaches it? Well naturally, that would be true of any body!

The event horizon by definition is the point at which an object can no longer possibly find the energy to escape, so how can it be in two places at the same time? It should be (and appear to be) in the same place for all objects.

Why should it "appear to be" in the same place for all objects? On the contrary, if a row of observers are falling in one after the other, then when an observer in the rear crosses the horizon the light he's seeing from the other observers at that moment is the light they emitted at the moment each of them crossed the horizon, but this observer will still see the other observer in a row in front of him, at a range of different apparent visual distances. It's really no different from the idea in SR that if you have a row of observers who enter the future light cone of some event E in succession (which would also mean they were crossing the Rindler horizon of some possible observer accelerating at a constant rate), then at the moment the observer in the rear first enters the future light cone, the light he's seeing from other observers at that moment is the light each of them emitted at the moment they entered the future light cone, and at this moment they still appear to be in a row at different distances. (And of course just as the Rindler horizon looks like the edge of a light cone when plotted in Minkowski coordinates, so the black hole event horizon looks like the edge of a light cone when plotted in Kruskal-Szekeres coordinates, which is a coordinate system where all light worldlines are straight lines at 45 degrees from the time axis, just like in Minkowski coordinates. But I'm putting this in parentheses because you may not want to think about the representation of the horizon and of light signals in some coordinate system--the paragraph above outside of the parentheses deals only with visual observations without explaining to explain why this is what the falling observer sees, so if you don't want to deal with coordinate systems you pretty much have to accept it without explanation).

Also if the event horizon continues to look far off as you approach it then light is escaping from the “actual” horizon.

No, it isn't. But again, if you just make wrong statements without providing your reasoning, I can't really help you to see where your error lies. It may help to remember that the apparent size of the horizon is based on the set of angles where no light from distant stars can reach your eyes, if there was a solid shell hovering just an infinitesimal fraction of a nanometer above the event horizon, I'm pretty sure the visual size of the solid shell would be different from the apparent size of the region where no distant stars can be seen (it seems like it would have to, since you would see the shell rush up to meet you and you'd punch through it a moment before you crossed the horizon, but at the moment you cross the horizon the black sphere where stars are blocked still looks like it's far away from you).

edit: actually, looking at the "through the horizon" section here, they say:

As you fall through the horizon, at 1 Schwarzschild radius, something quite unexpected happens. You thought you were going to fall through the red grid that supposedly marks the horizon. But no. The red grid still stands off ahead of you.
Instead, the horizon splits into two as you pass through it. Click on Penrose diagrams to understand more about why the horizon splits in two.

And the Penrose diagram, which is basically just like a compacted version of a Kruskal-Szekeres diagram, shows why the horizon appears to split in two as you cross it in this animation. If there was a spherical shell eternally hovering just above the event horizon, in this graph it would look like a hyperbola that hugs extremely close to both the antihorizon (really the white hole event horizon) on the bottom and the regular horizon on top, so I think up until just before the moment you reached it, the shell's apparent size would look about the same size as the black region (since on the page the red grid marking the antihorizon is the same size as the black region obscuring stars), and then suddenly it would split off from the black sphere and rush up to meet you (or perhaps just part of it would rush up to meet you, like a spike or bump rising out of the sphere) like the white grid representing the horizon (see the third animation on the main page, or the white 'Schwarzschild bubble' in the similar animations on http://casa.colorado.edu/~ajsh/singularity.html).

 

[A-wal]

There is no notion of "relative velocity" (or gravitational time dilation) outside of coordinate systems, they are frame-dependent concepts in GR.

Relative velocity still applies in curved space-time.

No, it's not obvious at all, so you need to explain it. "An external observer can never witness an object crossing the horizon" has no clear connection to the statement "Length contraction and time dilation force anything that does cross the horizon to do it when the black hole exists at a single point in time and space". How does not seeing things cross the horizon have anything to do with length contraction or time dilation or the notion that "the black hole exists at a single point in time and space"?

The reason an external observer can never witness an object crossing the horizon is length contraction/time dilation. But they will have to see everything cross the horizon eventually if the black hole has a limited life span. From this (any external) perspective it would have to mean that it happens when the black hole exists at a single point in time and space at the very end of its life.

I'm fine with not using them as long as you don't try to sneak in inherently coordinate-dependent notions like "relative velocity" or "length contraction" (and we can only talk about 'time dilation' in a coordinate-independent way if we're talking about a twin-paradox-like situation where two observers meet twice and compare the elapsed time on their clocks between meetings, any notion of 'time dilation' other than this is also inherently coordinate-dependent).

If gravitational time dilation applies when two objects with different world lines meet up then it always applies. If not, then at what distance would it suddenly not apply?

OK, if you just want to talk about apparent visual size, then for a Schwarzschild black hole both will see the BH as being the same apparent size at the moment they each cross the horizon.

I take it this assumes an ever-lasting black hole?

I thought you were talking about what "really" happens in some frame, you didn't specify that you wanted to talk about visual appearances.

Same thing isn't it?

I don't know of any way to "explain" visual phenomena like this except by calculating which events have light beams that all converge on the observer at the moment he crosses the horizon, which would require using a coordinate system to work out the timing of events and the light travel times (and in any cases the calculations for the apparent size and the horizon would be above my head I think, though the appendix of this paper discusses angular size of spheres in GR, and http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000078000002000204000001&idtype=cvips&gifs=yes&ref=no probably goes into more detail but isn't available for free online).

Okay so there is the actual horizon and an apparent one when viewing an object that crossed at an earlier time, and it appears to be beyond the actual horizon. That would mean that there's always an actual horizon and an apparent one because things like that never just start happen, it's always a matter of degree. That's why I questioned it earlier when it seemed you were implying there was a jump. So the two horizons appear closer the further you are away? Presumably if you were to move away from the black hole then the object that appears beyond your event horizon would appear to move back outwards?

"Grown" in what sense? Remember, the only notion of size you wanted to talk about was apparent visual size--do you just mean the apparent visual size grows as the falling observer approaches it? Well naturally, that would be true of any body!

I meant that object that crossed earlier had a larger size relative to the black hole than you do now, meaning the black hole has grown since (assuming both objects are the same size of course).

Why should it "appear to be" in the same place for all objects? On the contrary, if a row of observers are falling in one after the other, then when an observer in the rear crosses the horizon the light he's seeing from the other observers at that moment is the light they emitted at the moment each of them crossed the horizon, but this observer will still see the other observer in a row in front of him, at a range of different apparent visual distances. It's really no different from the idea in SR that if you have a row of observers who enter the future light cone of some event E in succession (which would also mean they were crossing the Rindler horizon of some possible observer accelerating at a constant rate), then at the moment the observer in the rear first enters the future light cone, the light he's seeing from other observers at that moment is the light each of them emitted at the moment they entered the future light cone, and at this moment they still appear to be in a row at different distances. (And of course just as the Rindler horizon looks like the edge of a light cone when plotted in Minkowski coordinates, so the black hole event horizon looks like the edge of a light cone when plotted in Kruskal-Szekeres coordinates, which is a coordinate system where all light worldlines are straight lines at 45 degrees from the time axis, just like in Minkowski coordinates. But I'm putting this in parentheses because you may not want to think about the representation of the horizon and of light signals in some coordinate system--the paragraph above outside of the parentheses deals only with visual observations without explaining to explain why this is what the falling observer sees, so if you don't want to deal with coordinate systems you pretty much have to accept it without explanation).

You cannot see an object cross the horizon until you do so the closer object could still find the energy to escape. From your perspective though it shouldn't be able to escape because it's beyond your horizon. Let's assume it can and does escape. It starts to head towards you. What then? If it pulls right up and sits along side you then your horizon is now its horizon, so it's still just outside it. It's moved to you but it's still the same distance away from the horizon?

No, it isn't. But again, if you just make wrong statements without providing your reasoning, I can't really help you to see where your error lies. It may help to remember that the apparent size of the horizon is based on the set of angles where no light from distant stars can reach your eyes, if there was a solid shell hovering just an infinitesimal fraction of a nanometer above the event horizon, I'm pretty sure the visual size of the solid shell would be different from the apparent size of the region where no distant stars can be seen (it seems like it would have to, since you would see the shell rush up to meet you and you'd punch through it a moment before you crossed the horizon, but at the moment you cross the horizon the black sphere where stars are blocked still looks like it's far away from you).

The horizon is just in front of you but you can see an object beyond it, meaning light is escaping the horizon. And what does the earlier in falling object see when looking back at the later one? Does that mean the apparent horizon can be in front or behind you, or would an object behind have to actually catch up to you to reach the horizon. That means their two views of each other would be contradictory. One says there's space between them and one doesn't.

edit: actually, looking at the "through the horizon" section here, they say:

And the Penrose diagram, which is basically just like a compacted version of a Kruskal-Szekeres diagram, shows why the horizon appears to split in two as you cross it in this animation. If there was a spherical shell eternally hovering just above the event horizon, in this graph it would look like a hyperbola that hugs extremely close to both the antihorizon (really the white hole event horizon) on the bottom and the regular horizon on top, so I think up until just before the moment you reached it, the shell's apparent size would look about the same size as the black region (since on the page the red grid marking the antihorizon is the same size as the black region obscuring stars), and then suddenly it would split off from the black sphere and rush up to meet you (or perhaps just part of it would rush up to meet you, like a spike or bump rising out of the sphere) like the white grid representing the horizon (see the third animation on the main page, or the white 'Schwarzschild bubble' in the similar animations on http://casa.colorado.edu/~ajsh/singularity.html).

Okay, but that doesn't apply to your view of an object that falls in at an earlier time because there can't be a jump, like we said. So this notion of a split horizon must be describing something else, yes?

 

[JesseM]

Relative velocity still applies in curved space-time.

No, there is no frame-independent notion of "relative velocity" in curved spacetime. Take a look at all the posters explaining this to Anamitra in this thread, for example.

The reason an external observer can never witness an object crossing the horizon is length contraction/time dilation.

It is true that visually, a clock falling towards a black hole will appear to run slower and slower as it approaches, and will also appear increasingly compressed in the radial direction. But usually "time dilation" is understood in terms of a slowdown in the rate a clock is ticking relative to coordinate time in some reference frame--visual slowdown is not the same as "time dilation". Consider the simple example of Doppler shift in SR, which causes a clock moving towards you to appear to be ticking faster than your clock despite the fact that it's ticking slow in your frame, and which also causes a clock moving away from you to appear to be ticking even slower than the time dilation factor. Likewise with "length contraction"--the visual apparent length of an object may be different than its length-contracted length in your frame due to the fact that the light you see at any given moment was actually emitted by different parts of the object at different times in your frame (the Penrose-Terrell effect). So if you're referring to visual appearances, you should use some other term besides "length contraction/time dilation" as people (like me in earlier posts) are liable to misinterpret your meaning.

But they will have to see everything cross the horizon eventually if the black hole has a limited life span. From this (any external) perspective it would have to mean that it happens when the black hole exists at a single point in time and space at the very end of its life.

Yes, in the case of an evaporating block hole it's apparently true that theoretically (assuming you could detect light of arbitrary redshift, and assuming light was emitted continuously rather than in discrete photons), you would see light from objects crossing the horizon at the moment the black hole finally evaporated to zero radius (see the section 'What about Hawking radiation? Won't the black hole evaporate before you get there?' on this page). But that's just a statement about when the light emitted by those observers as they crossed the horizon finally reaches the outside observer, it doesn't imply that the black hole actually was point-sized for the infalling observers when they crossed the horizon.

I'm fine with not using them as long as you don't try to sneak in inherently coordinate-dependent notions like "relative velocity" or "length contraction" (and we can only talk about 'time dilation' in a coordinate-independent way if we're talking about a twin-paradox-like situation where two observers meet twice and compare the elapsed time on their clocks between meetings, any notion of 'time dilation' other than this is also inherently coordinate-dependent).

If gravitational time dilation applies when two objects with different world lines meet up then it always applies. If not, then at what distance would it suddenly not apply?

Your language is overly vague, I don't know what it means for it to "apply" or "not apply". To be precise about what I mean, suppose you have two observers who compare clock readings at a single point, fly apart, and then reunite and compare clock readings again. Pick any point A on the worldline of one observer, and any point B on the worldline of the second observer. Then it can be true that different coordinate systems disagree on whether the clock at point A was ticking slower than the clock at point B or vice versa, and likewise it can be true that other observers with different paths through space time disagree about the exact ration between the visual rate of ticking of the clock at point A vs. the clock at point B. Nevertheless, all frames/observers agree about the total elapsed time on each clock when they finally reunite at a single point in spacetime. Would you say this means time dilation "applies" or "doesn't apply" at points A and B on each clock's worldline?

OK, if you just want to talk about apparent visual size, then for a Schwarzschild black hole both will see the BH as being the same apparent size at the moment they each cross the horizon.

I take it this assumes an ever-lasting black hole?

Right, I'm talking about the Schwarzschild solution. For a black hole that was shrinking due to Hawking radiation or growing due to infalling matter, I'm pretty confident the apparent visual size would be different for observers who cross the horizon at different times (i.e. in terms of visual appearances, neither sees the other as right next to themselves at the moment they're crossing the horizon, even though they fall in along the same radial direction)

I thought you were talking about what "really" happens in some frame, you didn't specify that you wanted to talk about visual appearances.

Same thing isn't it?

No, but my way of talking may be confusing, when I said "really" I wasn't talking about frame-invariant physical facts, but rather the "real" coordinates of events in a given frame, as distinguished from visual appearances. As I mentioned above, even in SR things like visual clock rates and visual lengths can be different from the "real" amount of time dilation and length contraction in the observer's inertial rest frame.

Okay so there is the actual horizon and an apparent one when viewing an object that crossed at an earlier time, and it appears to be beyond the actual horizon. That would mean that there's always an actual horizon and an apparent one because things like that never just start happen, it's always a matter of degree. That's why I questioned it earlier when it seemed you were implying there was a jump.

Well, normally the horizon isn't something you can "see" directly, but if we imagine that somehow every event on the horizon generated photons which traveled in all directions of the future light cone of that event, so that the horizon was visible in its own right (as opposed to our just being able to see a black area where light from distant stars is blocked), then you would only begin to "see" the horizon at the moment you crossed it, since none of the photons generated at the horizon would make it outside the horizon, but they could reach observers falling into the horizon. Meanwhile for a Schwarzschild black hole, there is also an "antihorizon" which consists of both a white hole event horizon in our universe and a black hole event horizon leading to a different universe (see the maximally extended Kruskal-Szekeres diagram for a discussion). So if we imagine all events on the antihorizon were also generating photons of a different color (say, red) which traveled in all directions in the future light cone of the event that generated them, then as you traveled towards the black hole you would see a red sphere representing the white hole's event horizon, and after you crossed the horizon you'd continue to see the red sphere, but now you'd be seeing light from the black hole horizon in the other universe. But after you crossed the horizon you'd also suddenly start seeing light from black hole horizon in your own universe--if you imagine that this light is white instead of red, then you should be able to follow what's depicted in the third movie from the top on this page, where we approach a black hole covered with red grid lines and then suddenly a new set of white grid lines appear (forming a sort of dome shape above us) at the moment we cross the horizon. If you read about the maximally extended Kruskal-Szekeres diagram, you should be able to follow this similar Penrose diagram that I linked to earlier, which shows which events on the horizon and antihorizon emitted the red and white light reaching you at each point in your fall. And I also linked to http://casa.colorado.edu/~ajsh/singularity.html which shows somewhat cruder animations of what a falling observer would see (again with the antihorizon in red and the white 'Schwarzschild bubble' that you see after crossing the horizon in white), with somewhat more detailed accompanying explanation (for example, he mentions that at the moment you cross the horizon the white Schwarzschild bubble would first appear as a straight line reaching from you to the red horizon, only later expanding into a bubble, and that at that moment other objects which fell into the black hole along the same axis would be arrayed along this line, all appearing as they did the moment they crossed the horizon).

So the two horizons appear closer the further you are away?

No, why do you say that?

"Grown" in what sense? Remember, the only notion of size you wanted to talk about was apparent visual size--do you just mean the apparent visual size grows as the falling observer approaches it? Well naturally, that would be true of any body!

I meant that object that crossed earlier had a larger size relative to the black hole than you do now

And are you using "size" to refer only to apparent visual size, i.e. how much of the visual field is taken up by the black region devoid of stars (or the red antihorizon which apparently would coincide with it)? If so then I think as long as the black hole wasn't significantly changing its coordinate size (in some appropriate coordinate system like Eddington-Finkelstein coordinates) due to evaporation or significant influxes of matter, than observers who fell in at different times also shouldn't see a difference in apparent visual size at the moment they cross the horizon. If you want "size" to refer to something more than apparent visual size, I don't see how you can define any other notion of size without reference to a coordinate system.

You cannot see an object cross the horizon until you do so the closer object could still find the energy to escape. From your perspective though it shouldn't be able to escape because it's beyond your horizon.

What do you mean "beyond your horizon"? If the object escapes, then you will always see it as outside the red antihorizon, and if you ended up crossing the horizon yourself you would presumably see it remain outside the white "Schwarzschild bubble" (which consists of light from events on the black hole event horizon) which you are underneath.

Let's assume it can and does escape. It starts to head towards you.

And what are "you" doing? Remaining outside the horizon too, or crossing through the black hole event horizon? If the latter, then you will see it pass by you and move up to a greater radius before you pass the event horizon and begin to see the "Schwarzschild bubble".

The horizon is just in front of you but you can see an object beyond it, meaning light is escaping the horizon.

If other objects fell into the black hole horizon from your universe, you will always see them above the red antihorizon, not "beyond" it (though you could potentially see objects beyond it that were in regions IV or III of the Kruskal diagram, in either the white hole interior region or the "other universe" respectively. Once you cross the horizon yourself and see yourself under the white Schwarzschild bubble, every other object that fell in before you will be under the bubble too, and you could look above you and see other objects entering the bubble from above.

And what does the earlier in falling object see when looking back at the later one?

The earlier object would also see a white Schwarzschild bubble that appeared at the moment it crossed the horizon (again assuming for the sake of visualization that each event on the event horizon emits white light in all possible directions, in reality of course the bubble wouldn't actually be a visible shape), and assuming light from your crossing the horizon had time to reach the earlier object before it hit the singularity, it would at some point see you cross through the top of this expanding bubble.

 

[Austin0]

.

Well, normally the horizon isn't something you can "see" directly, but if we imagine that somehow every event on the horizon generated photons which traveled in all directions of the future light cone of that event, so that the horizon was visible in its own right (as opposed to our just being able to see a black area where light from distant stars is blocked), then you would only begin to "see" the horizon at the moment you crossed it,

.

Since this thread seems to be winding down I hope it is alright to ask a somewhat off topic question.
With the Rindler horizon; is there any visual image that can be associated with it as viewed from inside the accelerating frame?
Or is it more a question of light that will never reach you but you were not aware of anyway?
SO a diminished field of relative obscurity?
Thanks

 

[JesseM]

Since this thread seems to be winding down I hope it is alright to ask a somewhat off topic question.
With the Rindler horizon; is there any visual image that can be associated with it as viewed from inside the accelerating frame?
Or is it more a question of light that will never reach you but you were not aware of anyway?
SO a diminished field of relative obscurity?
Thanks

Well, note that the Rindler horizon has an upper and lower component (or maybe it would be better to say there are two horizons?), as seen in this diagram from the wikipedia article which plots both the horizons and lines of constant Rindler position and time in a Minkowski diagram. The upper component behaves like a black hole horizon in that the Rindler observers can't see anything that crosses it (though an observer on the "Rindler wedge" can cross it herself if she doesn't remain at rest in Rindler coordinates), but the lower component behaves like a white hole horizon in that Rindler observers can see light from events "below" it (the light rays would be traveling parallel to the upper horizon in the diagram), and it's impossible for any observer on the Rindler wedge to cross it (in Minkowski coordinates it's moving away from anyone on the Rindler wedge with a speed of c). So, in terms of Minkowski coordinates this means an observer on the Rindler wedge can see light from events that occurred arbitrarily far in either direction along the x-axis, including events much further than the current distance of the Rindler horizon. So certainly an inertial observer on the Rindler wedge would see nothing unusual, and at any given moment an accelerating Rindler observer should see the same thing as an inertial observer at the same point in spacetime who is instantaneously at rest relative to themselves.

 

[A-wal]

No, there is no frame-independent notion of "relative velocity" in curved spacetime. Take a look at all the posters explaining this to Anamitra in this thread, for example.

That's not what I meant. If something has a relative velocity of .5c for example then that velocity doesn't disappear in curved space-time. All space-time is at least a little bit curved so there'd be no such thing as relative verlocity at all.

It is true that visually, a clock falling towards a black hole will appear to run slower and slower as it approaches, and will also appear increasingly compressed in the radial direction. But usually "time dilation" is understood in terms of a slowdown in the rate a clock is ticking relative to coordinate time in some reference frame--visual slowdown is not the same as "time dilation". Consider the simple example of Doppler shift in SR, which causes a clock moving towards you to appear to be ticking faster than your clock despite the fact that it's ticking slow in your frame, and which also causes a clock moving away from you to appear to be ticking even slower than the time dilation factor. Likewise with "length contraction"--the visual apparent length of an object may be different than its length-contracted length in your frame due to the fact that the light you see at any given moment was actually emitted by different parts of the object at different times in your frame (the Penrose-Terrell effect). So if you're referring to visual appearances, you should use some other term besides "length contraction/time dilation" as people (like me in earlier posts) are liable to misinterpret your meaning.

I stand by my claim that the reason an external observer can never witness an object crossing the horizon is length contraction/time dilation. Gravitational time dilation/length contraction doesn't make it any less real.

Yes, in the case of an evaporating block hole it's apparently true that theoretically (assuming you could detect light of arbitrary redshift, and assuming light was emitted continuously rather than in discrete photons), you would see light from objects crossing the horizon at the moment the black hole finally evaporated to zero radius (see the section 'What about Hawking radiation? Won't the black hole evaporate before you get there?' on this page). But that's just a statement about when the light emitted by those observers as they crossed the horizon finally reaches the outside observer, it doesn't imply that the black hole actually was point-sized for the infalling observers when they crossed the horizon.

You're still sure about that then. I think it does.

Your language is overly vague, I don't know what it means for it to "apply" or "not apply". To be precise about what I mean, suppose you have two observers who compare clock readings at a single point, fly apart, and then reunite and compare clock readings again. Pick any point A on the worldline of one observer, and any point B on the worldline of the second observer. Then it can be true that different coordinate systems disagree on whether the clock at point A was ticking slower than the clock at point B or vice versa, and likewise it can be true that other observers with different paths through space time disagree about the exact ration between the visual rate of ticking of the clock at point A vs. the clock at point B. Nevertheless, all frames/observers agree about the total elapsed time on each clock when they finally reunite at a single point in spacetime. Would you say this means time dilation "applies" or "doesn't apply" at points A and B on each clock's worldline?

It's just easier to compare them when there's no distance between them because "all frames/observers agree about the total elapsed time on each clock when they finally reunite at a single point in spacetime" like you said. Use common sense instead of thinking in graphs. They can never ocupy the same exact space, but that doesn't mean they can't compare watches. If they can do that then how far away from each other do they have to be before they can't?

No, but my way of talking may be confusing, when I said "really" I wasn't talking about frame-invariant physical facts, but rather the "real" coordinates of events in a given frame, as distinguished from visual appearances. As I mentioned above, even in SR things like visual clock rates and visual lengths can be different from the "real" amount of time dilation and length contraction in the observer's inertial rest frame.

Yes, okay Doppler shift is a good example of visual appearance being different to what's actually happening, but we're ignoring Doppler shift.

So the two horizons appear closer the further you are away?

No, why do you say that?

Just because when would the horizon split? I just assumed it was always split and it's only noticable when you get close.

Well, normally the horizon isn't something you can "see" directly, but if we imagine that somehow every event on the horizon generated photons which traveled in all directions of the future light cone of that event, so that the horizon was visible in its own right (as opposed to our just being able to see a black area where light from distant stars is blocked), then you would only begin to "see" the horizon at the moment you crossed it, since none of the photons generated at the horizon would make it outside the horizon, but they could reach observers falling into the horizon. Meanwhile for a Schwarzschild black hole, there is also an "antihorizon" which consists of both a white hole event horizon in our universe and a black hole event horizon leading to a different universe (see the maximally extended Kruskal-Szekeres diagram for a discussion). So if we imagine all events on the antihorizon were also generating photons of a different color (say, red) which traveled in all directions in the future light cone of the event that generated them, then as you traveled towards the black hole you would see a red sphere representing the white hole's event horizon, and after you crossed the horizon you'd continue to see the red sphere, but now you'd be seeing light from the black hole horizon in the other universe. But after you crossed the horizon you'd also suddenly start seeing light from black hole horizon in your own universe--if you imagine that this light is white instead of red, then you should be able to follow what's depicted in the third movie from the top on this page, where we approach a black hole covered with red grid lines and then suddenly a new set of white grid lines appear (forming a sort of dome shape above us) at the moment we cross the horizon. If you read about the maximally extended Kruskal-Szekeres diagram, you should be able to follow this similar Penrose diagram that I linked to earlier, which shows which events on the horizon and antihorizon emitted the red and white light reaching you at each point in your fall. And I also linked to http://casa.colorado.edu/~ajsh/singularity.html which shows somewhat cruder animations of what a falling observer would see (again with the antihorizon in red and the white 'Schwarzschild bubble' that you see after crossing the horizon in white), with somewhat more detailed accompanying explanation (for example, he mentions that at the moment you cross the horizon the white Schwarzschild bubble would first appear as a straight line reaching from you to the red horizon, only later expanding into a bubble, and that at that moment other objects which fell into the black hole along the same axis would be arrayed along this line, all appearing as they did the moment they crossed the horizon).

And are you using "size" to refer only to apparent visual size, i.e. how much of the visual field is taken up by the black region devoid of stars (or the red antihorizon which apparently would coincide with it)? If so then I think as long as the black hole wasn't significantly changing its coordinate size (in some appropriate coordinate system like Eddington-Finkelstein coordinates) due to evaporation or significant influxes of matter, than observers who fell in at different times also shouldn't see a difference in apparent visual size at the moment they cross the horizon. If you want "size" to refer to something more than apparent visual size, I don't see how you can define any other notion of size without reference to a coordinate system.

What do you mean "beyond your horizon"? If the object escapes, then you will always see it as outside the red antihorizon, and if you ended up crossing the horizon yourself you would presumably see it remain outside the white "Schwarzschild bubble" (which consists of light from events on the black hole event horizon) which you are underneath.

And what are "you" doing? Remaining outside the horizon too, or crossing through the black hole event horizon? If the latter, then you will see it pass by you and move up to a greater radius before you pass the event horizon and begin to see the "Schwarzschild bubble".

If other objects fell into the black hole horizon from your universe, you will always see them above the red antihorizon, not "beyond" it (though you could potentially see objects beyond it that were in regions IV or III of the Kruskal diagram, in either the white hole interior region or the "other universe" respectively. Once you cross the horizon yourself and see yourself under the white Schwarzschild bubble, every other object that fell in before you will be under the bubble too, and you could look above you and see other objects entering the bubble from above.

The earlier object would also see a white Schwarzschild bubble that appeared at the moment it crossed the horizon (again assuming for the sake of visualization that each event on the event horizon emits white light in all possible directions, in reality of course the bubble wouldn't actually be a visible shape), and assuming light from your crossing the horizon had time to reach the earlier object before it hit the singularity, it would at some point see you cross through the top of this expanding bubble.

Wft are you on about. Keep it simple.

Let's go back to the two horizons. One just in front of you and one some distance away just in front of another object that very proberbly crossed the horizon earlier. How could it have if it's always possible for it to escape. This isn't just visual, it's real. If it can escape then in what sense is the horizon in front of you (which the other object is inside) real? What happens if you mantain your distance and the other object moves back out to your possition? It would mean it's moved but the event horizon has followed it. Just ignore the absolute horizon. Why would there even be one? It would be subject to gravitational length contraction which would mean its size would decrease the closer you got, going all the way up to infinity at the horizon. If it's infinite in one "frame" then it's infinite in all of them. You can never reach it in exactly the same way you can never reach c. The black hole becomes the singularity at 0 range, like the whole universe becomes a singularity at c.

 

[JesseM]

That's not what I meant. If something has a relative velocity of .5c for example then that velocity doesn't disappear in curved space-time.

But in fact it does "disappear" in the sense of there no longer being a unique objective truth about "relative velocity". You can define a notion of relative velocity that's coordinate-dependent, or a notion of relative velocity based on parallel transport over a particular path through spacetime, but there are always multiple possible coordinate systems and multiple possible paths, so there's no "objective" physical notion of relative velocity for objects at different points in spacetime in GR. For example, take a look at p. 167 of this textbook which discusses the notion of the relative velocity 4-vector in SR, and then in a footnote says "The concept of relative four-vector cannot be extended to theories of physics formulated over a non-linear space, i.e., curved spaces (e.g., General Relativity)" Likewise see this page from the site of physicist John Baez, who says:

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

If you still don't believe that there is no coordinate- and path-independent notion of "relative velocity" for objects at different points in spacetime in GR, feel free to start a new thread asking the resident GR experts in the forum about this.

All space-time is at least a little bit curved so there'd be no such thing as relative verlocity at all.

No such thing in GR, at least if you want a single uniquely correct objective "relative velocity" as opposed to a coordinate-dependent or path-dependent one.

I stand by my claim that the reason an external observer can never witness an object crossing the horizon is length contraction/time dilation. Gravitational time dilation/length contraction doesn't make it any less real.

OK, but you completely failed to address my question about whether you were talking about visuals or something else. I can think of only 3 senses in which we can talk about clocks ticking at different rates in GR:

1. Visual appearances--how fast the an observer sees the image of another clock ticking relative to his own clock
2. Local comparisons of elapsed times, as in the twin paradox where each twin looks at how much time each has aged between two meetings
3. Coordinate-dependent notions of how fast each clock is ticking relative to coordinate time at a particular moment (which depends on the definition of simultaneity in your chosen coordinate system)

If you are confident there is some other sense in which we can compare the rates of different clocks, please spell it out with some reference to the technical definition you are thinking of in GR. If you have some kind of hunch or intuition that there should be some "real truth" about relative clock rates but can't back it up with any technical details, please consider that hunches and intuitions are often untrustworthy in modern physics, and thinking intuitions can take precedence over precise mathematical definitions is a common feature of physics crackpots (see items 12-14 on Are you a quack? from physicist Warren Siegel). Finally, if you agree that those three are the the only ways of comparing clock rates that make sense in GR, please tell me which you are referring to when you talk about "time dilation" near the event horizon being the explanation for why an external observer can never witness anything crossing it.

Either way, please give me a clear answer to this question, don't just keep talking about "time dilation" without explaining what you mean by that phrase.

But that's just a statement about when the light emitted by those observers as they crossed the horizon finally reaches the outside observer, it doesn't imply that the black hole actually was point-sized for the infalling observers when they crossed the horizon.

You're still sure about that then. I think it does.

Unless this is just a vague hunch you need to explain why you think it does with some sort of detailed non-handwavey argument.

It's just easier to compare them when there's no distance between them because "all frames/observers agree about the total elapsed time on each clock when they finally reunite at a single point in spacetime" like you said. Use common sense instead of thinking in graphs.

Again, trusting "common sense" over precise mathematical definitions is a sure path to becoming a crackpot. A little thought shows that believing there is a "real truth" about the relative rate of ticking of clocks in different regions of spacetime is equivalent to believing there must be some "real truth" about simultaneity--for example, if I say clock A is running half as fast as clock B, that's equivalent to saying that if clock A showing a time T is simultaneous with clock B showing a time T', then clock A showing a time T + delta-t must be simultaneous with clock B showing a time T' + 2*delta-t. But as long as there is no objective truth about simultaneity, what's to stop you from picking a different definition of simultaneity where clock A showing T is still simultaneous with clock B showing T', but clock A showing T + delta-t is now simultaneous with B showing T + 0.5*delta-t or T + 3*delta-t?

They can never ocupy the same exact space, but that doesn't mean they can't compare watches. If they can do that then how far away from each other do they have to be before they can't?

Any distance greater than zero means there is no basis for comparing rates other than the 3 I mentioned earlier.

Yes, okay Doppler shift is a good example of visual appearance being different to what's actually happening, but we're ignoring Doppler shift.

If you're not talking about visual appearances I say your only remaining options are #2 and #3 in my list above. If you disagree, please explain exactly what your own fourth option would be, or if you just have a hunch there should be a fourth option but can't think of any technical way to define it in GR.

Just because when would the horizon split? I just assumed it was always split and it's only noticable when you get close.

No, the second (white) horizon doesn't appear until the moment you cross the horizon. Please read carefully the section "At the horizon, the Schwarzschild surface" of http://casa.colorado.edu/~ajsh/singularity.html which I directed you to earlier, particularly this bit:

The small white dot indicates our point of entry through the horizon. Remarkably, the Schwarzschild surface, the red grid, still appears to stand off at some distance ahead of us. The white dot is actually a line which extends from us to the Schwarzschild surface still ahead, though we only ever see it as a dot, not as a line. The dot-line marks the formation of the Schwarzschild bubble (see below), and our entry into that bubble. Persons who fell through the Schwarzschild surface at this precise point before us would lie arrayed along this dot-line. At this instant, as we pass through the horizon into the Schwarzschild bubble, we see all the other persons who passed through this location before us also pass through the horizon into the bubble.

Wft are you on about. Keep it simple.

If you're going to get hostile, and refuse to think about details in the name of some illusory "simplicity", I won't continue this conversation.

Let's go back to the two horizons.

"Get back to"? Defining the detailed meaning of the "two horizons" was exactly what I was doing in the sections you quoted and responded to with the dismissive "wtf" comment.

One just in front of you and one some distance away just in front of another object that very proberbly crossed the horizon earlier.

No, the reason I gave such detailed descriptions was in hopes that you'd follow along and not just jump back into relying on your own vague intuitions and hunches. Read what I said again (which is just a summary of the section of the page I quoted above, which I recommend reading in full):

he mentions that at the moment you cross the horizon the white Schwarzschild bubble would first appear as a straight line reaching from you to the red horizon, only later expanding into a bubble, and that at that moment other objects which fell into the black hole along the same axis would be arrayed along this line, all appearing as they did the moment they crossed the horizon

And I also said:

If the object escapes, then you will always see it as outside the red antihorizon, and if you ended up crossing the horizon yourself you would presumably see it remain outside the white "Schwarzschild bubble" (which consists of light from events on the black hole event horizon) which you are underneath.

So you would never see the second (white in the diagrams) visual horizon as "in front" of you as you suggested, instead it first appears as a straight line extending from you to the first (red in the diagrams) visual horizon at the moment you actually cross the horizon, and then it immediately expands into a bubble which you are underneath, as is any other object whose light you are seeing from a moment after that object crossed the horizon.

How could it have if it's always possible for it to escape. This isn't just visual, it's real. If it can escape then in what sense is the horizon in front of you (which the other object is inside) real?

For an eternal black hole, the red horizon is actually a physically separate horizon, the "antihorizon" one that borders the bottom of "our" exterior region I and the top of the alternate exterior region III in the maximally extended Kruskal-Szekeres diagram. The falling object genuinely never crosses this horizon, it's a white hole horizon in our universe and a black hole horizon in another exterior universe inaccessible from our own.

For a more realistic black hole that formed at some finite time from a collapsing star, you wouldn't actually be able to "see" any horizon from the outside, in the sense that light emitted from events on an event horizon would never reach anyone outside, at least not unless the black hole evaporated away. However, this section of the other site on falling into a black hole I linked to earlier also seems to say that if you could see the highly redshifted image of the collapsing star long after the black hole had formed, it would occupy almost exactly the same visual position as the red antihorizon of an eternal black hole:

The Penrose diagram shows that the horizon is really two distinct entities, the Horizon, and the Antihorizon. The Horizon is sometimes called the true horizon. It's the horizon you actually fall through if you fall into a black hole. The Antihorizon might reasonably called the illusory horizon. In a real black hole formed from the collapse of the core of a star, the illusory horizon is replaced by an exponentially redshifting image of the collapsing star. As the collapsing star settles towards its final no-hair state, its appearance tends to that of a no-hair black hole.

What happens if you mantain your distance and the other object moves back out to your possition? It would mean it's moved but the event horizon has followed it. Just ignore the absolute horizon.

I don't know what you mean by "absolute horizon"--are you talking about the white horizon in the diagram (the Schwarzschild bubble which can only be seen once you cross the horizon yourself), the red horizon in the diagram (the antihorizon), or something else? If either of those, your statement that "the event horizon has followed it" doesn't make sense, as long as you remain outside the horizon you'll never see the white horizon, and if you maintain a constant radius the red horizon should maintain a constant visual size.

Why would there even be one?

One what?

It would be subject to gravitational length contraction which would mean its size would decrease the closer you got, going all the way up to infinity at the horizon.

What would be subject to gravitational length contraction, the object or the horizon? And just as with "time dilation", please specify whether by "length contraction" you mean visual appearances, or frame-dependent length, or something else.

If it's infinite in one "frame" then it's infinite in all of them.

Nope, that's just flat-out wrong. I already told you many times that time dilation and length contraction don't go to infinity at the horizon in Kruskal-Szekeres coordinates, and also that in ordinary Minkowski spacetime you do have infinite time dilation and length contraction at the Rindler horizon if you use Rindler coordinates, but obviously this is a purely coordinate-based effect which disappears if you use ordinary inertial coordinates in the same spacetime.

You can never reach it in exactly the same way you can never reach c.

Obviously you have a powerful intuition that this is true, but you continue to fail to provide any detailed argument as to why anyone else should believe this, and you seem unwilling to consider that your own intuitions might be wrong (beware the Dunning-Kruger effect, another major cause of crackpotism IMO). Do you think you can "never reach" the Rindler horizon just because time dilation and length contraction go to infinity as you approach the horizon in Rindler coordinates, and a Rindler observer can never see anything reach the horizon? If not please explain what makes the black hole event horizon different.

The black hole becomes the singularity at 0 range, like the whole universe becomes a singularity at c.

Neither of these claims makes any sense in relativity, if you relied more on math and less on intuitions you might figure out why.

 

[A-wal]

But in fact it does "disappear" in the sense of there no longer being a unique objective truth about "relative velocity". You can define a notion of relative velocity that's coordinate-dependent, or a notion of relative velocity based on parallel transport over a particular path through spacetime, but there are always multiple possible coordinate systems and multiple possible paths, so there's no "objective" physical notion of relative velocity for objects at different points in spacetime in GR. For example, take a look at p. 167 of this textbook which discusses the notion of the relative velocity 4-vector in SR, and then in a footnote says "The concept of relative four-vector cannot be extended to theories of physics formulated over a non-linear space, i.e., curved spaces (e.g., General Relativity)" Likewise see this page from the site of physicist John Baez, who says:

If you still don't believe that there is no coordinate- and path-independent notion of "relative velocity" for objects at different points in spacetime in GR, feel free to start a new thread asking the resident GR experts in the forum about this.

That's still not what I meant. If something has a relative velocity of .5c for example then that velocity definitely doesn't disappear in curved space-time. Objects still move relative to each other in curved space-time is all I meant.

No such thing in GR, at least if you want a single uniquely correct objective "relative velocity" as opposed to a coordinate-dependent or path-dependent one.

I didn't say there was.

OK, but you completely failed to address my question about whether you were talking about visuals or something else. I can think of only 3 senses in which we can talk about clocks ticking at different rates in GR:

1. Visual appearances--how fast the an observer sees the image of another clock ticking relative to his own clock
2. Local comparisons of elapsed times, as in the twin paradox where each twin looks at how much time each has aged between two meetings
3. Coordinate-dependent notions of how fast each clock is ticking relative to coordinate time at a particular moment (which depends on the definition of simultaneity in your chosen coordinate system)

If you are confident there is some other sense in which we can compare the rates of different clocks, please spell it out with some reference to the technical definition you are thinking of in GR. If you have some kind of hunch or intuition that there should be some "real truth" about relative clock rates but can't back it up with any technical details, please consider that hunches and intuitions are often untrustworthy in modern physics, and thinking intuitions can take precedence over precise mathematical definitions is a common feature of physics crackpots

Funny, I thought letting precise mathematical definitions take precedence over intuitions is a common feature of physics crackpots. Maths only describes, it can't explain anything.

(see items 12-14 on Are you a quack? from physicist Warren Siegel).

I'm a quack?

Finally, if you agree that those three are the the only ways of comparing clock rates that make sense in GR, please tell me which you are referring to when you talk about "time dilation" near the event horizon being the explanation for why an external observer can never witness anything crossing it.

Either way, please give me a clear answer to this question, don't just keep talking about "time dilation" without explaining what you mean by that phrase.

The mass of the singularity determines the diameter of the event horizon. Say 100 s-units at my current range of say 100 d-units. I now halve the distance to 50 d-units. Make sense? It shouldn't. If I measure where half the distance is from my starting position (we'll say there's a marker keeping a constant distance from the event horizon from it's own perspective) and then travel to that position and measure my new current distance to the event horizon then it wouldn't be 50 d-units. It would be more because I'm now in space-time that was contracted from my previous perspective. I'd have to get closer to be 50 d-units away, but then I'd have the same problem. The closer you get the further you need to go, but this obviously can't go on for ever. It stops at the singularity. What happens to the s-units?

Unless this is just a vague hunch you need to explain why you think it does with some sort of detailed non-handwavey argument.

That's exactly what I've been doing! Non-technical does not = handwavey!

Again, trusting "common sense" over precise mathematical definitions is a sure path to becoming a crackpot. A little thought shows that believing there is a "real truth" about the relative rate of ticking of clocks in different regions of spacetime is equivalent to believing there must be some "real truth" about simultaneity--for example, if I say clock A is running half as fast as clock B, that's equivalent to saying that if clock A showing a time T is simultaneous with clock B showing a time T', then clock A showing a time T + delta-t must be simultaneous with clock B showing a time T' + 2*delta-t. But as long as there is no objective truth about simultaneity, what's to stop you from picking a different definition of simultaneity where clock A showing T is still simultaneous with clock B showing T', but clock A showing T + delta-t is now simultaneous with B showing T + 0.5*delta-t or T + 3*delta-t?

Nothing!

Any distance greater than zero means there is no basis for comparing rates other than the 3 I mentioned earlier.

I never said there was.

If you're not talking about visual appearances I say your only remaining options are #2 and #3 in my list above. If you disagree, please explain exactly what your own fourth option would be, or if you just have a hunch there should be a fourth option but can't think of any technical way to define it in GR.

#2.

No, the second (white) horizon doesn't appear until the moment you cross the horizon. Please read carefully the section "At the horizon, the Schwarzschild surface" of http://casa.colorado.edu/~ajsh/singularity.html which I directed you to earlier, particularly this bit:

So there is a jump? This is getting silly again.

If you're going to get hostile, and refuse to think about details in the name of some illusory "simplicity", I won't continue this conversation.

Hostile? You've lead a very sheltered life my friend. Please don't end the conversation. If I'm wrong then I'd like to understand why.

"Get back to"? Defining the detailed meaning of the "two horizons" was exactly what I was doing in the sections you quoted and responded to with the dismissive "wtf" comment.

You were starting to sound like a crackpot.

No, the reason I gave such detailed descriptions was in hopes that you'd follow along and not just jump back into relying on your own vague intuitions and hunches. Read what I said again (which is just a summary of the section of the page I quoted above, which I recommend reading in full):

I'm confused. I'm approaching the horizon and I can't see any object cross the horizon from the outside. I now cross the horizon and I see those same objects suddenly jump to some point along a line and the time since they crossed determines how far along the line they jump to?

And I also said:

So you would never see the second (white in the diagrams) visual horizon as "in front" of you as you suggested, instead it first appears as a straight line extending from you to the first (red in the diagrams) visual horizon at the moment you actually cross the horizon, and then it immediately expands into a bubble which you are underneath, as is any other object whose light you are seeing from a moment after that object crossed the horizon.

Hmm, I'm still not buying it.

For an eternal black hole, the red horizon is actually a physically separate horizon, the "antihorizon" one that borders the bottom of "our" exterior region I and the top of the alternate exterior region III in the maximally extended Kruskal-Szekeres diagram. The falling object genuinely never crosses this horizon, it's a white hole horizon in our universe and a black hole horizon in another exterior universe inaccessible from our own.

For a more realistic black hole that formed at some finite time from a collapsing star, you wouldn't actually be able to "see" any horizon from the outside, in the sense that light emitted from events on an event horizon would never reach anyone outside, at least not unless the black hole evaporated away. However, this section of the other site on falling into a black hole I linked to earlier also seems to say that if you could see the highly redshifted image of the collapsing star long after the black hole had formed, it would occupy almost exactly the same visual position as the red antihorizon of an eternal black hole:

I can honestly see no need for the "true" horizon.

I don't know what you mean by "absolute horizon"--are you talking about the white horizon in the diagram (the Schwarzschild bubble which can only be seen once you cross the horizon yourself), the red horizon in the diagram (the antihorizon), or something else? If either of those, your statement that "the event horizon has followed it" doesn't make sense, as long as you remain outside the horizon you'll never see the white horizon, and if you maintain a constant radius the red horizon should maintain a constant visual size.

I'm just having trouble with the transition from what's observed before reaching the horizon and how it maintains continuity if it can be crossed.

One what?

Absolute horizon. As in the one that doesn't recede but maintains a constant distance from the singularity despite the fact that distance is meant to be relative.

What would be subject to gravitational length contraction, the object or the horizon? And just as with "time dilation", please specify whether by "length contraction" you mean visual appearances, or frame-dependent length, or something else.

I meant the distance between the event horizon and singularity depends on distance it's viewed from.

Nope, that's just flat-out wrong. I already told you many times that time dilation and length contraction don't go to infinity at the horizon in Kruskal-Szekeres coordinates, and also that in ordinary Minkowski spacetime you do have infinite time dilation and length contraction at the Rindler horizon if you use Rindler coordinates, but obviously this is a purely coordinate-based effect which disappears if you use ordinary inertial coordinates in the same spacetime.

In the same space-time? When comparing objects at different distance from an event horizon they can't possibly be in the same space-time.

Obviously you have a powerful intuition that this is true, but you continue to fail to provide any detailed argument as to why anyone else should believe this, and you seem unwilling to consider that your own intuitions might be wrong (beware the Dunning-Kruger effect, another major cause of crackpotism IMO). Do you think you can "never reach" the Rindler horizon just because time dilation and length contraction go to infinity as you approach the horizon in Rindler coordinates, and a Rindler observer can never see anything reach the horizon? If not please explain what makes the black hole event horizon different.

Stop calling me names! It's mean.

Neither of these claims makes any sense in relativity, if you relied more on math and less on intuitions you might figure out why.

And if you relied more on intuitions and less on maths then you might question the things you're told a bit more. I thought the universe did become a singularity at c (as far as light, all energy for that matter is concerned).


Maybe I am being a cock, because I'm getting frustrated. First the name calling isn't nice. Secondly I told you that I don't like all the coordinate stuff but you continue to ask question like: "Do you think you can "never reach" the Rindler horizon just because time dilation and length contraction go to infinity as you approach the horizon in Rindler coordinates, and a Rindler observer can never see anything reach the horizon? If not please explain what makes the black hole event horizon different." I have no idea and honestly couldn't care less. I'd never even heard of Rindler coordinates or Schwarzschild coordinates or half the stuff that's been mentioned here until this conversation so it's not a reason for me thinking anything. All I care about is understanding what happens and why. And finally, I don't think all of your recent explanations are an understanding in any meaningful sense of the word, though I do appreciate the time and effort you've put in. I think they're what happens when you let maths lead the "understanding" instead of doing it the other way round. There, I said it.


P.S. Sorry about the delays between posts. I've written all these posts while at work and I've been off for three nights.

 

[YLW]

Hi, I seldom post so I'm really, really so, so sorry if I've posted in the wrong thread. But I've just been reading through a Discovery magazine article Back From The Future and here are excerpts. (http://discovermagazine.com/2010/apr/01-back-from-the-future/article_view?b_start:int=1&-C=)
________
"A series of quantum experiments shows that measurements performed in the future can influence the present. Does that mean the universe has a destiny—and the laws of physics pull us inexorably toward our prewritten fate?..."

"The boat trip has been organized as part of a conference sponsored by the Foundational Questions Institute to highlight some of the most controversial areas in physics. Tollaksen’s idea certainly meets that criterion. And yet, as crazy as it sounds, this notion of reverse causality is gaining ground. A succession of quantum experiments confirm its predictions—showing, bafflingly, that measurements performed in the future can influence results that happened before those measurements were ever made.

As the waves pound, it’s tough to decide what is more unsettling: the boat’s incessant rocking or the mounting evidence that the arrow of time—the flow that defines the essential narrative of our lives—may be not just an illusion but a lie."

"Just last year, physicist John Howell and his team from the University of Rochester reported success. In the Rochester setup, laser light was measured and then shunted through a beam splitter. Part of the beam passed right through the mechanism, and part bounced off a mirror that moved ever so slightly, due to a motor to which it was attached. The team used weak measurements to detect the deflection of the reflected laser light and thus to determine how much the motorized mirror had moved.

That is the straightforward part. Searching for backward causality required looking at the impact of the final measurement and adding the time twist. In the Rochester experiment, after the laser beams left the mirrors, they passed through one of two gates, where they could be measured again—or not. If the experimenters chose not to carry out that final measurement, then the deflected angles measured in the intermediate phase were boringly tiny. But if they performed the final, postselection step, the results were dramatically different. When the physicists chose to record the laser light emerging from one of the gates, then the light traversing that route, alone, ended up with deflection angles amplified by a factor of more than 100 in the intermediate measurement step. Somehow the later decision appeared to affect the outcome of the weak, intermediate measurements, even though they were made at an earlier time.

This amazing result confirmed a similar finding reported a year earlier by physicists Onur Hosten and Paul Kwiat at the University of Illinois at Urbana-Champaign. They had achieved an even larger laser amplification, by a factor of 10,000, when using weak measurements to detect a shift in a beam of polarized light moving between air and glass."
____________
So, asking as a layman, do the experiments confirm backward or retrocausality and that the arrow of time can move backwards as well?

Once again, so so sorry if I've posted wrongly.

 

[Austin0]

Well, note that the Rindler horizon has an upper and lower component (or maybe it would be better to say there are two horizons?), as seen in this diagram from the wikipedia article which plots both the horizons and lines of constant Rindler position and time in a Minkowski diagram. The upper component behaves like a black hole horizon in that the Rindler observers can't see anything that crosses it (though an observer on the "Rindler wedge" can cross it herself if she doesn't remain at rest in Rindler coordinates), but the lower component behaves like a white hole horizon in that Rindler observers can see light from events "below" it (the light rays would be traveling parallel to the upper horizon in the diagram), and it's impossible for any observer on the Rindler wedge to cross it (in Minkowski coordinates it's moving away from anyone on the Rindler wedge with a speed of c). So, in terms of Minkowski coordinates this means an observer on the Rindler wedge can see light from events that occurred arbitrarily far in either direction along the x-axis, including events much further than the current distance of the Rindler horizon. So certainly an inertial observer on the Rindler wedge would see nothing unusual, and at any given moment an accelerating Rindler observer should see the same thing as an inertial observer at the same point in spacetime who is instantaneously at rest relative to themselves.

Hi When you refer to the upper and lower components are you talking about the lower deceleration part of the diagram and the upper or acceleration half of the diagram??

When you say light from events arbitrarily far , I assume this also means distant in time also ,,is this correct??

That the events from much further than the horizon would then have to have occurred further in the past, yes??
If so this is more or less what I thought.

So if I am understanding you correctly, both Rindler and CMIRF observers would visually see both Doppler shift and aberration but nothing identifiable beyond that. yes?

Thanks for your explication

 

[JesseM]

(response to post #167, part 1)

That's still not what I meant. If something has a relative velocity of .5c for example then that velocity definitely doesn't disappear in curved space-time. Objects still move relative to each other in curved space-time is all I meant.

No such thing in GR, at least if you want a single uniquely correct objective "relative velocity" as opposed to a coordinate-dependent or path-dependent one.

I didn't say there was.

If you're talking about a purely coordinate-dependent or path-dependent notion of "relative velocity", then you must agree that just because two objects at distant locations have a relative velocity of 0.5c in some coordinate system or with velocities parallel-transported along some path, there may be some other coordinate system or path where the relative velocity is 0.9c or 0.1c or even 0. So it seems to me the relative velocity of 0.5c does disappear in curved spacetime, in the sense that it there is no objective sense in which two objects can be said to have a relative velocity of 0.5c as opposed to 0.9c or 0.

Funny, I thought letting precise mathematical definitions take precedence over intuitions is a common feature of physics crackpots. Maths only describes, it can't explain anything.

No physics theory has ever "explained" why matter/energy/particles behave in the way they do, it just gives equations describing their behavior. And anyone with experience dealing with physics crackpots will recognize that quite a lot of them are motivated by being unsatisfied with such abstract mathematical models, and mistakenly think that the equations should be understood in terms of some intuitive concrete model. Consider for example a few items from physicist John Baez's famous crackpot index:

15. 10 points for each statement along the lines of "I'm not good at math, but my theory is conceptually right, so all I need is for someone to express it in terms of equations".

...17. 10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn't explain "why" they occur, or fails to provide a "mechanism".

Or consider some of the characteristics of crackpot arguments in A brief field guide to scientific crackpots:

6. Criticisms of existing theories which rely on “common sense”. This particular branch of’ ‘crackpottery’ reminds me of a personal anecdote. Years ago I was wandering through the lamp department of a Service Merchandise store, when I noticed bright red signs prominently displayed: “CAUTION! Light bulbs are hot! Do not touch!”

Common sense is evidently a terribly inaccurate method of understanding the world! I’ve mentioned in previous posts numerous modern “common sense” ideas which were anything but when first proposed, among them: Newton’s laws, heliocentrism, the germ theory of disease, the brain as the center of intelligence, the theory of atoms.

For relativity denialists, the idea that space and time are intermingled violates common sense. They have no strong quantitative criticism of the theory (which they probably don’t understand anyway), only a mushy notion that it “feels funny”. Relativity felt funny to a lot of people, who raised objections to the theory in the form of apparent paradoxes — all of which were resolved successfully. The lesson physicists learned from this is that “common sense” is an artifact of our limited perceptions and place in the universe. In fact, modern science was really born when people realized that the universe might work differently in circumstances different from those experienced in daily life.

7. A complete absence of quantitative analysis. In a 165-page “primer” on geocentricity (discussed in another post), the long-disavowed notion that the Earth is the center of the universe, a crank author uses no calculations to back up his extravagant alternative theory of celestial motion. He claims the theory works just fine, but does it? Without calculations which can be checked for errors and compared to experiment, there’s no immediate way to tell. The relativity denialists mentioned earlier also present no quantitative results to back up their ideas, only appeals to “common sense”. The Templeton Foundation, a group dedicated to promoting the links of science and religion, once asked for proponents of intelligent design (dressed-up creationism) to submit research proposals. “They never came in,” said the Foundation’s senior vice president.
These are some of the characteristics and arguments of crackpot scientists, and are things which should raise a red flag if you come across them. Anybody have any other characteristics to add to the list?

(see items 12-14 on Are you a quack? from physicist Warren Siegel).

I'm a quack?

To me words like "crackpot" and "quack" don't really refer to characteristics of a person but rather common styles of argument used by people who think they have found conceptual flaws in mainstream scientific theories that they have never studied in any great detail--and having had experience arguing with denialists of a bunch of different mainstream theories on the internet, it does ring true that there are quite lot of commonalities to their style of argument regardless of what theory it is they're attacking, with an over-emphasis on intuitions and "common sense" being one prominent one (think of global warming denialists who have arguments like 'if the Earth is getting warmer, why did we have record snowstorms last year' or evolution deniers who say 'if man evolved from monkeys, how can monkeys still be around today'?) Warren Siegel attempts to compile a bunch of these commonalities, and I directed you to 12-14 above because they also dealt with the favoring of common sense over math:

12. "My theory doesn't need any complicated math."

Then how do you calculate anything? Science is not just knowing "what goes up must come down", but when and where it comes down.

Note: Quacks come in slightly different levels of sophistication in math. Some use only words, and no numbers whatsoever, but lots of pictures. The worst one I ever corresponded with claimed that dimensions did not physically exist, but were just abstract mathematical concepts, and you could never prove the existence of anything unless you could do it without equations. After giving him the examples of directions, he claimed that "up" and "down" did not physically exist.

Better ones actually know arithmetic, but no algebra, so even E=mc2 is usually beyond them. They will quote lots of numbers, which they "predicted" by some numerology, but never functions (like cross sections). They don't understand units, or conventions, and will not appreciate that some constants of nature may be more natural with extra factors of 2π or so, or that some are actually not constants (like running couplings).

13. "Numbers aren't important in science."
I guess you can throw out your clock.

14. "How you explain something is more important than the numbers."
Try that the next time you pay a bill.

For a more detailed discussion of the necessity to abandon common-sense intuitions about what a good model of physics should look like, and accept that physics is ultimately just about coming up with elegant mathematical models which accurately predict observation, consider this section from Richard Feynman's book The Character of Physical Law:

On the other hand, take Newton's law for gravitation, which has the aspects I discussed last time. I gave you the equation:

F=Gmm'/r^2

just to impress you with the speed with which mathematical symbols can convey information. I said that the force was proportional to the product of the masses of two objects, and inversely as the square of the distance between them, and also that bodies react to forces by changing their speeds, or changing their motions, in the direction of the force by amounts proportional to the force and inversely proportional to their masses. Those are words all right, and I did not necessarily have to write the equation. Nevertheless it is kind of mathematical, and we wonder how this can be a fundamental law. What does the planet do? Does it look at the sun, see how far away it is, and decide to calculate on its internal adding machine the inverse of the square of the distance, which tells it how much to move? This is certainly no explanation of the machinery of gravitation! You might want to look further, and various people have tried to look further. Newton was originally asked about his theory--'But it doesn't mean anything--it doesn't tell us anything'. He said, 'It tells you how it moves. That should be enough. I have told you how it moves, not why.' But people are often unsatisfied without a mechanism, and I would like to describe one theory which has been invented, among others, of the type you migh want. This theory suggests that this effect is the result of large numbers of actions, which would explain why it is mathematical.

Suppose that in the world everywhere there are a lot of particles, flying through us at very high speed. They come equally in all directions--just shooting by--and once in a while they hit us in a bombardment. We, and the sun, are practically transparent for them, practically but not completely, and some of them hit. ... If the sun were not there, particles would be bombarding the Earth from all sides, giving little impuleses by the rattle, bang, bang of the few that hit. This will not shake the Earth in any particular direction, because there are as many coming from one side as from the other, from top as from bottom. However, when the sun is there the particles which are coming from that direction are partially absorbed by the sun, because some of them hit the sun and do not go through. Therefore the number coming from the sun's direction towards the Earth is less than the number coming from the other sides, because they meet an obstacle, the sun. It is easy to see that the farther the sun is away, of all the possible directions in which particles can come, a smaller proportion of the particles are being taken out. The sun will appear smaller--in fact inversely as the square of the distance. Therefore there will be an impulse on the Earth towards the sun that varies inversely as the square of the distance. And this will be the result of a large number of very simple operations, just hits, one after the other, from all directions. Therefore the strangeness of the mathematical relation will be very much reduced, because the fundamental operation is much simpler than calculating the inverse of the square of the distance. This design, with the particles bouncing, does the calculation.

The only trouble with this scheme is that it does not work, for other reasons. Every theory that you make up has to be analysed against all possible consequences, to see if it predicts anything else. And this does predict something else. If the Earth is moving, more particles will hit it from in front than from behind. (If you are running in the rain, more rain hits you in the front of the face than in the back of the head, because you are running into the rain.) So, if the Earth is moving it is running into the particles coming towards it and away from the ones that are chasing it from behind. So more particles will hit it from the front than from the back, and there will be a force opposing any motion. This force would slow the Earth up in its orbit, and it certainly would not have lasted the three of four billion years (at least) that it has been going around the sun. So that is the end of that theory. 'Well,' you say, 'it was a good one, and I got rid of the mathematics for a while. Maybe I could invent a better one.' Maybe you can, because nobody knows the ultimate. But up to today, from the time of Newton, no one has invented another theoretical description of the mathematical machinery behind this law which does not either say the same thing over again, or make the mathematics harder, or predict some wrong phenomena. So there is no model of the theory of gravity today, other than the mathematical form.

If this were the only law of this character it would be interesting and rather annoying. But what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics.

...[A] question is whether, when trying to guess new laws, we should use seat-of-the-pants feelings and philosophical principles--'I don't like the minimum principle', or 'I do like the minimum principle', 'I don't like action at a distance', or 'I do like action at a distance'. To what extent do models help? It is interesting that very often models do help, and most physics teachers try to teach how to use models and to get a good physical feel for how things are going to work out. But it always turns out that the greatest discoveries abstract away from the model and the model never does any good. Maxwell's discovery of electrodynamics was made with a lot of imaginary wheels and idlers in space. But when you get rid of all the idlers and things in space the thing is O.K. Dirac discovered the correct laws for relativity quantum mechanics simply by guessing the equation. The method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way.

It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities. But this speculation is of the same nature as those other people make--'I like it', 'I don't like it',--and it is not good to be too prejudiced about these things.

The mass of the singularity determines the diameter of the event horizon.

Only if you are talking about some coordinate-dependent notion of "diameter", like the diameter in Schwarzschild coordianates or Eddington-Finkelstein coordinates. Please, if you are going to talk about "distance" then please tell me if you are talking about distances in some coordinate system, or apparent visual distances and sizes, or if you think there is some third option.

Say 100 s-units at my current range of say 100 d-units. I now halve the distance to 50 d-units. Make sense? It shouldn't. If I measure where half the distance is from my starting position (we'll say there's a marker keeping a constant distance from the event horizon from it's own perspective) and then travel to that position and measure my new current distance to the event horizon then it wouldn't be 50 d-units. It would be more because I'm now in space-time that was contracted from my previous perspective.

I don't know what you mean by "perspective". Are you talking about visual appearances at 100 d-units vs. 50 d-units, or are you imagining that the observer will use different coordinate systems depending on his distance, or what?

Unless this is just a vague hunch you need to explain why you think it does with some sort of detailed non-handwavey argument.

That's exactly what I've been doing! Non-technical does not = handwavey!

If it wouldn't be obvious to a physicist well-versed in the mathematics how to translate your verbal argument into a detailed mathematical one, then hell yes it's "handwavey". In physics verbal arguments are only meaningful insofar as they can be understood as shorthand for a technical argument, where the meaning of the shorthand is clear enough that you don't have to bother laboriously spelling everything out in technical terms. This is not to say that physicists don't appeal to physical intuitions in situations where they're groping for the correct answer to some problem they don't know how to solve in a rigorous mathematical way, but I think they always do so with an eye towards developing a technical mathematical argument, I think you'd be hard-pressed to find a physicist who believes that some verbal argument is sufficient in itself to demonstrate some claim even if they don't know how to translate it into technical terms (and don't think other physicists would be able to either).

 

[JesseM]

(response to post #167, part 2)

A little thought shows that believing there is a "real truth" about the relative rate of ticking of clocks in different regions of spacetime is equivalent to believing there must be some "real truth" about simultaneity--for example, if I say clock A is running half as fast as clock B, that's equivalent to saying that if clock A showing a time T is simultaneous with clock B showing a time T', then clock A showing a time T + delta-t must be simultaneous with clock B showing a time T' + 2*delta-t. But as long as there is no objective truth about simultaneity, what's to stop you from picking a different definition of simultaneity where clock A showing T is still simultaneous with clock B showing T', but clock A showing T + delta-t is now simultaneous with B showing T + 0.5*delta-t or T + 3*delta-t?

Nothing!

So does that mean you are retracting all your earlier statements suggesting there is some objective sense in which we can talk about which clock experiences more time dilation at different distances from the horizon, apart from the question of which clock experiences more total elapsed time between two meetings which each occur at a single point in space time? For example, this comment from post #161:

(and we can only talk about 'time dilation' in a coordinate-independent way if we're talking about a twin-paradox-like situation where two observers meet twice and compare the elapsed time on their clocks between meetings, any notion of 'time dilation' other than this is also inherently coordinate-dependent).

If gravitational time dilation applies when two objects with different world lines meet up then it always applies. If not, then at what distance would it suddenly not apply?

Or this one from post #165:

To be precise about what I mean, suppose you have two observers who compare clock readings at a single point, fly apart, and then reunite and compare clock readings again. Pick any point A on the worldline of one observer, and any point B on the worldline of the second observer. Then it can be true that different coordinate systems disagree on whether the clock at point A was ticking slower than the clock at point B or vice versa, and likewise it can be true that other observers with different paths through space time disagree about the exact ration between the visual rate of ticking of the clock at point A vs. the clock at point B. Nevertheless, all frames/observers agree about the total elapsed time on each clock when they finally reunite at a single point in spacetime. Would you say this means time dilation "applies" or "doesn't apply" at points A and B on each clock's worldline?

It's just easier to compare them when there's no distance between them because "all frames/observers agree about the total elapsed time on each clock when they finally reunite at a single point in spacetime" like you said. Use common sense instead of thinking in graphs. They can never ocupy the same exact space, but that doesn't mean they can't compare watches. If they can do that then how far away from each other do they have to be before they can't?

If you don't retract these comments, then how does this square with your agreement that depending on our choice of simultaneity convention we can reach different conclusions about whether clock B was ticking faster or slower than clock A between the events of clock A reading a time T and clock A reading a time T + delta-t?

If you're not talking about visual appearances I say your only remaining options are #2 and #3 in my list above. If you disagree, please explain exactly what your own fourth option would be, or if you just have a hunch there should be a fourth option but can't think of any technical way to define it in GR.

#2.

OK, #2 was "Local comparisons of elapsed times, as in the twin paradox where each twin looks at how much time each has aged between two meetings". But in this case you can't say anything about time dilation except when talking about total elapsed time between two local comparisons, in particular you can't say one clock was ticking slower when it was closer to the horizon. If you agree with that, it seems to me you are changing your tune from your comments in post #161 and #165 when the statements of mine you were disagreeing with were just statements that we can only talk about differences in total elapsed time and nothing else.

No, the second (white) horizon doesn't appear until the moment you cross the horizon. Please read carefully the section "At the horizon, the Schwarzschild surface" of the page which I directed you to earlier, particularly this bit:

So there is a jump? This is getting silly again.

In visual appearances there is a "jump" in the sense that the visual horizon which is shown in white in the diagrams is completely absent for any observer who hasn't yet crossed the event horizon, then at the moment the observer crosses it they will suddenly see it appear as a straight line reaching from their position down to the antihorizon depicted in red, after which it expands into a bubble enclosing them. I already explained this in previous posts, like the long explanation in post #162 which you gave the dismissive "wtf" response to:

But after you crossed the horizon you'd also suddenly start seeing light from black hole horizon in your own universe--if you imagine that this light is white instead of red, then you should be able to follow what's depicted in the third movie from the top on this page, where we approach a black hole covered with red grid lines and then suddenly a new set of white grid lines appear (forming a sort of dome shape above us) at the moment we cross the horizon.

...

And I also linked to http://casa.colorado.edu/~ajsh/singularity.html which shows somewhat cruder animations of what a falling observer would see (again with the antihorizon in red and the white 'Schwarzschild bubble' that you see after crossing the horizon in white), with somewhat more detailed accompanying explanation (for example, he mentions that at the moment you cross the horizon the white Schwarzschild bubble would first appear as a straight line reaching from you to the red horizon, only later expanding into a bubble, and that at that moment other objects which fell into the black hole along the same axis would be arrayed along this line, all appearing as they did the moment they crossed the horizon).

Hostile? You've lead a very sheltered life my friend. Please don't end the conversation. If I'm wrong then I'd like to understand why.

Of course I am not particularly offended by the phrase "wtf", it's the dismissive attitude it suggests that bothers me. The rules of this forum don't allow people to use it as a platform to try to make arguments against mainstream ideas in relativity, though they do allow for people who are confused by some aspect of relativity to ask questions about how some apparent problem or contradiction can be avoided. The crucial difference here is one of attitude--in the second case the person is willing to take as a default hypothesis that there almost certainly is some way around the problem they see and they genuinely want to learn what it is, in the first case they are confident that they really have discovered a genuine problem which shows that the mainstream view is wrong. Your dismissive comment about my detailed explanation, along with other comments suggesting you probably hadn't read it carefully and tried to visualize what I was talking about, makes it seem like your attitude is closer to that of an anti-relativity type who is trying to prove there is a flaw, as opposed to someone who trusts that there is most likely an error in their thinking and is making an honest effort to listen to others who are knowledgeable about the subject and trying to explain where the error might lie. If you do want to learn, then please pay attention to what I say, ask questions about the aspects of my explanation you don't understand, and don't make dismissive comments like tossing aside long explanations with "wtf", or comments like this:

You were starting to sound like a crackpot.

"Crackpot" is not just a generic insult for any argument that sounds weird to you, again it refers to a specific style of trying to argue against mainstream views. If you think anything I was telling you doesn't match what mainstream physicists (like the one who wrote up the webpages I was basing my comments on) would say about visual appearances for an observer falling into a black hole, please point out where, as far as I know I was just summarizing the explanations on those pages.

I'm confused. I'm approaching the horizon and I can't see any object cross the horizon from the outside. I now cross the horizon and I see those same objects suddenly jump to some point along a line and the time since they crossed determines how far along the line they jump to?

No, there is no jump in the apparent position of the objects who appear to be arrayed at various distances below you both before and after you cross the horizon, and likewise there is no jump in the position of the red antihorizon which all the objects appear to remain above, it's just that there is a sudden appearance of the white horizon itself, which first appears as a line going from your position, through all the objects below you falling on the same radial path, down the red antihorizon (at the moment it appears, the light from all the objects below you that lie along it is the light they emitted at the moment they crossed the horizon themselves). Then it immediately expands into a bubble which encloses both you and these objects below you, with the top rising above you so that objects may cross into the bubble above you (and at the moment you see any object above you reach the boundary of the white bubble, you are seeing light from that object at the moment it crossed the event horizon). The reason it all works this way would be a lot easier to understand if you familiarized yourself with the details of the Kruskal-Szekeres diagram and then looked carefully at this diagram, imagining adding some worldlines of objects which fell through the horizon at earlier and later times than the object with the blue worldline. Again if there's some aspect of these diagrams you don't understand, or you don't see the relevance to my explanation of the visual appearances above, please ask questions about this stuff rather than just dismissing it.

So you would never see the second (white in the diagrams) visual horizon as "in front" of you as you suggested, instead it first appears as a straight line extending from you to the first (red in the diagrams) visual horizon at the moment you actually cross the horizon, and then it immediately expands into a bubble which you are underneath, as is any other object whose light you are seeing from a moment after that object crossed the horizon.

Hmm, I'm still not buying it.

Not a substantive response. Do you have some specific reason to doubt that this is what GR would predict we'd see when falling into a Schwarzschild black hole if light was being emitted by events on the antihorizon and also from events on the horizon? If not, do you think there is some inconsistency or other problem with GR's predictions?

For an eternal black hole, the red horizon is actually a physically separate horizon, the "antihorizon" one that borders the bottom of "our" exterior region I and the top of the alternate exterior region III in the maximally extended Kruskal-Szekeres diagram. The falling object genuinely never crosses this horizon, it's a white hole horizon in our universe and a black hole horizon in another exterior universe inaccessible from our own.

For a more realistic black hole that formed at some finite time from a collapsing star, you wouldn't actually be able to "see" any horizon from the outside, in the sense that light emitted from events on an event horizon would never reach anyone outside, at least not unless the black hole evaporated away. However, this section of the other site on falling into a black hole I linked to earlier also seems to say that if you could see the highly redshifted image of the collapsing star long after the black hole had formed, it would occupy almost exactly the same visual position as the red antihorizon of an eternal black hole:

I can honestly see no need for the "true" horizon.

I didn't use the words "true horizon", what part of my above explanation are you referring to? Do you mean the white horizon as opposed to the red antihorizon? Do you doubt that there are solutions to the Einstein field equations involving both eternal Schwarzschild black holes as well as black holes that form from collapsing stars, and that when you consider the "maximally extended" version of these spacetimes (the meaning of 'maximally extended' is discussed in the Kruskal-Szekeres article, ask questions if you don't understand something about it), they include a region of spacetime where anything inside the region will inevitably hit the singularity, and a region where it's possible to avoid the singularity, with the boundary between the two regions defined as the "event horizon"?

I'm just having trouble with the transition from what's observed before reaching the horizon and how it maintains continuity if it can be crossed.

All physical objects behave in a continuous way visually, but the white horizon behaves in a discontinuous way visually as it goes from completely absent before you cross the horizon to suddenly appearing as a straight line which expands to a bubble after you cross the horizon. Again the reason for the sudden appearance should be fairly understandable if you follow how light behaves in Kruskal-Szekeres coordinates and also follow what's going on in this diagram. And remember that this sudden appearance is just based on the assumption (made for the sake of visualization) that every point in spacetime which lies on the event horizon emits white light, it would similarly be true in Minkowski spacetime that if you had some event E and assumed that every event on its future light cone emitted white light, you wouldn't actually see any of this white light until the moment you yourself entered the future light cone of E, at which moment you would see a straight white line reaching from you to the event E itself (with objects that entered the future light cone earlier along the same radial path to the position of E appearing arrayed along this line), which would then expand into an ellipsoid enclosing yourself and the objects that entered the future light cone earlier (and expanding to enclose objects which entered the future light cone of E after you). If you're having trouble understanding why GR would predict what I told you about the appearance of the black hole horizon, see if you can at least understand why this would be true for the appearance of a future light cone in SR, and ask questions if you don't.

Absolute horizon. As in the one that doesn't recede but maintains a constant distance from the singularity despite the fact that distance is meant to be relative.

Do you mean "constant distance from the singularity" in terms of some coordinate system like Schwarzschild coordinates, or in terms of visual appearances (in which case it's not correct to say the visual appearance of the horizon maintains a constant distance from the singularity, both visual horizons appear to change in apparent size as you approach and anyway the singularity has no apparent visual position, it can't be 'seen' any more than the Big Crunch singularity in a contracting universe could be 'seen' in advance), or do you imagine there is some other notion of "distance" besides coordinate distance and apparent visual distance?

What would be subject to gravitational length contraction, the object or the horizon? And just as with "time dilation", please specify whether by "length contraction" you mean visual appearances, or frame-dependent length, or something else.

I meant the distance between the event horizon and singularity depends on distance it's viewed from.

Any comment about "distance" is meaningless unless you specify what you mean by that word. Please answer my question: are you referring to apparent visual distance, or to distance in some coordinate system, or do you claim there is some third notion of "distance" aside from these? (I suppose you could also talk about the integral of ds^2 along some specific spacelike path, like the worldline of a hypothetical tachyon, which would have a coordinate-independent value just like proper time along a timelike worldline).

Nope, that's just flat-out wrong. I already told you many times that time dilation and length contraction don't go to infinity at the horizon in Kruskal-Szekeres coordinates, and also that in ordinary Minkowski spacetime you do have infinite time dilation and length contraction at the Rindler horizon if you use Rindler coordinates, but obviously this is a purely coordinate-based effect which disappears if you use ordinary inertial coordinates in the same spacetime.

In the same space-time? When comparing objects at different distance from an event horizon they can't possibly be in the same space-time.

You seem to have some confused idea about what "spacetime" means, it just refers to the continuous curved 4D manifold consisting of every possible point in space and time where a physical event could occur, including events at different distance from the horizon, along with a definite geometry (curvature at every point, defined by the metric) assigned to this manifold. "Same spacetime" just means we are talking about the same geometry--events which both occur on the same solution to the Einstein field equations (say, the Schwarzschild metric as opposed to the Minkowsk metric or a FLRW metric)

 

[JesseM]

(response to post #167, part 3)

Stop calling me names! It's mean.

Again terms like "crackpot" are meant to refer to a style of argument which you are skirting dangerously close to, not a comment on your personality or intellect.

And if you relied more on intuitions and less on maths then you might question the things you're told a bit more. I thought the universe did become a singularity at c (as far as light, all energy for that matter is concerned).

No, see the various threads on how it's meaningless to talk about the "perspective" of an observer moving at c, like this one. If you consider what some inertial landmarks look like for an observer moving at v relative to them in the limit as v approaches c, some quantities do approach infinity in this limit, but in any case this would only be approaching a coordinate singularity as opposed to a genuine physical singularity (where some quantity approaches infinity at a given point in all coordinate systems which approach arbitrarily close to that point, like the curvature singularity at the center of a black hole.)

Maybe I am being a cock, because I'm getting frustrated. First the name calling isn't nice. Secondly I told you that I don't like all the coordinate stuff but you continue to ask question like: "Do you think you can "never reach" the Rindler horizon just because time dilation and length contraction go to infinity as you approach the horizon in Rindler coordinates, and a Rindler observer can never see anything reach the horizon? If not please explain what makes the black hole event horizon different."

Please try to keep track of the context. That comment of mine was in response to your own statement from the end of post #165 where you were talking about "frames" (coordinate systems) too:

Let's go back to the two horizons. One just in front of you and one some distance away just in front of another object that very proberbly crossed the horizon earlier. How could it have if it's always possible for it to escape. This isn't just visual, it's real. If it can escape then in what sense is the horizon in front of you (which the other object is inside) real? What happens if you mantain your distance and the other object moves back out to your possition? It would mean it's moved but the event horizon has followed it. Just ignore the absolute horizon. Why would there even be one? It would be subject to gravitational length contraction which would mean its size would decrease the closer you got, going all the way up to infinity at the horizon. If it's infinite in one "frame" then it's infinite in all of them. You can never reach it in exactly the same way you can never reach c. The black hole becomes the singularity at 0 range, like the whole universe becomes a singularity at c.

So that's why I responded with the comment about Rindler coordinates, because presumably you don't think in that case that if time dilation is "infinite in one frame then it's infinite in all of them".

I'd never even heard of Rindler coordinates or Schwarzschild coordinates or half the stuff that's been mentioned here until this conversation so it's not a reason for me thinking anything.

Uh, I never said it was a "reason for you thinking anything", your thinking seems to be based on vague intuitions about things like "distance" and "time dilation" that are based on statements you've read by physicists (or physics popularizers) in various places, without realizing that physicists generally use such terms in the context of some specific coordinate system. So your failure to realize this means you try to imitate the way physicists would use such terms but without sufficient understanding of how they are just verbal shorthand for coordinate-dependent statements, and this causes a lot of your arguments to be not even wrong (also see Feynman's essay about http://www.lhup.edu/~DSIMANEK/cargocul.htm, where people imitate some of the external forms of scientific explanation but without the underlying technical substance). I'm trying to teach you to think more clearly about the terms you're throwing around--that's why I ask these questions offering you lists of different possible meanings that might be assigned to terms like "distance" and "time dilation" (like coordinate distance, apparent visual distance, etc.) and emphasize that your claims don't really make sense under any of these specific meanings.

I think they're what happens when you let maths lead the "understanding" instead of doing it the other way round. There, I said it.

That, fundamentally, is what modern physics is all about, and the sooner you realize that the sooner you'll be on track to thinking like physicists do. Any self-consistent mathematical model is potentially an accurate model of the universe, there are no real criteria that should cause you to reject any mathematical model besides the fact that it doesn't give accurate predictions. Think of it this way, if you had access to a computer of the Gods that could simulate any arbitrarily large collection of fundamental particles or points in spacetime or other basic entities, all constrained to obey some general mathematical law that could be programmed into the simulation, then any possible rules that could be programmed into the computer that would lead to the evolution of simulated beings whose empirical observations match our own should be considered a viable candidate for the "program" governing the behavior of our own universe. To argue otherwise would be to argue that we have some totally non-empirical reasons for believing the laws of physics must take one type of form (one that has some kind of appeal to our 'common sense' intuitions) as opposed to other types of forms that might still produce correct predictions about all empirical observations.

 

[JesseM]

Hi When you refer to the upper and lower components are you talking about the lower deceleration part of the diagram and the upper or acceleration half of the diagram??

I said "the Rindler horizon has an upper and lower component"--I was talking only about the horizon, which would be the two straight red lines at 45 degrees in the diagram which form the upper and lower boundary of the "Rindler wedge". These are analogous to the black hole event horizon and white hole event horizon that form the upper and lower boundaries of the exterior region I in the Kruskal-Szekeres diagram for a Schwarzschild black hole.

When you say light from events arbitrarily far , I assume this also means distant in time also ,,is this correct??

Right, for example if at some point an accelerating Rindler observer receives light from an event that happened 1000000 light years away in some inertial frame (perhaps far beyond the Rindler horizon), then this event would have happened 1000000 years earlier in that inertial frame. Note that if this event happened outside the Rindler horizon, it must have happened somewhere in the past light cone of the event at the center of that diagram, the one at the "tip" of the Rindler wedge where the two components of the horizon meet. In the Kruskal-Szekeres diagram this past light cone of the event at the center would correspond to the white hole interior region.

That the events from much further than the horizon would then have to have occurred further in the past, yes??
If so this is more or less what I thought.

Right.

So if I am understanding you correctly, both Rindler and CMIRF observers would visually see both Doppler shift and aberration but nothing identifiable beyond that. yes?

Right again.

 

[Anamitra]

Two gravitating objects may be observed at some point of time,{t=t_{0}}
A few seconds ago they were at a greater distance and were definitely attracting each other[and not repelling]. Another few seconds ago they were still attracting each other[and the force was not of repulsive nature].
Let's see it mathematically:
{F_{G}}{=}{\frac{dp}{dt}}
For t-->-t
We seem to have,
{F_{G}}{=}{-}{\frac{dp}{dt}}
But
{p}{=}{m}{\frac{dx}{dt}}, for a constant mass particle
Here also we need to change the sign of t
Finally we have[after time reversal],
{F_{G}}{=}{\frac{dp}{dt}}

For a particle with variable mass,

{F}{=}{m}{\frac{dv}{dt}}{+}{v}{\frac{dm}{dt}{=}{m}{\frac{dv}{dt}}{+}{\frac{dx}{dt}}{\frac{dm}{dt}

In the last term to the right we need to bring two changes [{t-->}{-}{t}] in time and so it does not change sign[assuming mass to increase in the forward direction of time].And the first term also does not change sign. So ,attraction remains attraction.
[Relativistically mass should increase with time with increasing speed.]

 

[A-wal]

If you're talking about a purely coordinate-dependent or path-dependent notion of "relative velocity", then you must agree that just because two objects at distant locations have a relative velocity of 0.5c in some coordinate system or with velocities parallel-transported along some path, there may be some other coordinate system or path where the relative velocity is 0.9c or 0.1c or even 0. So it seems to me the relative velocity of 0.5c does disappear in curved spacetime, in the sense that it there is no objective sense in which two objects can be said to have a relative velocity of 0.5c as opposed to 0.9c or 0.

Only if you are talking about some coordinate-dependent notion of "diameter", like the diameter in Schwarzschild coordianates or Eddington-Finkelstein coordinates. Please, if you are going to talk about "distance" then please tell me if you are talking about distances in some coordinate system, or apparent visual distances and sizes, or if you think there is some third option.

I don't know what you mean by "perspective". Are you talking about visual appearances at 100 d-units vs. 50 d-units, or are you imagining that the observer will use different coordinate systems depending on his distance, or what?

Do you mean "constant distance from the singularity" in terms of some coordinate system like Schwarzschild coordinates, or in terms of visual appearances (in which case it's not correct to say the visual appearance of the horizon maintains a constant distance from the singularity, both visual horizons appear to change in apparent size as you approach and anyway the singularity has no apparent visual position, it can't be 'seen' any more than the Big Crunch singularity in a contracting universe could be 'seen' in advance), or do you imagine there is some other notion of "distance" besides coordinate distance and apparent visual distance?

Any comment about "distance" is meaningless unless you specify what you mean by that word. Please answer my question: are you referring to apparent visual distance, or to distance in some coordinate system, or do you claim there is some third notion of "distance" aside from these? (I suppose you could also talk about the integral of ds^2 along some specific spacelike path, like the worldline of a hypothetical tachyon, which would have a coordinate-independent value just like proper time along a timelike worldline).

All I ever meant (as I keep telling you) is that objects can obviously move relative to each other in curved space-time, and that the velocity can be measured. Obviously if you measure it differently you’ll get a different result. But as long as we keep assuming they stick to the same method then what’s the problem? Besides can’t we just assume the shortest path? In fact, from now on I’ll always use the shortest path between any specified objects or phenomena unless I expressly specify otherwise. Now you’ll never have to ask me that again.

No physics theory has ever "explained" why matter/energy/particles behave in the way they do, it just gives equations describing their behavior.

Plenty of theories explain why matter/energy/particles behave in the way they do. In fact that's what every physics theory attempts to do. General relativity explains Newtons laws for example.

If it wouldn't be obvious to a physicist well-versed in the mathematics how to translate your verbal argument into a detailed mathematical one, then hell yes it's "handwavey". In physics verbal arguments are only meaningful insofar as they can be understood as shorthand for a technical argument, where the meaning of the shorthand is clear enough that you don't have to bother laboriously spelling everything out in technical terms. This is not to say that physicists don't appeal to physical intuitions in situations where they're groping for the correct answer to some problem they don't know how to solve in a rigorous mathematical way, but I think they always do so with an eye towards developing a technical mathematical argument, I think you'd be hard-pressed to find a physicist who believes that some verbal argument is sufficient in itself to demonstrate some claim even if they don't know how to translate it into technical terms (and don't think other physicists would be able to either).

I am not a physicist! It’s not fair for you to expect me to know how to put it in technical terms. It doesn't mean I don't get it. Equasions are the shorthand for whatever it is they represent.

So does that mean you are retracting all your earlier statements suggesting there is some objective sense in which we can talk about which clock experiences more time dilation at different distances from the horizon, apart from the question of which clock experiences more total elapsed time between two meetings which each occur at a single point in space time? For example, this comment from post #161:

Or this one from post #165:

If you don't retract these comments, then how does this square with your agreement that depending on our choice of simultaneity convention we can reach different conclusions about whether clock B was ticking faster or slower than clock A between the events of clock A reading a time T and clock A reading a time T + delta-t?

When I said use your common sense I was just saying that people can still compare watches in curved space-time. You disagree?

OK, #2 was "Local comparisons of elapsed times, as in the twin paradox where each twin looks at how much time each has aged between two meetings". But in this case you can't say anything about time dilation except when talking about total elapsed time between two local comparisons, in particular you can't say one clock was ticking slower when it was closer to the horizon. If you agree with that, it seems to me you are changing your tune from your comments in post #161 and #165 when the statements of mine you were disagreeing with were just statements that we can only talk about differences in total elapsed time and nothing else.

Of course they can say one clock was ticking slower when it was closer to the horizon. One clock was ticking slower when it was closer to the horizon if one spent all that time closer to the horizon and less time has passed for it.

I didn't use the words "true horizon", what part of my above explanation are you referring to?

It was from something you either quoted or linked.

Do you mean the white horizon as opposed to the red antihorizon? Do you doubt that there are solutions to the Einstein field equations involving both eternal Schwarzschild black holes as well as black holes that form from collapsing stars, and that when you consider the "maximally extended" version of these spacetimes (the meaning of 'maximally extended' is discussed in the Kruskal-Szekeres article, ask questions if you don't understand something about it), they include a region of spacetime where anything inside the region will inevitably hit the singularity, and a region where it's possible to avoid the singularity, with the boundary between the two regions defined as the "event horizon"?

I believe there is a region of space-time where anything inside would inevitably hit the singularity. I also believe it’s analogous to saying that there is a velocity that exists that is greater than c. I also believe this velocity can’t ever be reached.

You seem to have some confused idea about what "spacetime" means, it just refers to the continuous curved 4D manifold consisting of every possible point in space and time where a physical event could occur, including events at different distance from the horizon, along with a definite geometry (curvature at every point, defined by the metric) assigned to this manifold.

I just see it as the distance between objects, which is relative and the difference is the curve. To our linier perspective it means that everything with relative velocity moves in straight lines but through curved space-time – gravity. If you want to create your own curve you accelerate.

"Same spacetime" just means we are talking about the same geometry--events which both occur on the same solution to the Einstein field equations (say, the Schwarzschild metric as opposed to the Minkowsk metric or a FLRW metric)

Oh okay. My point was that one is in space-time that’s more length contracted/time dilated than the other. I don’t see how infinite time dilation/ length contraction can disappear if you change coordinate systems. You can’t change reality by measuring differently (yes I know you can in quantum mechanics). Change the value of something yes, but not get it to or from infinity. That still seems contradictory.

No, see the various threads on how it's meaningless to talk about the "perspective" of an observer moving at c, like this one. If you consider what some inertial landmarks look like for an observer moving at v relative to them in the limit as v approaches c, some quantities do approach infinity in this limit, but in any case this would only be approaching a coordinate singularity as opposed to a genuine physical singularity (where some quantity approaches infinity at a given point in all coordinate systems which approach arbitrarily close to that point, like the curvature singularity at the center of a black hole.)

I though it’s meaningless because the universe would be a singularity at c. I thought photons have a very short life span but it doesn’t matter because they’re infinitely time dilated meaning there frozen in time. It’s that’s true than surely the universe would be perceived as a singularity at c, if you reach c, but you can’t.

Uh, I never said it was a "reason for you thinking anything", your thinking seems to be based on vague intuitions about things like "distance" and "time dilation" that are based on statements you've read by physicists (or physics popularizers) in various places, without realizing that physicists generally use such terms in the context of some specific coordinate system. So your failure to realize this means you try to imitate the way physicists would use such terms but without sufficient understanding of how they are just verbal shorthand for coordinate-dependent statements, and this causes a lot of your arguments to be not even wrong (also see Feynman's essay about http://www.lhup.edu/~DSIMANEK/cargocul.htm, where people imitate some of the external forms of scientific explanation but without the underlying technical substance). I'm trying to teach you to think more clearly about the terms you're throwing around--that's why I ask these questions offering you lists of different possible meanings that might be assigned to terms like "distance" and "time dilation" (like coordinate distance, apparent visual distance, etc.) and emphasize that your claims don't really make sense under any of these specific meanings.

Read the blogs I wrote ages ago and tell me if I’m under the wrong impression about anything. You think I don’t understand the concepts just because I don’t know how to speak your language. If I am wrong about anything then by all means tell me but stop just telling me I don’t get it. Don’t get what? The fact that length contraction and time dilation have to be expressed in specific coordinate systems to make sense? No they don’t. Maybe they do on paper.

That, fundamentally, is what modern physics is all about, and the sooner you realize that the sooner you'll be on track to thinking like physicists do.

I really don’t want to think like physicists do. I want to get this straight in my head and still think the way I do. I don’t know how you can do it like that. Would you watch your favorite dvd in binary code? The code is necessary but no one cares what it looks like.

Any self-consistent mathematical model is potentially an accurate model of the universe, there are no real criteria that should cause you to reject any mathematical model besides the fact that it doesn't give accurate predictions. Think of it this way, if you had access to a computer of the Gods that could simulate any arbitrarily large collection of fundamental particles or points in spacetime or other basic entities, all constrained to obey some general mathematical law that could be programmed into the simulation, then any possible rules that could be programmed into the computer that would lead to the evolution of simulated beings whose empirical observations match our own should be considered a viable candidate for the "program" governing the behavior of our own universe. To argue otherwise would be to argue that we have some totally non-empirical reasons for believing the laws of physics must take one type of form (one that has some kind of appeal to our 'common sense' intuitions) as opposed to other types of forms that might still produce correct predictions about all empirical observations.

Of course you can't just assume intuition is right. That's stupid. But it always starts and ends with that. It decides what you test for and what conclusions you draw from the results. Without it you're trying to paint in the dark.


I don’t mind if I’m wrong. My ego isn’t tied up in this and I have nothing to prove. I find it difficult to accept what I don’t understand and I’m not convinced by what I’ve been told. How the hell can an object that can never reach the horizon from any external perspective ever cross the horizon from its own perspective? Is not just the light from those objects that’s frozen. How could it be if they could always escape? They’re moving slower and slower through time relative to you because time in that region is moving slower and slower relative to you. If the time dilation/length contraction go up to infinity then no given time can ever long enough and no distance can ever be short enough locally if it’s infinitely length contracted from a distance! Are those inertial coordinates you use to describe an object crossing the event horizon even relative? Does it take into account the fact that you’re constantly heading into an ever increasingly sharpening curve?

ObserverA measures the distance between ObserverB (who is much closer) and the horizon using some coordinate system or other. Then after moving right next to ObserverB measures it to be more than it seemed before in the same coordinate system. ObserverA: "You've moved!" ObserverB: "No I haven't!" (Because it's not length contracted when you're actually there). You have to un-length contract the space, making the distance longer, right? But if it’s infinitely length contract at the horizon then there’s going to be an infinite amount of space (and time) between you and it before you get there.

 

[timetravel_0]

OMG... Owe Emm Gee! Oh My Freaking Gosh! You two really went on for this long when the original post had a logical fallacy to begin with!

If Time is reversed everywhere, then Cause and Effect are reversed. So every effect results to it's cause. If two objects are drawn towards each other by gravity, then if time is reversed -the objects move away from each other. They are moving back to the CAUSE, not repelling! The cause of the attraction? Falling into each others gravitational field. If a ball is thrown in the air, it goes up, then comes down. Time reversed... well... it goes up and the comes down - back to the cause. Really this was a silly topic and if I was googleing something and this topic came up I'd be ticked...

 

[JesseM]

If Time is reversed everywhere, then Cause and Effect are reversed.

Time reversal symmetry doesn't imply time travel or the idea that any given system will suddenly start running backwards. Basically it just means that the laws of physics work in such a way that if you take a movie of a given system and play it backwards, there's no way for a physicist who sees the movie to be sure it's playing backwards rather than forwards, because the laws of physics would allow for a different system with different initial conditions to evolve in the forward direction in a way that's identical to the backward movie of the original system.

Even with time-symmetric laws of physics there may be cases where a given movie is a lot less likely to be playing forwards rather than backwards, but those are always cases where entropy is changing. And the fact that entropy is more likely to increase than decrease is understood in statistical terms (it's similar to the fact that if you shuffle a deck of cards starting with the cards in some order, the deck is likely to get more disordered, whereas if you start with a disordered deck it's unlikely that shuffling them will randomly happen to put them in order, though it's not impossible).

If two objects are drawn towards each other by gravity, then if time is reversed -the objects move away from each other.

Gravity is a time-symmetric theory, so any backwards movie of a gravitational system is consistent with the same laws. Think of a comet which moves in close to the Sun from far away, then swings around and travels away from the Sun again--nothing would seem particularly strange about this if you played it backwards! And if you assume there are no changes in entropy so all collisions are perfectly elastic, then even when collisions are involved the movie would still make sense in reverse--think of a ball which falls from height H down to the ground, if it collides with the ground in an elastic way it'll bounce right back up to height H, then fall again, bounce up again, etc., with the height never decreasing.

 

[A-wal]

A ship is deliberately pulled into a black hole. It crosses the horizon (arh, that just can't be right) when the black hole is a certain size and there's a second observer who follows close behind but doesn't allow themselves to be pulled in. There's a very strong rope linking the two. The second observer can never witness the first one reaching the event horizon so it can never be too late for them to find the energy to pull the first observer away from the black hole even after the it's shrunk to a smaller than when the first one crossed from it's own perspective. If the closer one always has the potential to escape the black hole under its own power from the further ones perspective then it should always be possible for the closer one to escape under the further ones power. So the first ship can't escape from it's own perspective but it does from the second ships perspective.

If that’s not a paradox then I don’t know what is. Get out of that one smeg head.

 

[Dale]

Get out of that one smeg head.

And you were complaining about my tone

 

[A-wal]

And you were complaining about my tone

Context dude. It's not what you say, it's how you say it, even when it's written down. I've gotten a lot more then I've given in this thread. Besides, smeg head is hardly a dirogatory term. I on the other hand have been repeatedly patronised for no sodding reason! Physicists

So I take it from the lack of responses that either I've outstayed my welcome or it's agreed that this is a guenuine paradox?

In case you'd like a self-consistant description of what happens when there's gravity with no force resisting it:

The event horizon expands outwards at c when a black hole forms. Anything caught within it when this happens is ejected at the singularity at the moment it forms as a gamma ray burst. You could look at it as going back in time but it will be spreading outwards with the black hole, so it's probably easier to imagine anything the event horizon touches as being converted to energy on the spot. There's absolutely no difference between these two ways of looking at it because the effect is exactly the same. Time is infinitely dilated at c so any distant observer (not caught inside while it's forming) will detect the grb and the initial gravity wave simultaneously either way. The black hole is the singularity because in its own frame it has zero size and exists for no time at all. It's only from a distance that it covers an area of space-time. I suppose you could also look at the initial expansion phase as the event horizon remaining fixed and anything within a certain range being pulled towards it at c. There's no way into a black hole once it stops expanding because the singularity covers less space-time the closer you get to it. It should gradually lose mass/energy just by exerting influence.

 

[JesseM]

All I ever meant (as I keep telling you) is that objects can obviously move relative to each other in curved space-time, and that the velocity can be measured.

So would you agree that if we are considering a segment of object A's worldline such that it never crosses the worldline of object B anywhere on that segment, we can always find a coordinate system where object B's speed relative to object A is zero throughout the time period of that segment, i.e. B is not moving relative to A during that period?

Obviously if you measure it differently you’ll get a different result. But as long as we keep assuming they stick to the same method then what’s the problem? Besides can’t we just assume the shortest path? In fact, from now on I’ll always use the shortest path between any specified objects or phenomena unless I expressly specify otherwise. Now you’ll never have to ask me that again.

You can only talk about the path of the shortest distance between two objects if you have a simultaneity convention, so you know which event on object A's worldline and which event on object B's worldline you are supposed to find the "shortest path" between.

No physics theory has ever "explained" why matter/energy/particles behave in the way they do, it just gives equations describing their behavior.

Plenty of theories explain why matter/energy/particles behave in the way they do. In fact that's what every physics theory attempts to do. General relativity explains Newtons laws for example.

That's not any sort of conceptual "explanation", it's just showing that the equations of theory #1 can be derived as some sort of approximation to the equations of some more accurate theory #2, but then the equations of theory #2 just have to be accepted with no explanation whatsoever. So, it's still correct to say that ultimately physics gives no explanations, it just gives equations.

If it wouldn't be obvious to a physicist well-versed in the mathematics how to translate your verbal argument into a detailed mathematical one, then hell yes it's "handwavey". In physics verbal arguments are only meaningful insofar as they can be understood as shorthand for a technical argument, where the meaning of the shorthand is clear enough that you don't have to bother laboriously spelling everything out in technical terms.

I am not a physicist! It’s not fair for you to expect me to know how to put it in technical terms. It doesn't mean I don't get it. Equasions are the shorthand for whatever it is they represent.

I didn't ask you to put anything in technical terms, I just said that if your argument couldn't be translated into mathematical terms by "a physicist well-versed in the mathematics" then it isn't physically meaningful. Also, in our exchanges I often give you various specific technical meanings that could be assigned to various vague phrases and ask you to think about those technical meanings and answer questions about your own meaning in terms of them, but although you occasionally pick from among these specific meanings you mostly just repeat the same vague formulations even after I have pointed out the ambiguity in them. For example, in post #166 I asked you:

OK, but you completely failed to address my question about whether you were talking about visuals or something else. I can think of only 3 senses in which we can talk about clocks ticking at different rates in GR:

1. Visual appearances--how fast the an observer sees the image of another clock ticking relative to his own clock
2. Local comparisons of elapsed times, as in the twin paradox where each twin looks at how much time each has aged between two meetings
3. Coordinate-dependent notions of how fast each clock is ticking relative to coordinate time at a particular moment (which depends on the definition of simultaneity in your chosen coordinate system)

If you are confident there is some other sense in which we can compare the rates of different clocks, please spell it out with some reference to the technical definition you are thinking of in GR ... if you agree that those three are the the only ways of comparing clock rates that make sense in GR, please tell me which you are referring to when you talk about "time dilation" near the event horizon being the explanation for why an external observer can never witness anything crossing it.

And in post #167 you did respond:

#2.

So, here you seem to claim that when you talk about "comparing clock rates" you are only talking about elapsed time on each clock between two local meetings. But then I pointed out at the start of post #171 that several of your earlier comments were pretty clearly talking about "comparing clocks" when they are far apart rather than just comparing their elapsed time between two local meetings, but instead of either retracting those earlier comments or explaining which meaning of 1-3 above they were referring to (or if you thought there was a different possible meaning that could be assigned to 'comparing clocks' besides 1-3), you just gave another completely ambiguous reply in your most recent response:

When I said use your common sense I was just saying that people can still compare watches in curved space-time. You disagree?

Why would I disagree, when I specified 3 different ways in which they could "compare watches in curved space-time"? The point is that "compare watches" is too vague since it could mean multiple different things, so I'd like to request that if you want to continue this conversation, please always specify which of the 3 you are talking about (or if you think there is some fourth option) any time you talk about "comparing watches".

Likewise in post #171 I talked about the ambiguity in talking about "distance":

Any comment about "distance" is meaningless unless you specify what you mean by that word. Please answer my question: are you referring to apparent visual distance, or to distance in some coordinate system, or do you claim there is some third notion of "distance" aside from these? (I suppose you could also talk about the integral of ds^2 along some specific spacelike path, like the worldline of a hypothetical tachyon, which would have a coordinate-independent value just like proper time along a timelike worldline).

That last comment about the "integral of ds^2" and tachyons may be overly confusing, but the idea is that in GR just as there is a coordinate-independent notion of "proper time" along the worldlines of slower-than-light objects (these worldlines are called 'timelike' ones), so there is a coordinate-independent notion of "proper distance" along a different kind of path through spacetime (a 'spacelike' one), a path where every point on the path occurs simultaneously according to some simultaneity convention (so with that choice of simultaneity convention, you are measuring the distance along a path at a single instant). So, if you want to know the distance between A and B at some time on A's clock, then given a choice of simultaneity convention you can talk about the "proper distance" along the shortest path between them at that moment. The choice of simultaneity convention is itself arbitrary since there are an infinite number of equally valid ways to define which set of events occurred "at the same time", but once you have fixed a choice of simultaneity convention, there is a coordinate-invariant notion of the shortest possible "proper distance" between two objects at any given moment.

So, just as I requested that you always specify which of the three sense 1-3 you mean when you talk about "comparing clocks", I would also request that if you want to continue the conversation you also always specify which of the following you mean (or if you think there is a fourth option) whenever you talk about "length" or "distance" or "size":

1. Apparent visual distance (angular diameter or something along those lines)

2. Coordinate distance in some arbitrary choice of coordinate system, taken at some coordinate time

3. Proper distance along the shortest path, given an arbitrary choice of simultaneity convention

Finally, please also specify which of the following (if any) you mean when you use phrases like "relative velocity":

1. Another visual definition, like how fast an object's visual position or angular diameter is changing with the observer's own proper (clock) time

2. Coordinate velocity in some choice of coordinate system

3. Given a choice of simultaneity convention, the rate at which "proper distance along the shortest path" is changing relative to some notion of time, like coordinate time in a coordinate system which uses that simultaneity convention, or the proper time of one of the two moving objects (if you pick #3, please specify which notion of time you want to use)

OK, #2 was "Local comparisons of elapsed times, as in the twin paradox where each twin looks at how much time each has aged between two meetings". But in this case you can't say anything about time dilation except when talking about total elapsed time between two local comparisons, in particular you can't say one clock was ticking slower when it was closer to the horizon. If you agree with that, it seems to me you are changing your tune from your comments in post #161 and #165 when the statements of mine you were disagreeing with were just statements that we can only talk about differences in total elapsed time and nothing else.

Of course they can say one clock was ticking slower when it was closer to the horizon. One clock was ticking slower when it was closer to the horizon if one spent all that time closer to the horizon and less time has passed for it.

No, #2 deals only with total elapsed times, it doesn't allow for comparison of rates during any segment of the trip that's shorter than the entire period from the first meeting to the second. Say A and B separated when B's clock read 0 seconds and they reunited when B's clock read 100,000 seconds, and A's clock also read 0 seconds when they separated but read 200,000 seconds when they reunited. If B spent the time between 10,000 and 90,000 seconds at some constant Schwarzschild radius close to the horizon, while the other 20,000 seconds were spent traveling from A to that closer radius and back, would you agree there's no way to decide whether B's clock was ticking faster or slower than A's during that period without having a definition of simultaneity to decide what A's clock read "at the same moment" that B's read 10,000, and what A's clock read "at the same moment" that B's read 90,000?

For an eternal black hole, the red horizon is actually a physically separate horizon, the "antihorizon" one that borders the bottom of "our" exterior region I and the top of the alternate exterior region III in the maximally extended Kruskal-Szekeres diagram. The falling object genuinely never crosses this horizon, it's a white hole horizon in our universe and a black hole horizon in another exterior universe inaccessible from our own.

For a more realistic black hole that formed at some finite time from a collapsing star, you wouldn't actually be able to "see" any horizon from the outside, in the sense that light emitted from events on an event horizon would never reach anyone outside, at least not unless the black hole evaporated away. However, this section of the other site on falling into a black hole I linked to earlier also seems to say that if you could see the highly redshifted image of the collapsing star long after the black hole had formed, it would occupy almost exactly the same visual position as the red antihorizon of an eternal black hole:

I can honestly see no need for the "true" horizon.

I didn't use the words "true horizon", what part of my above explanation are you referring to?

It was from something you either quoted or linked.

Even if that were true, how would it in any way relate to/refute the two paragraphs of mine quoted above (the ones starting with 'For an eternal black hole...'), which you were ostensibly responding to?

I believe there is a region of space-time where anything inside would inevitably hit the singularity. I also believe it’s analogous to saying that there is a velocity that exists that is greater than c. I also believe this velocity can’t ever be reached.

According to relativity it's not analogous, since objects reach that region in finite proper time, whereas no one could ever accelerate to c in finite proper time.

Nope, that's just flat-out wrong. I already told you many times that time dilation and length contraction don't go to infinity at the horizon in Kruskal-Szekeres coordinates, and also that in ordinary Minkowski spacetime you do have infinite time dilation and length contraction at the Rindler horizon if you use Rindler coordinates, but obviously this is a purely coordinate-based effect which disappears if you use ordinary inertial coordinates in the same spacetime.

In the same space-time? When comparing objects at different distance from an event horizon they can't possibly be in the same space-time.

You seem to have some confused idea about what "spacetime" means, it just refers to the continuous curved 4D manifold consisting of every possible point in space and time where a physical event could occur, including events at different distance from the horizon, along with a definite geometry (curvature at every point, defined by the metric) assigned to this manifold.

I just see it as the distance between objects,

Please specify which notion of "distance" 1-3 you mean, or if you think there is some other well-defined notion of distance.

which is relative and the difference is the curve. To our linier perspective it means that everything with relative velocity moves in straight lines but through curved space-time – gravity. If you want to create your own curve you accelerate.

How does this notion of "spacetime" as the "distance between objects" relate to your earlier comment "When comparing objects at different distance from an event horizon they can't possibly be in the same space-time"? Are you saying the two observers will define the "distance" differently? If so, then again, please specify which of the three notions of "distance" I gave is the best match for what you mean, if any.

My point was that one is in space-time that’s more length contracted/time dilated than the other.

What does it mean for "space-time" to be length contracted/time dilated? Only objects can be length contracted, and only clocks can be time-dilated. Are you saying the length of the guy near the black hole is shorter, and his clock is running slower? If so then as always I need to know which meaning 1-3 of "length" matches yours, and what "running slower" means in terms of the the 3 possible ways of "comparing clocks".

I don’t see how infinite time dilation/ length contraction can disappear if you change coordinate systems.

I still don't know what you claim there is infinite time dilation/length contraction near the horizon. For example, if you're talking about visual appearances (option #1 in both cases), it's true that a distant observer sees something approaching the event horizon become more squashed in apparent visual length and sees its clock appear to run slower, but the same would be true for the visual appearance of something approaching the Rindler horizon as seen by an accelerating observer at rest in Rindler coordinates.

You can’t change reality by measuring differently

I have no idea what "reality" you think you are referring to!

No, see the various threads on how it's meaningless to talk about the "perspective" of an observer moving at c, like this one. If you consider what some inertial landmarks look like for an observer moving at v relative to them in the limit as v approaches c, some quantities do approach infinity in this limit, but in any case this would only be approaching a coordinate singularity as opposed to a genuine physical singularity (where some quantity approaches infinity at a given point in all coordinate systems which approach arbitrarily close to that point, like the curvature singularity at the center of a black hole.)

I though it’s meaningless because the universe would be a singularity at c. I thought photons have a very short life span but it doesn’t matter because they’re infinitely time dilated meaning there frozen in time. It’s that’s true than surely the universe would be perceived as a singularity at c, if you reach c, but you can’t.

Nope, none of that is correct according to relativity. Again I really recommend reading some of the many threads on the subject, like this more recent one or this older one, if you want to correct your misunderstandings.

 

[JesseM]

(continued)

Read the blogs I wrote ages ago and tell me if I’m under the wrong impression about anything.

Yes, though some of what you say there is correct you still were making the same sort of confused statements about concepts like "length contraction" and "time dilation". For example, your explanation of the curved path of the light beam:

The light will be following a curved path from their perspective despite the fact that light always moves in a straight line. This is because gravitational pull is the equivalent to acceleration. So it distorts space-time, in this case through length contraction. ... Downwards momentum in freefall is caused by unchallenged length contraction.

In fact the curved path of the light beam in a room at rest in a gravitational field can just be understood in terms of the equivalence principle and the fact that a free-falling observer in a gravitational field will make the same observations as an inertial observer in flat spacetime, so since the observer at rest on the surface of the planet is accelerating relative to a freefalling observer near him, his observations should be equivalent to those of an accelerating observer in flat spacetime, who sees light paths as curved for reasons that have nothing to do with length contraction or time dilation. http://www.phy.syr.edu/courses/modules/LIGHTCONE/equivalence.html has some helpful gifs:

[PLAIN]http://www.phy.syr.edu/courses/modules/LIGHTCONE/anim/equv-m.gif

You think I don’t understand the concepts just because I don’t know how to speak your language.

No, I think you don't understand the concepts because your "conceptual" ideas lead you to reach conclusions which actively contradict the conclusions of GR. Anyway, I'm not asking you to figure out how I would say things, that's why I do things like give various ways I might interpret an ambiguous phrase and ask you to pick which one (if any) matches what you mean, and perhaps think more carefully about the distinct meanings I offer in case your own phrasing may have been equivocal without your realizing it.

If I am wrong about anything then by all means tell me but stop just telling me I don’t get it. Don’t get what? The fact that length contraction and time dilation have to be expressed in specific coordinate systems to make sense? No they don’t.

I say they do, unless you mean one of the other options I mentioned above, like using "time dilation" only to talk about elapsed time between two local comparisons without any notion of comparing the clock rates at any shorter time interval in between these comparisons when the clocks were far apart.

Maybe they do on paper.

What does that even mean, "on paper"? I'm just saying it's meaningless to talk about "length contraction" or "time dilation" unless you have some quantitative way to define those concepts, whether in terms of a coordinate system or something else. Do you disagree? Do you think we can talk about such things without a quantitative definition, that we can just sort of have a gut instinct that "that ruler's length is contracted" or "that clock is running slow" even if we have no way to measure these things, and don't even know what it would mean to measure them?

I really don’t want to think like physicists do. I want to get this straight in my head and still think the way I do. I don’t know how you can do it like that. Would you watch your favorite dvd in binary code? The code is necessary but no one cares what it looks like.

There are all sort of specific experiences I get out of watching a DVD that I wouldn't get out of seeing the binary code, like images and sounds. In contrast "length contraction" and "time dilation" aren't things we experience in any such direct way distinct from various specific technical meanings like the ones I gave.

I don’t mind if I’m wrong. My ego isn’t tied up in this and I have nothing to prove. I find it difficult to accept what I don’t understand and I’m not convinced by what I’ve been told. How the hell can an object that can never reach the horizon from any external perspective ever cross the horizon from its own perspective?

By "never reach the horizon from any external perspective" do you just mean what's seen visually by external observers? But it's similarly true in a visual sense that observers who remain outside the Rindler horizon (like the accelerating Rindler observers) will never ever see anything reach the Rindler horizon, I bet in that case you don't have any problem believing that the objects approaching it can experience crossing it though. Am I wrong? (please note that here I am not making any reference to Rindler coordinates, I'm purely talking about visual appearances for an observer with constant proper acceleration, who will have a visual Rindler horizon and won't ever see the light from events on or beyond it)

Is not just the light from those objects that’s frozen. How could it be if they could always escape?

It's also true that for an accelerating Rindler observer who is watching something approach the horizon, no matter how close he sees it get, he can never be 100% sure that it won't turn around and come back to him at some point. But I bet in this case you have no problem believing it is just a matter of light, that the object approach the horizon can always accelerate to avoid it at the last minute, and the closer it was to the horizon before it accelerated, the longer it will be before light from the moment of acceleration can catch up to the distant Rindler observer.

They’re moving slower and slower through time relative to you because time in that region is moving slower and slower relative to you.

Are you referring to one of the senses 1-3 of comparing the rate of their clocks with the rate of my clock if I am far from the horizon? If not, I see no reason to believe that "they're moving slower and slower through time relative to me" is even a meaningful claim, I don't have some sort of religious faith that there is some "true" time dilation distinct from any of those 3 senses, any more than I believe that objects have a "true" velocity or "true" x-coordinate.

Are those inertial coordinates you use to describe an object crossing the event horizon even relative? Does it take into account the fact that you’re constantly heading into an ever increasingly sharpening curve?

What "inertial coordinates" are you talking about? In a large region of curved spacetime, like a black hole spacetime, no coordinate system can really be "inertial". Kruskal-Szekeres coordinates aren't, even if they have some features in common with inertial coordinates, like the fact that light always moves at the same speed everywhere in these coordinates. Or are you talking about the Rindler horizon rather than an event horizon? When you talk about "heading into an ever increasingly sharpening curve" are you talking about the accelerating Rindler observers whose worldlines (as defined in an inertial frame) are hyperbolas that get ever closer to the diagonal Rindler horizon, as seen in the image from this page which I've posted in the past?

[URL]http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/Coords.gif[/URL]

ObserverA measures the distance between ObserverB (who is much closer) and the horizon using some coordinate system or other. Then after moving right next to ObserverB measures it to be more than it seemed before in the same coordinate system.

If you don't even have a specific coordinate system in mind, why do you think this would be true? If they're using Schwarzschild coordinates, and ObserverB is hovering at a constant distance above the horizon in these coordinates (which in coordinate-independent terms means he's experiencing a constant G-force, and seeing the apparent visual size of the black hole remaining costant) then by definition if ObserverA uses the same Schwarzschild coordinate system to define the distance between ObserverB and the horizon, he'll get the same answer regardless of his own position.

(Because it's not length contracted when you're actually there). You have to un-length contract the space, making the distance longer, right?

Nope, what you're saying here makes absolutely no sense to me. I really think you have some fundamentally wrong ideas about "length contraction" and "time dilation", if you continue to just confidently assume all your intuitions make sense without considering the possibility they might be a mistaken way of thinking about relativity, then we probably aren't going to be able to make any progress here.

 

[JesseM]

A ship is deliberately pulled into a black hole. It crosses the horizon (arh, that just can't be right) when the black hole is a certain size and there's a second observer who follows close behind but doesn't allow themselves to be pulled in. There's a very strong rope linking the two. The second observer can never witness the first one reaching the event horizon so it can never be too late for them to find the energy to pull the first observer away from the black hole

Yes, it can. After a certain time, any attempt by them to pull the other observer up using the rope will cause the rope to break (in fact if the outside observer waits long enough he'll see the rope break even if he doesn't try to pull it). Again, this is just the same as if you were talking about a scenario where one observer was accelerating at a constant rate and another observer, connected to the first by a rope, crossed the first observer's Rindler horizon. The first observer would never see the second observer reaching the horizon, but hopefully you don't think that means that no matter how long he waited, he'd still be able to pull the first observer back without the first ever experiencing crossing the horizon!

If the closer one always has the potential to escape the black hole under its own power from the further ones perspective then it should always be possible for the closer one to escape under the further ones power.

The closer one doesn't "always" have the potential to escape under their own power, they have only a very short proper time before they've reached the horizon and it's too late. But as long as we idealize that the closer one has the power to accelerate away an arbitrarily short amount of proper time before actually reaching the horizon--a billionth of a second before, a trillionth of a second before, 10^-1000 of a second before, etc.--then the outside observer can never rule out the possibility that he will wait until the "last minute" (i.e. last nanosecond or whatever) to accelerate, and thus that the outside observer might see it take years or centuries or millennia before the falling observer accelerates to turn around.

The situation with pulling him up isn't the same. If you pull on one end of a rope, the person on the other end doesn't feel the pull instantaneously, instead he won't feel the pull until a sound wave has traveled the length of the rope. If you pull your end of the rope a short time after the other guy starts falling in, then you may be able to see the sound wave from your pull catch up with him before he reaches the horizon (and thus for him to experience the tug before crossing the horizon), but if you wait too long then the sound wave can never ever catch up with him outside the horizon, you'll just see the wave going more and more slowly as it gets closer to the horizon. Remember I already spend a long time explaining to you how past a certain point a signal sent by an observer far from the horizon would never be able to catch up with a falling observer who appeared (visually) to be very close to the horizon from the perspective of a far observer, and you finally seemed to get it in post #149...well, this is just another application of the same idea, since a pull on one end of a rope also creates a sort of signal that can't affect the other end until the signal traverses the rope.

 

[A-wal]

Wow! Thanks for the time and effort again.

So would you agree that if we are considering a segment of object A's worldline such that it never crosses the worldline of object B anywhere on that segment, we can always find a coordinate system where object B's speed relative to object A is zero throughout the time period of that segment, i.e. B is not moving relative to A during that period?

Yes. I can also draw you a map to heaven if you'd like. The fact that it won't work if they can meet up and compare watches is to me a clue that it's a pointless statement.

You can only talk about the path of the shortest distance between two objects if you have a simultaneity convention, so you know which event on object A's worldline and which event on object B's worldline you are supposed to find the "shortest path" between.

That's not what I meant by shortest path. It's hard to explain what I mean because I don't think using coordinates in that way but there should surely be a way of using the simplest coordinate system. The shortest path between events coinciding with the same percentage of elapsed proper time maybe.

That's not any sort of conceptual "explanation", it's just showing that the equations of theory #1 can be derived as some sort of approximation to the equations of some more accurate theory #2, but then the equations of theory #2 just have to be accepted with no explanation whatsoever. So, it's still correct to say that ultimately physics gives no explanations, it just gives equations.

Curved space-time is the conceptual explanation.

I didn't ask you to put anything in technical terms, I just said that if your argument couldn't be translated into mathematical terms by "a physicist well-versed in the mathematics" then it isn't physically meaningful.

Lol. What I say is meaningless unless it makes sense to you? I know what you're getting at, it just sounds funny.

No, #2 deals only with total elapsed times, it doesn't allow for comparison of rates during any segment of the trip that's shorter than the entire period from the first meeting to the second. Say A and B separated when B's clock read 0 seconds and they reunited when B's clock read 100,000 seconds, and A's clock also read 0 seconds when they separated but read 200,000 seconds when they reunited. If B spent the time between 10,000 and 90,000 seconds at some constant Schwarzschild radius close to the horizon, while the other 20,000 seconds were spent traveling from A to that closer radius and back, would you agree there's no way to decide whether B's clock was ticking faster or slower than A's during that period without having a definition of simultaneity to decide what A's clock read "at the same moment" that B's read 10,000, and what A's clock read "at the same moment" that B's read 90,000?

No I wouldn't agree. This is what I meant by common sense. If B spent all that time at a closer radius and less time has passed for B than it has for A when they meet again then time was moving slower for B than it was for A. Why would two observers need to meet up to compare watches anyway? They can do it from a distance. It would just be more complicated. In fact couldn't they just send messages to each other and see how fast/slow there talking relative to each other?

Even if that were true, how would it in any way relate to/refute the two paragraphs of mine quoted above (the ones starting with 'For an eternal black hole...'), which you were ostensibly responding to?

The red horizon is the one that doesn't move right? I'm having trouble keeping up because you keep moving the goal posts. To start with there was only one horizon.

According to relativity it's not analogous, since objects reach that region in finite proper time, whereas no one could ever accelerate to c in finite proper time.

But I don't see how they could reach that region in finite proper time!

Please specify which notion of "distance" 1-3 you mean, or if you think there is some other well-defined notion of distance.

No no, I just mean distance. The Sun is further away than the Moon. You could come up with a coordinate system where it isn't, but it obviously still is. How much acceleration is needed to get there is a fairly tight description, or energy * time required. Same thing really. How about an average of all possible coordinate systems? Or use the background radiation as I said earlier, or use a rope between the two. It would still work the same without using any of that. I think you're placing WAY too much importance on this. Nothing changes if these aren't used and it's not necessary for what I'm saying. There's an area filled with radiation that's a purpleoid lightyears in every direction and before every thought expirement starts everyone accelarates to the frame where this radation is evenly spread throughout that area. Happy?

How does this notion of "spacetime" as the "distance between objects" relate to your earlier comment "When comparing objects at different distance from an event horizon they can't possibly be in the same space-time"? Are you saying the two observers will define the "distance" differently? If so, then again, please specify which of the three notions of "distance" I gave is the best match for what you mean, if any.

That's exactly what I meant, and see above.

What does it mean for "space-time" to be length contracted/time dilated? Only objects can be length contracted, and only clocks can be time-dilated. Are you saying the length of the guy near the black hole is shorter, and his clock is running slower? If so then as always I need to know which meaning 1-3 of "length" matches yours, and what "running slower" means in terms of the the 3 possible ways of "comparing clocks".

Yes, again that's exactly what I'm saying. See above.

I still don't know what you claim there is infinite time dilation/length contraction near the horizon. For example, if you're talking about visual appearances (option #1 in both cases), it's true that a distant observer sees something approaching the event horizon become more squashed in apparent visual length and sees its clock appear to run slower, but the same would be true for the visual appearance of something approaching the Rindler horizon as seen by an accelerating observer at rest in Rindler coordinates.

So?

I have no idea what "reality" you think you are referring to!

This one! You seem to be implying that you can change it by using a different measurement system.

Nope, none of that is correct according to relativity. Again I really recommend reading some of the many threads on the subject, like this more recent one or this older one, if you want to correct your misunderstandings.

Maybe I explained it badly. Time would in fact move infinitely fast for an object moving at c because it would be frozen in time from the perspective of an observer that it's moving at c relative to, which it can't so it doesn't really matter, and it would have to work both ways so it's paradoxical anyway.

Yes, though some of what you say there is correct you still were making the same sort of confused statements about concepts like "length contraction" and "time dilation". For example, your explanation of the curved path of the light beam:

In fact the curved path of the light beam in a room at rest in a gravitational field can just be understood in terms of the http://www.einstein-online.info/spotlights/equivalence_principle has some helpful gifs:

[PLAIN]http://www.phy.syr.edu/courses/modules/LIGHTCONE/anim/equv-m.gif[/QUOTE]How is "Downwards momentum in freefall is caused by unchallenged length contraction." a confused statement? I think that's a nice way of looking at it, I did explain it in terms of the equivalence principle as well. I do think the other part you quoted needs rewording though. I should have said "This is because resisting a gravitational pull is the equivalent to acceleration.", sorry. There's constructive criticism, then there's just being picky.

No, I think you don't understand the concepts because your "conceptual" ideas lead you to reach conclusions which actively contradict the conclusions of GR. Anyway, I'm not asking you to figure out how I would say things, that's why I do things like give various ways I might interpret an ambiguous phrase and ask you to pick which one (if any) matches what you mean, and perhaps think more carefully about the distinct meanings I offer in case your own phrasing may have been equivocal without your realizing it.

Why do you keep using "s in such a patronising way? Are you lonely or depressed at all? The last thing I'm trying to do is contradict GR. I'm trying to "conceptualise" it.

I say they do, unless you mean one of the other options I mentioned above, like using "time dilation" only to talk about elapsed time between two local comparisons without any notion of comparing the clock rates at any shorter time interval in between these comparisons when the clocks were far apart.

Why can clocks only be compared at short distances? This is what I meant earlier when I said that two people can never be in exactly the same space-time but can still compare watches and at what range does the universe stop them from doing it?

What does that even mean, "on paper"? I'm just saying it's meaningless to talk about "length contraction" or "time dilation" unless you have some quantitative way to define those concepts, whether in terms of a coordinate system or something else. Do you disagree? Do you think we can talk about such things without a quantitative definition, that we can just sort of have a gut instinct that "that ruler's length is contracted" or "that clock is running slow" even if we have no way to measure these things, and don't even know what it would mean to measure them?

No, I'm just saying that it's not necessary to define every little thing in order to explain something.

There are all sort of specific experiences I get out of watching a DVD that I wouldn't get out of seeing the binary code, like images and sounds. In contrast "length contraction" and "time dilation" aren't things we experience in any such direct way distinct from various specific technical meanings like the ones I gave.

You would get the images and sounds if you could read the code, you just wouldn't see or hear them, which is kind of my point.

By "never reach the horizon from any external perspective" do you just mean what's seen visually by external observers? But it's similarly true in a visual sense that observers who remain outside the Rindler horizon (like the accelerating Rindler observers) will never ever see anything reach the Rindler horizon, I bet in that case you don't have any problem believing that the objects approaching it can experience crossing it though. Am I wrong? (please note that here I am not making any reference to Rindler coordinates, I'm purely talking about visual appearances for an observer with constant proper acceleration, who will have a visual Rindler horizon and won't ever see the light from events on or beyond it)

There's no point with an accelerating observer at which they can't turn round and come back though is there? It should by the same with a black hole.

It's also true that for an accelerating Rindler observer who is watching something approach the horizon, no matter how close he sees it get, he can never be 100% sure that it won't turn around and come back to him at some point. But I bet in this case you have no problem believing it is just a matter of light, that the object approach the horizon can always accelerate to avoid it at the last minute, and the closer it was to the horizon before it accelerated, the longer it will be before light from the moment of acceleration can catch up to the distant Rindler observer.

See above.

Are you referring to one of the senses 1-3 of comparing the rate of their clocks with the rate of my clock if I am far from the horizon? If not, I see no reason to believe that "they're moving slower and slower through time relative to me" is even a meaningful claim, I don't have some sort of religious faith that there is some "true" time dilation distinct from any of those 3 senses, any more than I believe that objects have a "true" velocity or "true" x-coordinate.

Like I said earlier: If B spent all that time at a closer radius and less time has passed for B than it has for A when they meet again then time was moving slower for B than it was for A.

What "inertial coordinates" are you talking about? In a large region of curved spacetime, like a black hole spacetime, no coordinate system can really be "inertial". Kruskal-Szekeres coordinates aren't, even if they have some features in common with inertial coordinates, like the fact that light always moves at the same speed everywhere in these coordinates. Or are you talking about the Rindler horizon rather than an event horizon? When you talk about "heading into an ever increasingly sharpening curve" are you talking about the accelerating Rindler observers whose worldlines (as defined in an inertial frame) are hyperbolas that get ever closer to the diagonal Rindler horizon, as seen in the image from this page which I've posted in the past?

[PLAIN]http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/Coords.gif[/QUOTE]If using a coordinate system the length of an object is contracted from the perspective of a distant observer then it doesn't imply the object has literally shrunk. From it's own perspective the distance between it's tip and tail for example remain the same regardless of the perspective of a distant observer. For the distant observer to see what it looks like from the closer observers perspective they would have to lengthen the other object to get its true size, but it's not objects themselves that get length contracted/ time dilated. It's that dimension, so you would also have to increase the distance between the closer object and the event horizon to see it from their perspective, yes?

If you don't even have a specific coordinate system in mind, why do you think this would be true? If they're using Schwarzschild coordinates, and ObserverB is hovering at a constant distance above the horizon in these coordinates (which in coordinate-independent terms means he's experiencing a constant G-force, and seeing the apparent visual size of the black hole remaining costant) then by definition if ObserverA uses the same Schwarzschild coordinate system to define the distance between ObserverB and the horizon, he'll get the same answer regardless of his own position.

^

Nope, what you're saying here makes absolutely no sense to me. I really think you have some fundamentally wrong ideas about "length contraction" and "time dilation", if you continue to just confidently assume all your intuitions make sense without considering the possibility they might be a mistaken way of thinking about relativity, then we probably aren't going to be able to make any progress here.

Too mean I think of "length contraction" and "time dilation" as I explained with that circle thing. It's the simplest way and my mind completely literal. ^^

 

[A-wal]

Yes, it can. After a certain time, any attempt by them to pull the other observer up using the rope will cause the rope to break (in fact if the outside observer waits long enough he'll see the rope break even if he doesn't try to pull it). Again, this is just the same as if you were talking about a scenario where one observer was accelerating at a constant rate and another observer, connected to the first by a rope, crossed the first observer's Rindler horizon. The first observer would never see the second observer reaching the horizon, but hopefully you don't think that means that no matter how long he waited, he'd still be able to pull the first observer back without the first ever experiencing crossing the horizon!

The closer one doesn't "always" have the potential to escape under their own power, they have only a very short proper time before they've reached the horizon and it's too late. But as long as we idealize that the closer one has the power to accelerate away an arbitrarily short amount of proper time before actually reaching the horizon--a billionth of a second before, a trillionth of a second before, 10^-1000 of a second before, etc.--then the outside observer can never rule out the possibility that he will wait until the "last minute" (i.e. last nanosecond or whatever) to accelerate, and thus that the outside observer might see it take years or centuries or millennia before the falling observer accelerates to turn around.

The situation with pulling him up isn't the same. If you pull on one end of a rope, the person on the other end doesn't feel the pull instantaneously, instead he won't feel the pull until a sound wave has traveled the length of the rope. If you pull your end of the rope a short time after the other guy starts falling in, then you may be able to see the sound wave from your pull catch up with him before he reaches the horizon (and thus for him to experience the tug before crossing the horizon), but if you wait too long then the sound wave can never ever catch up with him outside the horizon, you'll just see the wave going more and more slowly as it gets closer to the horizon. Remember I already spend a long time explaining to you how past a certain point a signal sent by an observer far from the horizon would never be able to catch up with a falling observer who appeared (visually) to be very close to the horizon from the perspective of a far observer, and you finally seemed to get it in post #149...well, this is just another application of the same idea, since a pull on one end of a rope also creates a sort of signal that can't affect the other end until the signal traverses the rope.

This it what it all boils down to. I don't get this at all. If we follow the wave along the rope then even if the wave will never reach the other ship, there will still be no point when it's too late to pull them back. And if the black hole doesn't last for ever the wave will eventually reach them. It's not just that you can't witness an object crossing the horizon from a distance. No object can cross the horizon from a distance. It should always be possible to have a rope strong enough to pull them out. There can't be an infinite amount of force on the rope surely. Besides, there shouldn't be any way for the distant observer to know that the closer one has crossed. This is what I need explaining. When you move between the two perspectives they contradict each other. And thinking about it, the further observer can always catch the closer one before they reach the horizon because you can never witness them cross until you do, so again the closer one can never have crossed from any distance apart from 0, which isn't really a distance anyway. Maybe I'm being thick but this conversation has done nothing but confirm to me that it's absolutely impossible to cross the event horizon of a black hole except maybe at the moment it forms (which, because of time dilation, is more than a moment from a distance because it can't "expand" (it's size is 0 from 0 distance because of length contraction) faster than c).

BTW why the speed of sound. I know it's the speed that mechanics works at, but why? I don't think this is like asking why c has the value it does because that's constant.

 

[PeterDonis]

This it what it all boils down to. I don't get this at all. If we follow the wave along the rope then even if the wave will never reach the other ship, there will still be no point when it's too late to pull them back.

PMFJI, since I'm not sure if this thread is still live, but after reading through the thread, I'm not clear on what you think the answer would be to the question you just posed (will there be a point where it's too late to pull the "lower" observer back) in the scenario described by Greg Egan on his Rindler Horizon page (which I think has already been linked to in this thread):

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

His diagram about halfway down the page, with Adam and Eve and their worldlines, shows clearly that there *is* a point along Eve's worldline where she can no longer send any signal to Adam (an impulse down the rope, or indeed any kind of signal at all, since no signal can travel faster than light) that will reach him before he crosses the horizon.

If you agree that this can happen in flat spacetime, then you should also agree that it can happen in the spacetime around a black hole, because the latter is the same as the former as far as the behavior of the horizon and objects near it goes. The difference around a black hole is that the curvature of the spacetime allows observers who are hovering at constant radial coordinate r (and accelerating--meaning that they're analogous to the "Rindler observers" Egan describes, who are accelerating along hyperbolas in Minkowski spacetime) to maintain a constant distance from observers very far away from the hole, whereas in flat spacetime any accelerating "Rindler observer" will eventually catch up with and pass an observer at rest in an inertial frame who is very far away at some earlier time (such as the time t = 0 in Egan's diagrams).

Throwing in black hole evaporation doesn't really change any of this; it just adds some further interesting phenomena to the "future" of events where observers cross the horizon.

 

[A-wal]

Look at it like this. How long would a hovering (maintaining a constant distance from the horizon) observer have to wait before the black hole evaporates? It would obviously depend on how close they were to the horizon, but surely the hovering time needed to witness the end of the black holes life would reach zero at the horizon? If you need to accelerate for zero time then you don't need to accelerate.

 

[PeterDonis]

Look at it like this. How long would a hovering (maintaining a constant distance from the horizon) observer have to wait before the black hole evaporates? It would obviously depend on how close they were to the horizon, but surely the hovering time needed to witness the end of the black holes life would reach zero at the horizon? If you need to accelerate for zero time then you don't need to accelerate.

You can't hover *at* the horizon, because the acceleration that would be required to do so diverges to infinity as the horizon is approached. (Or, alternatively, you could say that hovering at the horizon requires moving at the speed of light, and no timelike observer can do that.)

It is true, however, that given any pair of events which are separated by a given Schwarzschild coordinate time t, the closer to the horizon you hover (while remaining outside it), the less proper time will pass for you between that pair of events. That quantity, the proper time that passes for you, does go to a limit of zero as the radial coordinate r at which you hover approaches the Schwarzschild radius, 2M. Of course, you would need to be able to endure extremely large accelerations to do this, since, as I said above, the acceleration required to hover goes to infinity as the radial coordinate r goes to 2M.

So if, say, an observer very, very far away from the black hole would say that it took 10^18 years for the black hole to evaporate (starting from some agreed event that you and the faraway observer both label as time zero), you could, if you hovered close enough to the horizon, experience only a year of proper time between the agreed starting event and the evaporation of the black hole. Now let's say that, six months into that year of your proper time, someone free-falls past you towards the horizon, and just as they pass, you toss them a rope. Within an interval of your proper time much *less* than the six months remaining until the black hole evaporates, you will not be able to prevent them from crossing the horizon by tugging on the rope, even if the rope has such a high tensile strength that the impulse of your tugging is propagated along it with the speed of light (the fastest possible speed). This is true even though, six months later according to your proper time, the black hole will evaporate.

So suppose you hover even closer to the horizon--so close that now only a single *day* of your proper time passes from the agreed starting point until the black hole evaporates. It will *still* be possible for a freely falling observer falling past you at noon on that day, say (supposing that the day "starts" at midnight by your clock), to whom you toss a rope as they pass, to fall such that, in an interval of your proper time much *less* than a day--say a minute--you will be unable to prevent them from crossing the horizon by tugging on the rope, just as above. And again, this is true even though, when your clock strikes midnight again, the black hole will evaporate.

I understand that it's hard to visualize how all this can work; but it's the clear and unambiguous prediction of GR, and the math behind it works the *same* (with a few technicalities that don't affect the argument here) as the math Greg Egan uses on the page I linked to to show that after a fairly short time by Eve's clock, she can't keep Adam from crossing her Rindler horizon by tugging on the rope. In Kruskal coordinates, the spacetime diagram of you, hovering above the horizon, and the person falling past you to whom you toss a rope, even *looks* the same as Egan's diagram (again, with a few technicalities that don't affect the argument)--you follow a hyperbola that looks just like Eve's, and the person falling past you follows a path that looks almost like Adam's (it curves inward--i.e., to the left in Egan's diagram--because of the way Kruskal coordinates scale, but again, that's just a technicality that doesn't affect the argument, it just makes the actual calculation of specific numbers a little more complicated). And none of it changes if the black hole evaporates at some future time. (In Egan's diagram, or the equivalent in Kruskal coordinates, the black hole's evaporation would appear very far up and to the right, and would not affect anything in the area he shows.)

 

[A-wal]

Yea I know you couldn't actually hover at the horizon itself because that would require infinite energy, just like reaching c. My point is you wouldn't have to. If the time needed to maintain distance from the black hole in order to outlive it reaches zero at the horizon then you don't need to accelerate at all. Length contraction and time dilation should mean the black hole has zero size and exists for no time at all at the horizon, meaning the black hole and the singularity are the same thing. Objects close to the horizon should appear to get further from it as an observer moves into more and more length contracted space when approaching the horizon.

Maybe I'm just being dense, but I don't think so.

 

[JesseM]

Yea I know you couldn't actually hover at the horizon itself because that would require infinite energy, just like reaching c. My point is you wouldn't have to. If the time needed to maintain distance from the black hole in order to outlive it reaches zero at the horizon then you don't need to accelerate at all.

Since people keep mentioning the Rindler horizon analogy to you, could you please show that you've given it some thought when making arguments like this one (or the observer-on-a-rope argument, or any of your other arguments) by discussing why you think the conclusions about an observer crossing the black hole event horizon should be any different than the conclusions about an observer crossing the Rindler horizon? Obviously a Rindler horizon doesn't evaporate, but the basic principle is the same--if you think an observer approaching the event horizon gets hurtled arbitrarily far into the future even if he's not accelerating to hover at constant distance, do you think the same is true for an observer approaching the Rindler horizon? For example, say I am hovering some distance from the BH and in 2010 A.D. I see a falling observer dropping towards the horizon, if I know the black hole will evaporate in 10 billion A.D. I guess you think I should predict that the falling observer will find himself hurtled forward in time to 10 billion A.D. before he can reach the horizon? If so, suppose in SR I am an accelerating Rindler observer, and in 2010 A.D. I see another observer moving (inertially) toward the Rindler horizon, and I know in 10 billion A.D. the universe is due to suddenly fill with purploid gas, do you think I should predict the inertial observer will find himself hurtled forward in time and suddenly choking on purploid gas before he reaches the Rindler horizon? If not, why do you think the two cases are different? For both horizons, it's true that the rate of ticking of a clock hovering very near the horizon, as seen by distant observers, approaches zero in the limit as the clock's distance from the horizon approaches zero. If this fact is not sufficient grounds to believe that a non-hovering observer approaching the horizon gets hurtled arbitrarily far into the future in the case of a Rindler horizon, there's no logical reason it's sufficient grounds to believe that's true in the case of a black hole event horizon either.

 

[Mentz114]

A ship is deliberately pulled into a black hole. It crosses the horizon (arh, that just can't be right) when the black hole is a certain size and there's a second observer who follows close behind but doesn't allow themselves to be pulled in. There's a very strong rope linking the two. The second observer can never witness the first one reaching the event horizon so it can never be too late for them to find the energy to pull the first observer away from the black hole even after the it's shrunk to a smaller than when the first one crossed from it's own perspective. If the closer one always has the potential to escape the black hole under its own power from the further ones perspective then it should always be possible for the closer one to escape under the further ones power. So the first ship can't escape from it's own perspective but it does from the second ships perspective.

If that’s not a paradox then I don’t know what is. Get out of that one smeg head.

No problem, novelty-eraser-head.

The second observer is bound to take an infinite time to pull the first one out.

 

[PeterDonis]

Since people keep mentioning the Rindler horizon analogy to you, could you please show that you've given it some thought when making arguments like this one (or the observer-on-a-rope argument, or any of your other arguments) by discussing why you think the conclusions about an observer crossing the black hole event horizon should be any different than the conclusions about an observer crossing the Rindler horizon?

I second this motion. In fact, on thinking this over after posting last night, I wanted to amplify the scenario I described then along these same lines, by constructing a scenario involving a Rindler horizon that is exactly analogous to the scenario I described involving a black hole horizon. I'll use Egan's Adam and Eve scenario as a starting point. Suppose there is another observer, Seth, who at time t = 0 is at an extremely large x-coordinate in Egan's diagram, *much* larger than s_0, the x-coordinate where Adam drops off Eve's ship at time t = 0. Seth's x-coordinate at time t = 0 is so large, in fact, that we can't tell for sure whether he is following an inertial trajectory (constant x-coordinate for all times t) or a hyperbolic trajectory like Eve's, but with a much, much smaller acceleration (so small as to be imperceptible). To make the analogy with the black hole scenario more exact, I'll treat Seth as traveling on the latter type of trajectory--i.e., a hyperbola, x^{2} - c^{2}t^{2} = s^{2}, but with a value of s so large that the acceleration c^{2} / s is negligible. This corresponds to an observer in the black hole scenario who is at a very, very large value of the radial coordinate r, so large that the acceleration required to "hover" at that radius is negligible, so the observer's proper time is basically the same as Schwarzschild coordinate time.

Now pick a very, very large interval of time in the global inertial frame; say, that between the inertial time coordinates t = - 1/2 * 10^18 years, and t = + 1/2 * 10^18 years. Consider the two events on Seth's worldline with those time coordinates; since Seth's acceleration is negligible, his proper time is basically the same as t, so he will experience 10^18 years between those two events. (I've picked these time values, of course, so that the event at time t = 0, when Adam drops off Eve's ship, is exactly halfway between them.) Now, by making Eve's acceleration large enough (and hence making s_0, her x-coordinate at time t = 0, small enough), we can make the proper time that Eve experiences between the events on *her* worldline with the two values of the t-coordinate above as small as we like; we can make it 1 year or even 1 day, by choosing her acceleration appropriately. But no matter how small we make her proper time between those two events, her proper time elapsed between when Adam drops off Eve's ship and when she can no longer prevent him from crossing the Rindler horizon (Egan calls this time \tau_{crit}) will be *much* smaller still.

 

[A-wal]

Since people keep mentioning the Rindler horizon analogy to you, could you please show that you've given it some thought when making arguments like this one (or the observer-on-a-rope argument, or any of your other arguments) by discussing why you think the conclusions about an observer crossing the black hole event horizon should be any different than the conclusions about an observer crossing the Rindler horizon? Obviously a Rindler horizon doesn't evaporate, but the basic principle is the same--if you think an observer approaching the event horizon gets hurtled arbitrarily far into the future even if he's not accelerating to hover at constant distance, do you think the same is true for an observer approaching the Rindler horizon? For example, say I am hovering some distance from the BH and in 2010 A.D. I see a falling observer dropping towards the horizon, if I know the black hole will evaporate in 10 billion A.D. I guess you think I should predict that the falling observer will find himself hurtled forward in time to 10 billion A.D. before he can reach the horizon? If so, suppose in SR I am an accelerating Rindler observer, and in 2010 A.D. I see another observer moving (inertially) toward the Rindler horizon, and I know in 10 billion A.D. the universe is due to suddenly fill with purploid gas, do you think I should predict the inertial observer will find himself hurtled forward in time and suddenly choking on purploid gas before he reaches the Rindler horizon? If not, why do you think the two cases are different? For both horizons, it's true that the rate of ticking of a clock hovering very near the horizon, as seen by distant observers, approaches zero in the limit as the clock's distance from the horizon approaches zero. If this fact is not sufficient grounds to believe that a non-hovering observer approaching the horizon gets hurtled arbitrarily far into the future in the case of a Rindler horizon, there's no logical reason it's sufficient grounds to believe that's true in the case of a black hole event horizon either.

Okay point finally taken. I think I've reached the limit of what I can understand as I go along. I'll look into the Rindler horizon in flat space-time and see if that helps me get it straight in my head. But before I do: I assume you can approach the Rindler horizon without infinite acceleration, but infinite acceleration (even if you're just drifting in) is effectively what you would experience at the event horizon of a black hole because you would need infinite acceleration to resist the pull and hover at the horizon if you could reach it.

No problem, novelty-eraser-head.

:)

The second observer is bound to take an infinite time to pull the first one out.

Why would it take an infinite amount of time? The closer observer definitely can't reach the horizon from the perspective of the further one, so it should always be possible to pull them out with a finite amount of energy in a finite amount of time using a finite strength rope.

I second this motion. In fact, on thinking this over after posting last night, I wanted to amplify the scenario I described then along these same lines, by constructing a scenario involving a Rindler horizon that is exactly analogous to the scenario I described involving a black hole horizon. I'll use Egan's Adam and Eve scenario as a starting point. Suppose there is another observer, Seth, who at time t = 0 is at an extremely large x-coordinate in Egan's diagram, *much* larger than s_0, the x-coordinate where Adam drops off Eve's ship at time t = 0. Seth's x-coordinate at time t = 0 is so large, in fact, that we can't tell for sure whether he is following an inertial trajectory (constant x-coordinate for all times t) or a hyperbolic trajectory like Eve's, but with a much, much smaller acceleration (so small as to be imperceptible). To make the analogy with the black hole scenario more exact, I'll treat Seth as traveling on the latter type of trajectory--i.e., a hyperbola, x^{2} - c^{2}t^{2} = s^{2}, but with a value of s so large that the acceleration c^{2} / s is negligible. This corresponds to an observer in the black hole scenario who is at a very, very large value of the radial coordinate r, so large that the acceleration required to "hover" at that radius is negligible, so the observer's proper time is basically the same as Schwarzschild coordinate time.

Now pick a very, very large interval of time in the global inertial frame; say, that between the inertial time coordinates t = - 1/2 * 10^18 years, and t = + 1/2 * 10^18 years. Consider the two events on Seth's worldline with those time coordinates; since Seth's acceleration is negligible, his proper time is basically the same as t, so he will experience 10^18 years between those two events. (I've picked these time values, of course, so that the event at time t = 0, when Adam drops off Eve's ship, is exactly halfway between them.) Now, by making Eve's acceleration large enough (and hence making s_0, her x-coordinate at time t = 0, small enough), we can make the proper time that Eve experiences between the events on *her* worldline with the two values of the t-coordinate above as small as we like; we can make it 1 year or even 1 day, by choosing her acceleration appropriately. But no matter how small we make her proper time between those two events, her proper time elapsed between when Adam drops off Eve's ship and when she can no longer prevent him from crossing the Rindler horizon (Egan calls this time \tau_{crit}) will be *much* smaller still.

Huh?


Whatever the closer observer does can be witnessed by the further observer. The closer observer spends one second shortening the distance between themselves and the horizon by one metre. The further observer sees the closer one shorten the distance between themselves and the horizon by a centimetre and it takes them one hundred seconds to do it. The black hole has one hundred years of life left from the perspective of the further observer. The closer one now has one year to reach the horizon before it's gone. The closer observer again spends one second shortening the distance between themselves and the horizon by one metre. The further one sees the closer one shorten the distance between themselves and the horizon by a millimetre and it takes them one thousand seconds to do it. The closer one now has just over a month to reach the horizon before it's gone. Better hurry!

At the horizon any amount of time for the further observer would be zero to the closer one. You could say that's why the closer one will never reach the horizon from the further ones perspective, but it's then a contradiction to say that the closer one can reach the horizon from their own perspective because that can't be witnessed by the further observer. Time dilation and length contraction can change the value of stuff but not whether it happens or not.

 

[JesseM]

Okay point finally taken. I think I've reached the limit of what I can understand as I go along. I'll look into the Rindler horizon in flat space-time and see if that helps me get it straight in my head. But before I do: I assume you can approach the Rindler horizon without infinite acceleration, but infinite acceleration (even if you're just drifting in) is effectively what you would experience at the event horizon of a black hole because you would need infinite acceleration to resist the pull and hover at the horizon if you could reach it.

It's effectively what you'd experience if you wanted to remain at the horizon, but just falling through it you wouldn't experience anything strange as you instantaneously crossed it--a falling observer experiences no proper acceleration at all (a given observer's proper acceleration is the same as the G-force they experience). What's true is that if you want to hover at some height H above the horizon in Schwarzschild coordinates, the proper acceleration needed to do so approaches infinity as H approaches 0. But then it's also true that if an observer wants to hover at a constant height H above the Rindler horizon in Rindler coordinates (which is the same as saying that in their own instantaneous inertial rest frame at each moment, the distance to the horizon should be H), then the proper acceleration needed to do so also approaches infinity as H approaches 0. Neither indicates that a non-hovering observer need experience any proper acceleration as they approach either type of horizon.

 

[A-wal]

No I know that you wouldn't experience acceleration if you were free-falling. I just meant that relative to someone in flatter space-time you would be effectively accelerating. You would experience time dilation and length contraction as you were accelerating and it would reach infinity at the horizon, so you shouldn't need to accelerate to outlive the black hole.

 

[JesseM]

No I know that you wouldn't experience acceleration if you were free-falling. I just meant that relative to someone in flatter space-time you would be effectively accelerating. You would experience time dilation and length contraction as you were accelerating and it would reach infinity at the horizon, so you shouldn't need to accelerate to outlive the black hole.

By someone in "flatter space-time" do you just mean someone further away from the black hole where the curvature is smaller? If so I don't know what you mean by "relative to" them, are you talking about how you are moving in some coordinate system they are using, how you appear to them visually, something else? Why do you think you are "effectively accelerating", and what does that even mean?

And can you please do like I asked and explain whether you think your arguments apply to the case of an observer falling through the Rindler horizon? Why do you think an observer falling through the black hole event horizon is "effectively accelerating" if you don't think the same is true about an observer crossing the Rindler horizon? Why do you think "you would experience time dilation and length contraction as you were accelerating and it would reach infinity at the horizon" in the case of a black hole event horizon, when presumably you don't say that about a Rindler horizon? In both cases, after all, an observer hovering at fixed distance above the horizon sees your time dilation approach infinity as you approach the horizon in a visual sense, but if you don't think this implies the time dilation is "really" going to infinity in the case of the Rindler horizon, what makes you so sure the time dilation is "really" going to infinity in the case of a black hole event horizon? What is the relevant difference for you?

 

[PeterDonis]

Whatever the closer observer does can be witnessed by the further observer. The closer observer spends one second shortening the distance between themselves and the horizon by one metre. The further observer sees the closer one shorten the distance between themselves and the horizon by a centimetre and it takes them one hundred seconds to do it. The black hole has one hundred years of life left from the perspective of the further observer. The closer one now has one year to reach the horizon before it's gone. The closer observer again spends one second shortening the distance between themselves and the horizon by one metre. The further one sees the closer one shorten the distance between themselves and the horizon by a millimetre and it takes them one thousand seconds to do it. The closer one now has just over a month to reach the horizon before it's gone. Better hurry!

You've left out one key factor in all of the above: *how* does the further observer see the closer observer doing these things? For him to see them, the light from the events happening down near the horizon has to "climb" all the way out to the further observer's radial coordinate. Because of the way the light cones are bent, it takes a *long* time for the light to get out. That means the further observer *sees* things much later not because they *actually happen* much later, but because the light takes so long to get to him. When the further observer tries to assign coordinates to the actual events that emitted the light, he has to correct for the light travel time; and when he does that, he finds that those events near the horizon did not *actually* happen close to the black hole evaporating; they happened a long time before, but it took the light that long to get to him.

This is one reason why Schwarzschild coordinates are bad ones to use when trying to relate what happens near the horizon to what happens far away. Kruskal coordinates (which, as I noted before, look a lot like the coordinates used in Egan's diagram for Adam and Eve) make it a lot more obvious what's going on. Suppose Adam emits a light ray the instant he drops off Eve's ship (at t = 0, x = s_0). When will that light ray reach Seth, way, way out at a huge x-coordinate? Just look at the diagram: the light has to go up and to the right at a 45 degree angle until it hits Seth's worldline. That means it won't reach Seth until almost 1/2 x 10^18 years have passed. Does that mean Adam didn't drop off Eve's ship until almost t = 1/2 x 10^18 years? Of course not; he dropped off at t = 0, but it took almost 1/2 x 10^18 years for the light to reach Seth. The same thing happens for events close to a black hole horizon.

 

[A-wal]

By someone in "flatter space-time" do you just mean someone further away from the black hole where the curvature is smaller?

Yep.

If so I don't know what you mean by "relative to" them, are you talking about how you are moving in some coordinate system they are using, how you appear to them visually, something else? Why do you think you are "effectively accelerating", and what does that even mean?

I meant that someone in a stronger gravitational field is time dilated relative to you and so doesn't have to be literally accelerating to reach infinite time dilation at the horizon.

And can you please do like I asked and explain whether you think your arguments apply to the case of an observer falling through the Rindler horizon? Why do you think an observer falling through the black hole event horizon is "effectively accelerating" if you don't think the same is true about an observer crossing the Rindler horizon? Why do you think "you would experience time dilation and length contraction as you were accelerating and it would reach infinity at the horizon" in the case of a black hole event horizon, when presumably you don't say that about a Rindler horizon? In both cases, after all, an observer hovering at fixed distance above the horizon sees your time dilation approach infinity as you approach the horizon in a visual sense, but if you don't think this implies the time dilation is "really" going to infinity in the case of the Rindler horizon, what makes you so sure the time dilation is "really" going to infinity in the case of a black hole event horizon? What is the relevant difference for you?

I haven't had a chance to look into an accelerating observer in flat space-time crossing the Rindler horizon yet. I'll let you know.

You've left out one key factor in all of the above: *how* does the further observer see the closer observer doing these things? For him to see them, the light from the events happening down near the horizon has to "climb" all the way out to the further observer's radial coordinate. Because of the way the light cones are bent, it takes a *long* time for the light to get out. That means the further observer *sees* things much later not because they *actually happen* much later, but because the light takes so long to get to him. When the further observer tries to assign coordinates to the actual events that emitted the light, he has to correct for the light travel time; and when he does that, he finds that those events near the horizon did not *actually* happen close to the black hole evaporating; they happened a long time before, but it took the light that long to get to him.

This is one reason why Schwarzschild coordinates are bad ones to use when trying to relate what happens near the horizon to what happens far away. Kruskal coordinates (which, as I noted before, look a lot like the coordinates used in Egan's diagram for Adam and Eve) make it a lot more obvious what's going on. Suppose Adam emits a light ray the instant he drops off Eve's ship (at t = 0, x = s_0). When will that light ray reach Seth, way, way out at a huge x-coordinate? Just look at the diagram: the light has to go up and to the right at a 45 degree angle until it hits Seth's worldline. That means it won't reach Seth until almost 1/2 x 10^18 years have passed. Does that mean Adam didn't drop off Eve's ship until almost t = 1/2 x 10^18 years? Of course not; he dropped off at t = 0, but it took almost 1/2 x 10^18 years for the light to reach Seth. The same thing happens for events close to a black hole horizon.

But what I said would presumably still apply no matter how long you had to wait for the light to reach you. I don't think the delay changes anything. After that's been taken into account what I said there still applies.

 

[PeterDonis]

But what I said would presumably still apply no matter how long you had to wait for the light to reach you. I don't think the delay changes anything. After that's been taken into account what I said there still applies.

As long as everything remains outside the horizon, yes, you can claim that in essence the delay doesn't change anything. But your logic breaks down at the horizon. You say:

At the horizon any amount of time for the further observer would be zero to the closer one.

This is not correct. What is correct is that a light ray emitted from an event exactly on the horizon (for example, by an observer falling into the black hole just as he crosses the horizon) will never reach the further observer--just as a light ray emitted by Adam just as he crosses the Rindler horizon in Egan's diagram (the line x = t) will never reach Eve. But that doesn't prevent Adam from crossing the Rindler horizon, and it doesn't prevent an observer from crossing a black hole's horizon.

It's also not correct to imagine an observer somehow "hovering" at the horizon, even as a limiting case of observers hovering closer and closer to the horizon. There can't be any such observer, because the acceleration required would be infinite, and the observer would have to move at the speed of light to stay at the horizon. Such observers are not allowed for the same reason they're not allowed anywhere in relativity; there is nothing special about a black hole horizon in this respect.

You could say that's why the closer one will never reach the horizon from the further ones perspective, but it's then a contradiction to say that the closer one can reach the horizon from their own perspective because that can't be witnessed by the further observer. Time dilation and length contraction can change the value of stuff but not whether it happens or not.

You're implicitly assuming here that anything that happens anywhere can always, in principle, be "witnessed" by any observer anywhere. There is no such requirement in relativity, and in fact, the cases we're discussing are cases where that assumption fails--there are events in the spacetimes we're discussing that simply can't be "witnessed" by certain observers. But those events are still perfectly real.

I do agree that saying "the closer one will never reach the horizon from the further one's perspective" is misleading, and I personally would not describe what's happening that way, because of the confusion it can lead to. I prefer to describe it as I have above, by simply saying that light emitted from events at the horizon will never reach observers outside the horizon--so the observers outside the horizon will never *see* anything crossing the horizon.

 

[A-wal]

It's also not correct to imagine an observer somehow "hovering" at the horizon, even as a limiting case of observers hovering closer and closer to the horizon. There can't be any such observer, because the acceleration required would be infinite, and the observer would have to move at the speed of light to stay at the horizon. Such observers are not allowed for the same reason they're not allowed anywhere in relativity; there is nothing special about a black hole horizon in this respect.

The acceleration required to resist the pull of gravity would be infinite at the horizon like trying to reach c as I said a few posts ago, but it wouldn't be infinite at any distance, no matter how small. So what's the problem with observers hovering closer and closer to the horizon?

You're implicitly assuming here that anything that happens anywhere can always, in principle, be "witnessed" by any observer anywhere. There is no such requirement in relativity, and in fact, the cases we're discussing are cases where that assumption fails--there are events in the spacetimes we're discussing that simply can't be "witnessed" by certain observers. But those events are still perfectly real.

As I understand it the only time certain events can't be witnessed is if an object crosses the horizon.

I do agree that saying "the closer one will never reach the horizon from the further one's perspective" is misleading, and I personally would not describe what's happening that way, because of the confusion it can lead to. I prefer to describe it as I have above, by simply saying that light emitted from events at the horizon will never reach observers outside the horizon--so the observers outside the horizon will never *see* anything crossing the horizon.

That's assuming you could reach the horizon in the first place, which is why all my examples take place at some distance away from the horizon, so it's not an issue anyway. The light from the closer observer will always reach the further one eventually.