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

[Dale]

If something gets longer or shorter, surely it does so regardless of the coordinate system used?

I have not been following the conversation very closely but it seems to me that you are talking about a measured length while JesseM is talking about a coordinate length. The results of measurements are indeed coordinate independent. Perhaps you could describe how you would measure the length of a black hole using a stationary measuring apparatus and a moving measuring apparatus.

 

[JesseM]

I have not been following the conversation very closely but it seems to me that you are talking about a measured length while JesseM is talking about a coordinate length. The results of measurements are indeed coordinate independent. Perhaps you could describe how you would measure the length of a black hole using a stationary measuring apparatus and a moving measuring apparatus.

But in a curved spacetime, since rigid rulers are impossible there are an infinite variety of different measurement procedures you could use, no? And for each coordinate system where objects at constant position coordinate have a timelike worldline, wouldn't there be a corresponding physical measurement procedure involving a network of measuring devices that were at rest in those coordinates, such that they would always give the same answer for lengths as the coordinate system itself? In this case there doesn't seem to be a lot of distinction between the notion of coordinate length and the notion of measured length.

 

[Dale]

But in a curved spacetime, since rigid rulers are impossible there are an infinite variety of different measurement procedures you could use, no?

Yes. That is why it would be important for A-wal to be explicit about the specific measurement procedure.

And for each coordinate system where objects at constant position coordinate have a timelike worldline, wouldn't there be a corresponding physical measurement procedure involving a network of measuring devices that were at rest in those coordinates, such that they would always give the same answer for lengths as the coordinate system itself?

I am not really sure one way or the other, but in any case his question (as I understand it) isn't if the number you get would match some coordinate system, it is if you would get a different result using a measuring procedure at rest wrt the black hole or moving wrt it. Of course, it is probably best if I let him speak for himself.

 

[A-wal]

If I measure the distance to the black hole to be 101 light years and the distance from the event horizon to the singularity is .001 light year and I accelerate to a speed where the distance to the black hole is now 10 light years (it took 1 light (in the original frame) to accelerate), what would I measure the event horizon to be? If it were a star instead of a black hole then the edge of the star would extend outward relative to the contracting space-time, but a black hole event horizon is space-time. It should contract inward. Or to put it another simpler way; what would the shape of the event horizon of a black hole look like if it was flying past us at very nearly c?

If the effect of gravity at the event horizon is equivalent to a relative verlocity of c, what would happen to the distance between the event horizon and the singularity as you approach it?

 

[JesseM]

If I measure the distance to the black hole to be 101 light years and the distance from the event horizon to the singularity is .001 light year

Measure it how? Using what coordinate system, or what physical measurement procedure? There is no single "natural" way to do measurements over large distances in curved spacetime because you can't have rigid measuring-rods in curved spacetime (and measuring distance also requires a simultaneity convention, since the idea is to measure the distance from one end to the other at a single moment in time).

There seems to be this presupposition in all your questions and arguments that "distance" and "measurement" have some unique well-defined meaning for a given observer in GR. They don't, so you need to understand that and either drop this line of argument or reframe it in terms of some particular procedure for defining "distance" out of the infinite variety of equally valid options that can be used in GR.

 

[ryuunoseika]

I definitely remember reading something official that said the laws of physics don't distinguish between the past and the future. I thinkit might have been A Brief HistoryOf Time. You could run it backwards and it would still work just as well. But now I've thought about it, there's something I can't resolve. Take two objects in space that are static relative to each other. They would gravitate towards each other. Now if time was running backwards then they would be moving away from each other. So gravity would be a repulsive force. But that doesn't work because if time was running backwards on Earth, we would still be pulled towards the planet, not pushed away. In other words it would work in freefall/at rest, but not when accelerating against gravity. How can it be both repulsive and attractive at the same distances?

Think about the ultimate final result of the two gravitating particles: they would ultimately collide and, provided that they are perfectly ellastic, bounce back to their original starting potential then fall again. They would repeat this pattern forever, and thus, at any point in time they could be reversed and it would look exactly the same, the only diffence is that, when viewed in a reversed arrow of time, a bounce is gravity pulling and gravity isa bounce.

 

[A-wal]

There seems to be this presupposition in all your questions and arguments that "distance" and "measurement" have some unique well-defined meaning for a given observer in GR. They don't, so you need to understand that and either drop this line of argument or reframe it in terms of some particular procedure for defining "distance" out of the infinite variety of equally valid options that can be used in GR.

I'm getting on your nerves now aren't I? Sorry but I just don't see how it makes any difference when comparing a change in length. If something extends or contracts then surely it does so no matter how it's measured. I'm not trying to be a twat and I do appreciate the responses but I still get the impression that you know what I mean and you're just trying to be awkward.

I am not really sure one way or the other, but in any case his question (as I understand it) isn't if the number you get would match some coordinate system, it is if you would get a different result using a measuring procedure at rest wrt the black hole or moving wrt it. Of course, it is probably best if I let him speak for himself.

Yes, and what the effect of gravity itself would be at the event horizon given that it's the equivalent to moving at c.

If I measure the distance to the black hole to be 101 light years and the distance from the event horizon to the singularity is .001 light year and I accelerate to a speed where the distance to the black hole is now 10 light years (it took 1 light (in the original frame) to accelerate), what would I measure the event horizon to be? If it were a star instead of a black hole then the edge of the star would extend outward relative to the contracting space-time, but a black hole event horizon is space-time. It should contract inward. Or to put it another simpler way; what would the shape of the event horizon of a black hole look like if it was flying past us at very nearly c?

I messed that up! It was late and I was in a hurry. I should have said 100 as the starting distance and then reverse 1 ly and then get a run up to cross the same point (marked by a nearby clump of matter before JesseyM says "What do you mean the same point?") at full speed. I was trying to avoid instant acceleration because you can't be moving if you haven't gone anywhere.

 

[Dale]

Sorry but I just don't see how it makes any difference when comparing a change in length.

It does make a difference. It is a question of definition. What do you mean by "length" of a black hole?

Yes, and what the effect of gravity itself would be at the event horizon given that it's the equivalent to moving at c.

That is not correct. There was a thread about this recently, and here was my comment in that thread:

If you have a very large black hole, such that the tidal forces at the event horizon are approximately zero, then that is equivalent to a Rindler accelerating observer in flat spacetime. Not an inertial observer at any speed.

 

[A-wal]

It does make a difference. It is a question of definition. What do you mean by "length" of a black hole?

I understand that if something is undefinable then it's meaningless. And I get that you can use any coordinate system for measurement. I still don't see the problem. If something is at rest (using energy to stay at a constant distance) relative to a black hole then the event horizon has a definite radius, yes? If the object then stops using energy to resist gravity then it will move towards the black hole, yes? Now, you could change coordinate system and say that you've moved away from the black hole. It's technically true and completely beside the point. Whether or not a black holes event horizon changes as you approach it is just as valid a question as whether you move towards or away from something exerting a gravitational force, yes? As long as keep using the same coordinate system, the question makes sense, yes?

That is not correct. There was a thread about this recently, and here was my comment in that thread:If you have a very large black hole, such that the tidal forces at the event horizon are approximately zero, then that is equivalent to a Rindler accelerating observer in flat spacetime. Not an inertial observer at any speed.

Are you saying that it's not equivalent to moving at c but only with black holes above a certain size? That doesn't seem right. If it's variable then it would only be equivalent to c if the black hole just happened to be exactly the right size, but that's not what I've heard/read (from multiple sources).

 

[Nabeshin]

Are you saying that it's not equivalent to moving at c but only with black holes above a certain size? That doesn't seem right. If it's variable then it would only be equivalent to c if the black hole just happened to be exactly the right size, but that's not what I've heard/read (from multiple sources).

Technically it holds for any size black holes, you simply need to make measurements over a much shorter time/distance interval so that the equivalence principle still holds. The point of making it a large black hole is that time scales are on the order of what is manageable to make measurements (seconds, minutes, etc.). Besides, there are no observers moving at c. So to talk about anything equivalent to that is meaningless in the first place.

The point is, an observer freely falling past the event horizon notices nothing special.

 

[A-wal]

I understand that there are no observers moving at c but I don't think it's meaningless to talk about the equivalent of moving at c. The time dilation/length contraction would be the same.

I know a freely falling observer would notice nothing special when crossing the event horizon, that's not the point. The point is I think they will be at the singularity (from their perspective of course) when they cross the horizon.

 

[Nabeshin]

The time dilation/length contraction would be the same.

How can you say this? The lorentz factor is undefined for v=c.

I know a freely falling observer would notice nothing special when crossing the event horizon, that's not the point. The point is I think they will be at the singularity (from their perspective of course) when they cross the horizon.

Are you trying to say that zero proper time elapses between when a freely falling observer crosses the event horizon and when he reaches the singularity? Because that is simply false.

 

[A-wal]

How can you say this? The lorentz factor is undefined for v=c.

It's not infinity then?

Are you trying to say that zero proper time elapses between when a freely falling observer crosses the event horizon and when he reaches the singularity? Because that is simply false.

Possibly, but I'm going to keep making a pest of myself until I understand why it's false.

 

[JesseM]

I'm getting on your nerves now aren't I? Sorry but I just don't see how it makes any difference when comparing a change in length. If something extends or contracts then surely it does so no matter how it's measured. I'm not trying to be a twat and I do appreciate the responses but I still get the impression that you know what I mean and you're just trying to be awkward.

No, you're simply totally wrong when you say "if something extends or contracts then surely it does so no matter how it's measured". "Extending" or "contracting" has no objective physical meaning, for any object that's extending in one coordinate system (or according to one measurement procedure), it's contracting in a different coordinate system (or according to a different measurement procedure), neither perspective is more "real" than than the other.

I understand that if something is undefinable then it's meaningless. And I get that you can use any coordinate system for measurement. I still don't see the problem. If something is at rest (using energy to stay at a constant distance) relative to a black hole then the event horizon has a definite radius, yes?

Not in any objective coordinate-independent sense, only if you choose to measure it in some particular coordinate system like Schwarzschild coordinates. Likewise, when you say the observer is staying "at constant distance" that has no coordinate-independent meaning either...usually when physicists talk about hovering at constant radius from a black hole they are assuming we are using Schwarzschild coordinates, but something hovering at constant radius in Schwarzschild coordinates would not be maintaining a constant distance in other systems like Kruskal-Szekeres coordinates. And in any case, just because an observer is hovering in a way that gives them a constant distance in Schwarzschild coordinates, that doesn't mean the observer himself can't use some totally different coordinate system to define his distance, or to define the radius of the event horizon.

If the object then stops using energy to resist gravity then it will move towards the black hole, yes? Now, you could change coordinate system and say that you've moved away from the black hole. It's technically true and completely beside the point.

For any short section of the object's worldline that doesn't include the event of the object crossing the horizon, there will be some coordinate systems that say its coordinate distance from the horizon is decreasing during that time interval, and others that say it's increasing during that time interval, neither perspective is more real than the other. On the other hand, while there are coordinate systems that say the object is temporarily moving away from the horizon during some section of its worldline, all coordinate systems should agree on local events like the object actually crossing the horizon (provided they actually cover the region of spacetime which includes that event), so if that happens they'll all have to agree the distance to the horizon does eventually decrease to zero. Likewise, I suppose the object could measure the local spacetime curvature (by measuring tidal forces) as it moved, and all coordinate systems would have to agree that this curvature was increasing as the object's own clock ticks forward, not decreasing. So, these might be ways in which you could give meaning to the idea that the object is "really" falling towards the horizon without trying to say its distance is "really" decreasing at all times. Do you have any analogous physical ways of defining your notion of whether an object is "really" expanding or contracting? If not, then why are you so sure there is any "real" truth about this question?

Whether or not a black holes event horizon changes as you approach it is just as valid a question as whether you move towards or away from something exerting a gravitational force, yes? As long as keep using the same coordinate system, the question makes sense, yes?

The question of whether an object is moving towards or away from the horizon is not a valid physical question unless you define it in some coordinate-independent way, and I suggested some ways of doing this without referring to the notion of "distance" above. Likewise, the question of whether or not a black hole event horizon changes is not a physical one unless you can define that in a coordinate-independent way. Of course the question of how the event horizon changes does have a coordinate-dependent answer within the context of a particular coordinate system, so if you're just looking for that sort of answer that's fine, but then you need to specify what kind of coordinate system you want--again, you can't use inertial coordinate systems to answer this question in a GR context, because any coordinate system that covers the entire region of spacetime containing the black hole would be too curved to be treated as equivalent to an inertial frame in flat SR spacetime.

 

[Dale]

Are you saying that it's not equivalent to moving at c but only with black holes above a certain size? That doesn't seem right. If it's variable then it would only be equivalent to c if the black hole just happened to be exactly the right size, but that's not what I've heard/read (from multiple sources).

No, it is not equivalent to moving at c for any size black hole.

An equivalence principle is always some limit of GR where the spacetime is flat and SR can be used to analyze the situation and make predictions. Usually the limit is a "small region" of spacetime where tidal effects are undetectable. In the case of the event horizon of a black hole the larger the black hole the less the tidal effects at the horizon, so for any fixed measuring apparatus over a fixed region of spacetime there is a black hole mass large enough that the tidal forces at the event horizon are undetectable. In that case, the spacetime is approximately flat in the region of the event horizon and you can analyze it using SR, and the equivalent SR situation is an accelerating observer (Rindler coordinates). It is not equivalent to an inertially moving observer at any speed.

For the same apparatus and region, if the black hole mass is smaller then the tidal forces are not-negligible at the event horizon and the curvature of spacetime is significant, and it cannot be analyzed as equivalent to anything in SR.

 

[A-wal]

No, you're simply totally wrong when you say "if something extends or contracts then surely it does so no matter how it's measured". "Extending" or "contracting" has no objective physical meaning, for any object that's extending in one coordinate system (or according to one measurement procedure), it's contracting in a different coordinate system (or according to a different measurement procedure), neither perspective is more "real" than than the other.

This is what I meant when I said a communication problem. I've already stated several times that I know you can use different coordinate systems and get different results. Don't change coordinate system and there's no problem!

Not in any objective coordinate-independent sense, only if you choose to measure it in some particular coordinate system like Schwarzschild coordinates. Likewise, when you say the observer is staying "at constant distance" that has no coordinate-independent meaning either...usually when physicists talk about hovering at constant radius from a black hole they are assuming we are using Schwarzschild coordinates, but something hovering at constant radius in Schwarzschild coordinates would not be maintaining a constant distance in other systems like Kruskal-Szekeres coordinates. And in any case, just because an observer is hovering in a way that gives them a constant distance in Schwarzschild coordinates, that doesn't mean the observer himself can't use some totally different coordinate system to define his distance, or to define the radius of the event horizon.

Hovering, but not maintaining the same distance? How else can you define hovering?

For any short section of the object's worldline that doesn't include the event of the object crossing the horizon, there will be some coordinate systems that say its coordinate distance from the horizon is decreasing during that time interval, and others that say it's increasing during that time interval, neither perspective is more real than the other. On the other hand, while there are coordinate systems that say the object is temporarily moving away from the horizon during some section of its worldline, all coordinate systems should agree on local events like the object actually crossing the horizon (provided they actually cover the region of spacetime which includes that event), so if that happens they'll all have to agree the distance to the horizon does eventually decrease to zero. Likewise, I suppose the object could measure the local spacetime curvature (by measuring tidal forces) as it moved, and all coordinate systems would have to agree that this curvature was increasing as the object's own clock ticks forward, not decreasing. So, these might be ways in which you could give meaning to the idea that the object is "really" falling towards the horizon without trying to say its distance is "really" decreasing at all times. Do you have any analogous physical ways of defining your notion of whether an object is "really" expanding or contracting? If not, then why are you so sure there is any "real" truth about this question?

If all coordinate systems agree then how can an observer away from a black hole say that an object will never cross the horizon while a local observer observes it crossing the horizon. How local do you have to be and what changes when you get that close to allow you to observe an object crossing the horizon?

The question of whether an object is moving towards or away from the horizon is not a valid physical question unless you define it in some coordinate-independent way, and I suggested some ways of doing this without referring to the notion of "distance" above. Likewise, the question of whether or not a black hole event horizon changes is not a physical one unless you can define that in a coordinate-independent way. Of course the question of how the event horizon changes does have a coordinate-dependent answer within the context of a particular coordinate system, so if you're just looking for that sort of answer that's fine, but then you need to specify what kind of coordinate system you want--again, you can't use inertial coordinate systems to answer this question in a GR context, because any coordinate system that covers the entire region of spacetime containing the black hole would be too curved to be treated as equivalent to an inertial frame in flat SR spacetime.

What about a binary system with a black hole and a real star – I know those exist. If you're free-falling and you compare the distance of the horizon to the star then can you say that it's defiantly increased or decreased?

No, it is not equivalent to moving at c for any size black hole.

That's not what I've been lead to believe. Also, everything I understand about special relativity seems intuitively to suggest that the event horizon is exactly equivalent to a relative velocity of c without actually moving anywhere. It's acceleration until you reach c. Surely you can't cross the event horizon of a black hole in the same way you can't accelerate to a relative velocity greater than c?

 

[JesseM]

This is what I meant when I said a communication problem. I've already stated several times that I know you can use different coordinate systems and get different results. Don't change coordinate system and there's no problem!

Then you shouldn't have said "if something extends or contracts then surely it does so no matter how it's measured", because different coordinate systems represent different ways of measuring length. And what do you mean by "don't change coordinate system"? You never said what coordinate system you wanted to use in the first place! If you're talking about Schwarzschild coordinates, then your comments about the event horizon changing size doesn't make sense, since they're aren't multiple Schwarzschild coordinate systems for different observers which assign different sizes to the horizon, for a given black hole there's just one Schwarzschild coordinate system, and the black hole is at rest with constant radius in that system.

Hovering, but not maintaining the same distance? How else can you define hovering?

I didn't just use the word "hovering", I was careful to say "hovering in a way that gives them a constant distance in Schwarzschild coordinates". The point is that there is no coordinate-independent way of defining the word "hovering"--if you are "hovering" at a constant distance in Schwarzschild coordinates, then in Kruskal-Szekeres coordinates you are not hovering because your distance is changing with time, and likewise if you are "hovering" at a constant distance in Kruskal-Szekeres coordinates, you would not be hovering in Schwarzschild coordinates.

For any short section of the object's worldline that doesn't include the event of the object crossing the horizon, there will be some coordinate systems that say its coordinate distance from the horizon is decreasing during that time interval, and others that say it's increasing during that time interval, neither perspective is more real than the other. On the other hand, while there are coordinate systems that say the object is temporarily moving away from the horizon during some section of its worldline, all coordinate systems should agree on local events like the object actually crossing the horizon (provided they actually cover the region of spacetime which includes that event), so if that happens they'll all have to agree the distance to the horizon does eventually decrease to zero. Likewise, I suppose the object could measure the local spacetime curvature (by measuring tidal forces) as it moved, and all coordinate systems would have to agree that this curvature was increasing as the object's own clock ticks forward, not decreasing. So, these might be ways in which you could give meaning to the idea that the object is "really" falling towards the horizon without trying to say its distance is "really" decreasing at all times. Do you have any analogous physical ways of defining your notion of whether an object is "really" expanding or contracting? If not, then why are you so sure there is any "real" truth about this question?

If all coordinate systems agree

Agree on what? I mentioned some specific things which different coordinate systems will agree on, I didn't say they'd agree on everything.

then how can an observer away from a black hole say that an object will never cross the horizon while a local observer observes it crossing the horizon.

It's not a disagreement between observers, it's a disagreement between coordinate systems--any observer is free to use any coordinate system they please (for example, an observer 'away from a black hole' is free to use Kruskal-Szekeres coordinates which predict the falling object crosses the horizon in finite coordinate time, even if this observer will never see the light from the crossing event). It's true that some coordinate systems say it takes infinite coordinate time for the falling object to reach the horizon while others say it takes finite coordinate time, but they all agree on the more physical point that it will only take a finite proper time (time as measured by a clock moving along with the falling object) for the object to reach the horizon.

How local do you have to be and what changes when you get that close to allow you to observe an object crossing the horizon?

To actually see the crossing event with your eyes, you have to follow the object through the horizon, and you won't be able to see it crossing until the moment you cross the horizon.

If you're free-falling and you compare the distance of the horizon to the star then can you say that it's defiantly increased or decreased?

Increased or decreased relative to what? And what coordinate system or measurement procedure are you using to measure this distance? Again, questions about distances in GR are meaningless unless you specify your choice of coordinate system/measurement procedure, there are an infinite variety of possible ones to choose from.

 

[Dale]

That's not what I've been lead to believe. Also, everything I understand about special relativity seems intuitively to suggest that the event horizon is exactly equivalent to a relative velocity of c without actually moving anywhere. It's acceleration until you reach c. Surely you can't cross the event horizon of a black hole in the same way you can't accelerate to a relative velocity greater than c?

Since in SR you cannot have a velocity of c then how could any situation in GR possibly be equivalent to it? The whole point of all equivalence principles is (in some limit) to replace a curved spacetime of GR with an equivalent situation in the flat spacetime of SR which is easier to analyze. It is not possible in SR to have a massive particle travel at c so it is not possible for any situation in GR to be equivalent to it. You cannot use the rules of SR to analyze a hypothetical situation that violates the rules of SR. I hope that is clear.

Here is the best page I have found on the topic:
http://www.gregegan.net/SCIENCE/Rindler/RindlerHorizon.html

The purpose of this web page, then, is to analyse in detail (using only special relativity) some interesting thought experiments that can be carried out by a constantly accelerating observer, who sees a “Rindler horizon” in spacetime that is very similar to the event horizon of a black hole. ... what it represents is an interesting limiting case: a black hole so massive that the spacetime curvature at its horizon is negligible.

 

[yuiop]

Perhaps A-wal is thinking of something like Gullstrand-Painleve coordinates sometimes called the river model.

In another thread https://www.physicsforums.com/showpost.php?p=2398459&postcount=7 Cleonis posted these two links describing GP coordinates.

http://arxiv.org/abs/gr-qc/0411060"

http://mitupv.mit.edu/wp/attach/4581/barry.pdf"

Both those linked documents describe free falling objects as being carried along in a river of spacetime flowing into the black hole and state that at the event horizon the river is flowing at speed of c. Below the event horizon falling objects are moving at greater than the speed of light, but this is OK because they are stationary with respect to the inflowing river. The limitation is that nothing can move at greater than the speed of light relative to the river.

 

[A-wal]

Then you shouldn't have said "if something extends or contracts then surely it does so no matter how it's measured", because different coordinate systems represent different ways of measuring length. And what do you mean by "don't change coordinate system"? You never said what coordinate system you wanted to use in the first place! If you're talking about Schwarzschild coordinates, then your comments about the event horizon changing size doesn't make sense, since they're aren't multiple Schwarzschild coordinate systems for different observers which assign different sizes to the horizon, for a given black hole there's just one Schwarzschild coordinate system, and the black hole is at rest with constant radius in that system.

But you're saying that length isn't absolute and extension can be viewed as contraction from a different coordinate system, which I've never questioned. You're saying that if there's length contraction from time A to time B in one coordinate system then you could change coordinate system and say it's extended again. I'm saying that if you pick a coordinate system in which the horizon contracts towards the black hole over time A to time B then there shouldn't be any coordinate system in which the length extends from time A to B. Maybe I'm wrong, but I feel like you didn't understand what I was asking. Maybe I'm wrong there too and you knew what I meant before, but you kept talking about changing coordinate systems which I'm trying to avoid.

For any short section of the object's worldline that doesn't include the event of the object crossing the horizon, there will be some coordinate systems that say its coordinate distance from the horizon is decreasing during that time interval, and others that say it's increasing during that time interval, neither perspective is more real than the other.

But if the object is in free-fall then couldn't it be be said that the ones in which the distance to the horizon is increasing are more real. Also what do you mean by short section of the wordline? If some don't work or change direction relative to the black hole over longer sections of the world line then can't they be considered as less real?

Agree on what? I mentioned some specific things which different coordinate systems will agree on, I didn't say they'd agree on everything.

On an object crossing the horizon. Sometimes all will agree on the object getting closer to the horizon and sometimes they wont?

It's not a disagreement between observers, it's a disagreement between coordinate systems--any observer is free to use any coordinate system they please (for example, an observer 'away from a black hole' is free to use Kruskal-Szekeres coordinates which predict the falling object crosses the horizon in finite coordinate time, even if this observer will never see the light from the crossing event). It's true that some coordinate systems say it takes infinite coordinate time for the falling object to reach the horizon while others say it takes finite coordinate time, but they all agree on the more physical point that it will only take a finite proper time (time as measured by a clock moving along with the falling object) for the object to reach the horizon.

And there's nothing paradoxical about that?

To actually see the crossing event with your eyes, you have to follow the object through the horizon, and you won't be able to see it crossing until the moment you cross the horizon.

So you see them crossing the horizon after you, and they see you crossing after them! Again, isn't that a paradox if they're at rest relative to each other?

Increased or decreased relative to what? And what coordinate system or measurement procedure are you using to measure this distance? Again, questions about distances in GR are meaningless unless you specify your choice of coordinate system/measurement procedure, there are an infinite variety of possible ones to choose from.

I was just thinking that you could use a percentage of the distance from the singularity to the star to define the size of the horizon, rather than a measurement of space which means nothing by itself.

You cannot use the rules of SR to analyze a hypothetical situation that violates the rules of SR. I hope that is clear.

...

Perhaps A-wal is thinking of something like Gullstrand-Painleve coordinates sometimes called the river model.

Apparently I can.

 

[JesseM]

But you're saying that length isn't absolute and extension can be viewed as contraction from a different coordinate system, which I've never questioned. You're saying that if there's length contraction from time A to time B in one coordinate system then you could change coordinate system and say it's extended again. I'm saying that if you pick a coordinate system in which the horizon contracts towards the black hole over time A to time B then there shouldn't be any coordinate system in which the length extends from time A to B.

I don't get it--aren't you directly contradicting yourself in the first and last sentence? You say you agree with me that "extension can be viewed as contraction from a different coordinate system", presumably meaning that you'd agree "contraction can be viewed as extension in a different coordinate system", but then you go on to say that if the black contracts over a given period of time in one coordinate system, there can't be another coordinate system where it extends. Isn't the idea that a contraction in one system can be viewed as an extension in another precisely what is meant by the statement "contraction can be viewed as extension in a different coordinate system"? Is the fact that we're dealing with the same time interval somehow relevant? Perhaps you thought that when I say "contraction can be viewed as extension in a different coordinate system", you think I am only saying that an object which contracts at one time in system A can also extend at a different time is system B, but never over the same section of the object's worldline? If so, that is definitely not what I meant.

You still seem not to understand that in GR coordinate systems are totally arbitrary ways of labeling the space and time coordinates of points in spacetime. If I pick two events--like events L and R on the worldlines of the left and right sides of a given object--then I can make up any damn coordinates for them I want. For example, I could label L as "x=2 cm, t=0 s" and R as "x=3 cm, t=0 s", so in this coordinate system the distance between the left and right ends of the the object at time t=0 must be 1 cm. But then I could define a different coordinate system where L is labeled with coordinates "x=0 light years, t=0 s" and R is labeled with "x=300 trillion light years, t = 0 s", so in this coordinate system the distance between left and right ends of the object at time t=0 s is 300 trillion light years. And this arbitrariness of labels applies just as well when we are dealing with events at different ends of a time interval. Suppose we have two events L1 and L2 on the worldline of the object's left end, and two events R1 and R2 on the worldline of the object's right end. I am free to totally arbitrarily choose my coordinate system #1 so that these events have the following coordinates:

L1: (x=0 cm, t=0 s) R1: (x=3 cm, t=0 s)
L2: (x=0 cm, t=5 s) R1: (x=90,000 km, t=5 s)

So, in coordinate system #1, from time t=0 s to time t=5 s the object's length expanded from 3 cm to 90,000 km. Now I can invent another arbitrary coordinate system where the same 4 events have the following coordinates:

L1: (x=10 light years, t=0 s) R1: (x=300 light years, t=0 s)
L2: (x=9 light years, t=5 s) R2: (x=9.001 light years, t=5 s)

So in coordinate system #2, from time t=0 s to time t=5 s (the same section of the worldlines of the right and left end, although the decision to define the ends of the sections as happening 5 seconds apart in both systems was another arbitrary choice), the object's length shrunk from 300 light years to 0.001 light years. None of this has any real physical meaning, it's just based on what labels I choose to arbitrarily apply to events.

Maybe I'm wrong, but I feel like you didn't understand what I was asking. Maybe I'm wrong there too and you knew what I meant before, but you kept talking about changing coordinate systems which I'm trying to avoid.

That doesn't make sense either, if you don't want to talk about multiple coordinate systems, why did you say "I'm saying that if you pick a coordinate system in which the horizon contracts towards the black hole over time A to time B then there shouldn't be any coordinate system in which the length extends from time A to B." Aren't you claiming here that the fact that the length contracts over that time interval in one coordinate system means it's somehow impossible to find a different coordinate system where the length expands in the same interval? If not, your use of the phrase "then there shouldn't be any coordinate system" is extremely confusing, I don't know how else to interpret it.

But if the object is in free-fall then couldn't it be be said that the ones in which the distance to the horizon is increasing are more real.

Not unless you can provide a rigorous physical definition of what it means for one coordinate system to be "more real" than another--otherwise it sounds more like a qualitative aesthetic assessment that you just find one coordinate system more intuitive or something. All definitions in physics must be mathematical, they can't be based on qualitative judgements.

Also what do you mean by short section of the wordline?

Just short enough so that it doesn't include the section of the worldline where the object crosses the horizon, since all coordinate systems must agree the distance between the object and the horizon goes to zero at the moment of the crossing.

If some don't work or change direction relative to the black hole over longer sections of the world line then can't they be considered as less real?

Again, only if you can define what it means for a coordinate system to be more or less "real".

Agree on what? I mentioned some specific things which different coordinate systems will agree on, I didn't say they'd agree on everything.

On an object crossing the horizon. Sometimes all will agree on the object getting closer to the horizon and sometimes they wont?

They will all agree the distance goes to zero at the moment it crosses the horizon, and all smooth coordinate systems should agree the distance varies in a continuous way rather than jumping, so they'll all have to agree there is some period before the crossing where the distance is shrinking. Still, for any point on the object's worldline where it hasn't yet crossed the horizon, no matter how close it is, you should be able to find a coordinate system where the distance doesn't start decreasing until after that point.

It's not a disagreement between observers, it's a disagreement between coordinate systems--any observer is free to use any coordinate system they please (for example, an observer 'away from a black hole' is free to use Kruskal-Szekeres coordinates which predict the falling object crosses the horizon in finite coordinate time, even if this observer will never see the light from the crossing event). It's true that some coordinate systems say it takes infinite coordinate time for the falling object to reach the horizon while others say it takes finite coordinate time, but they all agree on the more physical point that it will only take a finite proper time (time as measured by a clock moving along with the falling object) for the object to reach the horizon.

And there's nothing paradoxical about that?

No, what would be the paradox? I could define a coordinate system where it would take infinite coordinate system for the clock in my room to reach noon tomorrow, since again coordinate systems are just arbitrary labeling conventions (for example, I could label the event of the clock reading 10 seconds before noon with time coordinate t=1 year, the event of the clock reading 1 second before noon with t=2 years, the event of the clock reading 0.1 seconds before noon with t=3 years, the event of the clock reading 0.01 seconds before noon with t=4 years, the event of the clock reading 0.001 seconds before noon with t=5 years, etc.) This wouldn't change the fact that it only takes a finite amount of proper time for the clock to reach noon--the only really physical statements about any sort of time are ones that are about proper time, since this is the only kind of time measured by real physical clocks.

To actually see the crossing event with your eyes, you have to follow the object through the horizon, and you won't be able to see it crossing until the moment you cross the horizon.

So you see them crossing the horizon after you, and they see you crossing after them!

No, I see the light from them crossing the horizon at the exact moment that I cross the horizon, not after. An analogy in an SR spacetime diagram would be that if we defined a certain boundary in spacetime as lining up with the right side of the future light cone of some event E in the past, and my friend crosses this boundary before I do (i.e. enters the future light cone of E before I do), then I will see the light from the event of their crossing it at the exact moment that I cross it myself.

Again, isn't that a paradox if they're at rest relative to each other?

Why would it be? In SR, two observers at rest relative to each other can enter the future light cone of some past event E at different moments, and if I'm the second one to enter, I'll see the light from the event of the first guy entering the light cone at the exact moment that I enter the light cone myself. If you're familiar with spacetime diagrams in SR, it shouldn't be too hard to see this...and the analogy with a black hole event horizon becomes more obvious if you draw it in Kruskal coordinates, where the event horizon looks just like a light cone.

I was just thinking that you could use a percentage of the distance from the singularity to the star to define the size of the horizon, rather than a measurement of space which means nothing by itself.

But proportions can differ between coordinate systems too. If C is halfway in between A and B in one coordinate system, that doesn't stop you from defining another coordinate system where C is closer to A than B (assuming we are talking about all possible coordinate systems rather than some specific subset like inertial coordinate systems), again because coordinate systems in GR are just arbitrary ways of labeling events that can be chosen in any way you like.

 

[Dale]

...Apparently I can.

No, the "river model" is an interpretation of GR, not SR. And even in the river model an observer must accelerate continuously to remain outside the event horizon.

Why don't you read the link I posted earlier? It has lots of very good information and you may actually learn something. Come back once you have done so if you have any questions.

 

[A-wal]

I wrote a response to JesseM and lost it all. It's an understatement to say I'm pissed off. I'll never be able to rewrite it as it was and it won't be as concise if I start again because I'll be trying to remember what I put before rather than writing freely.

What about using background radiation to define the coordinate system? Make it the same density in all directions at any given distance.

No, the "river model" is an interpretation of GR, not SR. And even in the river model an observer must accelerate continuously to remain outside the event horizon.

I know it's GR. That's what I've been talking about. SR doesn't prevent v=c. It prevents v>c. You could argue that there's an infinite value for the strength of gravity at the event horizon. You could even move at v>c inside the event horizon because you're completely cut off from relative movement to anything outside it. Or another way of looking at it is v>c= an event horizon. What would an observer experience in their proper time as they cross it? I think they won't experience anything because they won't cross it from their own perspective, just as they won't cross it from the perspective of an outside observer.

 

[Dale]

SR doesn't prevent v=c. It prevents v>c.

Yes, SR does prevent v=c too (timelike four-vectors cannot be lightlike in any frame).

Read the link I posted, it is very useful.

 

[A-wal]

I don't get it--aren't you directly contradicting yourself in the first and last sentence? You say you agree with me that "extension can be viewed as contraction from a different coordinate system", presumably meaning that you'd agree "contraction can be viewed as extension in a different coordinate system", but then you go on to say that if the black contracts over a given period of time in one coordinate system, there can't be another coordinate system where it extends. Isn't the idea that a contraction in one system can be viewed as an extension in another precisely what is meant by the statement "contraction can be viewed as extension in a different coordinate system"? Is the fact that we're dealing with the same time interval somehow relevant? Perhaps you thought that when I say "contraction can be viewed as extension in a different coordinate system", you think I am only saying that an object which contracts at one time in system A can also extend at a different time is system B, but never over the same section of the object's worldline? If so, that is definitely not what I meant.

That doesn't make sense either, if you don't want to talk about multiple coordinate systems, why did you say "I'm saying that if you pick a coordinate system in which the horizon contracts towards the black hole over time A to time B then there shouldn't be any coordinate system in which the length extends from time A to B." Aren't you claiming here that the fact that the length contracts over that time interval in one coordinate system means it's somehow impossible to find a different coordinate system where the length expands in the same interval? If not, your use of the phrase "then there shouldn't be any coordinate system" is extremely confusing, I don't know how else to interpret it.

I'm not saying this is right but I'll explain what I meant. If there's length contraction from time A to time B in one coordinate system then you could change coordinate system and show length extension from time B in the second one compared to time B in the first. I was saying that if you pick a coordinate system in which the horizon contracts towards the black hole over time A to time B then you shouldn't be able to find any coordinate system in which the length extends from time A to B. But you've already said that you can.

Not unless you can provide a rigorous physical definition of what it means for one coordinate system to be "more real" than another--otherwise it sounds more like a qualitative aesthetic assessment that you just find one coordinate system more intuitive or something. All definitions in physics must be mathematical, they can't be based on qualitative judgements.

I know that. But gravity is an attractive force so why can't we say that the views where the distance is decreasing (I meant decreasing before, not increasing) are more real? More real are your words btw.

They will all agree the distance goes to zero at the moment it crosses the horizon, and all smooth coordinate systems should agree the distance varies in a continuous way rather than jumping, so they'll all have to agree there is some period before the crossing where the distance is shrinking. Still, for any point on the object's worldline where it hasn't yet crossed the horizon, no matter how close it is, you should be able to find a coordinate system where the distance doesn't start decreasing until after that point.

If the distance doesn't start decreasing until after the object has crossed the horizon then it must jump and it's not continuous. Or if it's destined to cross the horizon at some time in the future then what did you mean by that point. The point when it crosses the horizon has already been defined. Also the object in one of these coordinate systems would change direction relative to the black hole for absolutely no reason.

No, what would be the paradox? I could define a coordinate system where it would take infinite coordinate system for the clock in my room to reach noon tomorrow, since again coordinate systems are just arbitrary labeling conventions (for example, I could label the event of the clock reading 10 seconds before noon with time coordinate t=1 year, the event of the clock reading 1 second before noon with t=2 years, the event of the clock reading 0.1 seconds before noon with t=3 years, the event of the clock reading 0.01 seconds before noon with t=4 years, the event of the clock reading 0.001 seconds before noon with t=5 years, etc.) This wouldn't change the fact that it only takes a finite amount of proper time for the clock to reach noon--the only really physical statements about any sort of time are ones that are about proper time, since this is the only kind of time measured by real physical clocks.

One says something does happen, while another says the same thing never happens. That's the definition of a paradox.

But proportions can differ between coordinate systems too. If C is halfway in between A and B in one coordinate system, that doesn't stop you from defining another coordinate system where C is closer to A than B (assuming we are talking about all possible coordinate systems rather than some specific subset like inertial coordinate systems), again because coordinate systems in GR are just arbitrary ways of labeling events that can be chosen in any way you like.

And the background radiation idea? Same thing I spose?

Yes, SR does prevent v=c too (timelike four-vectors cannot be lightlike in any frame).

SR says it would take an infinite amount of energy to accelerate something with mass to c. Energy moves at c because it has no mass, so no energy is required. If you view an object at rest relative to yourself then you observe it moving at c through time. An object that's infinitely time dilated at the event horizon of a black hole is the equivalent of moving through space at c.

 

[Dale]

An object that's infinitely time dilated at the event horizon of a black hole is the equivalent of moving through space at c.

I'm tired of repeating myself, so I will try a different tack.

Using your "equivalent" SR scenario of an observer traveling at c, what region of spacetime is equivalent to the event horizon, i.e. what defines the boundary between the region of spacetime from which your observer moving at c can send and receive signals and the region of spacetime from which the observer cannot receive signals?

Using your "equivalent" SR scenario of an observer traveling at c, how can you explain how it can take an infinite amount of time according to an observer at rest wrt the event horizon and yet a finite amount of proper time for a free-falling observer to cross the horizon?

Using your "equivalent" SR scenario of an observer traveling at c, can you explain why it is not possible for an observer stationary wrt the event horizon to let a rope down into the event horizon?

Unless your "equivalent" scenario allows you to make useful physics predictions and calculations in SR that apply to the GR limiting case then it is not an equivalent scenario. I post the link again for your reference: http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

 

[JesseM]

But if the object is in free-fall then couldn't it be be said that the ones in which the distance to the horizon is increasing are more real.

Not unless you can provide a rigorous physical definition of what it means for one coordinate system to be "more real" than another--otherwise it sounds more like a qualitative aesthetic assessment that you just find one coordinate system more intuitive or something. All definitions in physics must be mathematical, they can't be based on qualitative judgements.

I know that. But gravity is an attractive force so why can't we say that the views where the distance is decreasing (I meant decreasing before, not increasing) are more real? More real are your words btw.

Actually it wasn't my words, as you can see from the previous statement I was responding to which I quoted again above (and I quoted it in the original post where I wrote the 'Not unless...' paragraph too...unfortunately when replying to a post the reply box doesn't show you the stuff the other person put in quotes, so it can be hard to follow the thread of the conversation, sometimes it helps to keep another window open to the post you're replying to when typing your reply). And like I said, it doesn't seem meaningful to say anything is more or less "real" in the context of physics unless some definition of that word is given. Anyway, just because gravity is an attractive force doesn't mean things can't be moving away from a gravitating body, even in an inertial coordinate system in Newtonian gravity.

They will all agree the distance goes to zero at the moment it crosses the horizon, and all smooth coordinate systems should agree the distance varies in a continuous way rather than jumping, so they'll all have to agree there is some period before the crossing where the distance is shrinking. Still, for any point on the object's worldline where it hasn't yet crossed the horizon, no matter how close it is, you should be able to find a coordinate system where the distance doesn't start decreasing until after that point.

If the distance doesn't start decreasing until after the object has crossed the horizon then it must jump and it's not continuous.

I didn't say anything about the distance not decreasing until after the object crossed the horizon. If you reread the paragraph of mine you were responding to above, I said they all agree the distance starts shrinking before the crossing of the event horizon, but I also said that if you pick any point on the object's wordline before it crosses the horizon--say, the point where it is only 1 nanosecond away from crossing the horizon according to its own proper time--then you can find a smooth coordinate system where the distance doesn't start decreasing until after that point (in this case, a coordinate system where the distance only begins shrinking in the last nanosecond of the object's proper time before it reaches the horizon).

Or if it's destined to cross the horizon at some time in the future then what did you mean by that point.

Exactly what I said, "any point on the object's worldline where it hasn't yet crossed the horizon", like the point on its worldline that happens exactly 1 nanosecond of proper time before the point where it crosses the horizon.

Also the object in one of these coordinate systems would change direction relative to the black hole for absolutely no reason.

Did you read my point about the fact that the allowable coordinate systems in GR are pretty much any arbitrary set of labels for events? If you understood that, why would you think there should be any problem with objects arbitrarily changing directions in a given system? I mentioned this in earlier posts, but I hope you took a careful look at the final animated diagram in http://www.aei.mpg.de/einsteinOnline/en/spotlights/background_independence/index.html discussing diffeomorphism invariance, where a variety of different totally arbitrary coordinate charts or drawn in relation to some colored shapes representing physical objects in space. If you replace the colored shapes with events and worldlines in spacetime, exactly the same is true spacetime coordinate systems, they can be drawn any way you please (as long as you respect some basic rules like smoothness and unique events being assigned unique coordinates). For example, if I take a Minkowski diagram showing various worldlines, and then over it I do a freehand drawing of a curvy line which in one section looks exactly like a profile of Mickey Mouse, I am free to take that curvy line and use it as the x=0 axis of a new coordinate system. A worldline of an inertial object which is just a straight line in Minkowski coordinates might have multiple crossing points with a curvy line like the one containing the Mickey Mouse profile (and this would even be true of a straight line drawn on top of the curvilinear coordinate systems shown in the animated diagram in the article I linked to), so in that non-inertial Mickey Mouse coordinate system the object's path would have to cross the x=0 axis multiple times, meaning it made multiple changes in direction in this system.

No, what would be the paradox? I could define a coordinate system where it would take infinite coordinate system for the clock in my room to reach noon tomorrow, since again coordinate systems are just arbitrary labeling conventions (for example, I could label the event of the clock reading 10 seconds before noon with time coordinate t=1 year, the event of the clock reading 1 second before noon with t=2 years, the event of the clock reading 0.1 seconds before noon with t=3 years, the event of the clock reading 0.01 seconds before noon with t=4 years, the event of the clock reading 0.001 seconds before noon with t=5 years, etc.) This wouldn't change the fact that it only takes a finite amount of proper time for the clock to reach noon--the only really physical statements about any sort of time are ones that are about proper time, since this is the only kind of time measured by real physical clocks.

One says something does happen, while another says the same thing never happens. That's the definition of a paradox.

Just because I use a classical coordinate system like the one I described above where it would take an infinite coordinate time for a clock to reach noon, that doesn't mean I am making any physical claim that the clock will "never" reach noon (i.e. that the event of the clock reaching noon is not one that occurs anywhere in real physical spacetime). It just means that if it does, it must do so in a region that lines outside the region of spacetime covered by the coordinate system (and not every coordinate system fills all of spacetime like inertial systems in SR, some just cover 'patches' of it). Similarly, the Schwarzschild coordinate system doesn't cover the region of spacetime where objects cross the horizon, but that doesn't mean that any physical claim is being made about the event of their crossing the horizon not happening anywhere in spacetime.

But proportions can differ between coordinate systems too. If C is halfway in between A and B in one coordinate system, that doesn't stop you from defining another coordinate system where C is closer to A than B (assuming we are talking about all possible coordinate systems rather than some specific subset like inertial coordinate systems), again because coordinate systems in GR are just arbitrary ways of labeling events that can be chosen in any way you like.

And the background radiation idea? Same thing I spose?

You could construct a coordinate system based on the average rest frame of the background radiation, but the laws of GR would obey the same tensor equations in this system as they do in every other system, so it wouldn't be a "preferred" coordinate system in the sense that physicists use the word.

If you view an object at rest relative to yourself then you observe it moving at c through time.

What do you mean by "moving at c through time"? Something like the mathematical trick used by Brian Greene which I talked about in post #3 of this thread which allows us to understand time dilation in terms of a tradeoff between "speed through space" and "speed through time"? But this trick seems to be specifically dependent on the way time dilation and 4-vectors work in SR, I don't know if there's any way to generalize it to a GR situation involving curved spacetime.

 

[A-wal]

Actually it wasn't my words, as you can see from the previous statement I was responding to which I quoted again above (and I quoted it in the original post where I wrote the 'Not unless...' paragraph too...unfortunately when replying to a post the reply box doesn't show you the stuff the other person put in quotes, so it can be hard to follow the thread of the conversation, sometimes it helps to keep another window open to the post you're replying to when typing your reply).

I know...

No, you're simply totally wrong when you say "if something extends or contracts then surely it does so no matter how it's measured". "Extending" or "contracting" has no objective physical meaning, for any object that's extending in one coordinate system (or according to one measurement procedure), it's contracting in a different coordinate system (or according to a different measurement procedure), neither perspective is more "real" than than the other.

...it's a pain isn't it?

Did you read my point about the fact that the allowable coordinate systems in GR are pretty much any arbitrary set of labels for events? If you understood that, why would you think there should be any problem with objects arbitrarily changing directions in a given system? I mentioned this in earlier posts, but I hope you took a careful look at the final animated diagram in this article discussing diffeomorphism invariance, where a variety of different totally arbitrary coordinate charts or drawn in relation to some colored shapes representing physical objects in space. If you replace the colored shapes with events and worldlines in spacetime, exactly the same is true spacetime coordinate systems, they can be drawn any way you please (as long as you respect some basic rules like smoothness and unique events being assigned unique coordinates). For example, if I take a Minkowski diagram showing various worldlines, and then over it I do a freehand drawing of a curvy line which in one section looks exactly like a profile of Mickey Mouse, I am free to take that curvy line and use it as the x=0 axis of a new coordinate system. A worldline of an inertial object which is just a straight line in Minkowski coordinates might have multiple crossing points with a curvy line like the one containing the Mickey Mouse profile (and this would even be true of a straight line drawn on top of the curvilinear coordinate systems shown in the animated diagram in the article I linked to), so in that non-inertial Mickey Mouse coordinate system the object's path would have to cross the x=0 axis multiple times, meaning it made multiple changes in direction in this system.

You need acceleration to create a non-inertial system, so let's just say that the only acceleration is coming form the black hole. I hate physicists; they're so dam picky!

Just because I use a classical coordinate system like the one I described above where it would take an infinite coordinate time for a clock to reach noon, that doesn't mean I am making any physical claim that the clock will "never" reach noon (i.e. that the event of the clock reaching noon is not one that occurs anywhere in real physical spacetime). It just means that if it does, it must do so in a region that lines outside the region of spacetime covered by the coordinate system (and not every coordinate system fills all of spacetime like inertial systems in SR, some just cover 'patches' of it). Similarly, the Schwarzschild coordinate system doesn't cover the region of spacetime where objects cross the horizon, but that doesn't mean that any physical claim is being made about the event of their crossing the horizon not happening anywhere in spacetime.

If the event of the object crossing the horizon is outside of the coordinate system then what's the point of it in this situation?

You could construct a coordinate system based on the average rest frame of the background radiation, but the laws of GR would obey the same tensor equations in this system as they do in every other system, so it wouldn't be a "preferred" coordinate system in the sense that physicists use the word.

I only know how to speak English.

What do you mean by "moving at c through time"? Something like the mathematical trick used by Brian Greene which I talked about in post #3 of this thread which allows us to understand time dilation in terms of a tradeoff between "speed through space" and "speed through time"? But this trick seems to be specifically dependent on the way time dilation and 4-vectors work in SR, I don't know if there's any way to generalize it to a GR situation involving curved spacetime.

It's a bit more than a trick, but it isn't as simple as that because length contraction isn't explained or needed in this explanation.

 

[JesseM]

Also the object in one of these coordinate systems would change direction relative to the black hole for absolutely no reason.

Did you read my point about the fact that the allowable coordinate systems in GR are pretty much any arbitrary set of labels for events? If you understood that, why would you think there should be any problem with objects arbitrarily changing directions in a given system? I mentioned this in earlier posts, but I hope you took a careful look at the final animated diagram in this article discussing diffeomorphism invariance, where a variety of different totally arbitrary coordinate charts or drawn in relation to some colored shapes representing physical objects in space. If you replace the colored shapes with events and worldlines in spacetime, exactly the same is true spacetime coordinate systems, they can be drawn any way you please (as long as you respect some basic rules like smoothness and unique events being assigned unique coordinates). For example, if I take a Minkowski diagram showing various worldlines, and then over it I do a freehand drawing of a curvy line which in one section looks exactly like a profile of Mickey Mouse, I am free to take that curvy line and use it as the x=0 axis of a new coordinate system. A worldline of an inertial object which is just a straight line in Minkowski coordinates might have multiple crossing points with a curvy line like the one containing the Mickey Mouse profile (and this would even be true of a straight line drawn on top of the curvilinear coordinate systems shown in the animated diagram in the article I linked to), so in that non-inertial Mickey Mouse coordinate system the object's path would have to cross the x=0 axis multiple times, meaning it made multiple changes in direction in this system.

You need acceleration to create a non-inertial system, so let's just say that the only acceleration is coming form the black hole. I hate physicists; they're so dam picky!

A non-inertial coordinate system is just one where the equations of SR (like the time dilation and length contraction equations) don't work, so any coordinate system in curved spacetime is non-inertial. In flat spacetime, it's usually true that an object at rest in a non-inertial coordinate system would be accelerating, but not always; for example, if you define a coordinate system where photons would be at rest, then this would be a non-inertial system in spite of the fact that an object at fixed position coordinate would be moving at a constant velocity of exactly c, not accelerating.

In any case, I don't understand how your comment is supposed to relate to my point that there's no well-defined sense in which a coordinate system where an object changes direction for no physical reason is less "real" than one where it does.

I hate physicists; they're so dam picky!

If by "picky" you mean that the terms you use have to be defined in precise mathematical terms, then yes. Otherwise, how is it science? There's nothing scientific about ill-defined qualitative judgments like "more real", they are just as subjective as aesthetic judgments like "more pretty".

If the event of the object crossing the horizon is outside of the coordinate system then what's the point of it in this situation?

Just that it refutes your claim that this coordinate system is making the definite prediction that the object never crosses the horizon, in contradiction with other coordinate systems where the object does cross the horizon at a well-defined coordinate time. A coordinate system cannot be used to predict anything one way or another about events which lie in regions of spacetime outside the region covered by the coordinate system, so you can't use Schwarzschild coordinates to predict that the object never crosses the horizon, all you can say is that it doesn't do so in the region of spacetime covered by this coordinate system (although it gets arbitrarily close to crossing the horizon in the limit as the Schwarzschild time coordinate goes to infinity).

You could construct a coordinate system based on the average rest frame of the background radiation, but the laws of GR would obey the same tensor equations in this system as they do in every other system, so it wouldn't be a "preferred" coordinate system in the sense that physicists use the word.

I only know how to speak English.

Do you have a question about the definition of "preferred frame" or are you just making a wisecrack? If you don't understand the definition of this term, it means a frame where the laws of physics obey different equations than in other frames, which is why I said that the microwave background radiation based coordinate system would not be preferred since "the laws of GR would obey the same tensor equations in this system as they do in every other system".

What do you mean by "moving at c through time"? Something like the mathematical trick used by Brian Greene which I talked about in post #3 of this thread which allows us to understand time dilation in terms of a tradeoff between "speed through space" and "speed through time"? But this trick seems to be specifically dependent on the way time dilation and 4-vectors work in SR, I don't know if there's any way to generalize it to a GR situation involving curved spacetime.

It's a bit more than a trick, but it isn't as simple as that because length contraction isn't explained or needed in this explanation.

Are you giving a mathematical definition of "speed through time" the way Greene did, or is this just some word-picture that seems intuitive to you even though you can't define it in any precise way?

 

[A-wal]

Using your "equivalent" SR scenario of an observer traveling at c, what region of spacetime is equivalent to the event horizon, i.e. what defines the boundary between the region of spacetime from which your observer moving at c can send and receive signals and the region of spacetime from which the observer cannot receive signals?

I don't understand the question. It's only possible for something with mass to travel under light speed unless it has access to an infinite amount of energy. If it could travel faster than c I suppose it'd collapse into a black hole. Is that what you're asking?

Using your "equivalent" SR scenario of an observer traveling at c, how can you explain how it can take an infinite amount of time according to an observer at rest wrt the event horizon and yet a finite amount of proper time for a free-falling observer to cross the horizon?

I'm saying that an observer shouldn't be able to cross the event horizon. How can you explain how it can take an infinite amount of time according to an observer at a distance from the event horizon and yet a finite amount of proper time for a free-falling observer to cross the horizon?

Using your "equivalent" SR scenario of an observer traveling at c, can you explain why it is not possible for an observer stationary wrt the event horizon to let a rope down into the event horizon?

Because it gets more length contracted and time dilated the closer it gets to the event horizon. Like approaching c.

@JesseM: I just don't see how using a coordinate system in which the object never crosses the horizon from any perspective can cast light on a hypothetical situation in which it does. My point is that there shouldn't be any coordinate system in which anything can cross an event horizon. It's always possible from the perspective of an outside observer that an object will have enough energy to escape from the black hole because it never crosses the horizon. I think the same should be true from the perspective of the faller because length contraction will always keep the event horizon some distance away until it's too late and they actually reach the singularity at the end of the black holes life. This doesn't contradict anything an outside observer sees because of time dilation. They'll both observe the same thing happening, but at different speeds and over different lengths.

p.s. It was just a wise crack. I never claimed it was a preferred frame. I don't see how changing coordinate systems makes any difference anyway.