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

[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.

 

[JesseM]

@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.

When did I say it casts light on this? I was just responding to your claim that it was a physical paradox that it crosses the horizon at finite time coordinate in some coordinate systems but not others. The point is, there is no genuine physical paradox, the coordinate systems where it doesn't cross the horizon (like Schwarzschild coordinates) are just incomplete ones which don't cover the entire spacetime manifold. There is a principle in general relativity called "geodesic completeness" which says that worldlines should never "end" at a finite value of proper time unless they run into singularities, if they do in the coordinate system you're using, that means the region of spacetime covered by the coordinate system is not geodesically complete, and can naturally be extended past the covered region.

My point is that there shouldn't be any coordinate system in which anything can cross an event horizon.

Why not?

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.

It's possible, but of course it's also possible that it does cross the horizon. Suppose I throw a ball at a wall, and I use a coordinate system which ends at a point on the ball's worldline before it has hit the wall...for example, I might be using Rindler coordinates in SR, and the ball might cross the Rindler horizon before it reaches the wall, which incidentally also means that no observer at rest in Rindler coordinates would ever see the ball reaching the Rindler horizon, the ball would seem to go slower and slower as it approached this horizon from the perspective of these observers (and just as with a black hole event horizon, they can never see the light from the ball crossing the Rindler horizon unless they cross the Rindler horizon themselves). In this case, of course it's possible that some other projectile knocks the ball off course in the region not covered by my coordinate system, but it's also possible that it does in fact hit the wall.

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.

Sorry, but it is pure nonsense to talk about "length contraction" without defining either the coordinate system the faller is using, or the measurement procedure they are using to define "length". Unless you can provide such a definition, your argument boils down to taking intuitions drawn from inertial coordinate systems in SR and trying to apply them to GR in a totally ill-defined and meaningless way. As Wolfgang Pauli said in another context, this is "not even wrong".

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.

Again you talk about "changing coordinate systems", but you still refuse to tell me what coordinate system you want to start with. Certainly it isn't Schwarzschild coordinates, since there aren't multiple versions of the Schwarzschild coordinate system for observers in different states of motion, and therefore it'd be meaningless to talk about "length contraction" seen by the falling observer if they were using Schwarzschild coordinates. And your suggestion about basing a coordinate system on the rest frame of the CMBR also would not result in multiple coordinate systems for different observers, it would just result in a single system which would naturally result in a single definition of "length" for all observers using this system.

 

[Dale]

You still have not read the link apparently.

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?

This is easy to explain using Rindler coordinates. Scroll down about half way to the section labeled http://www.gregegan.net/SCIENCE/Rindler/RindlerHorizon.html" .

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

How so? The rope is not being let out at relativistic speeds, so it is not significantly length contracted at all from the observer's perspective.

My point is that there shouldn't be any coordinate system in which anything can cross an event horizon.

But there are many such coordinate systems, all describing the same spacetime around a static spherically symmetric mass. One example is Eddington-Finkelstein coordinates. The event horizon is a coordinate singularity, not a physical singularity.

 

[A-wal]

When did I say it casts light on this? I was just responding to your claim that it was a physical paradox that it crosses the horizon at finite time coordinate in some coordinate systems but not others. The point is, there is no genuine physical paradox, the coordinate systems where it doesn't cross the horizon (like Schwarzschild coordinates) are just incomplete ones which don't cover the entire spacetime manifold. There is a principle in general relativity called "geodesic completeness" which says that worldlines should never "end" at a finite value of proper time unless they run into singularities, if they do in the coordinate system you're using, that means the region of spacetime covered by the coordinate system is not geodesically complete, and can naturally be extended past the covered region.

I'm saying the event horizon and the singularity are the same thing for someone crossing the horizon. They singularity and the horizon get closer the closer you get to the black hole.

My point is that there shouldn't be any coordinate system in which anything can cross an event horizon.

Why not?

Because it never happens from one perspective so it shouldn't from another.

It's possible, but of course it's also possible that it does cross the horizon. Suppose I throw a ball at a wall, and I use a coordinate system which ends at a point on the ball's worldline before it has hit the wall...for example, I might be using Rindler coordinates in SR, and the ball might cross the Rindler horizon before it reaches the wall, which incidentally also means that no observer at rest in Rindler coordinates would ever see the ball reaching the Rindler horizon, the ball would seem to go slower and slower as it approached this horizon from the perspective of these observers (and just as with a black hole event horizon, they can never see the light from the ball crossing the Rindler horizon unless they cross the Rindler horizon themselves). In this case, of course it's possible that some other projectile knocks the ball off course in the region not covered by my coordinate system, but it's also possible that it does in fact hit the wall.

But for an outside observer it's meaningless to speak of whether or not the object has crossed the horizon. It hasn't from this perspective, and it never will. Saying it does from it's own perspective is a contradiction.

Sorry, but it is pure nonsense to talk about "length contraction" without defining either the coordinate system the faller is using, or the measurement procedure they are using to define "length". Unless you can provide such a definition, your argument boils down to taking intuitions drawn from inertial coordinate systems in SR and trying to apply them to GR in a totally ill-defined and meaningless way. As Wolfgang Pauli said in another context, this is "not even wrong".

Not even wrong? Oh, I like knowing I was wrong. It means I've learned something. I'm not saying I'm right but I can't just take your word for it either. I need to understand, not just memorise facts.

Again you talk about "changing coordinate systems", but you still refuse to tell me what coordinate system you want to start with. Certainly it isn't Schwarzschild coordinates, since there aren't multiple versions of the Schwarzschild coordinate system for observers in different states of motion, and therefore it'd be meaningless to talk about "length contraction" seen by the falling observer if they were using Schwarzschild coordinates. And your suggestion about basing a coordinate system on the rest frame of the CMBR also would not result in multiple coordinate systems for different observers, it would just result in a single system which would naturally result in a single definition of "length" for all observers using this system.

You're the one who keeps talking about coordinate systems. I think it doesn't matter! I think length will contract the closer you get to the black hole within any single coordinate system. That's what gravity is.

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

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.

Read the link I posted, it is very useful.

I post the link again for your reference: http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

You still have not read the link apparently. This is easy to explain using Rindler coordinates. Scroll down about half way to the section labeled Free fall.

That would be cheating!

How so? The rope is not being let out at relativistic speeds, so it is not significantly length contracted at all from the observer's perspective.

See above.

But there are many such coordinate systems, all describing the same spacetime around a static spherically symmetric mass. One example is Eddington-Finkelstein coordinates. The event horizon is a coordinate singularity, not a physical singularity.

And again.

 

[Dale]

A-wal, if you are too lazy to even read the excellent reference I have provided and repeatedly emphasized then there is no point in continuing the discussion. Read the reference, then we will have something to discuss. Until then I will consider you a troll, not someone with an honest misunderstanding.

 

[A-wal]

I'm not a troll. I just prefer to have a two or or more way conversation rather than just reading.

Okay I've read it and I understand very little from it. I knew this would happen. Something just needs to click in my head and I'll understand what I've just read perfectly.


After a time of τcrit has passed for Eve, she must concede that it's too late for her to send Adam a message asking him to hitch a ride and catch up with the ship, since every signal she now sends will be received by him on the other side of the horizon.

WFT? It's always possible from Eve's perspective that Adam won't cross the horizon. He can always turn round and come back, so how can this make sense?


Suppose Adam decides to tie a rope around his waist when he steps off the ship, but Eve agrees to feed out the rope in such a way that Adam remains in free fall. Is this possible? Clearly it is, because we can imagine a rope of arbitrary length sitting motionless in our (t,x) coordinates, and all Eve has to do to keep her and Adam's rope slack is to feed it out in such a way that it matches that reference rope. This will require Eve to give the section of rope she is dispensing a velocity equal and opposite to her own ordinary velocity in the (t,x) frame, which is tanh(τ/s0). If Eve sticks to her notion of simultaneity then she'll never admit that Adam has passed through the horizon, so her task is endless (and the velocity she needs to give the rope will asymptotically approach the speed of light)...

Ha, I knew it! The last past in the brackets backs me up, I think. Adam approuches the speed of light relative to Eve and therefore length becomes contracted until he finally reaches the event horizon and a velocity of c. The length of the whole universe in the direction he's traveling in becomes 0, but that's not a problem because now he's at the horizon and can't escape. He's whole universe is the black hole, which is now just the singularity because the event horizon has contracted away.


...,but if she takes a more sensible approach and concedes that after a time of τcrit has elapsed there's no hope of hauling him back on to the ship, she will have fed out a length of just s0 [cosh(τcrit/s0) - 1] = s0/4 before reaching that point. The velocity at which she will be dispensing the rope at τcrit will be tanh(τcrit/s0) = 3/5.

Don't get it!

 

[Dale]

Okay I've read it and I understand very little from it. I knew this would happen. Something just needs to click in my head and I'll understand what I've just read perfectly.

Thanks for the effort. It is OK that you didn't understand it all, and I also expected it which is why I made the offer to answer questions about it. At least now we have a basis for a productive discussion.

It's always possible from Eve's perspective that Adam won't cross the horizon. He can always turn round and come back, so how can this make sense?

If he turned around and came back then he would no longer be inertial. So yes, it is possible, but that is not the scenario that was being described here. IF Adam remains inertial then at time τcrit it is too late for Eve to send Adam a message that will reach him prior to his crossing the event horizon.

Ha, I knew it! The last past in the brackets backs me up, I think. Adam approuches the speed of light relative to Eve and therefore length becomes contracted until he finally reaches the event horizon and a velocity of c.

This is certainly one way to measure speed in Eve's non-inertial reference frame (Rindler coordinates), but not the only way. This is one example why specifying the details is so important. However, even with this definition of speed nobody ever reaches c in any frame. Adam asymptotically approaches c in Eve's non-inertial reference frame and Eve asymptotically approaches c in any inertial reference frame. In Eve's frame Adam never reaches the event horizon so it doesn't make sense to talk about him reaching the event horizon and a velocity of c. In Adam's frame he reaches the event horizon at a velocity of 0 (i.e. the horizon moves towards him at c).

...,but if she takes a more sensible approach and concedes that after a time of τcrit has elapsed there's no hope of hauling him back on to the ship, she will have fed out a length of just s0 [cosh(τcrit/s0) - 1] = s0/4 before reaching that point. The velocity at which she will be dispensing the rope at τcrit will be tanh(τcrit/s0) = 3/5.

Before that time if she had a perfectly strong rope (speed of sound = c and unbreakable) she could pull him back to the ship. But after that time even a perfectly strong rope will be unable to pull him back.

 

[JesseM]

Because it never happens from one perspective so it shouldn't from another.

You seem to think the claim by an observer that "it never happens" is equivalent to the claim that the observer never sees it happen, but that's just silly. For example, there is a finite radius to the observable universe because light from sufficiently distant regions of space would not have had time to reach us even if it had been emitted immediately after the Big Bang, but that doesn't mean we believe that the universe actually ends outside this radius!

But for an outside observer it's meaningless to speak of whether or not the object has crossed the horizon.

It isn't meaningless, he just can't see it. What's more, he could easily see it happen at any time by diving in after it.

Did you read the link about the Rindler horizon seen by an observer experiencing constant acceleration in flat SR spacetime? The situation is quite analogous--as long as the observer continues his acceleration he will never see anything beyond the Rindler horizon, but he easily could just by ceasing to accelerate and crossing the Rindler horizon himself (note that the Rindler horizon is just a type of future light cone). Do you think it's meaningless for him to talk about whether something crosses the Rindler horizon, or that there is a physical contradiction between his perspective and that of inertial observers?

It hasn't from this perspective, and it never will. Saying it does from it's own perspective is a contradiction.

There aren't multiple "perspectives" on spacetime in relativity, just one objective truth. It's true that different observers can only see portions of the entire spacetime, but that doesn't imply they are making differing predictions. You might as well say that there is a "contradiction" between me today and me 5 years ago, because today there are events in my past light cone which were not part of the past light cone of my past self, and thus were impossible for him to see at that point.

You're the one who keeps talking about coordinate systems. I think it doesn't matter!

But you don't seem to understand that length is only defined in terms of coordinate systems or particular measurement procedures--it's meaningless to even use the word "length" outside of this context. Until you are willing to either 1) acknowledge this point and explain what coordinate system or measurement procedure you want to use, or 2) explain some alternate definition of "length" that does not depend on specifying a coordinate system or measurement procedure, then your arguments will continue to be "not even wrong", just based on a vague uninformed analogy with SR. So if you want to continue this discussion, please either pick option #1 or option #2, otherwise there seems to be little point in continuing.

 

[A-wal]

This is certainly one way to measure speed in Eve's non-inertial reference frame (Rindler coordinates), but not the only way. This is one example why specifying the details is so important. However, even with this definition of speed nobody ever reaches c in any frame. Adam asymptotically approaches c in Eve's non-inertial reference frame and Eve asymptotically approaches c in any inertial reference frame. In Eve's frame Adam never reaches the event horizon so it doesn't make sense to talk about him reaching the event horizon and a velocity of c. In Adam's frame he reaches the event horizon at a velocity of 0 (i.e. the horizon moves towards him at c).

That's not very relative. If the horizon is moving towards him at c then he is moving towards the horizon at c in that frame.

Before that time if she had a perfectly strong rope (speed of sound = c and unbreakable) she could pull him back to the ship. But after that time even a perfectly strong rope will be unable to pull him back.

Speed of sound?

You seem to think the claim by an observer that "it never happens" is equivalent to the claim that the observer never sees it happen, but that's just silly. For example, there is a finite radius to the observable universe because light from sufficiently distant regions of space would not have had time to reach us even if it had been emitted immediately after the Big Bang, but that doesn't mean we believe that the universe actually ends outside this radius!

But it's not just a trick of light is it. It's caused by time dilation through acceleration. It's real! Nothing can ever cross the event horizon from the perspective of an outside observer. It can't be claimed that it actually does because it's always possible the object will find the energy to break free like I said before. It can't even be claimed that the object will, for the same reason.

It isn't meaningless, he just can't see it. What's more, he could easily see it happen at any time by diving in after it.

That's changing frames and I don't see how it says anything about whether or not something happens in a frame not approaching infinite time dilation.

Did you read the link about the Rindler horizon seen by an observer experiencing constant acceleration in flat SR spacetime? The situation is quite analogous--as long as the observer continues his acceleration he will never see anything beyond the Rindler horizon, but he easily could just by ceasing to accelerate and crossing the Rindler horizon himself (note that the Rindler horizon is just a type of future light cone). Do you think it's meaningless for him to talk about whether something crosses the Rindler horizon, or that there is a physical contradiction between his perspective and that of inertial observers?

That's different because it involves the time light takes to move. It's a delay in what is seen so it does make sense to talk about what's really happening beyond his view point. I don't think the same applies to the black hole situation.

There aren't multiple "perspectives" on spacetime in relativity, just one objective truth.

That's my whole point. Yet you're saying that there are two very different truths. At least that's how I'm forced to interpret it.

It's true that different observers can only see portions of the entire spacetime, but that doesn't imply they are making differing predictions. You might as well say that there is a "contradiction" between me today and me 5 years ago, because today there are events in my past light cone which were not part of the past light cone of my past self, and thus were impossible for him to see at that point.

It's different when there's a separation in space time between events. This argument again doesn't apply to a black hole when you can get as close as you like and still nothing will cross the horizon.

But you don't seem to understand that length is only defined in terms of coordinate systems or particular measurement procedures--it's meaningless to even use the word "length" outside of this context. Until you are willing to either 1) acknowledge this point and explain what coordinate system or measurement procedure you want to use, or 2) explain some alternate definition of "length" that does not depend on specifying a coordinate system or measurement procedure, then your arguments will continue to be "not even wrong", just based on a vague uninformed analogy with SR. So if you want to continue this discussion, please either pick option #1 or option #2, otherwise there seems to be little point in continuing.

If it wasn't vague and uniformed then I wouldn't need to be here. I'd be writing a paper on it. I don't think anything I've said in this post requires a specific coordinate system.

 

[JesseM]

You seem to think the claim by an observer that "it never happens" is equivalent to the claim that the observer never sees it happen, but that's just silly. For example, there is a finite radius to the observable universe because light from sufficiently distant regions of space would not have had time to reach us even if it had been emitted immediately after the Big Bang, but that doesn't mean we believe that the universe actually ends outside this radius!

But it's not just a trick of light is it. It's caused by time dilation through acceleration. It's real!

Time dilation at a given moment is no more "real" than length contraction, both are entirely dependent on what coordinate system you use, they have no unique "real" value. In any case, if you're talking about the horizon of the observable universe I don't know what you mean by "time dilation through acceleration", in the standard cosmological coordinate system (comoving coordinates) all galaxies are treated as being at rest and clocks in all galaxies run at the same rate.

Nothing can ever cross the event horizon from the perspective of an outside observer.

What does "from the perspective of" mean, if you're not talking about what they see? If you would call it a "trick of light" that an accelerating observer in flat SR spacetime never sees anything beyond the Rindler horizon, then I would say it is equally just a trick of light that an observer outside the black hole's event horizon never sees anything at or beyond the event horizon. Perhaps you are using "perspective" in analogy with an observer's inertial rest frame in SR, but in GR there is no single coordinate system that uniquely qualifies as the "rest frame"; it's true in Schwarzschild coordinates that an object never reaches the event horizon at any finite coordinate time, but you could pick other coordinate systems for the observer outside the horizon to use where it does reach it at finite coordinate time, neither coordinate system uniquely qualifies as the observer's "perspective".

If "perspective of" refers neither to what the observer sees nor what happens in some coordinate system that you are referring to when you say "perspective", then I have no idea what you mean by this phrase, you'll have to explain it.

It can't be claimed that it actually does because it's always possible the object will find the energy to break free like I said before. It can't even be claimed that the object will, for the same reason.

Well, exactly the same is true for the accelerating observer about whether or not an object crosses the Rindler horizon--this observer will never see it reach the horizon, so he'll never know for sure if something didn't deflect it at the last moment. But again, there's no reason this accelerating observer can't say there is an objective truth about whether it crossed the horizon, even if he'll never know it as long as he continues to accelerate.

That's changing frames and I don't see how it says anything about whether or not something happens in a frame not approaching infinite time dilation.

Why is diving in after it "changing frames"? There's no reason he can't use the same coordinate system (which is all that 'frame' means in relativity) to analyze both the time he was outside the horizon and the time he dived in. Again, you seem to be drawing on some vague analogy to SR, but in SR we are talking about inertial frames, so "changing frames" just means the object accelerates and so its inertial rest frame is different before and after the acceleration. In GR there's no analogous sense where some physical motions involve "changing frames" while others don't, for any motion you can pick some coordinate systems where the object is at rest in that coordinate system throughout the motion, and other coordinate systems where the object starts at rest and then begins to move.

That's different because it involves the time light takes to move. It's a delay in what is seen so it does make sense to talk about what's really happening beyond his view point. I don't think the same applies to the black hole situation.

There are plenty of coordinate systems where light at the horizon is moving and just never reaches the observer outside the horizon, like Kruskal-Szekeres coordinates (In fact I believe there's a very close mathematical analogy between the analysis of the black hole in Kruskal-Szekeres coordinates and the analysis of the accelerating observers and the Rindler horizon in inertial coordinates). Likewise, if you use Rindler coordinates to analyze the area where the accelerating Rindler observers are located, I believe it's true in this system that light on the horizon is frozen, and time dilation becomes infinite as you approach the horizon.

It's different when there's a separation in space time between events.

What does "a separation in space time between events" mean? Would you not say there is a separation in spacetime between the accelerating observers and events on the other side of the Rindler horizon in SR, since as long as the observers continue to accelerate they will never get any signals from these events (they will never enter their future light cone)? What kind of "separation" is present between observers on the inside and outside of the black hole event horizon that is not also present between observers on the inside and outside of the Rindler horizon?

This argument again doesn't apply to a black hole when you can get as close as you like and still nothing will cross the horizon.

No matter how close you get the Rindler horizon you'll never see anything cross it, not unless you cross it yourself. Same with the black hole event horizon.

If it wasn't vague and uniformed then I wouldn't need to be here. I'd be writing a paper on it.

But then why do you keep resisting people's efforts to correct you on these points? Why not trust that people like me and DaleSpam know what we're talking about, and just ask questions about aspects you find confusing rather than try to argue you think we're wrong?

I don't think anything I've said in this post requires a specific coordinate system.

All comments about time dilation require a specific coordinate system just like comments about length, there is no objective truth about how slow a clock ticks as it nears the horizon that doesn't depend on your choice of coordinate system. In any case, unless you never plan to bring up the subject of "length" again, I would appreciate it if you would answer my previous question:

But you don't seem to understand that length is only defined in terms of coordinate systems or particular measurement procedures--it's meaningless to even use the word "length" outside of this context. Until you are willing to either 1) acknowledge this point and explain what coordinate system or measurement procedure you want to use, or 2) explain some alternate definition of "length" that does not depend on specifying a coordinate system or measurement procedure, then your arguments will continue to be "not even wrong", just based on a vague uninformed analogy with SR. So if you want to continue this discussion, please either pick option #1 or option #2, otherwise there seems to be little point in continuing.

 

[Dale]

That's not very relative. If the horizon is moving towards him at c then he is moving towards the horizon at c in that frame.

Certainly, you can define a "closing speed" as the difference in velocities in some frame. That value does not correspond to the speed of any physical object and is not limited to speeds less than c and does not induce length contraction or time dilation nor does it require infinite energy etc.

Speed of sound?

Yes, any mechanical disturbance in an object propagates through the object at the speed of sound. If Eve pulls on her end of the rope the pull travels towards the other end of the rope at the speed of sound in the rope.

 

[A-wal]

Time dilation at a given moment is no more "real" than length contraction, both are entirely dependent on what coordinate system you use, they have no unique "real" value. In any case, if you're talking about the horizon of the observable universe I don't know what you mean by "time dilation through acceleration", in the standard cosmological coordinate system (comoving coordinates) all galaxies are treated as being at rest and clocks in all galaxies run at the same rate.

You were talking about the time light takes to travel distances and using that as a way of saying that things happen beyond what we can see. It's not the same. There's a delay.

What does "from the perspective of" mean, if you're not talking about what they see? If you would call it a "trick of light" that an accelerating observer in flat SR spacetime never sees anything beyond the Rindler horizon, then I would say it is equally just a trick of light that an observer outside the black hole's event horizon never sees anything at or beyond the event horizon. Perhaps you are using "perspective" in analogy with an observer's inertial rest frame in SR, but in GR there is no single coordinate system that uniquely qualifies as the "rest frame"; it's true in Schwarzschild coordinates that an object never reaches the event horizon at any finite coordinate time, but you could pick other coordinate systems for the observer outside the horizon to use where it does reach it at finite coordinate time, neither coordinate system uniquely qualifies as the observer's "perspective". If "perspective of" refers neither to what the observer sees nor what happens in some coordinate system that you are referring to when you say "perspective", then I have no idea what you mean by this phrase, you'll have to explain it.

Really? An outside observer can objectively claim that an object has crossed the event horizon of a black hole?

Well, exactly the same is true for the accelerating observer about whether or not an object crosses the Rindler horizon--this observer will never see it reach the horizon, so he'll never know for sure if something didn't deflect it at the last moment. But again, there's no reason this accelerating observer can't say there is an objective truth about whether it crossed the horizon, even if he'll never know it as long as he continues to accelerate.

That's different, because of the delay in the time it takes for the light to reach the accelerator. It's happening because the observer is lengthening the distance between themselves and the source. You don't need to do that with the black hole example. You can get as close as you like.

Why is diving in after it "changing frames"? There's no reason he can't use the same coordinate system (which is all that 'frame' means in relativity) to analyze both the time he was outside the horizon and the time he dived in. Again, you seem to be drawing on some vague analogy to SR, but in SR we are talking about inertial frames, so "changing frames" just means the object accelerates and so its inertial rest frame is different before and after the acceleration. In GR there's no analogous sense where some physical motions involve "changing frames" while others don't, for any motion you can pick some coordinate systems where the object is at rest in that coordinate system throughout the motion, and other coordinate systems where the object starts at rest and then begins to move.

I've already stated a coordinate system. Make sure the background radiation is as uniform as possible, then don't alloy any coordinate system in which objects change direction or velocity for no sodding reason!

There are plenty of coordinate systems where light at the horizon is moving and just never reaches the observer outside the horizon, like Kruskal-Szekeres coordinates (In fact I believe there's a very close mathematical analogy between the analysis of the black hole in Kruskal-Szekeres coordinates and the analysis of the accelerating observers and the Rindler horizon in inertial coordinates). Likewise, if you use Rindler coordinates to analyze the area where the accelerating Rindler observers are located, I believe it's true in this system that light on the horizon is frozen, and time dilation becomes infinite as you approach the horizon.

And length contraction? Making the observer that is approaching the event horizon witness the horizon merge with the singularity the moment they reach it?

What does "a separation in space time between events" mean? Would you not say there is a separation in spacetime between the accelerating observers and events on the other side of the Rindler horizon in SR, since as long as the observers continue to accelerate they will never get any signals from these events (they will never enter their future light cone)? What kind of "separation" is present between observers on the inside and outside of the black hole event horizon that is not also present between observers on the inside and outside of the Rindler horizon?

I meant it the other way round. The distance is constantly changing in proportion to the acceleration in inertial frames.

No matter how close you get the Rindler horizon you'll never see anything cross it, not unless you cross it yourself. Same with the black hole event horizon.

I don't understand. How can you approach the event horizon when the horizon itself is being caused by acceleration. Unless you mean acceleration towards something. Wont the horizon always behind the accelerator though though?

But then why do you keep resisting people's efforts to correct you on these points? Why not trust that people like me and DaleSpam know what we're talking about, and just ask questions about aspects you find confusing rather than try to argue you think we're wrong?

I told you that I need to understand. Accepting what people have me isn't the same as understanding it. I could memorise every single know physical fact if my memory was that good. I wouldn't have any greater understanding of the universe than I've ever had.

All comments about time dilation require a specific coordinate system just like comments about length, there is no objective truth about how slow a clock ticks as it nears the horizon that doesn't depend on your choice of coordinate system. In any case, unless you never plan to bring up the subject of "length" again, I would appreciate it if you would answer my previous question:

I already have, haven't I?

Certainly, you can define a "closing speed" as the difference in velocities in some frame. That value does not correspond to the speed of any physical object and is not limited to speeds less than c and does not induce length contraction or time dilation nor does it require infinite energy etc.

I thought it corresponded to the event horizon?

Yes, any mechanical disturbance in an object propagates through the object at the speed of sound. If Eve pulls on her end of the rope the pull travels towards the other end of the rope at the speed of sound in the rope.

I never knew that!

 

[JesseM]

You were talking about the time light takes to travel distances and using that as a way of saying that things happen beyond what we can see. It's not the same. There's a delay.

So your point is just that we'll eventually be able to see it? Well, as it turns out if the universe has a rate of expansion that isn't slowing down and approaching zero (and it seems that the expansion rate in the real universe is actually accelerating), there can actually be regions where the light will never reach us because the space continues to expand between us and them faster than the light emitted from the distant region can bridge the distance.

Nothing can ever cross the event horizon from the perspective of an outside observer.

What does "from the perspective of" mean, if you're not talking about what they see? If you would call it a "trick of light" that an accelerating observer in flat SR spacetime never sees anything beyond the Rindler horizon, then I would say it is equally just a trick of light that an observer outside the black hole's event horizon never sees anything at or beyond the event horizon. Perhaps you are using "perspective" in analogy with an observer's inertial rest frame in SR, but in GR there is no single coordinate system that uniquely qualifies as the "rest frame"; it's true in Schwarzschild coordinates that an object never reaches the event horizon at any finite coordinate time, but you could pick other coordinate systems for the observer outside the horizon to use where it does reach it at finite coordinate time, neither coordinate system uniquely qualifies as the observer's "perspective". If "perspective of" refers neither to what the observer sees nor what happens in some coordinate system that you are referring to when you say "perspective", then I have no idea what you mean by this phrase, you'll have to explain it.

Really? An outside observer can objectively claim that an object has crossed the event horizon of a black hole?

No, how did you get that from what I said above? Of course you can't objectively claim that any event outside your light cone actually happened, because no information confirming that it happened can possibly have reached you. But that's not the same as a positive prediction that it didn't happen, it's just an acknowledgment of uncertainty about what really happened. Similarly, we have no positive evidence to support the empirical claim that Alpha Centauri still exists in 2009 (according to the definition of simultaneity in the solar system's rest frame), but that doesn't mean we are claiming it doesn't exist, and in fact we have good reason to think it extremely likely it does (we just won't know for sure until we get light from Alpha Centauri in 2013). An observer outside a black hole is in the same position--he has very good reason to believe an object did cross the horizon if he sees it getting extremely close with no other visible objects nearby to deflect it, he just can't be sure that it did based on empirical evidence available to him.

Well, exactly the same is true for the accelerating observer about whether or not an object crosses the Rindler horizon--this observer will never see it reach the horizon, so he'll never know for sure if something didn't deflect it at the last moment. But again, there's no reason this accelerating observer can't say there is an objective truth about whether it crossed the horizon, even if he'll never know it as long as he continues to accelerate.

That's different, because of the delay in the time it takes for the light to reach the accelerator.

But exactly the same is true in Kruskal-Szekeres coordinates, where any object at fixed Schwarzschild radius is accelerating away from the event horizon in these coordinates, and light emitted by an object at the moment it crosses the horizon is actually moving outward at a fixed coordinate speed that looks like a line at a 45 degree angle in a Kruskal-Szekeres diagram, just as a light ray moves at a 45 degree angle in a Minkowski diagram. Please take a look at post #4 of mine on another thread for a description of how Kruskal-Szekeres coordinates work, and see if you can follow it. Then compare the diagrams in that post (particular the third one) with the second diagram of the accelerating observers and the Rindler horizon here, you will see they look basically identical.

It's happening because the observer is lengthening the distance between themselves and the source. You don't need to do that with the black hole example. You can get as close as you like.

You are thinking in terms of Schwarzschild coordinates where the horizon has a fixed radius and light on the horizon is frozen. But this is only one of an infinite number of ways of viewing things, in Kruskal-Szekeres coordinates the event horizon is moving outward at a constant speed (so any observer who is not moving outwards themselves will eventually cross it), just as the Rindler horizon is moving outward at a constant speed in inertial coordinates (and if you choose to use the non-inertial Rindler coordinate system in flat spacetime, then the Rindler horizon is also at a constant position and the observers who are 'accelerating' in the inertial frame are now seen as being at rest in Rindler coordinates, so in Rindler coordinates you can also 'get as close as you like' to the Rindler horizon without moving outwards in these coordinates, and you'll still never be able to see anything beyond it).

I've already stated a coordinate system. Make sure the background radiation is as uniform as possible, then don't alloy any coordinate system in which objects change direction or velocity for no sodding reason!

Too vague and handwavey. There is a "reason" for all changes in direction or velocity, they can be understood in terms of the metric in that coordinate system, which gives you the spacetime curvature at each coordinate and determines what a geodesic path will look like in that coordinate system. If you're not satisfied with that answer, can you explain what the "reason" is for all the changes in direction and velocity for an object orbiting a source of gravity in Schwarzschild coordinates?

"Make sure the background radiation is as uniform as possible" is only really clear if we are talking about a universe with uniform curvature everywhere like the class of universes described by the Friedmann–Lemaître–Robertson–Walker metric, if the universe is lumpy on a local scale it's less clear. After all, radiation is affected by gravity just like anything else, so if we imagine a universe initially filled with uniform radiation near the Big Bang before any significant "lumps" had formed, then evolve it forwards a few billion years, I'd guess (though I'm not sure) that in the vicinity of a massive object like a black hole the only observers who would see the radiation in their neighborhood as being uniform in all directions (as opposed to redshifted in one direction and blueshifted in the other) would be ones in freefall along with the radiation, but you probably don't want a coordinate system where an observer falling into a black hole is treated as being at rest, do you?

Finally, if you are talking about only one coordinate system rather than a family of related coordinate systems like inertial frames, then I have no idea what you could mean when you talk about "length contraction" in this context. After all, length contraction in SR is tied to the idea of different observers having different rest frames, so an object can be shorter in the frame of an observer with a higher velocity relative to it than in the frame of an observer with a lower velocity. So what can you mean when you say that an observer falling into the black hole sees its length as shorter than one at constant radius, if you aren't talking about each observer having their own separate coordinate system for defining length?

There are plenty of coordinate systems where light at the horizon is moving and just never reaches the observer outside the horizon, like Kruskal-Szekeres coordinates (In fact I believe there's a very close mathematical analogy between the analysis of the black hole in Kruskal-Szekeres coordinates and the analysis of the accelerating observers and the Rindler horizon in inertial coordinates). Likewise, if you use Rindler coordinates to analyze the area where the accelerating Rindler observers are located, I believe it's true in this system that light on the horizon is frozen, and time dilation becomes infinite as you approach the horizon.

And length contraction? Making the observer that is approaching the event horizon witness the horizon merge with the singularity the moment they reach it?

I don't understand how this response has anything to do with the paragraph you were responding to. Is this a question or an argument? Is it meant to have anything to do with my statements about what's true in Kruskal-Szekeres coordinates or Rindler coordinates above, or are you just changing the subject? Since both KS coordinates and Rindler coordinates are single coordinate systems rather than a family of different ones, the notion of "length contraction" makes little sense if we are talking about either of them.

I meant it the other way round. The distance is constantly changing in proportion to the acceleration in inertial frames.

The distance between the Rindler horizon and the accelerating observers, you mean? Of course here the distance is constantly shrinking as seen in the inertial frame, because even though the accelerating observers are accelerating away from it, in the inertial frame the Rindler horizon is moving outward at light speed while the accelerating observers are always going slower than light. And anyway, exactly the same is true for the distance between the BH event horizon and observers outside the horizon in Kruskal-Szekeres coordinates, so this fails as an argument for saying there is some fundamental distinction between the two situations that would explain why you think an observer outside the BH predicts that objects "never" cross the horizon but you don't say the same for accelerating observers outside the Rindler horizon.

No matter how close you get the Rindler horizon you'll never see anything cross it, not unless you cross it yourself. Same with the black hole event horizon.

I don't understand. How can you approach the event horizon when the horizon itself is being caused by acceleration. Unless you mean acceleration towards something. Wont the horizon always behind the accelerator though though?

You can take any path through spacetime you like, not just one of the constant-acceleration paths shown in the second diagram from the Rindler horizon page--as long as your path doesn't actually cross the horizon you won't see the light from any other object crossing the horizon. For example, you might cut off your engines for a while, or even point your engines in the opposite direction so you're approaching the horizon even faster than if you were moving inertially, but then at the last minute before reaching the horizon, point your engines in the opposite direction and begin accelerating away again. In this situation, for a time you were "approaching the horizon" from the perspective of both the inertial frame (where everything is 'approaching the horizon' in the sense that its distance to the horizon is constantly shrinking, since the horizon moves outward at c) and from the perspective of Rindler coordinates (where the horizon is treated as being at rest, and the constant-acceleration paths seen in that second diagram from the Rindler webpage are also treated as being at rest), but as long as you avoided crossing it, no matter how close you got you won't have been able to see anything crossing it.

I told you that I need to understand. Accepting what people have me isn't the same as understanding it.

Obviously just accepting and not asking further questions won't help you understand things, but trying to prove people wrong may not be the best way either. Why not take the attitude of assuming that what people tell you is likely to be correct and to make sense, and to the extent that you think there are conflicts between what they tell you and other things you think you know about physics, accept that most likely the seeming conflicts are due to mistakes in understanding on your part, and try to figure out what these mistakes are by asking further questions and pointing to the seeming conflicts you see.

All comments about time dilation require a specific coordinate system just like comments about length, there is no objective truth about how slow a clock ticks as it nears the horizon that doesn't depend on your choice of coordinate system. In any case, unless you never plan to bring up the subject of "length" again, I would appreciate it if you would answer my previous question:

But you don't seem to understand that length is only defined in terms of coordinate systems or particular measurement procedures--it's meaningless to even use the word "length" outside of this context. Until you are willing to either 1) acknowledge this point and explain what coordinate system or measurement procedure you want to use, or 2) explain some alternate definition of "length" that does not depend on specifying a coordinate system or measurement procedure, then your arguments will continue to be "not even wrong", just based on a vague uninformed analogy with SR. So if you want to continue this discussion, please either pick option #1 or option #2, otherwise there seems to be little point in continuing.

I already have, haven't I?

Where do you think you did that? When I described option #1 and option #2, your only response was:

If it wasn't vague and uniformed then I wouldn't need to be here. I'd be writing a paper on it. I don't think anything I've said in this post requires a specific coordinate system.

Are you just saying that you don't know whether you agree that "length" can only be defined relative to a particular coordinate system or measurement procedure (option #1) or whether there could be some other way of defining it (option #2)? If you have no coherent idea of any other way of defining it, and no argument or authority that suggests there should be any other way, why not just trust me that this is in fact the only meaningful way to define it in physics and look at what the consequences would be for the rest of your argument?

 

[Dale]

I thought it corresponded to the event horizon?

No, let's say in some reference frame rocket A is moving inertially at 0.9 c from the left and B is moving inertially at 0.9 c from the right. In that case their "closing speed" would be 1.8 c, but they would still be able to communicate with each other by sending light or radio signals etc. There would be no event horizon because they are both moving inertially.

 

[A-wal]

Okay. I've thinking about it intently since your last post. What's the time? Holy crap! Actually I completely forgot about this. Let's start again. I appreciate the time taken to reply to my earlier posts. Please don't take my way of thinking as arrogance. It probably is but it works well for me.

From the perspective of someone a good distance away from the black hole it looks like nothing ever reaches the event horizon. They keep moving towards it at slower and slower rate. From the perspective of the person approaching the event horizon the rest of the universe seems to speed up as the approach the horizon. That's time dilation but there's also length contraction, and again the person approaching the black hole won't notice anything different in themselves because everything's relative but they will notice it if they look at the rest of the universe.

From the distant persons perspective again it will seem like the approaching (approaching the black hole) ship has changed shape because it's stretched along a straight line between it and the black hole. So to correct that from the approachers perspective we need to squish the dimension between the approaching ship and the black hole making the ship the right shape again. In doing that the event horizon also looses length in a straight line between it and the ship. The closer the ship gets, the more pronounced the length contraction becomes so that no matter how close it gets, it can never actually reach the event horizon.

I don't believe anything in that example is dependant on the coordinate system used to define it. In fact I never saw how it could make a difference. It it happens in one system, it should happen in all of them, it will just look different. Reaching the event horizon is exactly like accelerating to a relative velocity of c. It can't happen, surely. Am I wrong yet?

 

[Dale]

It it happens in one system, it should happen in all of them, it will just look different. Reaching the event horizon is exactly like accelerating to a relative velocity of c. It can't happen, surely. Am I wrong yet?

Instead of talking about the event horizon of a black hole let's talk about the event horizon in Rindler coordinates. Does the event of an inertial observer crossing the event horizon happen in an inertial frame? Does it happen in the Rindler coordinates?