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

[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?

 

[A-wal]

If those two questions have a different answer as I'm assuming they do (no to the first and yes to the second) then I don't see how they can be consistent with each other, unless they're describing different horizons. I also don't see how changing coordinate system can make the same object behave differently with respect to the black hole (or anything for that matter). Surely all coordinate systems that are accurate have to tell the exact same story!

The event horizon of a black hole is something that has a definite radius around the singularity providing an object has no inertial velocity relative to it. If the object doesn't expend energy to counteract the gravity from the black hole then it will accelerate towards it. The closer the object gets to the event horizon, the slower time moves from the perspective of that object, making it perceive everything else to be speeding up of course. If it were to reach the horizon it would be frozen in time, but it can't because time moves slower the closer it gets, just like an accelerating object comparing itself to an object in its original frame in flat space-time.

Besides, length contraction would make the black hole smaller from the objects perspective in the dimension of a straight line between the two, giving it an oblong event horizon. But time dilation would mean the black hole would evaporate before anything could cross the horizon even if it wasn't for length contraction. I think to tell if an object did cross the horizon you would have to see the black hole die. Any objects that disappear with it obviously did cross the horizon, but not until then.

I don't see how anything changes if you use a different coordinate system?

 

[Dale]

Surely all coordinate systems that are accurate have to tell the exact same story!

Actually, no. Some coordinates do not cover all of spacetime. The Rindler coordinates and Schwarzschild coordinates share the fact that there are some events in their respective spacetimes which are outside of the coordinate system. For example, what is the Schwarzschild coordinate of something halfway between the event horizon and the singularity?

 

[A-wal]

I'm liking the Schwarzschild one more and more. It says an object can't cross the horizon judging by what you just said, correct? How can an object reach the horizon when it would be infinitely time dilated from any other frame?

 

[Dale]

I'm liking the Schwarzschild one more and more. It says an object can't cross the horizon judging by what you just said, correct? How can an object reach the horizon when it would be infinitely time dilated from any other frame?

It would only be infinite time dilated for an object at rest in the Schwarzschild coordinates. For a free-falling observer the time dilation is finite.

 

[A-wal]

That's paradoxical! In one it does and in the other it doesn't. How can two different coordinate systems that give two different outcomes both be right? What if someone at rest in Schwarzschild coordinates waited till the end of the black holes life? What would someone who was falling in see when looking at a more distant object? You're saying the situation is non equivalent to looking at a closer object? That doesn't make sense!

I'm not giving up until I've got this straight in my head.

 

[Dale]

That's paradoxical! In one it does and in the other it doesn't. How can two different coordinate systems that give two different outcomes both be right?

There is no outcome which is disagreed-upon by different coordinate systems. The difference is not the coordinate systems, it is the motion of the observer. A free falling observer detects a finite redshift in any coordinate system and an accelerating observer (stationary in Schwarzschild coordinates) detects an infinite redshift in any coordinate system.

What if someone at rest in Schwarzschild coordinates waited till the end of the black holes life?

Then by definition it is not Schwarzschild coordinates. The Schwarzschild metric is stationary.

You really would be better off looking into Rindler coordinates. You can learn everything you need to know about the event horizon without the confusion of spacetime curvature.

 

[A-wal]

Apparently the Rindler coordinates are to show the relativistic effects of an object under acceleration. Like the fact that the back of the object has to accelerate faster than the front from the perspective of a non-accelerating observer because of length contraction. And apparently this can also be used to show the effects of curved space-time instead of acceleration, which makes sense because they're the same thing. An event horizon appears that the accelerating object can't reach for an observer using these coordinates, so an event horizon that can't be reached is created when applied to curved space-time as well.

I can't see any self-consistent way of resolving this without the event horizon receding as it's approached. A distant observer would see the in-falling object undergo length contraction and time dilation as it approaches and it would reach infinity at the horizon, so it can never get there, and conversely the object itself would observe the rest of the universe speeding up as it got closer, becoming infinitely reverse time dilated at the event horizon, again meaning it can never actually get there.

I'm assuming that I'm going wrong when I switch it round and look at it from the perspective of the in-falling observer. So I need to understand how the situation can be both non-symmetric and self-consistent at the same time (without moving the goal posts).

 

[JesseM]

That's paradoxical! In one it does and in the other it doesn't.

Schwarzschild coordinates don't specifically predict it doesn't cross the horizon, it's just that no finite time coordinate can be assigned to the event of crossing the horizon. And you can actually assign a finite set of coordinates to events on the object's worldline once it's inside the horizon, see the left diagram here (from Gravitation by Misner/Thorne/Wheeler):

 

[Dale]

Apparently the Rindler coordinates are to show the relativistic effects of an object under acceleration. Like the fact that the back of the object has to accelerate faster than the front from the perspective of a non-accelerating observer because of length contraction. And apparently this can also be used to show the effects of curved space-time instead of acceleration, which makes sense because they're the same thing. An event horizon appears that the accelerating object can't reach for an observer using these coordinates, so an event horizon that can't be reached is created when applied to curved space-time as well.

Essentially correct. I wouldn't say that the horizon cannot be reached, I would say that it cannot be reached in a finite amount of coordinate time. You can reach it in a finite amount of proper time simply by falling.

conversely the object itself would observe the rest of the universe speeding up as it got closer, becoming infinitely reverse time dilated at the event horizon, again meaning it can never actually get there.

The infalling observer doesn't see any of these effects. They pass through the event horizon without incident and without any observable effect other than they can no longer send a signal to the accelerating observer.

So I need to understand how the situation can be both non-symmetric and self-consistent at the same time (without moving the goal posts).

The two observres are physically non-symmetric, so you expect non-symmetric results and observations.

 

[A-wal]

If no finite time coordinate can be assigned to the event of crossing the horizon then it's meaningless to describe what happens after it has crossed. In that diagram what happen to the observers view of the rest of the universe? As they undergo more and more time dilation won't their view of the rest of the universe speed up? It would reach infinate speed at the horizon but black holes con't last for an infinate amount of time, so if you try to cross the event horizon af a black hole then you'll never be able to get there in time.

The infalling observer doesn't see any of these effects. They pass through the event horizon without incident and without any observable effect other than they can no longer send a signal to the accelerating observer.

This is a contradiction to me, sorry.

The two observres are physically non-symmetric, so you expect non-symmetric results and observations.

Same again.

 

[JesseM]

If no finite time coordinate can be assigned to the event of crossing the horizon then it's meaningless to describe what happens after it has crossed.

Why? Suppose you have an ordinary inertial coordinate system to describe your movements today, and at t=10350 seconds you step outside of your house. Now, let's transform your movements into a different non-inertial coordinate system:

x' = x
y'=y
z'=z
t' = t^2/(t-10350)

In this coordinate system, no finite time can be assigned to the event of your stepping outside the house. But that doesn't mean it can't be used to label events that happen to you after you step outside, say at t=10400 seconds in the original inertial coordinate system.

In that diagram what happen to the observers view of the rest of the universe? As they undergo more and more time dilation won't their view of the rest of the universe speed up?

No, not for a nonrotating black hole--if they are watching some clock hovering outside as they fall in, they will only see it reading some finite time as they cross the horizon (see the 'Will you see the universe end?' section of this page from the Usenet Physics FAQ). On the other hand, a rotating black hole has both an inner and an outer event horizon, and an observer crossing the inner horizon would see the entire infinite future of the outside universe, which also means that light falling into the black hole becomes blueshifted to infinity at the inner horizon and the energy density goes to infinity there, which is why general relativity's predictions about the inside of rotating black holes are treated as suspect and physicists think a theory of quantum gravity is needed to understand what happens at the inner horizon.

It would reach infinate speed at the horizon but black holes con't last for an infinate amount of time,

In pure general relativity a black hole would last an infinite time, at least unless the universe collapsed in a big crunch. Quantum gravity will probably give them a finite lifetime due to Hawking radiation though.

so if you try to cross the event horizon af a black hole then you'll never be able to get there in time.

There is no physical significance to the fact that it takes an infinite Schwarzschild coordinate time to reach the horizon, that's just a bug in how the coordinate system is defined, not really any different than the coordinate system I defined at the start of the post which cannot assign a finite time to the event of your stepping out your door. Always remember, coordinate systems are just arbitrary ways of labeling events, they don't have any deep physical significance.

 

[Dale]

This is a contradiction to me, sorry.

What is contradictory about it? Remember I was speaking about the event horizon in Rindler coordinates on flat spacetime, which I think will be a better starting point.

 

[A-wal]

First I’m told that my question has no basis unless it’s defined within a coordinate system and I’m not even wrong unless I do that, then coordinate systems are meaningless and don’t describe physical reality.


If an object is in a stronger gravitational field then it will be moving through time slowly relative to someone in a lower gravitational field. As an object approaches the event horizon it will become infinitely time dilated so no amount of time will be enough to observe an object crossing it.

So if we reverse that then the person falling in will see the distant observer as moving quickly through time, and as no amount of time is enough to see the observer cross the horizon; any given amount of time will pass for the observer approaching the horizon before they actually reach it, surley?

 

[Dale]

If an object is at rest in a stronger gravitational [STRIKE]field[/STRIKE] potential then it will be moving through time slowly relative to someone at rest in a lower gravitational [STRIKE]field[/STRIKE] potential. As an object approaches the event horizon it will become infinitely time dilated so no amount of time will be enough to observe an object crossing it.

So if we reverse that then the person [STRIKE]falling in[/STRIKE] at rest in a stronger gravitational potential will see the distant observer as moving quickly through time

I have corrected your statement to make it correct.

 

[JesseM]

First I’m told that my question has no basis unless it’s defined within a coordinate system and I’m not even wrong unless I do that, then coordinate systems are meaningless and don’t describe physical reality.

No one said that coordinate systems are "meaningless", they are well-defined entities which are very useful for making calculations that try to figure out the answer to questions about coordinate-invariant physical facts, like the proper time along a given worldline between two events on that worldline. However, coordinate systems are quite arbitrary, i.e. there is no physical reason why you must use one as opposed to any other, and coordinate-dependent claims like the claim that a clock's rate of ticking approaches zero as it approaches the horizon (true in Schwarzschild coordinates but not in Kruskal-Szekeres coordinates) don't really "describe physical reality", they just describe properties of the coordinate system.

If an object is in a stronger gravitational field then it will be moving through time slowly relative to someone in a lower gravitational field.

In Schwarzschild coordinates this is true, but not in any coordinate-independent sense. Indeed the question of how fast an object is "moving through time" is an inherently coordinate-dependent one, just as much so as the question of how quickly an object's x-coordinate is changing.

As an object approaches the event horizon it will become infinitely time dilated so no amount of time will be enough to observe an object crossing it.

If you're talking about observation, this can be defined in a coordinate-independent way, since it only concerns pairs of events which occur at the same localized point in spacetime (for example, if the event of my clock reading T=20 seconds coincides with the event of my being hit by the light from your clock reading T=15 seconds, this is something all coordinate systems agree on). But the fact that the hovering observer sees the image of the falling observer slow down as the falling observer approaches the horizon doesn't imply that time is "really" going slower for the falling observer. To see this, just consider the case of the Rindler horizon in flat SR spacetime. If you have a family of accelerating observers who are at rest in Rindler coordinates, and then graph their worldlines from the perspective of an ordinary inertial frame, you get something like this:

Note that none of these curved paths will ever cross the diagonal dotted line which bounds them, which is the path of a light beam and which also represents the "Rindler horizon"; and since all light paths are diagonal in a diagram of an inertial frame, you can see why no event on the left side of the Rindler horizon can ever send a signal that will intersect the worldlines of any of the accelerating observers, thus they will never see anything that happens in the region of spacetime that's on the left side of the horizon in the diagram.

Now consider an ordinary inertial observer at rest in this inertial frame, whose worldline would just be a vertical line. Any such observer who starts out on the right side of the Rindler horizon will eventually cross to the left side. Just as with an observer hovering outside of a black hole, the accelerating observers will visually see the inertial observer's clock slow down more and more as he approaches the horizon (try drawing diagonal lines representing light signals emanating from points on the inertial observer's worldline just before crossing the horizon, and you'll see that they take longer and longer to intersect the worldline of an accelerating observer), and will see it take an infinite time for him to actually cross. But from the point of view of the inertial frame this diagram is drawn in, you can also see that nothing special is happening to the inertial observer as he approaches and crosses the horizon, and since he's at rest in this frame his clock always runs at the same rate in this frame.

So if we reverse that then the person falling in will see the distant observer as moving quickly through time, and as no amount of time is enough to see the observer cross the horizon; any given amount of time will pass for the observer approaching the horizon before they actually reach it, surley?

No, again consider drawing an inertial observer with a vertical worldline in the above diagram. If you consider the point he crosses the Rindler horizon, and then draw a diagonal line sloping downward from that point and see where it intersects with one of the accelerating worldlines, then whatever point on the accelerating worldline it intersects with, that will be the event on the accelerating observer's worldine that the inertial observer is seeing as he crosses the horizon.

There is a very close analogy between lines of constant Rindler coordinate drawn in an inertial frame (as in the diagram above) and lines of constant Schwarzschild radius drawn in a Kruskal-Szekeres diagram--see here. So, everything I'm saying about observers outside the Rindler horizon vs. observers who cross it can be translated into statements about the black hole event horizon in the context of a Kruskal-Szekeres diagram.