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

[timetravel_0]

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

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

 

[JesseM]

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

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

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

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

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

 

[A-wal]

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

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

 

[Dale]

Get out of that one smeg head.

And you were complaining about my tone

 

[A-wal]

And you were complaining about my tone

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

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

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

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

 

[JesseM]

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

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

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

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

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

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

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

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

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

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

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

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

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

And in post #167 you did respond:

#2.

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

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

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

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

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

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

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

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

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

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

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

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

2. Coordinate velocity in some choice of coordinate system

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

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

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

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

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

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

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

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

It was from something you either quoted or linked.

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

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

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

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

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

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

I just see it as the distance between objects,

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

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

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

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

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

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

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

You can’t change reality by measuring differently

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

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

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

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

 

[JesseM]

(continued)

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

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

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

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

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

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

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

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

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

Maybe they do on paper.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[JesseM]

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

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

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

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

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

 

[A-wal]

Wow! Thanks for the time and effort again.

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

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

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

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

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

Curved space-time is the conceptual explanation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

See above.

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

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

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

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

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

^

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

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

 

[A-wal]

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

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

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

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

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

 

[PeterDonis]

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

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

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

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

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

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

 

[A-wal]

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

 

[PeterDonis]

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

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

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

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

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

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

 

[A-wal]

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

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

 

[JesseM]

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

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

 

[Mentz114]

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

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

No problem, novelty-eraser-head.

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

 

[PeterDonis]

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

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

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

 

[A-wal]

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

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

No problem, novelty-eraser-head.

:)

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

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

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

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

Huh?


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

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

 

[JesseM]

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

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

 

[A-wal]

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

 

[JesseM]

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

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

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

 

[PeterDonis]

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

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

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

 

[A-wal]

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

Yep.

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

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

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

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

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

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

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

 

[PeterDonis]

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

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

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

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

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

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

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

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

 

[A-wal]

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

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

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

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

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

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

 

[PeterDonis]

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

There isn't one, as long as you aren't trying to reason about what happens *at* the horizon, or whether observers can or cannot cross it. But as soon as you try to reason about events *at* (or below) the horizon (as for example when you say "time dilation becomes infinite" at the horizon, or "the closer observer can't reach the horizon because the further observer can never observe it"), the implicit assumptions you are making, which are valid above the horizon, break down, and hence your reasoning breaks down.

You seem to think that the hypothetical "observer at the horizon" is somehow a "limiting case" of observers hovering closer and closer to the horizon, so you can use something like "acceleration goes to infinity" to argue that the horizon can't be reached, in the same way you argue that the speed of light can't be reached because it would require infinite energy. That's not correct; the two situations are not analogous, even though there are some apparent similarities. For example, the horizon is a "null surface"--light beams emitted exactly at the horizon stay at the horizon forever--so it does appear to be "moving outward at the speed of light" to freely falling observers that pass through the horizon, just as the Rindler horizon in Egan's diagram, which is simply the line t = x, appears to be moving outward--in the positive x-direction--at the speed of light to Adam, who is following the worldline x = s_0, and crosses the horizon at the event x = s_0, t = s_0. However, the horizon does *not* appear to be moving outward at the speed of light to observers hovering at a constant radial coordinate r outside the horizon--it appears to be staying in the same place forever, the Schwarzschild radius r = 2M, just as the Rindler horizon in Egan's diagram appears to stay in the same place forever to Eve, whose worldline is the hyperbola $x^{2} - t^{2} = s_{0}^{2}$--to her, the horizon is always at a distance s_0 below her.

(By the way, I should emphasize that the Schwarzschild radial coordinate r is *not* a measure of actual radial distance, so, for example, if I am hovering at r = 2.5M, that does *not* mean I am an actual physical distance 0.5M above the horizon at r = 2M. This is one way in which the Adam and Eve situation is *not* exactly analogous to the situation outside a black hole; the x-coordinate s_0 where Eve's and Adam's worldlines meet *is* an actual physical distance, so the horizon does appear to be at a distance s_0 below Eve. This also means, by the way, that the radial coordinate r = 2M does *not* represent the physical "radius" of the horizon or the black hole. Physically, the Schwarzschild r coordinate is defined such that the *area* of the 2-sphere at a given r is $4 \pi r^{2}$, and the *circumference* of any great circle on such a 2-sphere is $2 \pi r$; so the physical meaning of the horizon being at r = 2M is that the *area* of the horizon, as a 2-sphere, is $16 \pi M^{2}$, and the circumference of a great circle at the horizon is $4 \pi M$. But because the spacetime is curved, the 2-sphere can have that area without having physical radius r. Good relativity texts are careful not to refer to r as a physical radius for that reason; Kip Thorne's *Black Holes and Time Warps*, for example, always refers to "circumference" instead of "radius".)

So if you want to restrict discussion solely to what happens outside the horizon, that's fine, as long as you are careful *not* to make any claims about what happens at or below the horizon, including whether or not observers can cross it.

 

[A-wal]

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

Correct me if I'm wrong but as I understand it an observer hovering closer to the horizon can be looked at undergoing constant acceleration (I'm not talking about the acceleration needed to resist gravity), and a free-falling observer can be looked at as accelerating at an ever increasing rate. But they don't feel as though they're accelerating because it's analogous traveling at a greater relative velocity in that sense. But they do feel the increase in rate of acceleration, as "proper acceleration" which can only be felt under under the influence of extremely strong gravitation. This is called tidal force. Correct?

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

Okay I've just looked it up and it seems that the Rindler horizon is just the point at which light (and therefore anything else) emitted from a greater certain relative velocity can't ever catch up to the accelerating observer. Is your point that in the flat space-time example the observer who stays behind will never see the accelerating observer disappear despite the fact that no signal sent will ever reach them, and this means that objects can disappear into a black hole despite the fact that observers further away can never witness this happening? For the black hole analogy it's just that there will be a point when light sent from a greater distance from the black hole can never reach a free-falling observer... before the black holes death anyway. That's because a free-falling observer in curved space-time is equivalent to an accelerating observer in flat space-time as I said above. This does nothing to address the issue of crossing an event horizon. Maybe I misunderstood the Rindler horizon though. This gives me an interesting thought. Acceleration can also be looked at as velocity relative to a non-constant c. Is that in any way meaningful or just complete bollocks? I can't tell any more. Too much having to find the words to define exactly what I'm getting at. Such a pain.

There isn't one, as long as you aren't trying to reason about what happens *at* the horizon, or whether observers can or cannot cross it. But as soon as you try to reason about events *at* (or below) the horizon (as for example when you say "time dilation becomes infinite" at the horizon, or "the closer observer can't reach the horizon because the further observer can never observe it"), the implicit assumptions you are making, which are valid above the horizon, break down, and hence your reasoning breaks down.

Of course my reasoning breaks down at the event horizon, along with common sense and anything resembling logic it seems. That's because my reasoning can't reach the event horizon. That's the whole point.

You seem to think that the hypothetical "observer at the horizon" is somehow a "limiting case" of observers hovering closer and closer to the horizon, so you can use something like "acceleration goes to infinity" to argue that the horizon can't be reached, in the same way you argue that the speed of light can't be reached because it would require infinite energy. That's not correct; the two situations are not analogous, even though there are some apparent similarities. For example, the horizon is a "null surface"--light beams emitted exactly at the horizon stay at the horizon forever--so it does appear to be "moving outward at the speed of light" to freely falling observers that pass through the horizon, just as the Rindler horizon in Egan's diagram, which is simply the line t = x, appears to be moving outward--in the positive x-direction--at the speed of light to Adam, who is following the worldline x = s_0, and crosses the horizon at the event x = s_0, t = s_0. However, the horizon does *not* appear to be moving outward at the speed of light to observers hovering at a constant radial coordinate r outside the horizon--it appears to be staying in the same place forever, the Schwarzschild radius r = 2M, just as the Rindler horizon in Egan's diagram appears to stay in the same place forever to Eve, whose worldline is the hyperbola $x^{2} - t^{2} = s_{0}^{2}$--to her, the horizon is always at a distance s_0 below her.

You've lost me a bit there. Can you express that in the form of a story?

(By the way, I should emphasize that the Schwarzschild radial coordinate r is *not* a measure of actual radial distance, so, for example, if I am hovering at r = 2.5M, that does *not* mean I am an actual physical distance 0.5M above the horizon at r = 2M. This is one way in which the Adam and Eve situation is *not* exactly analogous to the situation outside a black hole; the x-coordinate s_0 where Eve's and Adam's worldlines meet *is* an actual physical distance, so the horizon does appear to be at a distance s_0 below Eve. This also means, by the way, that the radial coordinate r = 2M does *not* represent the physical "radius" of the horizon or the black hole. Physically, the Schwarzschild r coordinate is defined such that the *area* of the 2-sphere at a given r is $4 \pi r^{2}$, and the *circumference* of any great circle on such a 2-sphere is $2 \pi r$; so the physical meaning of the horizon being at r = 2M is that the *area* of the horizon, as a 2-sphere, is $16 \pi M^{2}$, and the circumference of a great circle at the horizon is $4 \pi M$. But because the spacetime is curved, the 2-sphere can have that area without having physical radius r. Good relativity texts are careful not to refer to r as a physical radius for that reason; Kip Thorne's *Black Holes and Time Warps*, for example, always refers to "circumference" instead of "radius".)

Now that makes sense. "But because the spacetime is curved, the 2-sphere can have that area without having physical radius r." That's why I love relativity.

So if you want to restrict discussion solely to what happens outside the horizon, that's fine, as long as you are careful *not* to make any claims about what happens at or below the horizon, including whether or not observers can cross it.

But if I don't see how it can be reached then I don't see how it can be crossed!


Look at a graph that shows the path of an observer past an event horizon and the proper time they experience. Say the starting point is roughly 1 light year away from the black hole and the starting speed is .5c. At half the speed of light it shouldn't take two years to get there as the graph suggests because the distance will increase as they enter more and more length contracted space. The exact opposite in fact of an accelerating observer taking less proper time to reach their destination than their original frame suggested because of length contraction shortening the distance.

 

[Dale]

Of course my reasoning breaks down at the event horizon, along with common sense and anything resembling logic it seems. That's because my reasoning can't reach the event horizon. That's the whole point.

That is only because you are restricting your reasoning to Schwarzschild coordinates and Schwarzschild coordinates do not cover the event horizon. There is nothing wrong with common sense, logic, reason, or math, it is just that the coordinate chart ends before you get to the horizon.

But if I don't see how it can be reached then I don't see how it can be crossed!

By using a coordinate system that includes the event horizon and even points inside the event horizon.

 

[PeterDonis]

Correct me if I'm wrong but as I understand it an observer hovering closer to the horizon can be looked at undergoing constant acceleration (I'm not talking about the acceleration needed to resist gravity), and a free-falling observer can be looked at as accelerating at an ever increasing rate. But they don't feel as though they're accelerating because it's analogous traveling at a greater relative velocity in that sense. But they do feel the increase in rate of acceleration, as "proper acceleration" which can only be felt under under the influence of extremely strong gravitation. This is called tidal force. Correct?

I'll let JesseM weigh in as well, but I wanted to comment on this because it seems to me to be getting at a fundamental misconception you have. The acceleration of an observer hovering at constant radial coordinate r in Schwarzschild coordinates *is* "the acceleration needed to resist gravity", in the sense you mean. Since the observer is near a gravitating body, he has to accelerate--fire rockets, stand on the surface of a solid planet, etc.--to keep himself from falling towards it. That acceleration is also his "proper acceleration", and he feels it, just as you feel the 1 g acceleration caused by the surface of the Earth pushing up on you and keeping you from falling towards the Earth's center. So no, hovering at constant r in Schwarzschild coordinates is *not* "analogous to traveling at a greater relative velocity". They're fundamentally different things, because one observer (the one hovering at constant r) feels an acceleration, where the other (one just moving at a greater relative velocity, but still moving inertially) does not.

Tidal force is something different, and I don't think we should get into it until we have the other things we're discussing clear.

You've lost me a bit there. Can you express that in the form of a story?

I kind of thought I was telling one, but let me restate it with some more details filled in.

Start with a global inertial frame; we'll use coordinates X, T to refer to the distance and time in that frame. (We'll leave out the other two spatial dimensions because they don't affect anything in this scenario.) In this frame we define the following objects and worldlines:

* A light beam is emitted from the origin of these coordinates in the positive X direction; therefore the worldline of that light beam is the line $X = T$.

* An observer, Eve, is in a rocket ship undergoing a constant proper acceleration (i.e., the acceleration she feels, would measure with an accelerometer, etc.) of $a$. At time T = 0 in the global inertial frame, she is at the X-coordinate $s_0 = 1/a$, and at that instant she is momentarily at rest in the global inertial frame. (All units are "geometric units" where the speed of light is 1.) Thus, Eve's worldline is the hyperbola $X^2 - T^2 = s_0^2$.

* At that same event where Eve is momentarily at rest in the global inertial frame (i.e., the event X = $s_0$, T = 0), another observer, Adam, drops off her rocket ship and follows an inertial (free-falling) path. Thus, Adam's worldline (for T >= 0) is simply the line $X = s_0$.

Now, given the above, we can make the following remarks. Note that all of these remarks are just simple deductions from the above; there's no mystery or complication about any of them.

(1) The light beam, T = X, will *never* catch up to Eve; i.e., it will never cross her worldline. This is what justifies the term "Rindler horizon" in reference to the light beam--Eve will never see it, nor will she ever receive any signals from events on the horizon, or in the region of spacetime on the other side of it from her worldline.

(2) Adam, however, will intersect the light beam T = X at the event (X = $s_0$, T = $s_0$). At that event, the light beam will appear to him to be moving outward (in the positive X-direction), at the speed of light, of course. This is what justifies us saying that the horizon is moving outward at the speed of light as seen by an observer freely falling past it.

(3) At any event (X, T) on Eve's worldline, we can draw a straight line through the origin (X = 0, T = 0) that intersects that event. Call this line L. Line L will have slope

$$\frac{T}{X} = \frac{ \sqrt{ X^{2} - s_{0}^{2} } }{X}$$

This slope gives Eve's velocity at that event, as measured in the global inertial frame. We can do a Lorentz transformation using this velocity to obtain the following:

(3a) Eve will perceive all the events lying on line L to be simultaneous with the event (X, T) where the line crosses her worldline--in other words, she will assign all events on this line a "time" by her clock equal to her proper time at the event (X, T) on her worldline.

(3b) The spatial distance along line L between the origin and the event (X, T) where it crosses Eve's worldline will be $s_0$. This is true for *every* event on Eve's worldline. Because the origin is on the horizon (the line X = T), this justifies us saying that, from Eve's point of view, the horizon is always a constant distance $s_0$ below her.

If you have any questions at all about the above, we should get them resolved before proceeding any further.

 

[PeterDonis]

Now that makes sense. "But because the spacetime is curved, the 2-sphere can have that area without having physical radius r." That's why I love relativity.

If this is just a comment, I agree it's amusing. But if you actually mean it to be an argument against the claim I'm making, then we need to go into more detail about what curvature means. The statement I made, that "the 2-sphere can have that area without having physical radius r", is no more mysterious or illogical than the statement, "a triangle drawn on the surface of the Earth can have three right angles, but still be a triangle"--i.e., it can be a triangle without the sum of the angles being 180 degrees. Would you agree with the latter statement?

 

[JesseM]

Correct me if I'm wrong but as I understand it an observer hovering closer to the horizon can be looked at undergoing constant acceleration (I'm not talking about the acceleration needed to resist gravity), and a free-falling observer can be looked at as accelerating at an ever increasing rate.

There are only two basic types of "acceleration" I know of, acceleration relative to some coordinate system and proper acceleration which corresponds to the G-forces you feel (and also represents your coordinate acceleration in the local inertial frame where you are instantaneously at rest). Depending on the choice of coordinate system, any observer can have a coordinate acceleration of zero or a nonzero coordinate acceleration. But proper acceleration is more "objective", and a free-falling observer always experiences zero proper acceleration, while a hovering observer has nonzero proper acceleration (I don't know what you mean by 'not talking about the acceleration needed to resist gravity' though).

But they don't feel as though they're accelerating because it's analogous traveling at a greater relative velocity in that sense.

No idea what you mean by "analogous traveling at a greater relative velocity", or why that analogy would mean they don't feel they're accelerating.

But they do feel the increase in rate of acceleration, as "proper acceleration" which can only be felt under under the influence of extremely strong gravitation. This is called tidal force. Correct?

None of that really makes sense to me. Proper acceleration is completely different from tidal force, a point particle can experience proper acceleration while tidal forces are felt by extended objects, you can think of them as being due to the fact that gravity isn't uniform so different parts of them are being pulled in different directions or with different proper accelerations (see the final animated gif in http://www.einstein-online.info/spotlights/equivalence_principle on the equivalence principle for example). And proper acceleration doesn't require "extremely strong gravitation", you can feel it when accelerating in the complete absence of gravity, or in any deviation from free-fall in a gravitational field

Okay I've just looked it up and it seems that the Rindler horizon is just the point at which light (and therefore anything else) emitted from a greater certain relative velocity can't ever catch up to the accelerating observer.

Don't understand what you mean by "from a greater certain relative velocity", any event beyond the Rindler horizon will be unable to send light to the accelerating observer, the velocity of the object whose worldline the event lies on (i.e. the one emitting the light) doesn't matter. If you're familiar with the basic ideas of how inertial spacetime diagrams work in SR (time on vertical axis, space on horizontal, light worldlines always have a slope of 45 degrees, slower-than-light objects always have a slope closer to vertical than 45 degrees) then you can understand the Rindler horizon by looking at the diagram on this page:

Here the diagonal 45-degree line is the Rindler horizon, the curved line is an observer with constant proper acceleration (constant G-force) whose worldline is a hyperbola that gets arbitrarily close to the diagonal line but never touches it. If you draw a dot anywhere to the left of the diagonal line, and draw a second 45-degree line emanating from that dot (representing light emitted by some event past the Rindler horizon), you can see that this line can never touch the worldline of the accelerating observer either.

Is your point that in the flat space-time example the observer who stays behind will never see the accelerating observer disappear despite the fact that no signal sent will ever reach them

The accelerating observer is the one who "stays behind" in the sense that he always stays on the right side of the Rindler horizon, never crossing it. If you draw a vertical line on the diagram above, representing the worldline of an inertial observer, you can see that this observer will cross the horizon at some point, and thus any event on the inertial observer's worldline that is on or past the Rindler horizon can never send a signal which will reach the accelerating the observer. Also, if you consider points on the inertial observer's worldline just before he crosses the Rindler horizon, and you draw 45 degree light rays from these points, you can see that they will take longer and longer to hit the worldline of the accelerating observer the closer they are to the Rindler horizon. So, the accelerating observer will never actually see the inertial observer cross the Rindler horizon, instead he'll see the image of the inertial observer going more and more slowly as the inertial observer approaches the Rindler horizon, the time dilation going to infinity as the inertial observer gets arbitrarily close to it.

The way to think of the analogy is this:

accelerating observer in flat spacetime, never crossing Rindler horizon <--> hovering observer in black hole spacetime, never crossing event horizon

(both feel a constant G-force, both are unable to see light from events on or past the horizon, both see objects appearing to get more and more time dilated as they approach the horizon)

inertial observer in flat spacetime <--> freefalling observer in black hole spacetime

(both feel zero G-force, both experience crossing the horizon in finite time according to their own clock)

If you're still having trouble understanding this I can take the above image and use a paint program to add the worldline of an inertial observer and light rays emanating from different points on his worldline, if that'd help. If you do understand, then hopefully you agree that the accelerating observer sees the time dilation of the inertial observer go to infinity as the inertial observer approaches the horizon, but that doesn't mean the inertial observer's time dilation is "really" going to infinity in any objective sense, from the perspective of his own clock (or from the perspective of the vertical time axis in the inertial frame that's being used to draw the diagram) it only takes finite time to cross the horizon. In that case, what would make you so sure the same couldn't be true for a black hole? (i.e. in spite of the fact that the hovering observer sees the time dilation of the falling observer go to infinity as the falling observer crosses the horizon, that doesn't mean the falling observer's time dilation is 'really' going to infinity in any objective sense, by his own clock it only takes a finite time to cross the horizon)

This gives me an interesting thought. Acceleration can also be looked at as velocity relative to a non-constant c. Is that in any way meaningful or just complete bollocks? I can't tell any more. Too much having to find the words to define exactly what I'm getting at. Such a pain.

Doesn't really make sense to me either, you'd have to elaborate on what you mean. If you don't know how to define "what you're getting at", why are you always so sure there is a meaningful idea you're getting at? Do you deny the possibility that sometimes you might just have confused intuitions which don't actually have any well-defined meaning or aren't pointing you in the right direction? I think you'd get a lot further in understanding this stuff if you didn't have so much invested in the idea that all these intuitions of yours must be "on the right track" and the problem is just that it's hard to explain them, rather than admitting the possibility that some intuitions which seem to make sense to you on the surface might ultimately turn out to be confused or plain wrong.

 

[A-wal]

That is only because you are restricting your reasoning to Schwarzschild coordinates and Schwarzschild coordinates do not cover the event horizon. There is nothing wrong with common sense, logic, reason, or math, it is just that the coordinate chart ends before you get to the horizon.

Isn't that just another way of saying that space-time never reaches the horizon?

By using a coordinate system that includes the event horizon and even points inside the event horizon.

You can't change reality by changing coordinate systems.

There are only two basic types of "acceleration" I know of, acceleration relative to some coordinate system and proper acceleration which corresponds to the G-forces you feel (and also represents your coordinate acceleration in the local inertial frame where you are instantaneously at rest). Depending on the choice of coordinate system, any observer can have a coordinate acceleration of zero or a nonzero coordinate acceleration. But proper acceleration is more "objective", and a free-falling observer always experiences zero proper acceleration, while a hovering observer has nonzero proper acceleration (I don't know what you mean by 'not talking about the acceleration needed to resist gravity' though).

No idea what you mean by "analogous traveling at a greater relative velocity", or why that analogy would mean they don't feel they're accelerating.

None of that really makes sense to me. Proper acceleration is completely different from tidal force, a point particle can experience proper acceleration while tidal forces are felt by extended objects, you can think of them as being due to the fact that gravity isn't uniform so different parts of them are being pulled in different directions or with different proper accelerations (see the final animated gif in this article on the equivalence principle for example). And proper acceleration doesn't require "extremely strong gravitation", you can feel it when accelerating in the complete absence of gravity, or in any deviation from free-fall in a gravitational field

I'll let JesseM weigh in as well, but I wanted to comment on this because it seems to me to be getting at a fundamental misconception you have. The acceleration of an observer hovering at constant radial coordinate r in Schwarzschild coordinates *is* "the acceleration needed to resist gravity", in the sense you mean. Since the observer is near a gravitating body, he has to accelerate--fire rockets, stand on the surface of a solid planet, etc.--to keep himself from falling towards it. That acceleration is also his "proper acceleration", and he feels it, just as you feel the 1 g acceleration caused by the surface of the Earth pushing up on you and keeping you from falling towards the Earth's center. So no, hovering at constant r in Schwarzschild coordinates is *not* "analogous to traveling at a greater relative velocity". They're fundamentally different things, because one observer (the one hovering at constant r) feels an acceleration, where the other (one just moving at a greater relative velocity, but still moving inertially) does not.

Tidal force is something different, and I don't think we should get into it until we have the other things we're discussing clear.

I know all that! A free-falling observer doesn't feel as though they're accelerating. A hovering observer feels the acceleration they're undergoing to resist being pulled closer. You feel your own weight on Earth is because you're undergoing proper acceleration because you're not moving closer to the centre of the Earth. I thought I'd said enough to show that I'm well aware of everything in that paragraph. Obviously not.

Two observers. One in a stronger gravitational field than the other but maintaining a constant distance between them (accelerating/hovering). The one in the stronger gravitational field is time dilated and length contracted from the perspective of the other one. Not because they're accelerating but just because they're in a stronger gravitational field. Constant acceleration is needed to maintain a constant distance between the two. So the gravitational pull is acceleration in the opposite direction to their proper acceleration in that sense, and if they're free-falling then they're moving into an area of higher gravitation/acceleration so they're accelerating at an ever increasing rate.

And I know what tidal force is. Why should we leave that for now? You can't feel gravitation because it effects the space-time that you're in and you get pulled along with it, but you can feel like you're accelerating when you're in free-fall if the increase in the strength of gravity is sharp enough as to produce greatly differing relative strengths within the same body. The part of you that's closer would literally pull the rest of you along and you'd feel that as acceleration. You feel less tidal force the bigger the black hole because if the gravity is spread over a greater area then it doesn't increase as sharply.

I kind of thought I was telling one, but let me restate it with some more details filled in.

Adam/Eve sees whatever hardly qualifies as a story. Take me on an adventure. Stories also don't tend to have words like hyperbola in them. I don't even know what a bola looks like.

Start with a global inertial frame; we'll use coordinates X, T to refer to the distance and time in that frame. (We'll leave out the other two spatial dimensions because they don't affect anything in this scenario.) In this frame we define the following objects and worldlines:

* A light beam is emitted from the origin of these coordinates in the positive X direction; therefore the worldline of that light beam is the line $X = T$.

* An observer, Eve, is in a rocket ship undergoing a constant proper acceleration (i.e., the acceleration she feels, would measure with an accelerometer, etc.) of $a$. At time T = 0 in the global inertial frame, she is at the X-coordinate $s_0 = 1/a$, and at that instant she is momentarily at rest in the global inertial frame. (All units are "geometric units" where the speed of light is 1.) Thus, Eve's worldline is the hyperbola $X^2 - T^2 = s_0^2$.

* At that same event where Eve is momentarily at rest in the global inertial frame (i.e., the event X = $s_0$, T = 0), another observer, Adam, drops off her rocket ship and follows an inertial (free-falling) path. Thus, Adam's worldline (for T >= 0) is simply the line $X = s_0$.

Now, given the above, we can make the following remarks. Note that all of these remarks are just simple deductions from the above; there's no mystery or complication about any of them.

(1) The light beam, T = X, will *never* catch up to Eve; i.e., it will never cross her worldline. This is what justifies the term "Rindler horizon" in reference to the light beam--Eve will never see it, nor will she ever receive any signals from events on the horizon, or in the region of spacetime on the other side of it from her worldline.

(2) Adam, however, will intersect the light beam T = X at the event (X = $s_0$, T = $s_0$). At that event, the light beam will appear to him to be moving outward (in the positive X-direction), at the speed of light, of course. This is what justifies us saying that the horizon is moving outward at the speed of light as seen by an observer freely falling past it.

(3) At any event (X, T) on Eve's worldline, we can draw a straight line through the origin (X = 0, T = 0) that intersects that event. Call this line L. Line L will have slope

$$\frac{T}{X} = \frac{ \sqrt{ X^{2} - s_{0}^{2} } }{X}$$

This slope gives Eve's velocity at that event, as measured in the global inertial frame. We can do a Lorentz transformation using this velocity to obtain the following:

(3a) Eve will perceive all the events lying on line L to be simultaneous with the event (X, T) where the line crosses her worldline--in other words, she will assign all events on this line a "time" by her clock equal to her proper time at the event (X, T) on her worldline.

(3b) The spatial distance along line L between the origin and the event (X, T) where it crosses Eve's worldline will be $s_0$. This is true for *every* event on Eve's worldline. Because the origin is on the horizon (the line X = T), this justifies us saying that, from Eve's point of view, the horizon is always a constant distance $s_0$ below her.

If you have any questions at all about the above, we should get them resolved before proceeding any further.

Don't understand what you mean by "from a greater certain relative velocity", any event beyond the Rindler horizon will be unable to send light to the accelerating observer, the velocity of the object whose worldline the event lies on (i.e. the one emitting the light) doesn't matter. If you're familiar with the basic ideas of how inertial spacetime diagrams work in SR (time on vertical axis, space on horizontal, light worldlines always have a slope of 45 degrees, slower-than-light objects always have a slope closer to vertical than 45 degrees) then you can understand the Rindler horizon by looking at the diagram on this page:

Here the diagonal 45-degree line is the Rindler horizon, the curved line is an observer with constant proper acceleration (constant G-force) whose worldline is a hyperbola that gets arbitrarily close to the diagonal line but never touches it. If you draw a dot anywhere to the left of the diagonal line, and draw a second 45-degree line emanating from that dot (representing light emitted by some event past the Rindler horizon), you can see that this line can never touch the worldline of the accelerating observer either.

The accelerating observer is the one who "stays behind" in the sense that he always stays on the right side of the Rindler horizon, never crossing it. If you draw a vertical line on the diagram above, representing the worldline of an inertial observer, you can see that this observer will cross the horizon at some point, and thus any event on the inertial observer's worldline that is on or past the Rindler horizon can never send a signal which will reach the accelerating the observer. Also, if you consider points on the inertial observer's worldline just before he crosses the Rindler horizon, and you draw 45 degree light rays from these points, you can see that they will take longer and longer to hit the worldline of the accelerating observer the closer they are to the Rindler horizon. So, the accelerating observer will never actually see the inertial observer cross the Rindler horizon, instead he'll see the image of the inertial observer going more and more slowly as the inertial observer approaches the Rindler horizon, the time dilation going to infinity as the inertial observer gets arbitrarily close to it.

The way to think of the analogy is this:

accelerating observer in flat spacetime, never crossing Rindler horizon <--> hovering observer in black hole spacetime, never crossing event horizon

(both feel a constant G-force, both are unable to see light from events on or past the horizon, both see objects appearing to get more and more time dilated as they approach the horizon)

inertial observer in flat spacetime <--> freefalling observer in black hole spacetime

(both feel zero G-force, both experience crossing the horizon in finite time according to their own clock)

If you're still having trouble understanding this I can take the above image and use a paint program to add the worldline of an inertial observer and light rays emanating from different points on his worldline, if that'd help. If you do understand, then hopefully you agree that the accelerating observer sees the time dilation of the inertial observer go to infinity as the inertial observer approaches the horizon, but that doesn't mean the inertial observer's time dilation is "really" going to infinity in any objective sense, from the perspective of his own clock (or from the perspective of the vertical time axis in the inertial frame that's being used to draw the diagram) it only takes finite time to cross the horizon. In that case, what would make you so sure the same couldn't be true for a black hole? (i.e. in spite of the fact that the hovering observer sees the time dilation of the falling observer go to infinity as the falling observer crosses the horizon, that doesn't mean the falling observer's time dilation is 'really' going to infinity in any objective sense, by his own clock it only takes a finite time to cross the horizon)

I meant that the speed of light isn’t constant to an accelerating observer. You could measure your acceleration as if it were a constant velocity relative to light. The Rindler horizon would happen when this velocity exceeds c.

If this is just a comment, I agree it's amusing. But if you actually mean it to be an argument against the claim I'm making, then we need to go into more detail about what curvature means. The statement I made, that "the 2-sphere can have that area without having physical radius r", is no more mysterious or illogical than the statement, "a triangle drawn on the surface of the Earth can have three right angles, but still be a triangle"--i.e., it can be a triangle without the sum of the angles being 180 degrees. Would you agree with the latter statement?

It was just a comment.

Doesn't really make sense to me either, you'd have to elaborate on what you mean. If you don't know how to define "what you're getting at", why are you always so sure there is a meaningful idea you're getting at? Do you deny the possibility that sometimes you might just have confused intuitions which don't actually have any well-defined meaning or aren't pointing you in the right direction? I think you'd get a lot further in understanding this stuff if you didn't have so much invested in the idea that all these intuitions of yours must be "on the right track" and the problem is just that it's hard to explain them, rather than admitting the possibility that some intuitions which seem to make sense to you on the surface might ultimately turn out to be confused or plain wrong.

You think I’m the one who needs to be more open-minded and willing to accept the possibility that I might be wrong? OMFG!


I don’t understand why an object that crosses a Rindler in flat space-time is analogous to the event horizon of a black hole. The Rindler horizon would be equivalent to a free-falling observer reaching the point at which no light from a more distant object will reach it before the black hole dies, unless the free-falling observer accelerates. The event horizon on the other hand marks the point at which nothing, not even light can reach before the black hole dies. Even if I'm wrong and it marks the point at which nothing, not even light can escape, then I still don't see how they're the same thing?

 

[Dale]

Isn't that just another way of saying that space-time never reaches the horizon?

No, that is another way of saying that the coordinate chart does not cover the entire manifold. Spacetime is the manifold (which includes the exterior, the event horizon, and the interior), not the coordinate chart (which covers only the exterior).

You can't change reality by changing coordinate systems.

Therefore if your statement depends on the choice of coordinate system it does not reflect reality. All of your statements about something taking forever to reach the event horizon depend on the choice of Schwarzschild coordinates and therefore do not reflect reality by your own logic.

 

[PeterDonis]

I know all that! A free-falling observer doesn't feel as though they're accelerating. A hovering observer feels the acceleration they're undergoing to resist being pulled closer. You feel your own weight on Earth is because you're undergoing proper acceleration because you're not moving closer to the centre of the Earth. I thought I'd said enough to show that I'm well aware of everything in that paragraph. Obviously not.

You say some things that make it seem like you are aware of it, but then you say other things that make it seem like you're not. For example:

Two observers. One in a stronger gravitational field than the other but maintaining a constant distance between them (accelerating/hovering). The one in the stronger gravitational field is time dilated and length contracted from the perspective of the other one. Not because they're accelerating but just because they're in a stronger gravitational field. Constant acceleration is needed to maintain a constant distance between the two. So the gravitational pull is acceleration in the opposite direction to their proper acceleration in that sense, and if they're free-falling then they're moving into an area of higher gravitation/acceleration so they're accelerating at an ever increasing rate.

If they're free-falling, then *they're not accelerating*, because they don't *feel* any acceleration. It doesn't matter that, relative to a hovering observer (or an observer at rest on, say, the surface of the Earth), the free-falling observer *looks* like he's accelerating; what matters is which observer *feels* acceleration, because that's the real physical effect, which is invariant (i.e., it doesn't depend on what coordinates we use--which you have said is the definition of "real" you're using).

And I know what tidal force is. Why should we leave that for now? You can't feel gravitation because it effects the space-time that you're in and you get pulled along with it, but you can feel like you're accelerating when you're in free-fall if the increase in the strength of gravity is sharp enough as to produce greatly differing relative strengths within the same body. The part of you that's closer would literally pull the rest of you along and you'd feel that as acceleration. You feel less tidal force the bigger the black hole because if the gravity is spread over a greater area then it doesn't increase as sharply.

No--once again, *if you're in free fall, you don't feel like you're accelerating*. Tidal "accelerations" apply to objects in free fall, and those objects do *not* feel any acceleration. To get into in what sense the term "tidal acceleration" *is* justified, relative to certain particular coordinate systems, is a whole different subject, which is why I said I wanted to table it for now.

Adam/Eve sees whatever hardly qualifies as a story. Take me on an adventure. Stories also don't tend to have words like hyperbola in them. I don't even know what a bola looks like.

If what I said about Adam and Eve doesn't qualify as a "story", then I'm not sure what you want, or if what you want will qualify, from my point of view at least, as a physical "explanation" of what's going on. Explanations are not just stories.

It was just a comment.

Sorry if I confused you, but as I noted above, you have made a number of statements which don't make sense to me when taken together (I've given some examples above), so I'm trying to understand the mental model you have of what's going on.

I don’t understand why an object that crosses a Rindler in flat space-time is analogous to the event horizon of a black hole. The Rindler horizon would be equivalent to a free-falling observer reaching the point at which no light from a more distant object will reach it before the black hole dies, unless the free-falling observer accelerates. The event horizon on the other hand marks the point at which nothing, not even light can reach before the black hole dies. Even if I'm wrong and it marks the point at which nothing, not even light can escape, then I still don't see how they're the same thing?

What makes you think that "nothing, not even light can reach" the event horizon? Again, I don't understand the mental model that's leading you to that conclusion.

 

[A-wal]

No, that is another way of saying that the coordinate chart does not cover the entire manifold. Spacetime is the manifold (which includes the exterior, the event horizon, and the interior), not the coordinate chart (which covers only the exterior).

An area that light can never reach from the outside is an area that nothing can reach.

Therefore if your statement depends on the choice of coordinate system it does not reflect reality. All of your statements about something taking forever to reach the event horizon depend on the choice of Schwarzschild coordinates and therefore do not reflect reality by your own logic.

No, I said the choice of coordinate systems can't change reality. So by my logic they can't both be right. You can either reach the horizon or you can't, and I don't see how anything possibly could.

If they're free-falling, then *they're not accelerating*, because they don't *feel* any acceleration. It doesn't matter that, relative to a hovering observer (or an observer at rest on, say, the surface of the Earth), the free-falling observer *looks* like he's accelerating; what matters is which observer *feels* acceleration, because that's the real physical effect, which is invariant (i.e., it doesn't depend on what coordinates we use--which you have said is the definition of "real" you're using).

I know! I didn't say proper acceleration.

No--once again, *if you're in free fall, you don't feel like you're accelerating*. Tidal "accelerations" apply to objects in free fall, and those objects do *not* feel any acceleration. To get into in what sense the term "tidal acceleration" *is* justified, relative to certain particular coordinate systems, is a whole different subject, which is why I said I wanted to table it for now.

I KNOW you don't feel like you're accelerating when in free-fall! That's at least the third time I've had to say that. But in free-fall you do feel the difference of gravity between the closer parts and the further parts of your body to the centre of gravity. It's normally marginal but in the case of a black hole the gravity increases so sharply that it becomes noticeable, then painful, then very painful, then death. This seems like a pretty good description of tidal force. Are you saying it's something else entirely?

If what I said about Adam and Eve doesn't qualify as a "story", then I'm not sure what you want, or if what you want will qualify, from my point of view at least, as a physical "explanation" of what's going on. Explanations are not just stories.

Never mind.

Sorry if I confused you, but as I noted above, you have made a number of statements which don't make sense to me when taken together (I've given some examples above), so I'm trying to understand the mental model you have of what's going on.

Welcome to my world.

What makes you think that "nothing, not even light can reach" the event horizon? Again, I don't understand the mental model that's leading you to that conclusion.

How about a game of let's pretend? Let's pretend that general relativity has just been released, by me, and it's my version where nothing can reach the event horizon. Presumably you'd know straight away that there's an error. Now the onus is on every one else to explain to me how an object can reach the event horizon. See how you like it.