md65536

There's no paradox. You're talking about a finite spacetime that seems to be flat. Or basically an object with a proper length of over ten billion light years. You're calling that "the universe", which is fine as long as you avoid attaching meaning to that label, like that everything else you specify must be within that. Not everything in the (flat, toy) universe gets length contracted. Only the stuff that is moving relative to you does. The pilot isn't moving relative to the ship. If the ship is part of the universe, the universe won't be contracted to 1m, only the stuff moving relative to it. If the ship's not part of the universe, the universe can be like a 1m-thick wall traveling past the ship at near c. If you want to talk about the ship being at rest inside the flat universe, and then accelerating "instantly" to near c, then simultaneity is important if the universe is not static. It sounds like the ship is implicitly Born rigid, and clocks on parts of it would become out of sync with each other (by billions of years??). I think you would see the far edge of the approaching 'wall' appearing to age over twenty billion years in the nano seconds it takes to pass you, due to the relativity of simultaneity and relativistic Doppler shift. Actually, that idea's more complicated than I thought. Say the ship starts in the middle of a toy universe, and instantly accelerates to near c. Ignoring simultaneity, you might conclude that the universe contracts to a wall in the middle of the ship, with the ship sticking out both ends. But that's impossible because the back of the ship never travels backward. No part of the ship ever enters the "back" half of the universe. But with relativity of simultaneity, the different parts of the (Born rigid) ship travel through the front half of universe at different times... eek is that right? I think a pilot in the middle of the ship could consistently conclude, "I'm in the middle of the "universe" which is 1m wide and is smaller than my ship would be at rest (which it currently is not, it doesn't share a single inertial frame), but the back of my ship has already passed through the front edge of the universe has not yet reached the same speed as me and the universe is not yet contracted for them."??? That's confusing, I doubt I got it right. However, relativity of simultaneity does resolve this part of the paradox if you do it right. It sounds like you're referring to the spacetime as 'the universe' and others are referring to all the moving stuff in it as the universe? If all the stuff was moving, the pilot would measure it as length-contracted.

These two statements aren't consistent with each other. If the bike rider accelerates, ie. changes inertial frame, there is a shift in relative simultaneity with respect to the distant source clock. If the rider accelerates towards it, the coordinate time of the distant clock can change from 12:34 to 13:44 in a small local time, as in your example. But if the bike rider then accelerates away and returns to its former reference frame, the coordinate time can change from 13:44 to 12:34, or even earlier if it accelerates away more. If the first change in relative simultaneity is "the source clock runs very fast", how is the second not running backwards? It's because of confusion like this that I don't like the phrasing that a change in relative simultaneity means a clock is running fast, because it also implies a clock can run backwards. They're both just changes in relative simultaneity, but one is accepted as intuitively reasonable but the other isn't, meaning it isn't an intuitive way to describe it.

Are you okay with saying that the source clock runs backward if/while the very distant rider accelerates away from it? If that's not okay then what you wrote could be phrased differently to avoid it (eg. relative time instead of how the clock itself runs).

Yes, you're correct that it's about distance. The effect still occurs at low speed where time dilation is negligible. Google "Andromeda paradox", the effect can happen at walking speeds if the distance is large enough. The reason direction matters is that if the distant location is far enough, the travel time of light is long enough that you can move a significant distance even at slow speeds. If a source is millions of light years away, you can walk on the order of light days between "now" and when light arrives. Time dilation still applies but with vanishing speed it approaches zero. At walking speeds it might contribute seconds where light travel time contributes days. Assuming a time dilation factor rounded to 1, suppose two people separated by two light days walked toward each other, and meet after a million years, at which point they both receive a signal from a source that is "now" 1 million light years away. The one walking away from it says "the source is moving away from me and this light took one million years minus a day to reach me; the signal was sent after we started walking" and the other says "this same light took one million years plus a day; it was sent before we started walking" and they disagree on what is "now" at the source just like they did when they started walking. That is pretty much their "experience" of the phenomena. The virginia.edu link mentions a causal definition of past/future: When you're talking about "switching between past and future", you're talking about events that are in each other's elsewhere... it's switching from one part of the elsewhere to another. "Now" far away is neither in the causal past nor future of "now" here. The two observers moving in different directions only "experience" such a switch after a million years in this example, involving measurements made when they're far apart from each other. Edit: That last part's misleading, the observers don't have to be separated. Rather... direction matters because the distant object is moving in different directions relative to the two observers, so the observers disagree on how far the same light signal from it has travelled.

Actually, I think you can prove to yourself that a moving mirror must be able to change the angle of reflection. Try this: Have a box with a mirror on one interior surface, and two holes such that if you shone a beam of light through one, you could reflect it off the mirror and out the other hole. Consider that the stationary frame. Now in a moving frame (in a direction of the mirror's normal), the box is moving while the light makes its way from one hole, to the mirror, and out the other hole. If you draw this on paper using 3 positions for the box for when light enters, reflects, and exits, you'll see that the angle must be different than in the stationary frame. You can try this with or without length contraction (you'd have other contradictions), the angle of reflection is not going to be the same in all frames.

I did a bit of "common sense" analysis, just enough to resolve any apparent paradox in my mind. The path of a photon in the moving R2 frame would look something like this: ___ ___ ___ -> \_/ \_/ \_/ The angled legs are symmetric, and would have a length (or light time) of gamma times their length in the stationary frame. Due to aberration of light, the different mirrors would appear skewed in the R2 frame, in different ways depending on where the observer is. Do you happen to know, if there was a stationary mirror in the R2 frame, angled so that it matched the appearance of a moving skewed mirror, would light take the same path if it hit that stationary mirror as it would hitting the moving mirror? It seems it would but I'm not sure (a problem with using only common sense). That would mean that the light always appears to reflect at the "correct" angle with nothing visually paradoxical. Edit: It seems it wouldn't. Differently positioned observers sharing the R2 frame would see a moving mirror appearing skewed differently, but see the path of light being the same. The angle of reflection would change somewhat like a ping pong ball hit by a moving paddle. Is it now off-topic to discuss the relativity-related aspects of the topic? We should focus on the speculation? Should we start a new topic in Relativity if we don't want the thread killed when OP says the wrong thing, or is discussion of relativity just generally discouraged? (And would a new thread be killed if new OP says the wrong thing, eventually leading to the ideal condition where only people who already understand relativity can start a topic, but none start topics, and the Relativity forum is perfectly devoid of any active topics?)

You have two frames whose origins coincide at time 0. It looks like frame B is moving relative to A at velocity v. The equation x_A = v t_A describes a particle that is fixed to frame B's origin. Alice describes the moving particle to Bob... but it doesn't matter to Bob what gamma is because the particle is at Bob's origin. Eg. whether the length contraction factor is 2 or 3, if the proper length is zero, the relativistic length is still going to be zero. It could be that the equations are used here to find the equation describing B's origin relative to A. However that seems weird to me, because that form of the Lorentz transformation is based on the specification of how the origins relate, so it's like solving for something already known???

Just to add to that, it's not like in SR where each observer also has a different notion of simultaneity, but each of those is physically meaningful. Eg. in flat spacetime, any two events that can be considered simultaneous by someone will have intersecting future light cones, where different future observers can agree or disagree on whether the events were simultaneous. In GR you must make a choice of how to define the surfaces of a foliation, that's not just based on a physically meaningful connection between its events. You'd choose it to make a useful tool, not a 'real' representation of simultaneity throughout the universe for a given observer.

A foliation is a slicing-up of all of 4D spacetime into such 3D hypersurfaces. Since you can slice it up in infinitely different ways, that implies there's no 'universal now'; such a thing would be arbitrarily chosen. It's a mathematical thing. A surface of a typical foliation corresponds to an instant in time on a local scale, but just because it's mathematically possible to slice up spacetime doesn't mean that an entire surface meaningfully represents a moment in time. As long as spacetime obeys some reasonable rules, it's possible to foliate it... eg. the surfaces can't intersect. But if spacetime *needed* intersecting surfaces, I think that would imply some really weird physical consequences? I don't know the other mathematical rules, just adding 2 cents. My understanding is that if you have causally disconnected regions of spacetime, you can foliate it however you want because you'll never get things out of order. Like, if you took two different books and pushed them together so their pages interleaved randomly, and then glued them together, you're not going to have any pages out of order no matter how you put them together. But by analogy, the relative order of pages in different books is generally meaningless, as with foliations of all of spacetime. I'm not sure, but a foliation might require that a spacetime is connected. In the case of a black hole, would that require that spacetime is multiply connected? Which is not prohibited by GR. Or can you just take partial foliations using the world lines of multiple observers (like a distant observer and an infalling one) and combine them into one like gluing books? However, this isn't an issue in this thread. I think OP's example can be completely described using a distant observer's coordinate time, and only events outside of the black hole's event horizon. I may have earlier misunderstood that the example was relating interior and exterior events.

rjbeery's example didn't require the two observers (infalling A and distant B) to agree on simultaneity. Though, B and C (observer at location of black hole in B's coordinates, after it has evaporated) agree. It was brought up to illustrate the idea of A existing "forever" in B's coordinates, never passing the Schwarzschild BH event horizon. I agree the topic really has nothing to do with simultaneity, and that's why I'm commenting on it. You've said Schwarzschild BHs don't evaporate, and it's not possible to determine simultaneity across extended regions of spacetime. If someone's not following the details of the thread, they might think those are equally problematic and that "GR says the example's not possible." But, not being able to unambiguously define simultaneity resolves nothing of rjbeery's paradox, while Schwarzschild BHs not evaporating completely destroys it. If anyone else is struggling to see and understand the resolution of the "paradox", simultaneity's not a problem, but evaporation is. (Because, rjbeery has both the event horizon disappearing, and existing forever for the infalling object to be caught above it, both in the distant observer's coordinates.)

Sure, but rjbeery is talking about multiple clocks (infalling A, far away B, etc). If the different observers remain able to communicate (A before reaching the event horizon), they must be able to relate their times to each other. Does "time is purely local" mean that GR just doesn't say anything about how the clocks relate, and doesn't depend on them relating? In reality, even using GR to model spacetime, observers still can relate their clocks in ways that GR doesn't care about. But you can do that, even if not in all cases. If "Pick a method of determining simultaneity" is understood to mean that you're making a choice of what you mean by simultaneity and how you define it, then for example a clock hovering above a black hole, at rest relative to a distant clock, can use "radar time" to define simultaneity of events at the two clocks' locations. In this example, they can agree on simultaneity. Eg. if the hovering clock is gravitationally time dilated so that its clock is ticking at half the rate of distant clock, the clocks can be set so that every tick of the hovering clock happens "at the same time" (by their choice of simultaneity definition) as every second tick of the distant clock, and both observers can agree, and the choice of simultaneity can remain consistent and useful indefinitely. In the case described (A is infalling), each observer can have their own notion of simultaneity, but they won't agree with each other. I don't see how this is a problem in this thread, it's not like any claimed physical effects are based on simultaneity??

That paragraph is talking about the apparent horizon, and you're talking about the event horizon. See https://en.wikipedia.org/wiki/Apparent_horizon : I don't know enough to see any problems here. We *need* to make assumptions to model the interior of a black hole (inside the event horizon) because we can't make any observations to test our models. But that's exactly why there are no real problems; if you say some model or assumption "logically" implies some physical phenomena or paradox, but it has no observable consequences, how can you claim it's a real problem? It's like the movie Interstellar, which made up a paradoxical imagination of the interior of a black hole. Yet Kip Thorne says something like that the movie doesn't break any scientific laws, but that's because there are no laws that say what is observed inside a black hole. The event horizon and evaporation are things that have physical significance outside of the black hole, including effects that can be observed. The apparent horizon can't be observed from outside. Oh, I see a little clearer the problem that you're describing. But it's easily resolved. As Markus pointed out, a Schwarzschild BH doesn't evaporate. An infalling object A gets stuck on the event horizon "forever" (in B's coordinates), but the event horizon continues to exist forever. If on the other hand the BH evaporates in finite time, the event horizon no longer exists when the BH has evaporated away. Observer B doesn't have events occurring at the event horizon at times when the event horizon doesn't exist. Either A falls in and the black hole evaporates and A's world line ends with a finite coordinate time (in B's coords), or the black hole doesn't evaporate and the event horizon lasts forever with A on it (in B's coords), but not both. If you describe A and B in terms of causal connections, or events involving the other that they can observe, they're going to agree, no matter what realistic thing you have them do. In terms of simultaneity alone, they don't need to agree, and there's nothing paradoxical about that. But yes, object A can't both be trapped on a static event horizon forever, and let the event horizon evaporate in finite time. The event horizon can't be both static and non-static in a given reference frame.

This is what I got from your posts: 1) A black hole and all the events in its interior can be described in the coordinates of an observer at infinity. 2) A Penrose diagram of an evaporating black hole shows that the formation and disappearance of a black hole have the same time coordinate. 3) If an event A has a coordinate time that is less than the coordinate time of an event B, then A happened before B (maybe even in B's past?). Problems with this: (1) The interior events do not have meaningful time coordinates for this observer. (2) If that's what the diagram really shows, then the coordinates used in that diagram can't be the same as for the observer in (1). (3) You're comparing coordinate times of events that have no causal connections, and their ordering is irrelevant, but you see "logic problems" by treating it as something physical.

I'm not seeing any problem, except maybe mixing of different time coordinates. In my understanding, the point of having the observer at infinity is that it is "shielded" from any effects of spacetime curvature. In its coordinates, you could say eg. the black hole formed very far away and at coordinate time (ie. observer's local time) t=0, remained at rest, had a lifetime of 100 units of coordinate time, and finished evaporating at t=100, then sometime later at t>100 some other event happened at the location of the black hole. The same could be said if instead of a black hole, you're talking about a snowball with negligible mass. Neither has any effect on the coordinate time of the observer at infinity. There is nothing contradictory in the coordinate times of this observer on its own. So clearly we're comparing the times of different observers???, but it's not clear to me what other times you're speaking about here. Also, it should be possible to choose foliations of spacetime such that any events in the interior of the black hole are assigned meaningless coordinate times anywhere in [0, 100]. However, they would have no physical significance to the observer, and there'd be no way to break causality or create a contradiction through your choice.

Earlier you wrote: Can you remind us of the situation you're describing (I've lost track)? Whose coordinate time are you talking about here? It sounds elsewhere like you're talking about a Schwarzschild black hole, at rest (and evaporating) in the coordinates of an observer at infinity. However it also sounds like the Penrose diagram doesn't show those time coordinates, and the statement above doesn't match those coordinates either??? When you speak of time, if you could mention in each case whose coordinates you're referring to, that might make it clearer.

You're speaking of a general case, but if you compared two observers in flat spacetime (eg. at infinity, or in the location of a single event), wouldn't it have to be the case? I didn't see it specified what observers are being compared (eg. infalling vs. one at infinity), but if you're given the choice of which observers to compare, it should be possible? I think this is related to https://en.wikipedia.org/wiki/Black_hole_complementarity I don't think the issue is settled in accepted science. However, seeing the infalling object reach the event horizon doesn't make sense (unless the BH has zero size). Maybe the object evaporates just like the BH does. Maybe the object simply fades out of existence as it's infinitely red-shifted to nothing.

Doesn't the null surface correspond with the event horizon, which has the same physical significance (re. light-like intervals, causality, etc.) regardless of coordinates? Whether or not the event horizon at the Schwarzschild radius is a singularity depends on choice of coordinates, but that doesn't determine its existence or behaviour. It's still an event horizon.

The null surface is the event horizon? Isn't it located at the same place in different coordinates, just with different numerical representations in the different coordinate systems? What are you asking is invariant? There are aspects of spacelike and timelike things that are relative, and other aspects that are invariant, right? Mordred, can you please answer my previous question so that I know I'm not just wasting my time here? Is rapidity some kind of acceleration?

Exploring the causal connections with black holes seems related enough to the topic. I'd say that causal connections are invariant. I think light cones are invariant. A light signal from one event either reaches (causes) another event, or it doesn't. That doesn't change depending on observer. I don't know what you mean by "experience" but of course different observers will see things appear differently, observe different parts of spacetime etc. Obviously if a particle can survive (as expected) falling into a BH, there can be causal connections from outside to in. As far as I know, it is not known exactly what an observer inside the black hole would observe, as that requires some speculative extrapolation of testable physics. It is only causal connections from inside to out that are prohibited.

I don't think that's correct. It did exist, and it evaporated in the observer's past. It's only the interior events that aren't causally connected to the observer in the moment described. How is this relevant to the thread? Your link confirms that rapidity is not acceleration. Quote: "rapidity η (v) ≡ tanh^−1 (v)". Constant rapidity implies constant velocity. Do you understand that rapidity is not acceleration? See https://en.wikipedia.org/wiki/Rapidity : If you still think rapidity is acceleration or requires acceleration or whatever, can you please provide a definition that you're using?

Sure, if you're using hyperbolic functions with the Lorentz transformation. Not relevant here. It isn't. Have you actually read your link? You should. None of these terms are relevant to this thread. They're worse than irrelevant when used incorrectly.

??? From https://en.wikipedia.org/wiki/Lorentz_transformation : "Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called boosts, and the relative velocity between the frames is the parameter of the transformation." Do you think a boost is an acceleration? Can you please provide references to the definitions you're using (including rapidity)? I don't trust what you're saying because it's not making any sense to me with respect to anything else I read.

You keep referring to rapidity as acceleration. The only definitions I've seen are that it's a measure of relativistic velocity. What definition are you using? Your statement makes as little sense as saying an infalling observer is undergoing velocity.

Are you saying that you get different light cones with different coordinate choices? But if you have light from one event reaching another event on its future light cone, that doesn't change depending on coordinates, so I don't know what you mean. This makes no sense to me. Isn't it the events that have light cones, not observers? For example, if you have an event where an infalling observer contacts a hovering observer as it passes, that event has a light cone. Are you saying it has different light cones for the different observers?

Is it fair to say your argument is basically, "if a black hole's entire existence is within an event's past light cone, then any interior events of the black hole are also within that event's past light cone"? I think the argument is false. The interior events are not in your past light cone. For one thing, a light cone is based on the paths of light from one event to another, and a black hole doesn't have such paths from interior events to exterior. I think what you're doing is using a mathematical definition of a light cone in "Your" flat spacetime. Then with a black hole placed in that past light cone, you're effectively assigning flat spacetime coordinates to events within the black hole, like you might do if the black hole wasn't there at all and the spacetime remained flat. You're effectively giving physically meaningless flat-spacetime coordinates to events in a curved spacetime, which I think is okay, but then you're drawing conclusions about those events based on physics that applies in flat spacetimes, which is not correct. In the curved spacetime, the black hole's interior events are geometrically outside of your past light cone, I think... or, I have no idea. Maybe another way to put it is that black holes tilt light cones, and you're not accounting for that. As an amateur, I think I'm missing the maths and vocab that would make this clear and precise. edit: Thinking more about tilted light cones... Say the event of the BH evaporating is in your past light cone. You can say there's a causal connection between you and it because its future light cone intersects your past light cone. But for an event within the BH's horizon, the event's future light cone is tilted more than 45 degrees such that its future light cone does not intersect your past light cone, and there is no causal connection. I suspect there's more to it than that.