md65536 337 Posted June 28 13 hours ago, Markus Hanke said: No, we have to take this locally. It’s Schwarzschild coordinate time, so this is what a far-away stationary clock measures locally in its own frame of reference. It is not what physically happens anywhere else. In GR, time is always a purely local concept. 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. 13 hours ago, Markus Hanke said: Again, this is not possible. Time is a purely local concept in GR; there is, in general, no notion of simultaneity across extended regions of curved spacetime, and you can’t map notions of space and time local to some far-away observer into anything that happens anywhere else. In particular not to test particles in free fall, which aren’t stationary. You can define static hypersurfaces of simultaneity based on the coordinate system you have chosen (in Schwarzschild, these will be nested spheres), but that is not the same thing. 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?? 0 Share this post Link to post Share on other sites

Markus Hanke 292 Posted June 29 (edited) 18 hours ago, Halc said: It seems that there is no coordinate system that foliates all of spacetime. This seems to be an interesting argument against any philosophy of time that posits an absolute coordinate system (a preferred frame of one sort or another). Presentism is only a subset of these philosophies. The inability to identify any coordinate system that can consistently map any pair of events as to which occurs first seems to me to be a fatal flaw in such a philosophy. Well, I am not an expert in philosophy, so I can’t really comment on that. So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold. However, there will always be infinitely many possible foliation schemes, so there is never any preferred notion of time. This is in keeping with the principle of relativity, of course. 15 hours ago, md65536 said: 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. The relationship between these clocks is simply the coordinate transformation that relates the metrics. For example, the stationary far-away observer can use the Schwarzschild metric, whereas the observer in free fall uses the Gullstrand-Painleve metric. These are simply related by a coordinate transformation, since both frames are of course in the same physical spacetime. But they use different notions of what ‘time’ means, so defining a notion of simultaneity is not in general possible if the observers are separated in time and/or space, unless there are certain very specific symmetries present. At best, it might be possible to foliate spacetime in a manner that both observers can agree upon, using a suitable coordinate system and foliation parameter; but this works only in certain highly symmetric cases, and the foliation parameter is not something that any physical clock would actually show in either of the two frames, so I don’t see how it is helpful here. 15 hours ago, md65536 said: 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 You may be able to do this in certain special cases that are highly symmetric, which is why I put the qualifier “in general” as part of my original comment. Flat Minkowski spacetime is a trivial example. I don’t think it is possible in Vaidya spacetime though, which is what we would be talking about when it comes to evaporating black holes. Crucially, I don’t think it is helpful to even consider the concept of simultaneity in curved spacetimes, since it is not a generally applicable concept - in my experience, it is bound to lead to more confusion than clarity. 15 hours ago, md65536 said: 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?? I agree, it has little to do with topic of the thread, so I’m not sure why it was brought up at all. Edited June 29 by Markus Hanke 0 Share this post Link to post Share on other sites

md65536 337 Posted June 29 6 hours ago, Markus Hanke said: At best, it might be possible to foliate spacetime in a manner that both observers can agree upon, using a suitable coordinate system and foliation parameter; but this works only in certain highly symmetric cases, and the foliation parameter is not something that any physical clock would actually show in either of the two frames, so I don’t see how it is helpful here. 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.) 0 Share this post Link to post Share on other sites

Halc 5 Posted June 30 16 hours ago, Markus Hanke said: So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold. However, there will always be infinitely many possible foliation schemes, so there is never any preferred notion of time. This is in keeping with the principle of relativity, of course. I'm talking about interpretations that deny the principle of relativity. There seem not to be infinitely many possible foliation schemes. As a matter of fact there doesn't seem to be any that foliate all of spacetime. The typical one suggested is the curved (non inertial) comoving frame corresponding locally to the inertial frame in which the CMB appears isotropic, but any frame like that does not properly foliate local deviations from flat space like black holes. If they did, then rjbeery would have grounds to stand on when trying to objectively determine if event X inside a black hole occurs before or after event Y somewhere outside it, particularly after the BH has evaporated. An objective foliation scheme should not in any way depend on an observer. Any two observers, no matter how separated and unable to communicate, should be able to sync their clocks simply by setting said clock to the current objective time, and then I suppose having the clock running at some rate which depends on the speed of the clock and its current gravitational potential. The latter requires a standard 'zero', which also seems undefined. For example, what is the gravitational potential at the surface of Earth? Nobody publishes that. They only publish the potential if Earth was in an otherwise empty universe, which it obviously isn't. Anyway, point is, there is no viable objective foliation scheme that includes all spacetime events. The lack of a viable scheme means that time and motion cannot be objective. The principle of relativity cannot be denied. Correct me if I'm wrong. 0 Share this post Link to post Share on other sites

MigL 1154 Posted June 30 17 hours ago, Markus Hanke said: So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold. Can you please explain your thoughts on this matter. I tend to agree with Halc, that a foliation ( Cauchy surface ) is a surface in space-time which is like an 'instant in time'. As such, a 'global' ( entire manifold ) foliation is non-sensical, as it implies a universal now. 0 Share this post Link to post Share on other sites

md65536 337 Posted Tuesday at 09:03 AM 4 hours ago, MigL said: Can you please explain your thoughts on this matter. I tend to agree with Halc, that a foliation ( Cauchy surface ) is a surface in space-time which is like an 'instant in time'. As such, a 'global' ( entire manifold ) foliation is non-sensical, as it implies a universal now. 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. 0 Share this post Link to post Share on other sites

Markus Hanke 292 Posted Tuesday at 02:34 PM (edited) 20 hours ago, md65536 said: 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.) Yes, well put. 20 hours ago, md65536 said: It was brought up to illustrate the idea of A existing "forever" in B's coordinates, never passing the Schwarzschild BH event horizon. Ok, I see. In Vaidya spacetime this issue never arises, since (unlike with Schwarzschild) this coordinate time for a far-away observer remains finite. 11 hours ago, Halc said: Correct me if I'm wrong. No, you are correct. What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer. There are infinity many possible observers, and each one of them will use a different foliation scheme; hence the foliation is never objective and shared by everyone, it is always observer-dependent, even if it spans the entire spacetime. There is no such thing as universal time, of course. BTW, slicing up 4D spacetime into an ordered sequence of 3D hypersurfaces is called the ADM formalism of GR. Edited Tuesday at 02:36 PM by Markus Hanke 0 Share this post Link to post Share on other sites

md65536 337 Posted Tuesday at 05:06 PM 2 hours ago, Markus Hanke said: No, you are correct. What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer. There are infinity many possible observers, and each one of them will use a different foliation scheme; hence the foliation is never objective and shared by everyone, it is always observer-dependent, even if it spans the entire spacetime. There is no such thing as universal time, of course. 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. 1 Share this post Link to post Share on other sites

Halc 5 Posted Tuesday at 05:06 PM 2 hours ago, Markus Hanke said: What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer. ... There is no such thing as universal time, of course. I am talking about universal time, and while I agree that there's no such thing, there seems to be no contradiction arising by postulating it. I'm proposing such a contradiction here. If the choice is relative to a particular observer, it's hardly objective. Another clock cannot be set to the universal time without agreeing on this privileged observer or privileged location in space. Points in space 50 billion light years away do not exist at all relative to a given observer, so his personal choice of coordinates do no in fact foliate all of spacetime. There is a choice that does foliate all points in an arbitrarily large scale, but as pointed out in this topic, it doesn't work for excessive local curvature such as black holes. There seems in fact to be no possible coordinate system that does, and that's the contradiction that arises from postulating universal (objective/absolute) time/space. 0 Share this post Link to post Share on other sites

Markus Hanke 292 Posted Wednesday at 05:05 AM 11 hours ago, md65536 said: 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. Very well put +1 12 hours ago, Halc said: There seems in fact to be no possible coordinate system that does, and that's the contradiction that arises from postulating universal (objective/absolute) time/space. True, if you have large local variations in curvature (as would be the case in the actual universe), such a global foliation will not in general be possible. 0 Share this post Link to post Share on other sites