Notions of simultaneity in strongly curved spacetimeby PAllen Tags: curved, notions, simultaneity, spacetime, strongly 

#19
Nov1912, 12:04 PM

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 associated with each map is a metric expression, such that each map expressed the same geometry  if one is talking about the same sphere using different maps, and one map doesn't cover all of the sphere, you use other maps to cover the rest, such that you are always describing the same complete sphere. Who decides what is a valid coordinate system? Differential geometry has a well defined, precise, answer to this (see any definition of topological manifold, refined further to become a pseudoriemannian manifold). If you think this is wrong, then what is your precise criteria for a valid coordinate system? If it is different from above, you have a new theory, not GR as understood by everyone else. And in this new theory, general covariance is rejected, because that requires that any coordinates allowed by the criteria in the prior paragraph or good. One possible analogy for our disagreement is:  Imagine a 2sphere in polar coordinates. Bob doesn't like what happens at or near the poles. So Bob decides to analyze only different object: a sphere missing a little disk around each pole. This is a valid, different geometric object. It is easy to demonstrate that you have holes using only polar coordinates with metric. Now in the case of OS collapse, the hole you are proposing 'must' be accepted as the correct prediction of GR is rather strange. A clock in the middle of collapsing dust ball stops for no reason. It stops in a strange sense  locally everything proceeds at a normal rate until it is declared to stop. Note that for Krauss, et. all, assuming their quantum simulation is correct, they have good physical justification for this  this central clock is not acatually stopping; it evaporates in finite local time. Then it makes sense to talk about chopping a classical model at similar point. If, instead, you accept the the interior clock proceeds normally, there is no escaping (using any coordinates), that the clock is proceeding for some time after an event horizon has formed around it. Any signals it sends will not escape, but it can readily receive signals from an external observer. 



#20
Nov1912, 12:10 PM

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#21
Nov1912, 12:17 PM

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Outside the frozen ball, you can apply SC coordinates and metric (or others). Inside (where matter is), you must do something else. Now, SC time slicing is a specific choice of simultaneity (of which there are infinite choices, including simple, physically defined ones: http://physicsforums.com/showpost.ph...0&postcount=23 ). If I pick a different choice, and use it throughout, then the external observer assigns well defined time coordinates to the dust ball collapsing inside the horizon, and even a well defined time to the formation of a singularity. 



#22
Nov1912, 12:20 PM

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#23
Nov1912, 12:42 PM

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And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates. 



#24
Nov1912, 01:18 PM

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The point of this thread, is that you can choose many other intuitive simultaneity conventions for a distant observe (and simultaneity convention identically equals time sicing), which don't have a hole, and assign finite time coordinates to events inside the EH, which itself labeled with a finite time coordinate. 



#25
Nov1912, 02:18 PM

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#26
Nov1912, 03:52 PM

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I've got several comments:
On the map issue. There ARE coordinates that are mildly "special" at any given point. These coordinates are the ones where your map is drawn to scale. The mathematical feature of such maps is that the metric is diag(c, 1, 1, 1). Because these maps are to scale, one can freely interchange coordinate distances (i.e. changes in coordinate) with physical distances. They're easy to work with. Discussions don't go off in as many strange tangents when you use maps that are to scale, and people don't get "lost" so much, I find. Some recent discussion reads to me something like this: "Your'e reading the map wrong  here , use this one, It's almost to scale  well, not really to scale, actually, but the distortions are at least finiite." "No  I like the one that's infinitely distorted better, because  nevermind why, I just like it better." People for some reason seem to have a really hard time dealing with distances in GR but if done correctly it's not that complex  just make sure your map is to scale (and if its not, learn enough math to covert your maps so they are to scale, this can reasonably be motivated with as little math as algebra). The only remaining issue with maps is the SR issue. This is understanding that time is not absolute, that simultaneity is relative, that as you change your notion of simultaneity your notion of distance also changes correspondingly because spacetime is a continuum, and space and time are fundamentally linked. Mathematically: the maps of SR and GR preserve the Lorentz interval. This is where I keep feeling the communication is lacking  but many people who say they "get" this point obviously don't :(. The notion of "now" is relative. The local notion of now is determined by backwards compatibility with Newton's laws, but said notion of now (necessary for this backwards compatibility) is purely local, and not universal. This relativity means that this, or any other , notion of "now" is not used to determine cause and effect, but rather one uses light cones. To put it succinctly, "now" is relative, light cones are absolute. The last issue. I think there are some people who believe in "white holes", but not in the context of classical GR. See the thread on Nikodem Poplawski I started 



#27
Nov1912, 04:33 PM

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There is a flatland country near the equator, and sailors are setting out on voyages to distant destinations. Now comes a cartographer, who decides to make a map that can serve as a travel guide. Based on intuition (inspiration?) he then develops a map of the world with the strange property that it has medians that converge when going away from the equator towards the North. As a result everything ends in a singularity, which he gives the name North Pole. Thus it appears that one cannot get further away than the North pole, which is intellectually unsatisfactory. Then comes along a different cartographer who thinks up a conformal transformation that results in a very similar map, but now with the medians running parallel. The funny thing of that map is that the singularity is gone; on that map people can continue beyond the North pole. Of course, the two maps can be transformed from one into the other without problems up to that singularity, but predictions definitely differ from that point onward. Now they have a problem; at best one of the two will match reality. Either Flatlander Earth physics law can tell them which one to choose as the most correct one, or Earth science is incomplete, so that the flatlanders don't know yet which mapping would best, even just in theory. 



#28
Nov1912, 04:47 PM

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Note: regretfully I can merely choose like a voter about a political issue: Yes, I do have an opinion, and No, I probably don't have enough GR expertise to make a case. So, please regard me as a science reporter who meddles in expert discussions and asks annoying questions. 



#29
Nov1912, 04:56 PM

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But in each of these cases, I see a clear reason to 'draw a line'. Tachyons have to be added to SR in violation of the causal interpretation of SR. White holes have no process by which they can form. Similarly, for most of the others, there is no known process they can form out of plausible initial conditions. Singularities I interpret as a clear sign that GR has broken in this domain. With black holes, treated classically, we have, instead, the singularity theorems: With almost any reasonable starting point, once collapse has gotten close to a critical radius, it must proceed all the way to a singularity. Further, all the approaches, classically, to try to avoid an event horizon formation amount to my example chopping the poles off a sphere because I don't like what they do to my coordinate preference. 



#30
Nov1912, 05:27 PM

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For example, the black hole information loss problem basically comes down to: which do you want to keep when push comes to shove, general relativity or quantum mechanics? GR predicts that gravitational collapse leads to the formation of regions of spacetime from which information can't get back out; that means a quantum state that falls in gets lost, so unitarity, a central principle of QM, is violated. Hawking's position (at least until about 2004ish; he seems to have switched sides in what Susskind calls the "Black Hole War" about then) was, so much the worse for QM. Susskind's and t'Hooft's position (there were others, but they seem to have been the primary ones who held to this position throughout) was, so much the worse for GR. They can't both be right. But nobody, as far as I can see, was arguing one way or the other based on which coordinates were valid and which weren't. 



#31
Nov1912, 11:29 PM

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And wordlines of particles do not end. They just extend toward infinite future. You would not claim that there is some problem with this expression, right? [tex]\lim_{t\to\infty}f(t)=a[/tex] Let's say that t is coordinate time and f(t) is proper time of infalling particle. And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime. 



#32
Nov1912, 11:36 PM

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I thought it would be useful to do something Peter Donis suggested in another thread. That is, to look classically at the complete geometry of a collapsing shell of matter, which is like the Krauss case. I found a couple of references, and have done some order of magnitude calculations based on the arxiv paper (adjusted for a matter shell rather than a null dust shell).
The follwoing gives a Kruskal diagram (and other coordinates) for a collapsing shell. This one is pressureless, and only the final stages (where the matter is almost light like) is shown (the r=0 line and the shell line need to slanted closer to vertical to consider a collapse from rest not too far from the SC radius): http://casa.colorado.edu/~ajsh/collapse.html This paper discusses collapse of a null dust shell, including Kruskal coordinates (section 4): http://arxiv.org/abs/grqc/0502040 While this is not quite the case of interest, the techniques shown for matching interior and exterior are general. I think the interesting case to consider is a clock sitting inertially at the center of a collapsing shell, under the further assumption that the shell is transparent. There are a number of interesting features:  The spacetime inside the shell is pure Minkowski flat spacetime; there are no tidal forces at all. There is no matter density at all. In the whole region inside the shell, SR physics applies. The analog of this is obviously true for Newtonian physics; it is also well known that this is exactly true for GR as well (that inside a spherical shell, physics is indistinguishable from SR with no gravity).  The clock inside operates according to SR physics, time locally flowing normally, until shell collapse reaches singularity. The clock at the center cannot detect the shell passing the event horizon. Nothing about physics within the shell changes on approach to or passing the event horizion  the interior is pure SR until the singularity.  The clock at the center sees the external 'universe' proceeding normally except highly (but finitely) blue shifted until the moment of singularity. Here, the clock world line ends for a very physical reason: the shell has collapsed to a singularity, bringing the clock with it. One of the things I wanted to get right here was the blueshift relation. Unlike an OS collapse, where an infaller (with right trajectory) can have very mild or even no blue shift relative to the outside universe, the boundary matching conditions for a shell collapse require that the whole interior (for observers stationary with respect to shell center) experiences (in the limit of zero thickness shell) the same blueshift as a clock riding exactly on the shell.  A distant observer, of course, never sees the shell reach the event horizon. Similarly, they see the inside clock stop at a time before the clock hits the singularity.  Later free fall clocks will be seen, from a distance, to stop on reaching the event horizon. Such a clock, itself, will experience no such thing, and its time will progress further until it hits the singularity (the shell is 'long gone' into the singularity). Classically, one may ask by what possible rationale should one declare that a clock operating according to pure SR physics, be declared to stop for no local physical reason? What does what a distant observer sees have anything to do with their local, pure SR physics? All of the above is indisputable for classical GR. The Krauss paper, by choice, only covered what the outside observer sees, for the classical case, because of what they discovered about the quantum case. The quantum analysis (if true) showed that, a lot of the above does not happen in our universe. It shows, instead, that (no matter what coordinates are used), the interior clock actually evaporates into not quite thermal radiation well before its classical end. The Kruskal diagram changes in significant ways as a result. Only with this justification, does it make any sense to 'chop' the classical analysis based on Schwarzschild time coordinate. Note, that in the classical analysis without the quantum chop, perfectly reasonable, physically based simultaneity conventions, establish simultaneity between the interior clock and exterior clocks well after the shell has crossed the event horizon. One simple example is: http://physicsforums.com/showpost.ph...0&postcount=23 



#33
Nov1912, 11:48 PM

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[tex]\lim_{t\to\infty} \int_{t_0}^t dt' f(t')[/tex] where [itex]t_0[/itex] is the time by the distant observer's clock that the infalling object starts falling. Substitute the correct function f(t'), for the proper time of the infalling object, do the integral, and take the limit. You will find that it gives a finite answer. This shows that "extending the worldline to infinity" by the distant observer's clock only extends it by a finite length. The SC time coordinate is so distorted at the horizon that it makes finite lengths look like infinite lengths. This also shows why the region of spacetime covered by the SC time slicing can't be the entire spacetime: what happens to the worldlines once they reach the horizon? They have only covered a finite length, and spacetime is perfectly smooth and wellbehaved at the horizon: the curvature is finite, there is nothing there that would stop the objects from going further. The only physically reasonable conclusion is that they *do* go further; that is, that there is a region of spacetime on the other side of the horizon, where the infalling objects go, but which can't be covered by the SC time slicing. 



#34
Nov1912, 11:52 PM

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As for flat spacetime, the Rindler example Dr. Greg has posted beautiful pictures of, is relevant. The belief that there is no hole in SC exterior coordinates is 100% equivalent to the belief that most of the universe doesn't exist because a uniformly accelerating rocket can't see it. 



#35
Nov1912, 11:53 PM

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Transformations between inertial SR coordinates preserve Lorentz interval. And they don't change metric just as well. But in GR transformations between coordinates don't have to preserve metric intact. That's how Lorentz interval is left the same, right? 



#36
Nov1912, 11:56 PM

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