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Notions of simultaneity in strongly curved spacetime 
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#55
Nov2012, 02:28 PM

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PF Gold
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Maybe I misunderstand your intent. It is absolutely possible for a distant observer to assign remote times in a consistent way such that they consider the object to have crossed the horizon in finite time. They can also consistently assign remote times so that never happens. It will never be possible to verify one assignment over another precisely because event horizon crossing will never be seen. 


#56
Nov2012, 02:53 PM

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#57
Nov2012, 03:06 PM

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However, because GR is unambiguous that a BH forms in finite time for the infalling matter, and new matter falls in in finite time for the infalling matter, it must be said that GR predicts black holes. That this prediction is not verifiable from the outside, doesn't make it not a prediction of GR. For example, QCD says quarks will never be detected in isolation. They are still a prediction of QCD. The real 'way out' is that quantum gravity changes the classical GR predictions. 


#58
Nov2012, 03:30 PM

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http://en.wikipedia.org/wiki/Kruskal...es_coordinates If you consider a timelike freefall trajectory that starts at the past singularity (the hyperbola at the bottom of region IV), emerges from the white hole (i.e., crosses from region IV into region I), rises to some finite radius r at Kruskal time V = 0, then falls back into the black hole (crosses from region I into region II), and finally ends up at the future singularity (the hyperbola at the top of region II): such an object's trajectory is timesymmetric; the part before V = 0 is the exact time reverse of the part after V = 0. If, however, you are referring to a spacetime where a BH forms from the collapse of a massive object, then evaporates away, I haven't seen a Kruskaltype diagram of that case, but I have seen Penrose diagrams of the most obvious way to model it (which not everyone agrees is the correct model, but it's a good starting point for discussion). See, for example, the diagram here: http://en.wikipedia.org/wiki/Black_h...mation_paradox Compare with the Penrose diagrams on this page: http://www.pitt.edu/~jdnorton/teachi...ure/index.html The Penrose diagram corresponding to the Kruskal diagram I linked to above is in the section "Conformal Diagram of a Fully Extended, Schwarzschild Black Hole". The Penrose diagram corresponding to the classical GR model of a collapsing massive object (like a star) is in the section "A Conformal Diagram of a Black Hole formed from Collapsing Matter". Note that in *none* of the diagrams, other than the Kruskal diagram and the Penrose diagram corresponding to it, does the white hole appear. In the evaporation diagram, Hawking radiation escapes as the hole evaporates, but there is still a black hole interior region and a singularity, and anything that gets inside the horizon is still doomed to be destroyed in the singularity, according to this model. The big open question is, if this model is *not* correct (which most physicists in the field now seem to think it is not, since it leads to the loss of quantum information), what replaces it? There are a lot of suggestions, but no good answer yet. 


#59
Nov2012, 03:40 PM

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#60
Nov2012, 04:42 PM

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This discussion is growing a bit over my head, especially concerning time (my time, not Schwartzschild t, although it's almost the same ); I intended to quickly move on from a simple illustration to show that there is an issue, to a concrete physics discussion involving clocks and light rays. However it is interesting for me and perhaps also for invisible onlookers. I'll try to group things piecewise and only discuss the essentials.
On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics. Thus you would need to make a strong case with the following if your intention is to convince me (but I hope that that is not what you are trying to do): Secondly, of course I considered the fact that OS map t>∞ to τ>a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees. I did not make a plot of it, but I did not see an issue with that. In the OS model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!), an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ_{0}. As I picture it (for I have not seen a description of it), for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. In fact, it was a discussion based on a blog including that aspect with more than 100 posts that was the first thing that I read about this topic (http://blogs.discovermagazine.com/ba...reallyexist/) 


#61
Nov2012, 05:59 PM

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Two coordinate systems are just two different sets of labels attached to an overall space time. It can happen that they don't cover all the same region of spacetime. However, they are just relabelings of the same geometry for coverage in common. You obviously can't use a particular coordinate system for a part of the geometry it doesn't cover. As for coodinate infinities, let me try an example. Start with a flat plane with Euclidean metric (distance given by ds^2= dx^2 + dy^2). Now define coordinates u and v as: u=1/x , v = 1/y ; the metric (distance formula) expressed in these will be different, such that all lengths, angles and areas computed in cartesian coordinates are the same with computed with u and v  using the transformed metric. Note that u and v become infinite as you approach the x or y axis. However, no computation or measurement is different from cartesian coordinates (when you use the transformed metric). But you can't directly do a computation involving any point on or line crossing the x or y axis in these coordinates. You can compute the length of a line approaching the x axis and get a finite value limit value; you can continue it on the other side and get a finite value for its length, limiting from the other side. The ininite value of u and v has no geometric meaning, because coordinates are interpreted through the metric. The behavior of the t coordinate in SC coordinates is just like this. It has meaning only through the metric for computation of 'proper time' which is what a clock measures. If you compute proper time for an infalling clock, you get a finite value for it to reach the EH. If you continue it over the EH using, e.g. interior SC coordinates, you get an additional finite proper time from the EH to the singularity. You will find, that for a static clock (stationary with respect to the spherical symmetry), very far from the center, SC coordinate time matches clock time for that clock. It doesn't match clock time for other clocks. The closer you get the the EH, the less this t coordinate has anything to do with what clocks measure. Just like with my u coordinate above, u becoming infinite says nothing about what a ruler will measure. 


#62
Nov2012, 07:56 PM

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This goes back to what I said above; you appear to be assuming that it must somehow be possible to assign a welldefined "t" value to every single event, everywhere in spacetime. You can't. That's just the way it is. If you want to describe events at or inside the horizon, you simply can't use the "t" that the distant observer uses. It just can't be done. If you can't admit or can't grok this possibility, then probably further discussion is useless unless/until you can. It's not easy, I agree; it took me quite some time to wrap my mind around it. But it's critical to understanding the standard classical GR model of black holes. Those events inside the horizon, the ones with [itex]\tau > \tau_0[/itex], do *not* have welldefined "t" values at all, if "t" is the time coordinate of a distant observer. They simply can't be mapped in the distant observer's chart. 


#63
Nov2012, 08:29 PM

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If my wife gives birth to Judy and Jill, and Jill stays nearby and Judy goes to Africa, and I never hear from Judy again (unless I think Judy died), I have the expectation that there is simultaneity between events for Judy and for Jill. Their mutual causal connection to me gives me this expectation. Even more so if I believe Judy is getting my birthday cards (damn that she doesn't respond). This concept can be formalized using the one of the procedures I outlined to say: I consider (though I can't verify it) that the singularity of that collapse formed at 3 pm today for me. It almost seems you are saying there is a physically preferred chart for the distant observer. I don't accept this. I only accept that locally, there clear preference for FermiNormal coordinates; but globally? None. And the specific simultaneity I proposed a few times corresponds exactly, locally, to Fermi normal coordinates for an asymptotic observer  it diverges from this further away. 


#64
Nov2012, 09:27 PM

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(1) The integral curves of the time coordinate are also integral curves of the timelike Killing vector field; (2) The surfaces of constant time are orthogonal to these integral curves. Painleve time has property #1, but not #2. Kruskal "time" has neither. I agree this is a weak sense of "preferred", but it's important to note that it is these two properties together that make Schwarzschild coordinate time seem so "natural"; it *seems* like the "natural" extension of Fermi Normal coordinates along the distant observer's worldline, because it *seems* like properties #1 and #2 are the "right" ones for a "natural" coordinate chart to have. Only when you start looking close to the horizon do you start running into problems with this "natural" extension. 


#65
Nov2012, 10:32 PM

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#66
Nov2012, 11:32 PM

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Not sure about GR but I am certain about my understanding of SR. And isn't Rindler coordinates (and horizon) more an analogue of eternal BH rather than collapsing mass (forming BH)? And besides you have to take into account that rocket can't remain in state of uniform acceleration for indefinite time. And observer on that rocket would not observe rather static picture of other matter in the same state of uniform acceleration. But as far as we know we can remain in a state of gravitational acceleration for indefinite time and we observe a lot of matter in the same state. 


#67
Nov2012, 11:34 PM

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#68
Nov2112, 02:50 AM

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If you imagine the surface of such particles infalling, and you don't allow it to proceed to Sc radius, you have, geometrically, a hole: proper time on this surface is finite, area is finite, but if you stop it from continuing, it is geometrically a hole. Geometry is defined by invariants, not coordinate quantities. Consider the example I gave to harrylin several posts back of a horizontal geodesic in the plane in (u,v) coordinates. u coordinate goes to infinity on both sides of coordinate singularity, but it is still nothing but a geodesic in the flat plane. 


#69
Nov2112, 02:51 AM

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So here's my thought process: Schwartzschild's solution is the one that I heard about in the literature, and it happens to be the one that I happened to stumble on in the first papers that I read about this topic, dating from 2007 and 1939. It is obviously a valid reference system according to GR, and it is "privileged" in the same sense as inertial frames and centre of mass systems are "privileged": it allows for the most simple mathematics, so simple that one doesn't need to be an expert to understand it. Thus it is a natural choice in a public discussion about predictions of GR. And the way I understand the first paper that I read about this, it's probably all I will ever need to understand this topic. 


#70
Nov2112, 03:21 AM

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I may not be able to fully catch up with this thread  I'm reading this at work while I should be doing something very different ... But here's a quick unrelated point:



#71
Nov2112, 04:23 AM

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#72
Nov2112, 05:54 AM

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another detail:
I estimated the intensity of starlight that hits the eye of the infaller when looking forward  thus what he literally will see; the prediction of an event. In contrast, the "see" in your sentence is probably a prediction of what the infaller will calculate. 


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