
#379
Dec1812, 11:59 PM

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However, the coordinate dependence of gravitational time dilation is shown by something like Lemaitre coordinates, which reproduce all measurements of any other coordinates, but display no gravitational time dilation (in the sense of dilation as a function of position). The key physical point is that measured time dilation is always done between two clocks; or between an emitter and receiver. It is thus a feature of two world lines and signals between them. The measurements are coordinate independent, and gravitational redshift or time dilation are not necessary to compute any such measurement. Note also that the general way to define clock and redshift comparison between two world lines applies perfectly well to the vicinity of coorbiting neutron stars. Meanwhile, there is no way to even define gravitational time dilation for such an inherently nonstatic field. Gravitational time dilation is a useful concept for static spacetime  but it is not a general feature of GR, and it is never necessary to use. J. L. Synge, in his classic book on general relativity, argued against using it at all  because one universal method may be used in all cases (kinematic, cosmologic, and in strong, nonstatic geometry) instead. 



#380
Dec1912, 06:31 AM

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#381
Dec1912, 07:07 AM

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Two different coordinate systems may disagree on the meaning of the measurement, e.g. they may disagree whether or not the rod is accurately measuring length, but they must agree on what value is measured. 



#382
Dec1912, 07:11 AM

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What you are telling is (mainstream) speculation. The closest thing to something like that as I can imagine is galactic collisions. I for example can ask what people experience when they die. So what? As I see speculations about BHs relies on one important thing that GR takes as a postulate. That is that laws of physics are independent of (Newtonian) gravitational potential. If we assume that this assumption holds without bonds then we have no reason to assume that anything will happen with a clock falling into the hypothetical BH. But I don't buy the idea about assumptions holding without bonds. And that takes it out of domain of GR. 



#383
Dec1912, 08:01 AM

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#384
Dec1912, 04:47 PM

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There's growing experimental evidence for the existence of event horizons. Basically, black hole candidates are very black, and don't appear to surface features.
WHen matter falls onto a neutron star, the surface heats up and reradiates. The spectra signature is rather distinctive, also there are "type 1 x ray bursts". Black hole candidates do not appear to have any such "surface" features, and it's already very difficult to explain by any means other than an event horizon how they can suck in matter without , apparently reradiating anything detectable. For the details, see See for instance http://arxiv.org/pdf/0903.1105v1.pdf and check for other papers by Naryan in particular. 



#385
Dec1912, 06:17 PM

P: 1,162

Quote by pervect
Quote by Austin0 View Post Where in the stated parameters : Quote by Austin0 View Post Quote by Austin0 View Post Quote by Austin0 View Post Quote by Austin0 View Post 



#386
Dec1912, 07:25 PM

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#387
Dec1912, 07:33 PM

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It sounds as if we still haven't clearly explained the Zeno time analogy... Let's try a different tack... When we say that there is time dilation between two observers, what are we saying in coordinateindependent terms? We are saying that: 1) There are two points on A's worldline; call them A1 and A2. Call the proper time between them ΔA. 2) There are two points on B's worldline; call them B1 and B2. call the proper time between them ΔB. 3) A claims, using some more or less reasonable definition of "simultaneous", that A1 and B1 are simultaneous and that A2 and B2 are simultaneous. 4) Now A calculates the ratio ΔA/ΔB. If that ratio comes out to be greater than unity, then A says that B's clock is running slow because of time dilation. Obviously this result depends on the simultaneity convention used to choose the endpoints B1 and B2 as well as the metric distance between them on B's worldline. The standard SR definition of time dilation is contained in this more general definition; you just use the obvious and only sensible simultaneity convention, namely that all events on a line of constant t in a given frame are simultaneous in that frame, and you'll get the SR time dilation formula. This is the only definition of time dilation that can be made to work in GR, although this is somewhat obscured by the need to do the calculations in SOME coordinate. Note that in GR the choice of simultaneity convention is arbitrary, and that if you cannot draw null geodesics from B1 to A1 and from B2 to A2 there's no reason to even prefer one convention over another. The same method even applies in a classical scenario (although it is trivial and uninteresting). There's only one possible simultaneity convention, that defined by the Newtonian absolute time, and the ratio ΔA/ΔB always comes out to one, so there's no reason to mess with any of this coordinateindependent description. But that is the point of the Zeno time analogy. We pick a deliberately absurd time coordinate instead of the obvious Newtonian one; it's so absurd that we cannot assign any time coordinate to event B2 ("the arrow hits the wall"), and then we calculate in this coordinate system that the arrow cannot hit the wall. Of course we know that the arrow does in fact hit the wall, so we know that something is wrong with the coordinate system and that the ratio of zeno time to arrow time is not telling us anything. And it's the same way with the Schwarzchild time coodinate. The ratio of A's Schwarzchild time coordinate to proper time on B's worldline serves only to mislead. The interesting quantity is the ratio of proper time between any two points on B's world line and any two points on A's worldline. 



#388
Dec1912, 08:09 PM

P: 1,162

Well of course given these conditions everything else is obvious. But then you have simply rewritten Zeno's paradox completely. Simply stuck Zeno's and Achilles names on the conditions of free fall in Sc coordinates. And those conditions are not derivable from the stated Zeno time as above ,alone. Since everyone basically agrees there is no merit in the logic in the classic Zeno argument, then by association and implication anyone considering the possible validity of the Sc case is obviously silly, right??? What other point was there as you simply made the scenarios identical (I.e. completely different from the classic argument).??? If those assumptions had been explicitly stated by Pervect then it would have been quite obvious that Zeno time was explicitly dilated and outside any classical context . 



#389
Dec1912, 08:22 PM

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Is there any reason to take the Schwarzchild time coordinate in spacetime more or less seriously than the Zeno time coordinate in classical space? (IMO the answer is "yes", but for a rather unsatisfying and unfundamental reason  there are some problems that are computationally easier if you choose to work them using the SC time coordinate, while AFAIK there are no interesting problems that are more easily solved by transforming into Zeno coordnates). 



#390
Dec1912, 08:30 PM

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Austin0 was asking for some more detailed math. I'd suggest looking at Caroll's GR lecture notes online.
I'll also add that while Caroll's online notes are perfectly fine (one can't trust everything online, Caroll's online notes are drafts of a book by a physics profesor that was later published. From my POV the main advantage of them is that they're free). I could also quote similar statements from some of my other GR textbooks, (i.e. Caroll is not an isolated occurrence among textbooks). However, I think it would be better from a pedagogical point of view if interested people went out and found their own textbooks if they don't like Caroll (though I can't think of any valid reason for not liking Caroll). But  onto Caroll: One can then observe directly that the event horizion is reached at a finite tau, even though t(tau) is infinite. If one is willing to just take the limit as r approaches the event horizon one can do this all in Schwarzschild coordinates. This is even observable. One can say that's it's possible to observe the limiting sequence of proper time as one approaches the event horizon from outside, and observe that the limit is finite. To go futher and carry the trajectory smoothly through the event horizon, one needs coordinates that are better behaved, which is what Caroll does next. The point of the Zeno analogy is to demonstrate a simple example of how a coordinate time can be infinite while the time actually measured on a clock is finite. Specifically, zeno time is infinite, while as far as Achilles is concrned, there's a finite time at which he passes the tortise. I'm afraid I don't understand the difficulties people are having in understanding the analogy. It could be my fault, sometimes I "leap ahead' too far when I write. The way you demonstrate that the proper time on an infalling clock is actually finite rigorously is that you calculate it. Post #12 in this thread http://www.physicsforums.com/showpos...4&postcount=12 (and a later post after it, #13) To verify this is a solution one needs to demonstrate that said trajectory satisfies the geodesic equations. You'll find them in my post #12, Caroll's GR lecture notes, for starters. The idea behind the Zeno analogy isn't to "prove" anything  that's what textbooks are for. The idea behind the Zeno analogy is to illustrate how t can be infinite and tau can be finite in a simple, easytounderstand example. WEll, the Zeno analogy does prove one thing. It demonstrates that just because you have a time coordinate t going to infinity doesn't prove that something doesn't happen. It's an example of how t going to infinity can be the result of a poor choice of coordinates. It's a counterexample to the argument "t goes to infinity, therefore it can't happen". Historically, I do believe that the "tortise coordinate" was named after the tortise in Zeno's paradox, but I haven't seen anything really detailed on this in textbooks. There was something in Scientific American about it a long time ago as well, I think. 



#391
Dec1912, 08:59 PM

P: 1,162

I really just meant that time dilation was in effect within the stated conditions and coordinates And yes I am quite aware of the meaning of dilation and it is the ratio of rates or intervals of two different clocks. In a recent post I made the simple statement that time dilation was inherently relative. Self evidently true for exactly this reason. It is meaningless applied to a single clock. Like the term length contraction or the word faster. It intrinsically requires and implies a comparison. But somehow I got a bunch of flack from several people telling me I was wrong. ??????????? I am not convinced that Sc coordinates are necessarily preferred or correct. I am still just learning their subtleties and details and trying to synthesize a logically coherent structure up to the horizon. My exception to this analogy was purely logical. You all may be ultimately right about Sc coords and the horizon but this use of Zeno added nothing of logical probative value to the debate and was actually misleading in it's subtle reframing of Zeno. 



#392
Dec1912, 09:52 PM

P: 1,162

I do understand the math processes and reasoning behind this. Integrating proper time is not difficult to grasp , certainly not after SR Now that I understand that your statement of Zeno time was with the expectation that it was assumed Achilles' proper velocity was constant even though it decreased in Zeno's frame then of course the situations are effectively identical. Of course this means that this adaptation is no clearer or more persuasive than the original Sc scenario. I have never said that the infaller doesn't reach the horizon in some relatively short proper time on its clock.I have questioned the assertion that this does not transform to some tremendously distant future time in the frame of the distant observer. This seems to call into question the Sc coordinates not only in the immediate vicinity of the horizon but effectively throughout the system. How or why a system which is empirically verified within a certain range of the domain would become totally unreliable (pathological ;) ) in another part. 



#393
Dec1912, 09:55 PM

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[tex]n=log_2 \left( \frac{100}{100vt} \right)[/tex] Taking the derivative of Zeno coordinate time wrt Achilles proper time we get [tex]\frac{dn}{dt}=\frac{v}{(100vt) ln(2)} \neq 1[/tex] So Achilles' clock does not run at the same rate as Zeno coordinate time. Taking the inverse transform we get [tex]t=\frac{100}{v}(12^{n})[/tex] so [tex]\lim_{n\to \infty } \, t = \frac{100}{v}[/tex] So as Zeno coordinate time goes to infinity Achilles proper time does not. 



#394
Dec1912, 10:27 PM

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#395
Dec1912, 10:33 PM

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#396
Dec1912, 10:37 PM

P: 1,162

so according to Nugatory I get that it was supposed to be understood implicitly that that was a given but everything i said was clearly within the context of what Pervect actually outlined. 


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