Any coordinate system in GR?
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From: JanPB on 25 Aug 2006 23:50 Ken S. Tucker wrote: > > Jan, I think you should study the original paper > Ed Green cited, there is no such thing as an > event horizon or Black-Hole's. BTW, Dr. Loinger > and I discussed this at length. In short, the > original Schwarzschild Solution has been > bastardized and mis-understood for simplicity. No, the "original" solution is the only one. This follows from basic ODE theory and the definition of tensor. > The bastardized version became popularized > and embraced by astronomers who now see > BH's under their beds. There is no other version, bastardized or not. How many solutions do you have to the following ODE: f'(x) = f(x) ....given the initial condition f(0) = 1 ? I can see f(t) = exp(t). Is there any other? How about another ODE, just slightly more complicated: -2 f(x) f'(x) + 1/x * (1 - f(x)^2) = 0 ....given the initial condition f(1) = 0 ? -- Jan Bielawski
From: I.Vecchi on 26 Aug 2006 09:04 Tom Roberts ha scritto: > I.Vecchi wrote: > > Your construct relies on the arbitrary duplication of the the horizon, > > Sure. As is easily seen in a Kruskal diagram, and as I said. Here you > seem to be in violent agreement with what I said (but later you get it > wrong). > > It is arbitrary which portion of the horizon you consider, but as both > are contained in the manifold it is only the analyst who is subject to > this choice, not the manifold itself (or the physical system for which > it is a model). Are you saying that for every black hole in the universe there is a corresponding white hole? > > > inventing two copies of what is actually a single > > physical/observational domain. > > Not really. Those regions of the complete (inextensible) manifold are > DIFFERENT. That is, considered from a given point in spacetime near but > outside the horizon, the past and future horizons are NOT "the same". > This difference between past and future is true in ANY spacetime, of course. > > That is, for any point in any manifold of GR, every locus > in the past lightcone of the point is disjoint from every > locus in the future lightcone of the point. This is true > in everyday life -- just think about how different is your > ability to observe events in the past from events in the > future. > > > > It remains a fact that the extension > > across the horizon is not unique and that each extension yields a > > distinct physical object. > > Not "physical object" but rather region of the manifold. Which corresponds to a physical/observational object. Otherwise we are talking about nothing. > Basically you > happened to choose E-F coordinates that do not cover the manifold, but > both sets of E-F coordinates cover the exterior region; which set of E-F > coordinates you choose will determine into which region of the manifold > you can extend. Yes, which set of coordinates I choose will determine it . And correspondily yield the black hole or the white hole solution. > BTW neither choice includes other regions of the > complete (inextensible) manifold, but Kruskal-Szerkes coordinates > include them all. Yes, it's mathematical construct useful for didactic purposes. > > > > A black hole is not a white hole. Period. > > Sure. And both are contained in the Kruskal diagram, and in the complete > (inextensible) manifold. > > Have you never looked at a Kruskal diagram? -- you seem rather > unknowledgeable about basic aspects of this manifold (yes, singular -- > you are discussing different regions of a single manifold). Any > reasonably modern textbook on GR will have it. The point here is its relevance to the topic of this discussion. As for extending my knowledge, which may indeed be limited, vigorous (and, ideally, polite) online discussion appears to be a fruitful approach. Beside the above, I surmise that there are other space-time extensions across the horizon corresponding to hybrid white hole/black hole solutions (*). This would be impossible according to your argument, right? Cheers, IV (*) In the corresponding charts the metric gets singular at one point on the horizon, but that does not make it more unphysical than, say, the Schwarzschild solution.
From: I.Vecchi on 26 Aug 2006 09:17 carlip-nospam(a)physics.ucdavis.edu ha scritto: > The Kruskal-Szerkes extension is the unique maximal analytic extension > of the Schwarzschild exterior geometry. Nice. Is there a proof for that? I mean a mathematically rigorous one, where the hypotheses are cleary stated, so that one can weigh their physical relevance. A reference would be appreciated. > Which part do you want to > give up? In any given case, the one that does not fit observation. You are warmly welcome to provide your feedback on the point I raise about "hybrid solutions" in my reply to Tom Roberts. IV
From: Edward Green on 26 Aug 2006 12:46 JanPB wrote: > Ken S. Tucker wrote: > > > > Jan, I think you should study the original paper > > Ed Green cited, there is no such thing as an > > event horizon or Black-Hole's. BTW, Dr. Loinger > > and I discussed this at length. In short, the > > original Schwarzschild Solution has been > > bastardized and mis-understood for simplicity. > > No, the "original" solution is the only one. This follows from basic > ODE theory and the definition of tensor. > > > The bastardized version became popularized > > and embraced by astronomers who now see > > BH's under their beds. > > There is no other version, bastardized or not. How many solutions do > you have to the following ODE: > > f'(x) = f(x) > > ...given the initial condition f(0) = 1 ? I can see f(t) = exp(t). Is > there any other? > > How about another ODE, just slightly more complicated: > > -2 f(x) f'(x) + 1/x * (1 - f(x)^2) = 0 > > ...given the initial condition f(1) = 0 ? Hi Jan, I don't see you ever got back to me on my humble request to show me where I went wrong in my heuristic. It may be, as Hobba suggested, that there is nothing corresponding to a dilation or strain of the manifold in GR. If that's true, then that removes the conflict between the explicit form of the metric trying to compensate for dilation and also for an arbitrary change of units in different regions of the manifold. Maybe that's the answer to my question. Anyway, since you are discussing black holes/white holes, let me jump ahead. I'd like to know: do black holes and white holes represent regions of a single complete inextensible manifold corresponding to the Schwarzschild metric, or are they result of something like a sign choice -- disconnected solutions? This seems like a rather important point. If the solutions are disconnected, then it is less damaging to discard one of the solutions as unphysical. If they are different branches of the same analytic extension, then discarding one is a much more serious theoretical problem: analytic extensions are what we obtain by creeping across the surface of the solution, so to speak, and saying "well, we have a solution up to here, what is required to extend the solution a little further, assuming the field equations to be satisfied". An arbitrary truncation of such a process because we obtain unpalatable results is tantamount to saying the theory is broken somewhere. (Theories are allowed to be broken somewhere, and still be beautiful and useful).
From: Edward Green on 26 Aug 2006 14:51
Ken S. Tucker wrote: Sorry, I missed your reply at first. > Basically s.p.r. has no expertise in GR so although > your question is well posed and penetrating it exceeded > the moderators understanding. Myself I'd be interested > in possible well informed opinions that occasionally > post to s.p.r. , that said I'll try. You are a gentleman, as always. > > The offhand claim is sometimes made that GR, which I take it means the > > formal machinery of GR, treats all coordinate systems equally. Is this > > true? > > Pardon my apparent evasiveness... And mine. :-) [snip gracious mini tutorial in GR] Ken, I seem to be subject to a mini-curse: nobody quite recognizes what I'm asking; and finally, given enough perserverence, I may come up with answers, still unrecognizable. Not recognizing the question, respondants are either sympathetic or insulting (or silent) according to their own personality, but not, from my point of view, on target. I asked how the metric -- particularly the explicit coordinate representation of the metric -- adapts to dilations in spacetime. Hobba's answer is, there is nothing corresponding to a dilation in spacetime. I'm not sure if I believe this or not. What about gravitaional time dilation? This phrase tends to indicate that a second as measured by a standard physical clock recorded over _here_ may not correspond to a second over _there_. Physical clocks and rulers, composed of standard arrangement of atoms, give what we might consider a natural local metric to spacetime. If we have reason to believe -- without the possibility of direct comparison -- that these metrics would not agree in different regions, then it seems plausible to say that there is a relative dilation of spacetime beween the regions. How does the formal representation of a metric handle this? Is the change in a time-like variable measured in local seconds prefixed with a different coefficient in region one than in region two? Or are local seconds always prefixed with the same constant coefficient in a given representation of a metric, as are, say, local centimeters. That doesn't seem right, otherwise all spacetimes would look like flat spacetime as seen through the lens of the explicit metric! Schwarschild's metric, e.g. http://en.wikipedia.org/wiki/Schwarzschild_metric trivially has the property that the prefix to dt is a function of r. The "dt" implicitly represents time as measured by a appropriate standard physical clock. So evidently not all such clocks are equal with respect to the metric -- time is "stretched" in some regions relative to others. Now, I can rephrase my question. Suppose we replaced "t" with t' = a(t)t , a( ) a positive increasing function. This can be interpreted as changing the units of time from "natural clock seconds", and not in a uniform way. How does the explicit form of the metric handle this? In this case, a possible answer seems trivial: we simply insert the inverse of the positive factor "a(t)"... better, a^-1(t') ... in the formal metric, and everything is fine. This is the point where I see a possible problem. We now have a function of variables labeled t',r,O (omega) which corresponds to the same spacetime as a different function of variables labeled t,r,O . So, what's the problem? We've just changed varables. The problem is, suppose we could demonstrate a second _different spacetime_ with the same variable labels t',r,O and the same formal metric. Is this a problem? I can't carry through my threat here, with the Schwarschild metric, but that was the point of my toy one dimensional manifold -- I tried to motivate such a possibility. OK, Ken, here is my current postulation for an answer: you _can't_ reconstruct a physical spacetime simply from a set of variable labels and an explicit form of a metric on such labels: there is an element missing. That element is a perscription of the relationship between the local differentials of the variables and standard local physical processes -- "so many wavelengths of a certain spectral transition", and so forth. If this perscription is the same everywhere, then it can simply be stated as "choice of units", as in, centimeters and seconds; if the perscription varies, then the additional information will be more complicated. But there is additional structure required to complete the model. Comment: the "standard local clock" may need clarification. Comment: my internal pendulum is again swinging towards supposing my question was close to being well-posed or at least well-posable, and that therefore those who merely dismissed it as containing elementary errors would have done better to remain silent, than open their mouth and remove all doubt. > GR treats all Frames of Reference's (FoR's) equally. > AE's law, G_uv = T_uv constrains the metrical > geometry, to solutions of the tensor g_uv satisfying > that law, that is why G_uv=T_uv is a called a field > equation. For example a simplistic solution follows > from G_uv=0 called the Schwarzchild Solution, that > I'm sure you're aware of, that provides metrics like, > > g_00 = 1 - 2m/r , g_rr = 1/g_00 etc. > > The reason for CS independence is because g_uv > is a tensor and can be properly transformed via > the usual procedure, > > g'_ab = (&x^u/&x'_a) (&x^v/&x'_b) g_uv > > to any other CS, using light-years, millimetres > polar, elliptical, cylindrical as you please, and > in time, seconds or dog-years, meaning the > physical reality is independent of units. Of course. I'm thinking however that the units must be stated, which is sufficiently trivial as to be invisible, and that if we allow arbitrary point to point fluctuations in the units, we may need some additional non-trivial statements to complete the specification, beyond the explicit form of the metric. As always, I could be wrong. Thanks, Ken. |