Any coordinate system in GR?
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From: I.Vecchi on 28 Aug 2006 14:42 Thanks to you and the others for your feedback. Without it it would have taken me a substantially larger amount of time, effort and wit to reach whatever understanding of the issues I may have now. One more consideration. A clock is, by its very nature, a massive, extended object. Beyond the horizon no information can be tranferred from the bottom to the top of the clock. It is not clear to me that any clock would keep working under such conditions. So, thinking about it, I am no longer sure that the statement that "nothing special" happens at the horizon is accurate. Clocks may stop working there. Actually, this is not a purely speculative statement, since the effect of a very strong gravitational field could be tested on physical clocks also in the exterior domain. If physical clocks stop, what happens to time? Cheers, IV
From: carlip-nospam on 28 Aug 2006 15:17 In sci.physics I.Vecchi <tttito(a)gmail.com> wrote: > 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. One reference are Hawking and Ellis, _The Large Scale Structure of Space-Time_, p. 155. No proof is given. If you look at De Felice and Clarke, _Relativity on curved manifolds_, section 10.4, you will find a careful discussion of maximal analytic extensions, along with a statement that if such an extension exists, it is unique. This is not my specialty, and I don't know off hand where to find the exact proofs, but I am certainly inclined to believe these authors. >> Which part do you want to give up? > In any given case, the one that does not fit observation. A black hole formed from collapsing matter is not, of course, described by the full Kruskal-Szekeres extension. That's because the spacetime is not empty -- the region containing matter can't be described by a solution of the vacuum field equations. If you solve the equations for collapsing matter, you'll find a portion of regions I and II of the Kruskal-Szekeres extension (the "black hole" part) glued to a non-vacuum solution, whose nature depends on exactly what sort of collapsing matter you're describing. Is this all you're worried about? Steve Carlip
From: I.Vecchi on 28 Aug 2006 15:44 I.Vecchi ha scritto: > One more consideration. A clock is, by its very nature, a massive, > extended object. Beyond the horizon no information can be tranferred > from the bottom to the top of the clock. It is not clear to me that any > clock would keep working under such conditions. So, thinking about it, > I am no longer sure that the statement that "nothing special" happens > at the horizon is accurate. Clocks may stop working there. Actually, > this is not a purely speculative statement, since the effect of a very > strong gravitational field could be tested on physical clocks also in > the exterior domain. If physical clocks stop, what happens to time? Nonsense, IV
From: Koobee Wublee on 28 Aug 2006 16:55 JanPB wrote: > No, it doesn't. I have just CALCULATED it from YOUR OWN metric. If you > want to claim otherwise, you have to find an error in my calculation > (you can't). The metric indicates a distortion in space. Thus, after computing the area in curved space, the radius does not necessarily follow the computation in flat space. Your mistake is to mix curved space with flat space. In the meantime, you still owe me answers on the following. With the following spacetime, ds^2 = c^2 dt^2 / (1 + r / K) - (1 + r / K) (K^4 / r^4) dr^2 - (K^4 / r^2) (1 + r / K)^2 dO^2 How do you specify a distance 2 AU from the center of the sun? Given the following two independent spacetime where K is an integration constant, ** ds^2 = c^2 dt^2 / (1 + K / r) - (1 + K / r) dr^2 - (r + K)^2 dO^2 ** ds^2 = c^2 (1 - K / r) dt^2 - dr^2 / (1 - K / r) - r^2 dO^2 How do you know which of the two is the first form? And why? Or better yet, let's write the above equations into a more general form, ** ds^2 = c^2 dt^2 / (1 + K / (r + Q)) - (1 + K / (r + Q)) dr^2 - (r + Q)^2 dO^2 where (K = 2 G M / c^2), (Q = any constant) How do you know Q has to be (- K) and not some other number?
From: I.Vecchi on 28 Aug 2006 17:01
carlip-nospam(a)physics.ucdavis.edu ha scritto: > In sci.physics I.Vecchi <tttito(a)gmail.com> wrote: > > > 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. > > One reference are Hawking and Ellis, _The Large Scale Structure of > Space-Time_, p. 155. No proof is given. If you look at De Felice > and Clarke, _Relativity on curved manifolds_, section 10.4, you will > find a careful discussion of maximal analytic extensions, along with > a statement that if such an extension exists, it is unique. This is > not my specialty, and I don't know off hand where to find the exact > proofs, but I am certainly inclined to believe these authors. > > >> Which part do you want to give up? > > > In any given case, the one that does not fit observation. > > A black hole formed from collapsing matter is not, of course, described > by the full Kruskal-Szekeres extension. That's because the spacetime > is not empty -- the region containing matter can't be described by a > solution of the vacuum field equations. If you solve the equations > for collapsing matter, you'll find a portion of regions I and II of > the Kruskal-Szekeres extension (the "black hole" part) glued to a > non-vacuum solution, whose nature depends on exactly what sort of > collapsing matter you're describing. > > Is this all you're worried about? Thanks for the feedback and references, which I will look up. A "careful discussio of maximal analytic extensions" sounds like something I should read. IV |