From: Chris Hillman Subject: Re: GR Challenges Date: 5 Feb 2001 19:04:31 GMT Newsgroups: sci.physics.research Summary: Solutions to Einstein's equations on spherical shells On Sun, 4 Feb 2001, Daryl McCullough wrote: > First, does anyone know of a solution to Einstein's equations > in which there is no matter anywhere except in a thin, spherical, > nonrotating, collapsing shell? Yes; indeed, I believe this is a standard example of Israel's thin shell formalism. I don't have any references for you off the top of my head (but Steve Carlip probably will, if he is reading this). However, I can suggest that anyone interested in collapsing thin shells first study two simpler types of models: 1. the simpler Darmois matching condition for an Oppenheimer-Snyder model (collapsing FRW dust ball region "matched" to an exterior Schwarzschild vacuum region), and then see if you can construct a collapsing spherical "thick shell" of FRW dust, i.e. with a Carter-Penrose diagram something like this (each point represents a two sphere of an appropriate radius) curvature singularity 88888888888 |** /\ |** / \ i^infty | ** / \ | *** / \ | ***/ \ i^0 ("r = infty") | **** / | ***** / | ***** / r= 0 | /****** / |/ ***** / | **** / | **** / | *** / | ** / | ** / i^-infty | * / | * / |* / |*/ |/ (For example, see the discussion of "Einstein vacuoles" in the textbook by Stephani.) (Hint: use the FRW dust with E^3 hyperslices for the shell of dust, and try to match to Schwarzschild vacuum at the outer boundary, and to Minkowksi vacuum at the inner boundary.) 2. static thin shells, for example a thin spherical shell matched to Minkowksi vacuum inside and Schwarzschild vacuum outside. (For example, see the problem book by Lightman et al.) > I know that outside the shell, things will look just like a > Schwarzschild geometry, but I'm particularly wanting to know what the > geometry *inside* the shell is like, especially after the shell has > collapsed past its Schwarzschild radius. For a perfectly spherical shell, I agree, it should be Minkowksi vacuum (it is of course best to use an appropriate polar spherical chart!) This doesn't mean it is inappropriate to continue the diagonal line "EH" above into the dust region and the Minkowksi region, because this has of course a global "teleological" definition! > Second, I'm wondering if there is any good guesses as to > how Hawking radiation modifies the Schwarzschild geometry > of a spherically symmetrical black hole. I'm not asking > for a full quantum-mechanical treatment, but instead just > a phenomenological model. I suspect the best phenomenological model at the -classical level- would be to ignore Hawking radiation entirely :-/ However, you might want to look for preprints by Frolov and others at LANL which investigate possible classical "phenomenological models" for black hole interiors and also "evaporating" holes. > My first guess would be that if the Hawking radiation is slow enough, > then we could approximate the geometry of a shrinking black hole by > letting the parameter "M" in the Schwarzschild geometry be a > slowly-varying function of time: > > ds^2 = (1-2M(t)/r) dt^2 - (1-2M(t)/r)^{-1} dr^2 If you had used an (ingoing) Eddington-Finkelstein chart here, you would have rediscovered the (ingoing) Vaidya solution! All you need to do is to start with such a chart and let m become an (arbitrary!) function of time! > But I don't really think that that can be right---the > parameter t means less and less as you approach > r = M, (because g_tt goes to zero there). > > Has there been any work on a metric describing a shrinking > black hole? Sure, the Vaidya solution is the one you want. That isn't suitable for a classical approximation to Hawking radiation, however; it models the classical emission of outgoing EM or other massless radiation (from a stellar model) or the contraction of spherical wavefronts of massless radiation onto a stellar model or black hole. See for example the archived collection of posts on this solution on my page http://www.math.washington.edu/~hillman/RelWWW/group.html Chris Hillman ============================================================================== From: Tom Roberts Subject: Re: GR Challenges Date: Tue, 6 Feb 2001 01:26:20 GMT Newsgroups: sci.physics.relativity,sci.physics.research Daryl McCullough wrote: > First, does anyone know of a solution to Einstein's equations > in which there is no matter anywhere except in a thin, spherical, > nonrotating, collapsing shell? Yes. Birkhoff's theorem applies. In any region of spacetime outside the collapsing shell the manifold is isometric to a region of Schwarzschild spacetime with M = constant; M is related to the total mass of the shell and its rate of collapse. In any region of the manifold inside the shell the manifold is isometric to Schw. spacetime with M=0, which is of course isometric to a region of Minkowski spacetime. Note that the boundary of each of these regions is changing with time. In essence inside the shell there is no warning about its approach (but one could see it using radar or other means). > I'm > particularly wanting to know what the geometry *inside* the > shell is like, especially after the shell has collapsed past > its Schwarzschild radius. The region inside the shell is flat. This holds even after the shell is smaller than its Schw. radius. In this case the inside region is of course inside the event horizon. Note this all holds even if the "mass shell" is replaced by a collapsing spherical wave of radiation. A hapless observer could sit in a flat region of spacetime and suddenly find herself crushed inside a black hole, with absolutely no warning (and no way to obtain any warning -- at least for the mass shell she could get warning via radar). > Second, I'm wondering if there is any good guesses as to > how Hawking radiation modifies the Schwarzschild geometry > of a spherically symmetrical black hole. My guess is that Bohr's correspondence principle holds for any observer in freefall who does not experience large tidal forces. Continuity implies this holds also for observers who do not experience enormous accelerations (say a few g or so, but likely this is so for accelerations of billions of g). This guess is supported by existing observations (but they are not at all exhaustive). > Has there been any work on a metric describing a shrinking > black hole? Not in GR -- there are strong theorems about black hole horizons. Basically no matter what happens, the total area of the event horizons of all black holes cannot decrease. That's a paraphrase; for the actual theorems see Hawking and Ellis, _The_Large_Scale_Structure_of_Space-Time_. Tom Roberts tjroberts@Lucent.com