Are Finkelstein/Kruskal "interior black hole solution" compatible with Einstein's GR?by harrylin Tags: eep, finkelstein, kruskal, rindler coordinates 

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Dec512, 02:27 PM

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http://www.physicsforums.com/showthread.php?p=4185579 



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#99
Dec512, 03:08 PM

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Typically, theories of physics tell you what follows from hypothesized initial conditions. Usually, the theories don't tell you what the initial conditions are, you have to find those out empirically. It would be pretty weird if GR only applied to our universe. 



#100
Dec512, 03:20 PM

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(First, a quick note: "near the black hole" is still vacuum. The black hole region is vacuum at the horizon, and all the way down to r = 0. But I think that's a minor point compared to what I'm going to say below.) Suppose we want to solve the Einstein Field Equation subject to the following conditions: (1) The spacetime is spherically symmetric. (2) The spacetime is vacuum everywherei.e., there is no matter *anywhere*, ever. The complete solution to the EFE under these conditions includes an exterior region (which I'll call region I), a black hole region (region II), a second exterior region (region III), and a white hole region (region IV). The solution doesn't "start near a black hole"; it doesn't "start near" anywhere. It's just the complete solution we get when we impose those conditions ("complete" meaning "including all possible regions which are indicated by the math, whether they are physically reasonable or not"). Suppose we want to solve the Einstein Field Equation subject to the following somewhat different conditions: (1') The spacetime is spherically symmetric. (2') On some spacelike slice, the spacetime is vacuum for radius > R_0 (where R_0 is some positive value), but is *not* vacuum for radius <= R_0; instead, the region r <= R_0 on this spacelike slice is filled with dust (where "dust" means "a perfect fluid with positive energy density and zero pressure") which is momentarily at rest. (3') We are only interested in the spacetime to the future of the spacelike slice given in #3. The complete solution we get when we impose these conditions is what I'll call the "modernized OppenheimerSnyder model" ("modernized" to avoid any concerns about whether or not it was the model OS originally proposed; this model is described, for example, in MTW). This spacetime has three regions: an exterior vacuum region (which I'll call region I'), a black hole interior vacuum region (region II'), and a nonvacuum collapsing region (region C'). There is no white hole region, and no second exterior region, in this spacetime. Now, in the vacuum regions I' and II', the solution of the EFE is the vacuum solution: that is, it is *exactly the same* as the solution in the corresponding portions of regions I and II. Another way of saying this: if I describe regions I and II in a suitable coordinate chart, and regions I' and II' in a suitable coordinate chart, I can identify an open set of coordinate values in regions I and II that meet the following conditions: (A) The coordinate values are exactly the same as the ones in regions I' and II'; and (B) The invariant quantities at each corresponding set of coordinate values (I <> I', and II <> II') are identical. A fairly common shorthand, I believe, for what I've said above is that region I' is isometric to a portion of region I, and region II' is isometric to a portion of region II. Or, speaking loosely, regions I' and II' can be thought of as "pieces" of regions I and II that have been "cut and glued" to region C'. Hopefully all this makes somewhat clearer how the term "solution" is being used, and what it means to say that "the same solution" appears in different models. 



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Dec512, 03:24 PM

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#102
Dec512, 03:35 PM

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Maybe I should expound a bit more on what I'm looking for here. The standard view of this scenario is that the two cases are exactly parallel: in both cases, the accelerated observer (Eve, Eve'), because of her proper acceleration, is unable to observe or explore a region of spacetime that the freefalling observer (Adam, Adam') can. The physical criterion that distinguishes them is clear, and is the same in both cases (zero vs. nonzero proper acceleration). You are claiming that, contrary to the above, the cases are different: Adam is "privileged" in the first case, but Eve' is in the second. So I'm looking for some criterion that picks out Adam in the first case, but picks out Eve' in the second; in other words, something that applies to Adam but not Eve, and applies to Eve' but not Adam'. The only criterion I have so far is "moves in a straight line according to my chosen coordinates", but that only pushes the problem back a step: what is it that applies to the coordinates of Adam but not Eve, *and* to those of Eve' but not Adam'? I haven't seen an answer yet. 



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Dec512, 04:52 PM

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#106
Dec612, 06:24 AM

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If you want to distinguish the two then I would suggest "constant of integration" for the posthoc constants and "boundary condition" for the apriori constants. Under that categorization (which I wouldn't use) I would agree that the curvature at the horizon arises from a constant of integration rather than a boundary condition. You can always change a constant of integration into a boundary condition by changing the order of operations. 



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Dec612, 08:52 AM

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Dec612, 09:15 AM

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