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#235
Oct1611, 12:07 AM

PF Gold
P: 1,376

You see the problem is because black hole does not form at once. It starts as a small black hole at the center of gravitating body and then grows larger. But what are conditions for formation of this small black hole? Say we can compare the same massive body in two different situations. in one case it is in empty universe and in the other case it is inside large cloud of dust. How gravity around the body changes in those two situations? EDIT: There other things as well. One is question about initial conditions (how we can arrive at situation we are considering). Other is changes caused by increase of density  heating up and consequent expansion and cooling of individual stars. But it's hard to discuss all questions at once. 


#236
Oct1611, 12:56 AM

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#237
Oct1611, 01:52 AM

PF Gold
P: 1,376

The other way how you can have "different infinities" is by redefining function under consideration. This is closer to different simultaneities as it seems to me. So I will describe this in more detail. Lets say that we have function f(x)=g(x)+h(x) and it approaches infinity as x>x0. So how we can define g(x) and h(x)? There are three possibilities: 1) g(x) and h(x) both approach infinity as x>x0. 2) g(x) approaches infinity as x>x0 but h(x) approaches some finite value h0. 3) h(x) approaches infinity as x>x0 but g(x) approaches some finite value g0. Obviously we are not redefining infinity across 1), 2) and 3) but functions g(x) and h(x). And this example is useful for another purpose as well. Let's say that f(x) is roundtrip time for signal moving at c toward point that is at radius x from center of black hole but g(x) and h(x) are time of forward trip and backward trip respectively. Then we can identify case 1) as frozen star case 2) as white hole and case 3) as black hole While we can extrapolate h(x) in case 2) and g(x) in case 3) even after x has reached x0 physically meaningful is only f(x) that gives roundtrip time. And that is cornerstone of relativity. (That might be the reason why Einstein was not taking black holes seriously.) 


#238
Oct1611, 07:58 AM

Mentor
P: 17,535

If this is what you meant then I agree, otherwise could you clarify your meaning? 


#239
Oct1611, 09:35 AM

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#240
Oct1611, 09:42 AM

P: 1,115

What 'essence' or geometric 'object' can be entirely absent >= r_{b}, yet there strongly for r_{b}>r>r_{a}, so as to explain it? And what's more it has to be shown to be cumulative in effect, and not a mere 'blip' that leaves no trace on exit past r<r_{a}, so to speak. Tall order indeed! Sole uniquely present identity I can think of might be divergence, but that seems most unlikely a solution, and in itself creates another issue. Namely, if divergence is truly absent exterior to r_{b}, this gives the lie to those claiming that in GR 'gravity truly gravitates'. What say you sir? [One final comment: in #222 you mentioned agreement between yourself and DrGreg's finding in http://www.physicsforums.com/showpos...5&postcount=10, but I read him there as saying interior length are as at infinity, once the metric is applied. A misunderstanding?] 


#241
Oct1611, 10:58 AM

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#242
Oct1611, 11:53 AM

PF Gold
P: 1,376

Look if you say that function asymptotically approaches value a as it's argument approaches infinity then it means a is the limit no matter what you do with the argument. Maybe there is some confusion with my argument that I can still clear up. I can explain my argument in two steps rather than one: 1) in Schwarzschild metric interior of black hole is completely disconnected from exterior because there is no future beyond infinite future and there is no past before infinite past (where you could hope to connect interior with exterior). 2) there can be any number of spacetime patches that are completely disconnected from our spacetime. There can be even any number of universes that are completely disconnected from our universe. As they do not affect our reality in any way it can be stated that they are not real or alternatively they do not exist. 


#243
Oct1611, 12:12 PM

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(See post #75, the "inside" is the blue region, the observer is the black line, the red and green lines specify Rindler coordinates.) 


#244
Oct1611, 12:46 PM

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It is true that the interior Schwarzschild *coordinate patch* is disconnected from the exterior Schwarzschild coordinate patch; that is what is meant by statements about the "infinite future" and whether anything is "beyond" it. But that statement does not support your argument, because it only applies to a particular coordinate system; it is not a statement about the underlying geometry, which is what is important for the physics. 


#245
Oct1611, 12:48 PM

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#246
Oct1611, 01:09 PM

PF Gold
P: 1,376

So I am stating that the same "infinite future" applies to set of observers that have parallel time dimension i.e. it applies to global coordinate system as a whole. I suppose that you can come up with some nasty example where I would have hard time defining "observers with parallel time dimension" but as we are talking about spherically symmetric coordinate systems centered on black hole I can always come up with Euclidean coordinate system after factoring out time dilation (and change in radial length unit if something like that shows up). 


#247
Oct1611, 01:36 PM

PF Gold
P: 1,376

But I have one question about Rindler coordinates. It seems to me that time dimension can not be arbitrarily extended for Rindler observer, is it right? There is certain point ahead of Rindler observer (in flat coordinates) where Rindler observer reaches speed of light and time stops for him. And because of this it's hard for me to associate real observers with Rindler observer. 


#248
Oct1611, 01:47 PM

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#249
Oct1611, 01:51 PM

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#250
Oct1611, 03:32 PM

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#251
Oct1611, 03:52 PM

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The existence of one congruence which share the same "infinite future" does not in any logical way forbid the existence of another set of congruences which share a different "infinite future". Your line of reasoning seems to be that there is a timelike congruence which ends up in the usual "infinite future" therefore all timelike congruences must end up in the same "infinite future". This is not sound logic. 


#252
Oct1611, 04:42 PM

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P: 6,248

* The entire stressenergy tensor is the "source" on the RHS of the Einstein Field Equation (EFE). The LHS is the Einstein tensor, which is defined in terms of the Ricci tensor, which is a contraction of the Riemann curvature tensor. * For the nonvacuum region in question, a reasonable stressenergy tensor would be a "perfect fluid" with nonzero pressure: [tex]T_{00} = \rho[/tex] [tex]T_{11} = T_{22} = T_{33} = p[/tex] where [itex]\rho[/itex] is the energy density and [itex]p[/itex] is the pressure. The offdiagonal components are all zero. For the spacetime to be static and spherically symmetric, both [itex]\rho[/itex] and [itex]p[/itex] must be functions of the radial coordinate r only. Also, [itex]\rho[/itex] should be positive everywhere in the nonvacuum region, but [itex]p[/itex] will not be; a negative pressure is a tension, and in order for the nonvacuum region to be stable, there will have to be tension somewhere, to keep it from falling apart (a positive pressure everywhere would be possible only if the nonvacuum region went all the way down to the center, r = 0, without any hollow interior). * The fact that the stressenergy tensor (and therefore the Einstein tensor) is divergenceless is one way of expressing local conservation of energy; the divergence of the stressenergy tensor, over some small piece of spacetime, is just the net energy going in or out of that piece of spacetime. The divergence being zero just means energy in equals energy out; i.e., energy is conserved. So you're right that this probably doesn't help much for this particular problem. * You can still define a "potential" in the nonvacuum region, because you can do that for any spherically symmetric geometry. And you can still view the gradient of this potential as being the "acceleration due to gravity". I *think* that the effects on the metric, including its tangential part, in the nonvacuum region can be expressed in terms of the potential, but I'm not certain; the metric coefficients may be more complicated than that in the nonvacuum region. The only metric coefficient that I'm pretty well certain can be expressed entirely in terms of the potential is [itex]g_{00}[/itex], the timelike coefficient. 


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