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On the nature of the infinite fall toward the EH 
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#487
Jan1013, 09:37 AM

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PF Gold
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The way I would put the point I think you're trying to make here is that if you want to claim that GR breaks down at the EH, you have to be relying on some *other* notion than "strong gravity" in the above sense. And since nobody has come up with any such notion that picks out the EH in all cases (i.e., regardless of the mass of the hole) *and* is generally covariant, it seems like any claim that GR always breaks down at the EH must violate general covariance; it must rely on properties of particular coordinate charts (such as the SC chart becoming singular at the EH). Whether or not the (nongenerally covariant) notion you pick deserves the name "strong gravity" (in some other sense than the covariant sense I gave above) seems to me to be a side issue. 


#488
Jan1013, 10:10 AM

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Anyway, suppose you tried to do that with gravity. You start off with Minkowsky spacetime and no gravity. Just particles floating around. Then pick a frame (to make this work, you have to choose a frame to serve as your standard for time). Write the field equations in this frame. Then modify the equations as follows: replace the constant G by a function G(t), which starts off at [itex]G(\infty) = 0[/itex] and smoothly increases to [itex]G(+\infty) = G_0[/itex], where [itex]G_0[/itex] is the current value of G. Now, these equations are no longer covariantthey have a preferred coordinate system. However, they are still legitimate differential equations. We can still solve them, numerically at least. What I would expect to be the case is that for very large values of [itex]t[/itex], the solutions would settle down to a solution of the unaltered Einstein Field Equations. However, it's not clear to me that you would ever get the interior of a black hole event horizon. So it would settle down to a solution of the EFE that's missing some regions. Or it seems possible that it would. It's sort of like the case with perturbation theory in physics. Certain solutions (bound states for example) can't be obtained perturbatively. 


#489
Jan1013, 10:15 AM

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#490
Jan1013, 10:54 AM

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PF Gold
P: 5,059

 closed manifolds are rejected; null infinity must be well defined.  any points on the manifold not connected to null infinity, or that are part of null infinity, are removed from any solution. Note, an open subset of a manifold is still a valid manifold. It will still everywhere satisfy the EFE, if the 'trial solution' did. This new law is strictly coordinate independent, thus manifestly generally covariant. Thus, I think you must add new rules to the EFE to remove horizons and interiors, but it can be done in a generally covariant way. It could be argued that this is in the same spirit as energy conditions that effectively reject mathematically valid Einstein tensors (= stress energy tensor). I, of course, feel that there is no physical basis for an additional law like this  it only serves to violate the equivalence principle. A less artificial way to change GR is to add evaporation to it in such a way as to guarantee that no event loses connection to null infinity before evaporation completes. 


#491
Jan1013, 11:17 AM

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PF Gold
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#492
Jan1013, 11:21 AM

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PF Gold
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#493
Jan1013, 11:35 AM

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PF Gold
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[edit: Here is a reference showing null infinity for De Sitter space: http://www.math.miami.edu/~galloway/...qg7_11_021.pdf which suggests my comment about 'closed' needs clarification. ] 


#494
Jan1013, 12:49 PM

Mentor
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#495
Jan1013, 12:55 PM

Mentor
P: 17,202

The only way to get around that is to modify the EFE or impose some sort of adhoc restriction to the set of admissible manifolds. 


#496
Jan1013, 03:48 PM

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PF Gold
P: 6,126

However, I'm not sure I was right to say that an open (or flat) FRW spacetime doesn't have a future null infinity; I've been trying to find a link to a Penrose diagram of that spacetime but haven't been able to. I'm not sure, though, that being spatially compact is equivalent to not having a future null infinity. That's what I think needs further thought. 


#497
Jan1013, 04:04 PM

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PF Gold
P: 6,126

I know that seems weird, since it's obviously possible to express the vacuum exterior region as a perturbation of Minkowski spacetime. But that region is not geodesically complete; so the region we're expressing as a perturbation is not a perturbation of *all* of Minkowski spacetime, it's only a perturbation of a *portion* of Minkowski spacetime; in the simplest case, it's the portion of Minkowski spacetime outside some radius r from a chosen central point. Which leaves the question of what is the initial condition of the region *inside* that radius? If the region inside radius r starts out in a nonvacuum initial state, then the complete initial state is no longer Minkowski spacetime. But if the region inside radius r starts out as vacuum, then as I said above, I don't think you can construct a solution that turns that vacuum interior into a black hole interior by varying G with time; but you could, perhaps, turn that "vacuum" interior (with particles floating around but no gravity) into a nonvacuum interior with a massive gravitating body in it (if the "particles" have enough mass to form such a body once gravity is "turned on"). 


#498
Jan1013, 11:33 PM

PF Gold
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#499
Jan1113, 05:00 AM

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#500
Jan1113, 06:19 AM

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#501
Jan1113, 09:26 AM

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PF Gold
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#502
Jan1113, 09:28 AM

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PF Gold
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#503
Jan1213, 12:50 AM

PF Gold
P: 1,376

So if one considers possibility that EH does not form then he has to add some parameter that can indicate nearness of EH. Basically it is gravitational potential that can do that. And only then one can make speculations like  maybe density of available quantum states goes down as we go down in gravitational potential or anything else like that. 


#504
Jan1213, 05:14 AM

P: 3,035

It is true that observing stars near the center of our galaxy at Sagittarius A*, orbiting at very high speeds around a common focus is highly suggestive of something very massive there, if we add that this very spot is relatively dark, it is reasonable to suspect there must be "something like a SMBH" there. And it is expected that in a not very long time we'll have more relevant data to help us discern between a black hole or "something else" that noone at this point has a reasonable theory for. One thing I don't understand very well is that given the huge mass (4.3 million suns) calculated, in a very compact space, why there seems to be no gravitational lensing effects on the stars closest to Sagittarius A*. We do observe this effects in clusters in wich the mass is much more disperse. 


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