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## On the nature of the "infinite" fall toward the EH

 Quote by harrylin everything that people currently have in mind has already been discussed several times and the last discussions appear to not have helped anyone with anything.
I agree. There simply is no substitute for actually learning the math. In the end, discussions on internet forums just can't provide a shortcut.

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 Quote by stevendaryl I have to disagree a little bit here. The field equations by themselves describe spacetime dynamics within a region of spacetime. They don't say anything about what regions must exist, do they? So in Schwarzschild coordinates, there is a region of spacetime described by Schwarzschild coordinates $2GM/c^2 < r < \infty$ $- \infty < t < \infty$ $0 \leq \theta \leq \pi$ $0 \leq \phi < 2 \pi$ The field equations by themselves don't say anything about the existence of other regions. Now, you can argue physically that there should be other regions besides this one, using the principle of geodesic completeness, or by considering how a star collapses, or something. But the field equations themselves don't say what regions of spacetime exist, they only describe how dynamics works within a region. Or at least, it seems that way to me.
Sure, but SC are not the only coordinates, and many of those other coordinates are equally valid solutions of the EFE which do cover regions inside the horizon. Due to the fact that a given chart maps an open subset of the manifold, the existence of any chart covering the interior implies that those events are part of the whole manifold, while the fact that SC doesn't cover them does not imply the opposite.

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

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 Quote by PAllen The technical definition of horizon uses null infinity.
Yes, I agree. I wasn't disputing your definition of a black hole event horizon; I was only saying that a black hole event horizon can't exist in a spacetime that doesn't have a null infinity. Other kinds of horizons can exist (such as cosmological horizons), but not black hole event horizons.

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.

 Quote by PAllen my comment about 'closed' needs clarification.
I think the proper term would be "compact", or more precisely "spatially compact"--i.e., any spacelike slice is a compact manifold. Spacelike slices of de Sitter spacetime are, I believe, not compact.

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.

 Quote by PAllen But you would need to add a rule that says any SET the produces a horizon is illegal. I call that modifying GR.
The rule couldn't be that simple, since a vacuum SET allows a horizon to form. I was actually thinking of something along the lines of: what if it were possible to prove that, when quantum effects are included, the "effective" SET at the classical level is such that a horizon is always prevented from forming (because the closer a horizon comes to forming, the larger the negative pressure is in the effective SET). This wouldn't require modifying the EFE or any of the postulates of GR; it would just be a (rather unexpected, and unlikely in my view, but possible) consequence of how the underlying quantum laws produce an effective SET at the classical level.

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 Quote by stevendaryl What I would expect to be the case is that for very large values of $t$, the solutions would settle down to a solution of the unaltered Einstein Field Equations.
That seems reasonable to me.

 Quote by stevendaryl 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.
Such a solution would be geodesically incomplete, whereas the initial state (Minkowski spacetime) is geodesically complete. So I'm not sure this would work. What I think would happen instead is that you would not be able to construct a solution with a time-varying G that contained the Schwarzschild exterior. See below.

 Quote by stevendaryl It's sort of like the case with perturbation theory in physics. Certain solutions (bound states for example) can't be obtained perturbatively.
It's true that the maximally extended Schwarzschild solution to the EFE is something like a soliton; I believe some physicists have actually used that term to describe it. That would mean it's not "reachable" as a perturbation of Minkowski spacetime.

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 non-vacuum 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 non-vacuum interior with a massive gravitating body in it (if the "particles" have enough mass to form such a body once gravity is "turned on").

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 Quote by DaleSpam There simply is no substitute for actually learning the math. In the end, discussions on internet forums just can't provide a shortcut.
It depends on what do you mean by "math" and for what purpose do you need it.

 Quote by PeterDonis I was actually thinking of something along the lines of: what if it were possible to prove that, when quantum effects are included, the "effective" SET at the classical level is such that a horizon is always prevented from forming (because the closer a horizon comes to forming, the larger the negative pressure is in the effective SET).
I have often considered this very "what if", but you have expressed it in a way clearer than anything I've managed to write myself.
 Quote by PeterDonis This wouldn't require modifying the EFE or any of the postulates of GR; it would just be a (rather unexpected, and unlikely in my view, but possible) consequence of how the underlying quantum laws produce an effective SET at the classical level.
Agreed. But it is refreshing to see that unlikely as it may be it is at least considered as a possibility something (a plausible way of preventing not only singularities but also event horizons from forming) that was not even admitted as a mathematically and/or physically valid scenario in previous discussions. (Even if it was in the literature as shown by PAllen's references by Krauss et al. in the first posts in this thread).

 Quote by PeterDonis what if it were possible to prove that, when quantum effects are included, the "effective" SET at the classical level is such that a horizon is always prevented from forming (because the closer a horizon comes to forming, the larger the negative pressure is in the effective SET). This wouldn't require modifying the EFE or any of the postulates of GR; it would just be a (rather unexpected, and unlikely in my view, but possible) consequence of how the underlying quantum laws produce an effective SET at the classical level.
It's hard for me to see how that could work, because locally, there is nothing that indicates that you're near an event horizon (in the case of a large enough black hole).

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 Quote by TrickyDicky unlikely as it may be it is at least considered as a possibility something (a plausible way of preventing not only singularities but also event horizons from forming) that was not even admitted as a mathematically and/or physically valid scenario in previous discussions. (Even if it was in the literature as shown by PAllen's references by Krauss et al. in the first posts in this thread).
Just to be clear, it's a "possibility", but I don't think it's given much consideration by mainstream physicists. Just having an SET that violates the energy conditions is not enough, as the rebuttals to the Krauss et al. paper show (the effective SET associated with black hole evaporation violates energy conditions, but the rebuttals show that evaporation by itself can't prevent a horizon from forming). You would need an SET that *grossly* violates the energy conditions, *and* the violation would need to be highly sensitive to how close a horizon was to forming, so to speak--meaning that the violation would need to be highly sensitive to a *nonlocal* property, since locally there is no way to tell how close a horizon is to forming, as stevendaryl pointed out (see my response to him for some further thoughts).

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 Quote by stevendaryl locally, there is nothing that indicates that you're near an event horizon (in the case of a large enough black hole).
True, and as I just responded to TrickyDicky, this makes the mechanism I was referring to highly unlikely. But since there are weird nonlocalities in quantum mechanics, I don't think one could make a blanket statement that it is impossible at our current state of knowledge about quantum gravity. Betting odds are another matter, of course.

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 Quote by stevendaryl It's hard for me to see how that could work, because locally, there is nothing that indicates that you're near an event horizon (in the case of a large enough black hole).
 Quote by PeterDonis True, and as I just responded to TrickyDicky, this makes the mechanism I was referring to highly unlikely. But since there are weird nonlocalities in quantum mechanics, I don't think one could make a blanket statement that it is impossible at our current state of knowledge about quantum gravity. Betting odds are another matter, of course.
In GR there is nothing that indicates that you're near an event horizon. And yet globally event horizon have rather observable consequences (black hole).

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.

 Quote by pervect But lets move a bit onto the observational side and away from the math for a little bit. There's clearly something very massive and rather dark at the center of our galaxy - we can see the orbits of stars around - something. http://arxiv.org/abs/astro-ph/0210426 "Closest Star Seen Orbiting the Supermassive Black Hole at the Centre of the Milky Way"
I've rescued this post from the beginning of this long thread because I agree its healthy sometimes to move away for a while from the purely mathematical side to what is actually observed, if only to put things in perspective.
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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 Quote by TrickyDicky I've rescued this post from the beginning of this long thread because I agree its healthy sometimes to move away for a while from the purely mathematical side to what is actually observed, if only to put things in perspective. 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.
I don't know the current status of observations of this, but there is apparently work being proposed.

See for instance http://arxiv.org/abs/1204.2103

 The massive black hole at the Galactic center Sgr A* is surrounded by a cluster of stars orbiting around it. Light from these stars is bent by the gravitational field of the black hole, giving rise to several phenomena: astrometric displacement of the primary image, the creation of a secondary image that may shift the centroid of Sgr A*, magnification effects on both images. The near-to-come second generation VLTI instrument GRAVITY will perform observations in the Near Infrared of the Galactic Center at unprecedented resolution, opening the possibility of observing such effects. Here we investigate the observability limits for GRAVITY of gravitational lensing effects on the S-stars in the parameter space [DLS,gamma,K], where DLS is the distance between the lens and the source, gamma is the alignment angle of the source, and K is the source apparent magnitude in the K-band. The easiest effect to be observed in the next years is the astrometric displacement of primary images. In particular the shift of the star S17 from its Keplerian orbit will be detected as soon as GRAVITY becomes operative. For exceptional configurations it will be possible to detect effects related to the spin of the black hole or Post-Newtonian orders in the deflection.