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


by rjbeery
Tags: fall, infinite, nature
PeterDonis
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Jan10-13, 09:37 AM
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Quote Quote by stevendaryl View Post
I'm specifically talking about case number (2).
Ah, ok.

Quote Quote by stevendaryl View Post
The event horizon ISN'T a region of strong gravity in KS coordinates.
I think this is a somewhat misleading way of stating it. Whether or not gravity is "strong" at the EH seems to me to depend on the mass of the BH; a BH with a small enough mass could indeed have "strong" gravity (in the sense of strong spacetime curvature) at the horizon. But "strong" in this sense *is* a generally covariant notion; curvature invariants are the same regardless of which chart you compute them in. You can even compute them in the SC chart at the horizon if you take limits as r -> 2m.

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 (non-generally 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.
stevendaryl
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Quote Quote by PeterDonis View Post
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 (non-generally 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.
I agree with that paraphrase. Here's a thought experiment about gravity that I think is interesting, even though it might have very little practical use. In quantum scattering, at least in one course I took on the subject years ago, a common mathematical technique is to by-hand add time-dependence to the coupling constants. That is, you imagine that in the distant past, the coupling constant was 0, and that very slowly its strength increased with time to the current value. An example of such a slowly-increasing function might be [itex]\lambda = \dfrac{1}{2}\lambda_0 (1+arctanh(kt))[/itex] for a very small value of [itex]k[/itex]. The point of having a slowly changing coupling constant is that it (hopefully, anyway) allows you to understand the states of the coupled system as perturbations of the states of the uncoupled system.

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 covariant--they 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.
harrylin
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Jan10-13, 10:15 AM
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Quote Quote by stevendaryl View Post
I would think that the conclusion should not be "further discussion with you on this topic is useless" but rather "I should make more of an effort to clarify what I mean, since people seem confused by it."
Once more, you seem to have understood (and without any confusion) what I think to be a main point of the recent discussions related to black holes and I suggested that perhaps you can explain it better than me (and so you did, although you approached it from a different angle). But IMHO 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. So, if you think that something useful can come out of further discussion of the same things, good luck with it.
PAllen
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Jan10-13, 10:54 AM
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Quote Quote by PeterDonis View Post
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 (non-generally 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.
To serve as an argument that you have to modify GR, I did at some point (I think in this thread) throw out a generally covariant addition to the field equations that I thought accomplished this. I called it, I think, the "universe boundary law". It is:

- 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.
PeterDonis
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Quote Quote by PAllen View Post
- closed manifolds are rejected; null infinity must be well defined.
The problem with this is that it doesn't just rule out closed manifolds; it rules out all manifolds that aren't asymptotically flat. You can only define null infinity in an asymptotically flat manifold. For example, an open FRW manifold such as the one currently used in the "best fit" model for our universe has no null infinity.

Quote Quote by PAllen View Post
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.
This doesn't necessarily have to change GR; you could (I believe) construct the classical limit of such an "evaporation" model by using an SET with a sufficiently large negative pressure. This would violate several energy conditions, so such SETs are usually considered "unphysical", but when quantum effects are included it's no longer clear that the energy conditions always have to hold anyway.
PeterDonis
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Jan10-13, 11:21 AM
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Quote Quote by PAllen View Post
I, of course, feel that there is no physical basis for an additional law like this - it only serves to violate the equivalence principle.
Not only that, it appears to require that local physics--whatever it is that, locally, prevents an event horizon from forming--must "know" the entire future of the spacetime, so that local events can "know" when they are getting close to losing connection with null infinity (more precisely, with *future* null infinity, which is the point).
PAllen
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Jan10-13, 11:35 AM
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Quote Quote by PeterDonis View Post
The problem with this is that it doesn't just rule out closed manifolds; it rules out all manifolds that aren't asymptotically flat. You can only define null infinity in an asymptotically flat manifold. For example, an open FRW manifold such as the one currently used in the "best fit" model for our universe has no null infinity.
I'm not sure that's right. The technical definition of horizon uses null infinity. I've seen claims in the literature that BH's are only technically undefinable for closed spacetimes. If there is no null infinity, then all singularities are technically naked.

[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. ]
Quote Quote by PeterDonis View Post

This doesn't necessarily have to change GR; you could (I believe) construct the classical limit of such an "evaporation" model by using an SET with a sufficiently large negative pressure. This would violate several energy conditions, so such SETs are usually considered "unphysical", but when quantum effects are included it's no longer clear that the energy conditions always have to hold anyway.
But you would need to add a rule that says any SET the produces a horizon is illegal. I call that modifying GR.
DaleSpam
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Quote Quote by harrylin View Post
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 Quote by stevendaryl View Post
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

[itex]2GM/c^2 < r < \infty[/itex]
[itex]- \infty < t < \infty[/itex]
[itex]0 \leq \theta \leq \pi[/itex]
[itex]0 \leq \phi < 2 \pi[/itex]

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.
PeterDonis
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Jan10-13, 03:48 PM
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Quote Quote by PAllen View Post
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 Quote by PAllen View Post
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 Quote by PAllen View Post
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.
PeterDonis
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Quote Quote by stevendaryl View Post
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.
That seems reasonable to me.

Quote Quote by stevendaryl View Post
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 Quote by stevendaryl View Post
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").
zonde
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Quote Quote by DaleSpam View Post
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.
TrickyDicky
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Quote Quote by PeterDonis View Post
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 Quote by PeterDonis View Post
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).
stevendaryl
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Quote Quote by PeterDonis View Post
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).
PeterDonis
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Quote Quote by TrickyDicky View Post
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).
PeterDonis
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Quote Quote by stevendaryl View Post
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.
zonde
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Quote Quote by stevendaryl View Post
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 Quote by PeterDonis View Post
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.
TrickyDicky
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Quote Quote by pervect View Post
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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