Graphical example of BH formation by PAllen

PAllen

Sure you can choose not to worry about the true horizon. Once a black hole is stable, with no more matter falling in for for quite a while, they coincide to any limit of measurement. Apparent horizons are more complex to derive for the scenarios under discussion. However, we can say the following:

1) For the collapsing spherical shell, by the time the the shell is inside its SC radius, the apparent horizon is at the SC radius. At this time (as observed by interior observers), there is not yet any singularity, nor (necessarily) any high density of matter (if the shell is enormous enough). Note, it is guaranteed that a singularity will form as the shell cannot stop collapsing at this point. (per GR of course).

2) For the collapsing star cluster, a similar observation is true. As soon as the cluster is within its SC radius, we know the apparent horizon is at the SC radius. There is no requirement that any stars have collided, nor any singularity exist yet (for interior observers). Again, per GR, it is guaranteed that a singularity will form.

The only thing I can't fill in (with my available time and resources) is the early history of the apparent horizon in these two scenarios. The true horizon is easier to derive general features of using general principles.

zonde

Let's take a closer look at SC solution and how much does it applies to collapsing body.
SC solution describes gravity around static (existing in equilibrium state) body. Now we take series of SC solutions with the same mass and progressively smaller radius. As mass is the same and radius shrinks it seems like we can claim that this series of SC solutions describes collapsing body.
But each solution for certain radius describes static body. And in order for the same body to go from larger radius to smaller radius and then reach equilibrium state at smaller radius it should release binding energy (reducing it's mass by appropriate amount). And that makes quite different series of SC solutions.

So in order to claim that this series of SC solutions with the same mass and progressively smaller radius describe collapsing body we have to assume equivalence between
compressed smaller body (less particles) that has not yet released binding energy
and
bigger body (more particles) at the same radius that has already released binding energy.

Does it make sense so far?

PAllen

There is no need for such complexity unless you reject pure math: Birkhoff's theorem. Assuming spherical symmetry, and any shell of matter just inside its SC radius, it is guaranteed that the true horizon is at the SC radius and the apparent horizon is inside it by some infinitesimal amount. If you don't want to accept this, you have no choice but to admit that you reject GR, because this is pure mathematical proof. Unlike the singularity theorems, Birkhoff's theorem makes no assumptions about 'reasonable matter states'. Nothing is assumed except the Einstein Field equations.

TrickyDicky

Well yes, that is true as long as you don't count as assumption an (rather unphysical) isotropic vacuum universe.

Come to think of it, maybe isotropic vacuum is a redundancy, is a vacuum that is not isotropic conceivable?

my_wan

I kind of like the way PAllen constructed the thought experiment to. The one thing that appears to be missing in many of these descriptions is from the perspective of a person entering the black hole. From this perspective the notion that there is an event horizon to cross dries up, like chasing a mirage. As you approach a super massive black your local metric of spacetime is distorted such that the event horizon will appear to shrink away from you. This is because locally the speed of light is always constant such that the notion of a local horizon cannot correspond to a point at which the speed of light is exceeded. That's what keeps you safe from tidal forces while entering a supermassive black hole.

If we mix PAllen's description with an apparently shrinking event horizon, and assume the internal structure is still present when entered, then once the event horizon shrinks enough, such that not enough mass remains within the event horizon to produce an event horizon, the black hole will effectively have evaporated from their perspective.

My question, if this holds, is: would the time dilation (relatively slowed time) of a crew entering be sufficient that when this time dilation is taken into account would enough time pass for the external observer for the black hole to have evaporated from that perspective also, such as from Hawking radiation? In fact a number of interesting questions can be formulated.

This assumption is not problematic with or without GR. Black holes were theoretical entities long before relativity. Basically the above assumption is the equivalent of:
1. Assume gravity is strong enough that photons cannot escape.

In Newtonian physics this was simply due to an assumed mass of the photon. GR only made the description more variable depending on the world line of the observer providing the description. Sonic black holes are another interesting phenomena used to model some of these effects.

zonde

I am not sure but isn't it result of Birkhoff's theorem that interior of spherical massive shell is flat spacetime?
In that case Birkhoff's theorem does not allow symmetrically collapsing shell as it would have to have curved spacetime inside it. Isn't it so?

PAllen

No (Birkhoff's theorem says nothing at all about interior of a shell); and No (Birkhoff's theorem in no way prevents or even says anything about a collapsing spherical shell except for the metric outside the shell.

It would really help to study basic GR before attempting to refute the understandings of those author's who have studied it for decades.

zonde

I think that the utility of examples with free falling observers dries up at the moment when you try to construct global coordinate system where some background stays more or less static, isotropic and homogenous.
Tidal forces are not exclusively associated with event horizon. Tidal forces are present in any field of gravity.

This assumption is problematic if you are trying to construct an argument about possible formation of black hole.
Look up Begging the question fallacy.

zonde

Birkhoff's theorem says that purely longitudinal gravity waves do not exist and so perfectly spherical gravity waves do not exist as well. Changes in gravitational potential inside perfectly spherically symmetric collapsing shell can propagate only as perfect spherically symmetric gravity waves that do not exist according to Birkhoff's theorem.

Let's make it clear. I see no problem with Birkhoff's theorem (so far). But I see problem with interpretation about what it implies.

We don't have perfect spherical symmetry in nature. As we go down the scale there is the level where granularity appears.

PAllen

Not only Birkhoff's theorem, but the most general spherically symmetric GR solutions simply have the result that a collapsing or oscillating matter that is spherically symmetric does not radiate, so there is no contradiction at all.
Of course there is no perfect spherical symmetry, but as with much of physics, we use a simple case to get at certain fundamentals. In this case, that both apparent horizon and true horizon exist may exist when there is no singularity (yet), and no great mass density. These conclusions are trivially provable per my argument given spherical symmetry. Do you argue that a slight deviation from such symmetry radically changes these conclusions? Then justify this absurd conclustion.

pervect

I think it's sufficient to argue that spherical symmetry could exist. It's not like having spherical symmetry breaks any physical laws.

zonde

You are adding that part about collapsing and oscillating on top of math. This is interpretation of math.

Yes, we do that all the time.

I will respond to pervect's comment. PAllen, if you think that your question is not addressed by my reply to pervect then please tell.

I would argue that perfect spherical symmetry breaks laws of quantum mechanics.
Let's say we have source of light that is approximately spherically symmetric. It can emit spherical light pulse.
Light can be polarized so it obviously can't be purely longitudinal. Now let's require that this approximately spherical light source is perfectly spherically symmetric. Then we can argue that such lightsource should emit perfectly spherical pulse of light but because perfectly spherical light can be only purely longitudinal wave we arrive at contradiction.

Naty1

Pallen, PeterDonis:
All the jibber jabber* about null surfaces [which you two agreed upon] got me thinking about some of the details of those....I did some checking in Wikipedia and found:

[*This is Penny's 'technical term' for physicsspeak in THE BIG BANG tv show]

I wasn't aware of this underlying distinction:

http://en.wikipedia.org/wiki/Penrose...arity_theorems

Do these two cases lead to different horizons with any different characteristics??




I think the definition I have seen is consistent with 'outward-pointing light rays are actually converging (moving inwards)..'...Do you guys agree??

Seems like other horizons maybe Rindler, might not meet this 'closed' definition?? Is that correct?? I'm thinking of a Rindler horizon that looks like these:

http://en.wikipedia.org/wiki/Rindler...dler_observers

Thank you

PAllen

Let me clarify what what is definition and what is math in the statement that "any spherically symmetric, asymptotically flat GR solution does not radiate energy via gravitational waves". First, no assumptions at all are needed about matter (e.g. no energy condition on the stress energy tensor. No assumptions are needed about vaccuum, other fields, existence of any static regions, etc.

Definition of gravitational radiation energy in an asymptotically flat pseudo-riemannian manifold: the difference between the ADM energy and the Bondi energy. Each of these is a strictly mathematically defined quantity. For example, for a mutually orbiting bodies, the ADM energy remains constant, the Bondi energy is a decreasing function of time, the difference being the energy carried away by the gravitational radiation.

Known theorem: given any asymptotically flat spherically symmetric pseudo-rieamannian manifold (could have non-vanishing Ricci curvature (= stress energy) everywhere, meaning no vaccuum[except in limit at infinity]; could be oscillating, collapsing, whatever ), the ADM energy = Bondi energy. Thus there is no gravitational radiation.

PAllen

Bringing in QM is a red herring to a discussion of predictions of classical theories. However, your argument is strictly classical, so we can ignore that. Trivially, who says we have to consider EM radiation at all (as previously argued, we already know that gravitational radiation won't exist given spherical symmetry)? Obviously, to talk about 'seeing' we need it, but then it can be introduced in the same approximate sense we talk about test bodies - light follows null geodesics, and we don't inquire into its details (e.g. we haven't been talking about the energy carried away from a collapsing body by the light allowing us to see it; we blithely assume we can make this as insignificant as desired).

To model light as an EM field in GR, we have to consider a stress energy tensor that is not vaccuum anywhere - E and B fields contribute to the stress energy tensor. So we are talking about something very different from your ideal SC case if these contributions are significant. Then, I believe it does follow that there are no exactly spherically symmetric solutions. However the deviations from spherical symmetry can be made as small as desired, and no conclusions we've been discussing would be affected.

In short, classically this is a red herring as well.


So far as I see, you have not offered an substantive argument against the conclusions from Birkhoff's theorem that a collapsing spherical shell could have an apparent horizon while the interior of the shell is still empty (and this would be true for any choices for surfaces of simultaneity that go inside the SC radius).

PeterDonis

There are no exactly spherically symmetric solutions for EM *radiation*; the lowest order radiation is dipole. The Wiki page on null dust solutions has a good overview of the types of spacetimes that contain "radiation":

http://en.wikipedia.org/wiki/Null_dust_solution

There is an exactly spherically symmetric solution with a nonzero EM field: Reissner-Nordstrom spacetime, which has a purely radial electric field. But there is no EM radiation in that spacetime; it is static.

PAllen

I thought it was clear that I was referring to solutions with radiation, since that was the issue Zonde raised. However, it never hurts to clarify.