Graphical example of BH formation by PAllen

[Austin0]

Huh? Of course there would. To an outside observer the object appears to freeze just outside of the horizon while the light from that object redshifts away. Sure, it'll very quickly become colder than the CMB, but it'll never become completely black.

OK we have an infalling object which approaches and passes through the horizon.
At the surface we imagine a certain number of photons which are permanently trapped.
Another quantity of photons which would not be trapped but whose coordinate speed is slowed down to the point of extremely delayed emergence to reach outside observers.

But from the outside, the whole passage occurred in an extremely short coordinate time interval. SO the number of these time release photons would actually be exceedingly small compared to the many billions of years of BH lifespan before possible evaporation.

Do you think a small number of photons making their way out per year would be in some way detectable or capable of discrimination from the background of radiation from infalling matter and Hawking radiation?
Do you think "seeing" is an apt descriptive term to apply to an image based on the assumption of these undetectable photons?

My assumption would be that in the real universe , infalling matter would disperse and/or redirect inward all of the retarded photons so that in a relatively short period of time there would be no trace image, even in abstract principle.
Just my opinion.

 

[zonde]

And who arranges the rocks? Why not just admit:

1) You think what GR predicts violates your sense of plausibility.

GR is not so monolithic as you are trying to imply by your phrase "GR predicts".

But yes, what you claim violates my sense of plausibility. But there might be different reasons for that. For example, we understand the same term differently and that leads to different conclusions.

2) As a result you think the GR is incorrect in this scenario.

I can't make consistent picture out of the things that you claim.

That would be honorable rather than claiming that the top experts in GR (not me, others who I study) are misinterpreting its equations.

Scepticism is important thing in science. That is basically what makes it differ from religion.

Further, except for the details of where it breaks down, you would then be in good company - many serious physicists think GR breaks down in the vicinity of horizons, and many more near the singularity. But that is different from disputing what GR predicts.

(Personally, I think, at least macroscopically, GR only breaks down near the singularity, and the horizon behavior you don't like actually occurs. I think, microscopically, a horizon may not be a true horizon, but macroscopically it behaves as GR predicts. Horizon behavior is likely to be subject to observational test within the next decades. )

I am interested what is out there. And I am interested how much GR can help in finding that out.

 

[PAllen]

I can't make consistent picture out of the things that you claim.

Does it boil down to: you cannot accept that a 2-sphere horizon forms around empty space with no matter inside (in the case of a collapsing shell of matter, or shell like distribution of black holes)?

If so, I note that while I give a logical argument for this (not a 'reference to authority' - though I could ask you to study the Vaidya dust analysis in Poisson's "A Relativists's Toolkit", where growth of horizon from inside out is discussed in great detail), your response appears to me to be absolutely nothing except "I don't like it". Who is being closed minded rather than logical here?

 

[zonde]

Does it boil down to: you cannot accept that a 2-sphere horizon forms around empty space with no matter inside (in the case of a collapsing shell of matter, or shell like distribution of black holes)?

Yes. Look you are using word "forms" and that rather clearly says: "I am thinking about this in a time ordered manner." But when I read definition of absolute horizon it's clear that this definition is very inconvenient for time ordered approach (you have to extrapolate things from present to infinite future and then back). So this definition makes more sense for blockworld and even then it is quite inconvenient (to me it seems non-local).

Another problem with time ordered approach is that you need definition of simultaneity. But relativity uses round trip of light for that and so that definition breaks down for black holes. We of course can extrapolate simultaneity from regions where it is defined. And in that case it seems that we should get Schwarzschild coordinates. (Or maybe we can make testable predictions for different simultaneity? In SR this obviously is not possible but in GR?)

 

[PeterDonis]

Yes. Look you are using word "forms" and that rather clearly says: "I am thinking about this in a time ordered manner." But when I read definition of absolute horizon it's clear that this definition is very inconvenient for time ordered approach (you have to extrapolate things from present to infinite future and then back). So this definition makes more sense for blockworld and even then it is quite inconvenient (to me it seems non-local).

This has already been addressed upthread. See post #37. PAllen is thinking about the EH "forming" in a time-ordered manner, but that doesn't mean the EH is *defined* that way. Its definition does not require any assumptions about time ordering or simultaneity. It is "non-local" in a sense, since its definition requires knowledge of the entire spacetime as a 4-D manifold; but that doesn't imply that information about the future has to "propagate" back into the past.

 

[PAllen]

This has already been addressed upthread. See post #37. PAllen is thinking about the EH "forming" in a time-ordered manner, but that doesn't mean the EH is *defined* that way. Its definition does not require any assumptions about time ordering or simultaneity. It is "non-local" in a sense, since its definition requires knowledge of the entire spacetime as a 4-D manifold; but that doesn't imply that information about the future has to "propagate" back into the past.

See also #53 and #66 where I give physically motivated definitions for this time ordering.

 

[zonde]

This has already been addressed upthread. See post #37. PAllen is thinking about the EH "forming" in a time-ordered manner, but that doesn't mean the EH is *defined* that way. Its definition does not require any assumptions about time ordering or simultaneity. It is "non-local" in a sense, since its definition requires knowledge of the entire spacetime as a 4-D manifold; but that doesn't imply that information about the future has to "propagate" back into the past.

Absolute horizon is defined in retrospective manner and that is incompatible with time ordered approach.

We can look at things from perspective of blockworld but then you have to take an extra step in sorting out which statements are scientific and which are not.

 

[zonde]

See also #53 and #66 where I give physically motivated definitions for this time ordering.

You can give definitions for time ordering but they relay on locally defined things (events). They don't work for retrospectively defined thing.
Retrospectively defined thing don't "form" and you can't "encounter" retrospectively defined thing.

 

[PeterDonis]

Absolute horizon is defined in retrospective manner and that is incompatible with time ordered approach.

No, it isn't. Any statement about the EH "existing" at a certain "time", or "forming" by a certain "time", given a specific simultaneity convention (and PAllen specified that), can be translated into a statement about a particular spacelike hypersurface intersecting a particular null surface in the 4-D spacetime. There's no problem at all.

 

[zonde]

Absolute horizon is defined in retrospective manner and that is incompatible with time ordered approach.

No, it isn't. Any statement about the EH "existing" at a certain "time", or "forming" by a certain "time", given a specific simultaneity convention (and PAllen specified that), can be translated into a statement about a particular spacelike hypersurface intersecting a particular null surface in the 4-D spacetime. There's no problem at all.

You are talking past the statement you quoted.

Definition of horizon is question about picking out particular null surface from other null surfaces as special.

 

[zonde]

PAllen,
How vital is concept of absolute horizon for this discussion (formation of black hole)?
Isn't it possible to define black hole using apparent horizon? At least in some specific cases if not all?

 

[PeterDonis]

You are talking past the statement you quoted.

Definition of horizon is question about picking out particular null surface from other null surfaces as special.

Yes. So what? It's still a null surface, and you can still translate any statement about the horizon "existing" at a particular "time" into a statement about the particular null surface that is the horizon intersecting a particular spacelike surface.

 

[PAllen]

PAllen,
How vital is concept of absolute horizon for this discussion (formation of black hole)?
Isn't it possible to define black hole using apparent horizon? At least in some specific cases if not all?

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]

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.

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]

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?

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]

Unlike the singularity theorems, Birkhoff's theorem makes no assumptions about 'reasonable matter states'. Nothing is assumed except the Einstein Field equations.

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.

I liked this graphical example of black hole formation posted by PAllen in another thread and I want to discuss it.

It is not unusual that arguments defending existence of black hole go like that:
1. Assume that BH exists.

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]

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.

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]

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?

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 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.

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.

That's what keeps you safe from tidal forces while entering a supermassive black hole.

Tidal forces are not exclusively associated with event horizon. Tidal forces are present in any field of gravity.

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.

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]

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.

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.

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

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]

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.

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.

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.

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]

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.

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

Of course there is no perfect spherical symmetry, but as with much of physics, we use a simple case to get at certain fundamentals.

Yes, we do that all the time.

Do you argue that a slight deviation from such symmetry radically changes these conclusions? Then justify this absurd conclustion.

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

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:

Space-like singularities are a feature of non-rotating uncharged black-holes, while time-like singularities are those that occur in charged or rotating black hole exact solutions. Both of them have the following property:
geodesic incompleteness: Some light-paths or particle-paths cannot be extended beyond a certain proper-time or affine-parameter (affine parameter is the null analog of proper time).
It is still an open question whether time-like singularities ever occur in the interior of real charged or rotating black holes, or whether they are artifacts of high symmetry and turn into spacelike singularities when realistic perturbations are added.

http://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems

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

A trapped null surface is a set of points defined in the context of general relativity as a closed surface on which outward-pointing light rays are actually converging (moving inwards). Trapped null surfaces are used in the definition of the apparent horizon which typically surrounds a black hole.

[edit] Definition

We take a (compact, orientable, spacelike) surface, and find its outward pointing normal vectors. The basic picture to think of here is a ball with pins sticking out of it; the pins are the normal vectors.

Now we look at light rays that are directed outward, along these normal vectors. The rays will either be diverging (the usual case one would expect) or converging. Intuitively, if the light rays are converging, this means that the light is moving backwards inside of the ball. If all the rays around the entire surface are converging, we say that there is a trapped null surface.

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_coordinates#The_Rindler_observers

Thank you

 

[PAllen]

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

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]

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.

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 vacuum 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]

To model light as an EM field in GR, we have to consider a stress energy tensor that is not vacuum 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.

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]

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.

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.

 

[PAllen]

This paper suggests it should be perfectly possible to have spherically symmetric collapse with outgoing null radiation (which can represent incoherent light):

http://arxiv.org/pdf/gr-qc/0504045v1.pdf

This particular construction specifies ingoing radiation (incoming Vaidya metric), but it seems very likely to me that you could match outgoing Vaidya to collapsing dust using similar methods. This would be a perfectly spherically symmetric solution.