Clarifying Black Hole Horizons: An Examination of Observer Perspectives

[PAllen]

As for the white hole I wasn't aware that there was no counterpart to it in a collapse model - so the solution can be extended only after the collapse ends, not before - weird and interesting, thanks for the explanation.

No, it cannot be extended after, either. The white hole quadrant is in the past of the BH quadrant. That past is occupied by the collapse. What you have to do is remove the collapse from the history of the universe, and replace it with a completely different (unrealizable) past.

 

[wabbit]

No, it cannot be extended after, either. The white hole quadrant is in the past of the BH quadrant. That past is occupied by the collapse. What you have to do is remove the collapse from the history of the universe, and replace it with a completely different (unrealizable) past.

Ah that is a sad state of affairs indeed. I guess I must get my white holes from Planck stars then : )

 

[JoeMath]

Perhaps a more important consideration is this. Assume a perfect collapsing stellar object. Massive, zero angular momentum, zero charge, isolated from all other objects by a million LY. What does the surface of this core look like as it approaches the Schwarzschild radius? At very tiny distances (nano meter? Pico meter? 10^-10 meter?) the surface of the core is likely composed of neutrons. But quantum mechanics does not allow a perfect geometric sphere. Quantum tunneling (or the uncertainty principle if you wish) must eject neutrons from the surface and form a foam or cloud that is constrained only by gravity and not by the strong nuclear force. QM also predicts that the energy density distribution could allow some of these “free” neutrons sufficient velocity to escape, thus reducing the mass of the core.
The point I’m trying to make is that the standard GR texts casually state that the core collapses and passes through the Schwarzschild radius and out of view. There is never a mention of basic QM effects, which could be sufficient to cause the mass of the core to decrease and prevent the surface of the core from ever reaching the Schwarzschild radius.

 

[wabbit]

The notion that you could simply excise the interior in classical GR doesn't make sense. This violates the equivalence principle, b/c the near horizon region is always Rindler space which is physically traversable

If we take the (open) exterior region as the whole of spacetime, it is still a valid GR solution, and I do not see where the EP is violated within that spacetime.

It is true that some observers trajectories end, in finite proper time, outside of spacetime, and that these trajectories can be extended to end later - still leaving spacetime, but only at an unavoidable curvature singularity.

It is perhaps unnatural to assume they vanish at the horizon when there is a natural extension, but this concerns events that in any case are in principle not observable from outside the horizon.

Should we take the view that physics is concerned only with predicting observable phenomena, the minimal assumption about what exists, for observers outside the horizon, sems to be that spacetime is just that exterior region.

I am not arguing in favor of this philosophical stand here, only that, as I understand it, it is not contradicted by physics (within GR).

This may of course be incorrect but at this point I strongly suspect that the existence of the interior region is (within GR and ignoring QM), physically undecidable for observers in the exterior region.

In any case, I want to thank you and other responders for the many explanations provided, this discussion has been very instructive even if, despite finally finding my feet a few posts ago, you may be wondering if I have now lost my head : )

 

[PeterDonis]

What does the surface of this core look like as it approaches the Schwarzschild radius?

Like the surface of a collapsing object.

At very tiny distances (nano meter? Pico meter? 10^-10 meter?) the surface of the core is likely composed of neutrons.

Possibly, but not necessarily. It depends on what is collapsing. If it's a star of a few solar masses or more, the collapsing matter might not have had time to be converted to neutrons by increasing pressure and density by the time the Schwarzschild radius is reached.

Btw, it's important to understand that this kind of collapse process, at the point where the horizon is approached, is not a "gradual" process--it's not like an object that is slowly moving between quasi-equilibrium states of gradually decreasing radius. No stable equilibrium is possible for a radius smaller than 9/8 of the Schwarzschild radius; so as soon as the collapse proceeds beyond that point, which is still a significant distance short of the horizon, the collapse will accelerate; it can no longer be envisioned, even in principle, as a "slow" progression from one radius to another.

quantum mechanics does not allow a perfect geometric sphere

For what? For the whole surface of the collapsing matter? For a collapsing mass of a few solar masses or more, the surface is way too large for quantum effects to be significant; everything should be firmly in the classical regime.

There is never a mention of basic QM effects, which could be sufficient to cause the mass of the core to decrease and prevent the surface of the core from ever reaching the Schwarzschild radius.

"Could be" is way, way, way different from "must be". It's also way, way, way different from "might be in a significant number of cases". As noted above, for a collapsing object of a few solar masses or more, i.e., any gravitational collapse that is realistically to be expected in our universe now or in the future, quantum effects should be entirely negligible; the spacetime geometry is well within the classical regime at and well below the horizon.

For possible collapse processes in the early universe (i.e., "primordial black holes"), this is not necessarily the case; but this is a very different case from the case of a collapsing isolated object. In the primordial case we are talking about density fluctuations in a medium that can be modeled as a continuous fluid; there is no exterior vacuum region, so the question of what the "surface" of the collapse looks like is meaningless. It's a different scenario requiring a different model.

 

[JoeMath]

My point about QM and the collapsing core is that the uncertainty applies to the radial component and not just to the entire surface area. How can you define the radius of the core so precisely and not violate the uncertainty principle? If the uncertainty principle holds in the radial direction, what then? How is the surface defined?

 

[PeterDonis]

How can you define the radius of the core so precisely and not violate the uncertainty principle?

You can't define it to infinite precision without violating the uncertainty principle, true. But how uncertain the horizon radius is depends on how massive the hole is--the more massive the hole, the smaller the uncertainty, at least with the "obvious" application of the uncertainty principle. Try calculating the uncertainty in horizon radius for a black hole of a few solar masses. If it's small enough (which it is), then the classical GR model is a perfectly good approximation--i.e., quantum effects are negligible, as I said.

 

[Dale]

the definition should rest on the physical information alone. In that sense, a spacetime that consists exactly of the Schwarzschild exterior region and nothing else

I am OK with this in principle, but then for consistency no spacetime should include anything outside a given observers past light cone, including other observers worldlines. This would lead to some weird stuff in SR flat spacetime.

Also, an argument by ignorance generally can't be used to make assertions about the points on which ignorance is claimed. So the most you could claim is that we don't know about the interior, not that the accepted models are wrong about it.

 

[wabbit]

I am OK with this in principle, but then for consistency no spacetime should include anything outside a given observers past light cone, including other observers worldlines. This would lead to some weird stuff in SR flat spacetime.

Outside any of this light cones. For an inertial observer in Minkowski spacetime, this is the whole of spacetime (not for an accelerated observer though).
Also I make no claim that "no spacetime should..." - only that it is possible to assume only a smaller spacetime, and that a given observer cannot in principle ever prove that something exists outside of the union of his past cones (including those he will have in his future).
Of course, if that observer is falling into the black hole, his "minimal spacetime" includes (some portion of?) the horizon and interior region.

Also, an argument by ignorance generally can't be used to make assertions about the points on which ignorance is claimed. So the most you could claim is that we don't know about the interior, not that the accepted models are wrong about it.

Absolutely. I make no claim whatsoever that accepted models of the interior are wrong - only that, unless we are prepared to jump into a black hole, we can legitimately assume, but not conclusively prove, that the interior exists. But a decisive experiment is possible, it just takes a strong commitment to the advancement of our knowledge, and accepting the fact that we will not publish the results : )

Also, it is clear that working with a larger spacetime can be convenient and useful. Even tough we cannot prove anything conclusively about the region of the universe outside our current and future observable universe, it would seem at best complicated to formulate cosmology under the assumption that nothing exists outside of it. Once we have say an LCDM model however, it is more a philosophical (or methodological) decision to ascribe existence to the part that is in principle forever unobservable.

 

[Dale]

Outside any of this light cones. For an inertial observer in Minkowski spacetime, this is the whole of spacetime (not for an accelerated observer though).

If we are limiting ourselves to "physical information alone" then you cannot use future past light cones as we have no information about them. You have to make assumptions to use future past light cones. If you allow such assumptions then you can easily make such assumptions that include the interior of a black hole.

unless we are prepared to jump into a black hole, we can legitimately assume, but not conclusively prove, that the interior exists. But a decisive experiment is possible, it just takes a strong commitment to the advancement of our knowledge, and accepting the fact that we will not publish the results : )

I agree 100%, and have made similar statements in the past.

One other thing that you might want to consider is that any null surface is an event horizon in the sense that once we cross it we can no longer send signals back to the other side. Every moment of every day we are crossing event horizons and can no longer send signals to certain regions of spacetime where we used to be. The results of any experiment that you perform today cannot be published to people who have not crossed "today's event horizon" with you.

 

[wabbit]

If we are limiting ourselves to "physical information alone" then you cannot use future past light cones as we have no information about them.

I don't really agree with this but perhaps I am abusing the term "physical information", and "potential physical information" could be better. For definiteness: I am using $M_p(\gamma)=\{x\in M|\exists y \in \gamma, x<y\}$ where $\gamma$ is the wordline of an observer in a spacetime $M$, "$<$" means "is in the causal past of", and $M_p(\gamma)$ is the proposed minimal spacetime of the observer.

If you allow such assumptions then you can easily make such assumptions that include the interior of a black hole.

Yes, but they are not part of this minimal spacetime. Of course it can be extended to the interior, this is just non-minimal then.

One other thing that you might want to consider is that any null surface is an event horizon in the sense that once we cross it we can no longer send signals back to the other side. Every moment of every day we are crossing event horizons and can no longer send signals to certain regions of spacetime where we used to be. The results of any experiment that you perform today cannot be published to people who have not crossed "today's event horizon" with you.

Interesting. This is a dynamic horizon (one surface associated to each point in my wordline), not a fixed surface in spacetime. It is unusual in that it is defined by who I can send messages to instead of who can send messages to me - a "reverse horizon" in a way.

If we are limiting ourselves to "physical information alone" then you cannot use future past light cones as we have no information about them. You have to make assumptions to use future past light cones. If you allow such assumptions then you can easily make such assumptions that include the interior of a black hole.

I agree 100%, and have made similar statements in the past.

One other thing that you might want to consider is that any null surface is an event horizon in the sense that once we cross it we can no longer send signals back to the other side. Every moment of every day we are crossing event horizons and can no longer send signals to certain regions of spacetime where we used to be. The results of any experiment that you perform today cannot be published to people who have not crossed "today's event horizon" with you.

 

[Dale]

I don't really agree with this but perhaps I am abusing the term "physical information", and "potential physical information" could be better. For definiteness: I am using $M_p(\gamma)=\{x\in M|\exists y \in \gamma, x<y\}$ where $\gamma$ is the wordline of an observer in a spacetime $M$, "$<$" means "is in the causal past of", and $M_p(\gamma)$ is the proposed minimal spacetime of the observer.
Yes, but they are not part of this minimal spacetime. Of course it can be extended to the interior, this is just non-minimal then.

If you include the "future past light cones" then you can always make even this minimal spacetime include part of the interior of the event horizon simply by assuming that the observer's worldline crosses the horizon some time in the future.

I believe that it is well known and well accepted that any open subset of a manifold is also a manifold. So you can certainly say that you are interested in only such-and-such submanifold, defined however you like. Therefore, I don't have any opposition to your idea itself, but I think that you are drawing a conclusion from it that isn't as strong as you seem to believe.

In order to exclude the EH you have to assume some priveliged observer's future worldline. That assumption seems no better to me than the alternative assumption that the spacetime is geodesically complete.

 

[wabbit]

I m not saying it is better than another, and yes it is relative to an observer or class of observer, that's its purpose - i find it interesting to understand where the line is o
between what isp rovable and what is in principle not, in the modelling from terrestrial observers, Some commented initially that what I was describing was inconsistent with GR or otherwise impossible - After clarifications I think it's actually reasonable, and I see some limitations, but it's still interesting, the exterior is not just any open submanifold. Its one where, if an observers worldine is entirely in the submanifold, then so is his whole M_p - which in essence just says that the exterior includes its complete own past. Not every open submanifold can say that : )
Well its just fun to explore a bit this as a tourist of GR : )

 

[PeterDonis]

the exterior is not just any open submanifold. Its one where, if an observers worldine is entirely in the submanifold, then so is his whole M_p - which in essence just says that the exterior includes its complete own past. Not every open submanifold can say that : )

Are you sure about that last statement? Can you give a non-trivial example of an open submanifold and a worldline entirely contained in that submanifold, where the entire M_p of the observer's worldline is not contained in the submanifold? (By "non-trivial" I mean excluding obviously contrived cases like only taking the open submanifold of events within some small radius of the worldline, which is a valid open submanifold that contains the entire worldline but obviously excludes almost all of the past light cone of any event on the worldline.)

 

[wabbit]

Are you sure about that last statement? Can you give a non-trivial example of an open submanifold and a worldline entirely contained in that submanifold, where the entire M_p of the observer's worldline is not contained in the submanifold?

Any open submanifold strictly contained in M_p, but containg the worldline, satifies that prescription. Thats a whole continuum of examples : )

 

[Dale]

i find it interesting to understand where the line is o
between what isp rovable and what is in principle not, in the modelling from terrestrial observers

Fair enough. Terrestrial observers would definitely be privileged observers, but for selfish reasons they would also be particularly interesting.

 

[wabbit]

Yes I must admit a parochial interest in that particular class of observers, though I will gladly share my toughts and compare notes with visiting aliens : )

 

[Dale]

(By "non-trivial" I mean excluding obviously contrived cases like only taking the open submanifold of events within some small radius of the worldline, which is a valid open submanifold that contains the entire worldline but obviously excludes almost all of the past light cone of any event on the worldline.)

I think what you want is not "non trivial" submanifold, but rather submanifolds bounded by a null surface. I think many such sub manifolds have the property mentioned by wabbit, if not all of them.

 

[wabbit]

I think what you want is not "non trivial" submanifold, but rather submanifolds bounded by a null surface. I think many such sub manifolds have the property mentioned by wabbit, if not all of them.

Ah that is possible, it would be an interesting characerization of this class of submanifolds. Hmm need to consider.

 

[PeterDonis]

Any open submanifold strictly contained in M_p, but containg the worldline, satifies that prescription.

Hm, yes, my definition of "non-trivial" wasn't restrictive enough, as DaleSpam pointed out. I agree with you that the restriction to manifolds bounded by a null surface, as he suggests, would be an interesting case to consider.

 

[wabbit]

To clarify, the property i mentionned as interesting for E (contains its own past) is stronger than the one quoted by PeterDonis (contains the past of one given worldline) or different if the latter is meant as "contains the past of a given maximally extended M-worldline".

The second just defines submanifolds S that contain M_p(gamma), and is not very special. I doubt that any submanifold containing M_p(gamma) must be bounded by anything special. In fact, take any closed subset of the interior of M\M_p(gamma), bounded by any surface you like, and M minus that subset fits the requirement, and its boundary includes that surface.

Hmm just reread you, not sure this answers it - you want a non trivial example of a submanifold that is bounded by a null surface, contains a maximally extended M-worldline, and doesn't contain the past of that worldline? If so this does seem exotic at first sight, I don't know if it's possible or not.

Here a wordline is not necessarily maximally extended, it can be just a segment.

The first property above says that E contains the past of any event within it, or equivalently E contains the past of any E-worldine. Among these E-wordlines are some maximally extended M-wordlines, and E of course also contains their past.
The interior region I, though bounded by a null surface, does not contain its own past, nor that of I-wordlines. (I don't think it contains any maximally extended M-wordline however)

 

[PAllen]

I don't find this category of manifolds very interesting. Suppose I define the manifold consisting of the past light cone of myself at 3 PM today, minus the null surface (and me at 3 PM). This manifold will contain the past of every event and every world line contained in it. I think most people would consider this an 'egocentric' manifold of little significance.

 

[wabbit]

This is true. Any submanifold defined as you did as the past of something, contains its own past. The converse isn't true I think, though the exterior region is the past of the horizon.

Anyway it's just one property of E that not everyone shares, call it "past-complete" or whatever, it's not the eighth wonder of the world for sure. The fact that I found it interesting as a tourist is no indication that it should be interesting to those better versed in GR.

 

[Haelfix]

It would also preclude a meaningful physical interpretation of things like DeSitter cosmologies in the static patch, where your friend Alice eventually falls off the end of the world.

 

[wabbit]

Not sure what you mean - but then again I have no idea what the static patch of deSitter cosmologies is.

 

[harrylin]

except that the coordinates you label as meaningful for an external observer of a BH correspond precisely to Rindler coordinates for a uniformly accelerating rocket. If you believe the one 'never' you must believe the other if you are logically consistent. The other logically consistent position - accepted by almost all physicists - is that both 'nevers' are coordinate artifacts, not something physically meaningful. [..].

An essential assumption is missing in that argument: you assume that the physical interpretation of the coordinates must be identical. However, that is not the case.

Probably most physicists believe that acceleration and gravitation are equivalent but not identical; for starters gravitation functions without a rocket engine. Consequently the physical interpretation differs. For example (to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates, while the same apparent difference in a gravitational field may be interpreted as physically real. The logically consistent conclusion is then the contrary from the one you advance.

 

[PeterDonis]

Probably most physicists believe that acceleration and gravitation are equivalent but not identical

More precisely, they believe that acceleration in free space and being at rest in a gravitational field are (locally) equivalent (assuming the proper acceleration in both cases is the same).

For example (to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates

No, it isn't. Two astronauts at the rear and front ends of the rocket can exchange repeated round-trip light signals and verify that the rear one's clock rate is slower (less elapsed time between successive signals). This is as "real" as the corresponding experimental result for two people at rest in a gravitational field at slightly different altitudes.

 

[harrylin]

More precisely, they believe that acceleration in free space and being at rest in a gravitational field are (locally) equivalent (assuming the proper acceleration in both cases is the same).

Yes indeed, thanks for the precision :-)

No, it isn't. Two astronauts at the rear and front ends of the rocket can exchange repeated round-trip light signals and verify that the rear one's clock rate is slower (less elapsed time between successive signals). This is as "real" as the corresponding experimental result for two people at rest in a gravitational field at slightly different altitudes.

That is erroneous; but I'm afraid that this is a permanent bug. I'll nevertheless clarify this once more.

Doppler effect and clock rate are different physical concepts; their definitions are unrelated. The elapsed time between successive signals coming from a distant clock is a function of both.

An astronaut in the rocket can measure light signals from a clock in the front with a clock in the rear; accounting for the rocket's acceleration (which she calculates from the thrust of the rocket engine and the rocket's mass) she will conclude that the rear clock rate is approximately the same (less elapsed time between successive signals as predicted by the Doppler effect).

 

[PeterDonis]

An astronaut in the rocket can measure light signals from a clock in the front with a clock in the rear; accounting for the rocket's acceleration (which she calculates from the thrust of the rocket engine and the rocket's mass) she will conclude that the rear clock rate is approximately the same (less elapsed time between successive signals as predicted by the Doppler effect).

What happens when light signals make a round trip? Or repeated round trips? Or does your qualifier "to first order" mean you were ignoring that?

Also, exactly the same logic can be applied in a local inertial frame in a gravitational field, for example to a light signal traveling between two observers at rest in the Earth's gravitational field at slightly different altitudes. When viewed in a local inertial frame, the frequency shift in the light signal can be entirely attributed to the Doppler effect. So if you are only looking at things "to first order", there is no difference between the two scenarios.

 

[PAllen]

An essential assumption is missing in that argument: you assume that the physical interpretation of the coordinates must be identical. However, that is not the case.

Probably most physicists believe that acceleration and gravitation are equivalent but not identical; for starters gravitation functions without a rocket engine. Consequently the physical interpretation differs. For example (to first order) the apparent difference in clock rates in an accelerating rocket is an artefact of using accelerating coordinates, while the same apparent difference in a gravitational field may be interpreted as physically real. The logically consistent conclusion is then the contrary from the one you advance.

I would argue that the local physics is identical. To hover above a large horizon (so tidal effects are not extreme), you would need a steadily firing rocket (or suspension from a more distant steadily firing rocket). Any local measurements you make, including the behavior of signals bounced of objects closer to the horizon, and the fact that you could send to, but not receive messages, from on object that fell through the horizon are identical to the equivalent experiments in a uniformly accelerating rocket [in 'empty space' far away from anything]. If you use any natural procedure for setting up coordinates around this hovering observer, you get coordinates identical (delta second order tidal effects) to the rocket in deep space (Rindler coordinates). Further, in both cases, the redshift between higher and lower altitudes in a lab is purely Doppler in both situations, if expressed in locally inertial coordinates.

Thus, per local physics, as well as mathematics, the coordinates are equivalent and the horizons are equivalent, and any sense of 'never happens' based on two way signal behavior is identical. In both cases, you can choose to stop your physically experienced acceleration and then immediately access the other side of the horizon.