Observing black holes inside black holes

[l0st]

Can an observer, freely falling into a black hole observe another black hole, falling with him, after crossing horizon?

I assume one should be, as for freely falling observer nothing special happens, when he crosses horizon. But just wanted to double-check.

 

[mfb]

I would assume the same. He won't be able to enjoy it for long, however.

 

[PeterDonis]

Can an observer, freely falling into a black hole observe another black hole, falling with him, after crossing horizon?

I would assume the same.

Unfortunately, that assumption, though it is a natural one, is not correct. A black hole is not an ordinary object, and intuitions about how ordinary objects work don't always apply to black holes.

The key question that exposes the problem is: what happens to the horizon of the smaller black hole when it falls inside the larger one? Remember that a black hole's horizon is a globally defined surface: it's the boundary of the region that can't send light signals to infinity. If you think about it, this means that you can't have one horizon inside another; if you're inside the larger horizon, you're already inside the region that can't send light signals to infinity, so there can't be a second horizon in that region.

Instead, what happens is that the two horizons merge. If you think of a spacetime diagram, with time vertical and space dimensions horizontal, then a single black hole will look like a cylinder, and a pair of black holes merging will look like a pair of trousers (the legs are the two holes before they merge, and the upper part is the single hole that is formed when they merge). If one hole is much smaller than the other, the trousers will be lopsided, with one leg much wider than the other.

An observer falling along with the smaller black hole will observe nothing unusual locally when he crosses the horizon, yes. But since the horizon itself is not visible, he won't be able to see when the holes merge anyway, so the fact that they merge does not create any local issues.

 

[mfb]

But since the horizon itself is not visible, he won't be able to see when the holes merge anyway, so the fact that they merge does not create any local issues.

Which means it still looks like a black hole for him (and the observer can see things like an accretion disk, redshift, ...), even if we outside do not assign an individual event horizon to it. Right?

 

[marcus]

That's interesting Mfb. I hadn't thought of it that way. What we see when we look at a black hole is the trapping horizon and the behavior surrounding it. The accretion disk, as you say, and the redshift of stuff falling in.

I guess we don't actually know that the original definition was a good one, involving the infinite future of the universe. What we think is a black hole may explode due to quantum effects, or be swallowed up in a larger one, or whatever. We don't know the future and our knowledge of quantum gravity is incomplete.
So the "future null infinity" definition is ill-posed and only leads to paradoxes. We have to think about the phenomena that could actually be witnessed and decide on an operational basis what to call a black hole. Or so I think anyway.

In some of the BH research literature I've been seeing, the BH is defined in terms of a trapping surface where all the future lightcones tilt inwards, their outer sides tangent to the trapping surface.

The T-surface is not assumed to be eternal, it occurs in an observer's present and endures for an unspecified length of time. The trapping effect might be impermanent and conceivably be canceled by some drastic geometric upheaval in the remote future : ^)

One does not have to speculate about the trapping possibly lasting forever. We could agree that if something walks and talks like a black hole, we call it a black hole.

So if you have a big one and a little one, and the big one is so big that tidal effects are not immediately a problem, then the little one can fall in thru the big one's horizon, and not immediately notice or lose its identity. An observer accompanying it might continue seeing that it looks and acts like a BH, and deem it to be one.

 

[PeterDonis]

Which means it still looks like a black hole for him (and the observer can see things like an accretion disk, redshift, ...),

I'm not sure. To the extent that those phenomena are connected to an apparent horizon (i.e., a surface at which radially outgoing light no longer moves outward--marcus uses the term "trapping surface", which means the same thing), rather than an event horizon, then it might be possible for them to persist even after the merger; apparent horizons, unlike event horizons, can be nested.

However, I'm still not sure this would work; I'm having difficulty seeing how even the apparent horizon associated with the smaller hole could persist once it's inside the apparent horizon of the larger one. The reason is that, once you're inside the apparent horizon of the larger hole, even radially outgoing light moves inward, and moves inward more and more quickly as you get further and further inside. If the apparent horizon of the smaller hole still persists, then somehow the spacetime curvature produced by the smaller hole has to reverse the behavior of radially outgoing light rays, so that there is a region between the two apparent horizons (larger and smaller) where outgoing light rays move outward again. I'm not sure I see how this could happen, and if it can't, then the apparent horizon of the smaller hole will simply merge with that of the larger hole, just as the event horizons do.

This does imply that the infalling observer will see changes when he falls inside the larger apparent horizon--basically, the phenomena associated with the smaller hole's apparent horizon (accretion disk, redshift, etc.) will disappear over some relatively short period of that observer's proper time as he crosses the larger apparent horizon. However, this does not contradict the fact that an event horizon is not locally observable, because that only applies to an event horizon, not an apparent horizon; you can measure the behavior of light rays locally to determine whether you are inside an apparent horizon.

 

[marcus]

... I'm having difficulty seeing how even the apparent horizon associated with the smaller hole could persist once it's inside the apparent horizon of the larger one. The reason is that, once you're inside the apparent horizon of the larger hole, even radially outgoing light moves inward, ... If the apparent horizon of the smaller hole still persists, then somehow the spacetime curvature produced by the smaller hole has to reverse the behavior of radially outgoing light rays, so that there is a region between the two apparent horizons (larger and smaller) where outgoing light rays move outward again...

Peter just for definiteness let's picture that the two Schwarzschild radii are one millimeter and one light year respectively. So we can be confident about neglecting tidal effects for a while.

I'll tell you what occurs to me. It is that the small BH is in free fall.

The equivalence principle tells me that in the reference frame of the small BH the gravitational field has been zeroed out. Like with the man doing experiments in the plummeting elevator. That reference frame might be applicable only in a small neighborhood surrounding the free fall small BH.
But from the standpoint of an observer in that neighborhood, freely falling with the BH, light would appear to be moving out from the small horizon in all directions. Although for another observer the ripples would be traveling inwards! Depends on perspective. Anyway that's how I picture it. : ^)

 

[PeterDonis]

just for definiteness let's picture that the two Schwarzschild radii are one millimeter and one light year respectively. So we can be confident about neglecting tidal effects for a while.

No problem here.

the small BH is in free fall.

More precisely: we can construct a local "inertial frame" in which the small BH is centered; within this local "inertial frame" we can treat the spacetime geometry as a Schwarzschild solution with mass equal to the small BH's mass; this Schwarzschild solution is asymptotically flat, so the "edge" of this local "inertial" frame matches up to a very good approximation with the almost flat geometry on that size scale (since we're neglecting the tidal gravity of the large BH) described by a Schwarzschild solution with mass equal to the large BH's mass.

This is OK as long as you're not concerned with any internal details of the small BH and can just treat it as a test object falling freely in the spacetime geometry of the large BH. However, the presence of an apparent horizon for the small BH is one of those internal details; test objects don't have apparent horizons. See below.

The equivalence principle tells me that in the reference frame of the small BH the gravitational field has been zeroed out.

Only if we treat the small BH as a test object, as above. But, as above, test objects don't have apparent horizons. If we want to ask whether the small BH still has an apparent horizon after it's fallen inside the large BH, we can't treat it as a test object, so we can't apply the equivalence principle on the size scale of a local inertial frame in the geometry of the large BH. We have to actually look at the geometry on the size scale of the small BH, and once we do that, we can no longer say that "the gravitational field has been zeroed out" in the local "inertial" frame described above, because that's only true on a size scale much larger than the size scale of the small BH.

from the standpoint of an observer in that neighborhood, freely falling with the BH, light would appear to be moving out from the small horizon in all directions.

Only if we ignore the geometry on the size scale of the small BH, and treat it as a test object. If we do that, then of course the observer will see light behaving normally for a local inertial frame; this would be true regardless of what kind of test object was falling in.

But if we look at the geometry on the size scale of the small BH, we can't treat what's going on as being in a single local "inertial" frame, so we can't apply the EP in the way you are suggesting. We would have to start with the small BH as a perturbation on the Schwarzschild geometry of the large BH, when the two holes are well separated, and evolve that perturbed geometry forward in time to see what happened when the two horizons got close together. AFAIK nobody has done that, but if someone has, I would love to see a link to the paper.

 

[marcus]

Can an observer, freely falling into a black hole observe another black hole, falling with him, after crossing horizon?

I assume one should be, as for freely falling observer nothing special happens, when he crosses horizon. But just wanted to double-check.

I would assume the same. He won't be able to enjoy it for long, however.

Some additional info from a University of Virginia Astronomy website:
http://www.astro.virginia.edu/~jh8h/ (webpage of Prof. John Hawley)
http://www.astro.virginia.edu/~jh8h/Foundations/Foundations_1/quest9.html (Astro FAQ)
==quote astro.virginia.edu==
Question:
We see black holes, e.g., the one in M87. ..., could one black hole exist inside another black hole?

Answer:
Actually a black hole could exist inside a black hole. Imagine turning a whole galaxy into a black hole by bringing all its stars extremely close together. All these stars might themselves be black holes. The individual black holes won't even be touching when they are all surrounded by a larger event horizon with a radius corresponding to the mass of the galaxy. However, once they are all within the event horizon they will end up merging together in a final collapse to a singularity.
==endquote==
Personally I have no interest in arguing about it, and don't offer the U.Virginia Black Hole FAQ as authority. I couldn't find anybody who said one couldn't, but here's somebody who says one could.
My own view is that for a much smaller small one the equivalence principle applies.

 

[PeterDonis]

Some additional info from a University of Virginia Astronomy website

I would really like to see some actual math to back up the statement that black holes can exist inside other black holes, as in the example of a galaxy filled with black holes being turned into a galaxy-sized black hole by bringing all the individual holes close enough together.

 

[marcus]

I think you need actual math to prove one couldn't. The default is that one could, as I see it.
Professor John Hawley, the chair of Astro at U of V (2006-2012) author of an Astro Textbook, who teaches Cosmology gave what seems like a simple argument for why. Here's background from his faculty webpage
http://www.astro.virginia.edu/~jh8h/
==quote Prof. Hawley==
Professional Data

• Associate Dean for the Sciences, College of Arts and Sciences, 2012-
• Chair, Department of Astronomy, 2006-2012
• Professor, 1999-
• Assistant Professor, 1987-93
• Associate Professor, 1993-99
  University of Virginia
• 2013 Shaw Prize in Astronomy
• 1993 Helen B. Warner Prize, American Astronomical Society
• Member, American Astronomical Society, American Physical Society, International Astronomical Union

Education

• B.A, Astronomy and Physics, 1980, Haverford College
• Ph.D. Astronomy, 1984, University of Illinois
• Bantrell Fellow, 1984-87, California Inst. of Technology

Teaching:
In the spring semester I teach ASTR 3480, Introduction to Cosmology. This is a survey of modern cosmology intended for both science and non-science majors.

The cosmology course uses my textbook Foundations of Modern Cosmology which is published by http://www.oup-usa.org/ .
==endquote==

Here's what he says:
==quote==
Actually a black hole could exist inside a black hole. Imagine turning a whole galaxy into a black hole by bringing all its stars extremely close together. All these stars might themselves be black holes. The individual black holes won't even be touching when they are all surrounded by a larger event horizon with a radius corresponding to the mass of the galaxy.
==endquote==

 

[PeterDonis]

what seems like a simple argument for why

Yes, this is the argument I was referring to in my previous post. I am assuming that when he says the individual black holes are "surrounded by a larger event horizon", he means the individual apparent horizons are inside the larger event horizon. If he means individual event horizons are inside the larger event horizon, that seems obviously wrong to me because you can't have one event horizon inside another, for the reasons I gave earlier in this thread.

If he means individual apparent horizons can be inside the larger event horizon (which also means, ISTM, they are going to be inside a larger apparent horizon), then, as I said, I'd like to see some math to back that up. I don't think one can say the "default" is that they can; without math it's just a question of whose intuition you believe.

I don't agree that the equivalence principle makes the "default" that they can either, because, as I said before, the EP assumes that the smaller object is a test object, and test objects don't have apparent horizons; they have, by hypothesis, no effect on the spacetime geometry at all, and that fact is what justifies treating a local inertial frame as being flat. If you're trying to decide whether one apparent horizon can be inside another, you can't treat the smaller object as a test object in a locally flat patch of spacetime--you have to actually look at its spacetime geometry, since the presence or absence of an apparent horizon on the smaller object's size scale is a question of spacetime geometry.

 

[atyy]

I think these papers make it hard to have a black hole in a black hole.

http://arxiv.org/abs/gr-qc/9410004
On the topology of stationary black holes
P.T. Chrusciel, R.M. Wald

http://arxiv.org/abs/gr-qc/9410023v2
Topology of Event Horizons and Topological Censorship
Ted Jacobson, Shankar Venkataramani

 

[l0st]

http://arxiv.org/abs/gr-qc/9410004
On the topology of stationary black holes
P.T. Chrusciel, R.M. Wald

At least this article talks about topology as seen by external observer.

 

[martinbn]

I think these papers make it hard to have a black hole in a black hole.

http://arxiv.org/abs/gr-qc/9410004
On the topology of stationary black holes
P.T. Chrusciel, R.M. Wald

http://arxiv.org/abs/gr-qc/9410023v2
Topology of Event Horizons and Topological Censorship
Ted Jacobson, Shankar Venkataramani

I don't think these are relevant. They are concerned with the topology of the event horizon, not whether you can have nested ones. PeterDonis said it a few times by definition it is impossible to have nested event horizons. But if I understand him correctly he suspect that you cannot even have an apparent horizon within an event horizon. It may look like one, so it will be an apparent apparent horizon, but if you do the maths it will turn out that it is not.

 

[tzimie]

I assume it is also possible to CREATE a BH inside bigger BH.

Scenario: almost-critical neutron star is falling into supermassive BH. Based on "no drama" assumption, it safely crosses the horizon of supermassive BH, and has few minutes left before the singularity. At this moment, and observer near the neutron star drops an object into it, making it mass critical and creates a BH inside another BH.

I think you can create BH inside another BH, it is a direct consequence of "no drama".

 

[PeterDonis]

almost-critical neutron star is falling into supermassive BH. Based on "no drama" assumption, it safely crosses the horizon of supermassive BH, and has few minutes left before the singularity. At this moment, and observer near the neutron star drops an object into it, making it mass critical and creates a BH inside another BH.

No, this won't work. As soon as the neutron star and the small object fall past the supermassive black hole's horizon, the mass of the supermassive hole increases slightly, and its horizon expands slightly. So the neutron star and the small object are already inside the event horizon of the expanded hole before the object falls into the neutron star. You can't create a second event horizon inside another event horizon, for the reasons I already gave in this thread.

As for what happens when the object is dropped into the neutron star, sure, it could collapse; but that collapse won't form a second event horizon inside the existing one (since that's impossible). Whether or not it forms a second apparent horizon is the question under discussion, and it's not sufficient just to propose thought experiments, since the whole point is whose thought experiment is actually backed up by the math.

 

[mfb]

As soon as the neutron star and the small object fall past the supermassive black hole's horizon, the mass of the supermassive hole increases slightly, and its horizon expands slightly.

So what?

So the neutron star and the small object are already inside the event horizon of the expanded hole before the object falls into the neutron star.

Depends on when we drop it.

We would have to start with the small BH as a perturbation on the Schwarzschild geometry of the large BH

What about the opposite direction? The small BH-like object has a larger influence on the local field. Consider spacetime in the frame of our observer, treat the large BH as perturbation to the field of our small one.

The reason is that, once you're inside the apparent horizon of the larger hole, even radially outgoing light moves inward, and moves inward more and more quickly as you get further and further inside.

Yes, but there are still two different inward speeds, corresponding to two different directions of light for our observer.

 

[PeterDonis]

Depends on when we drop it.

True, but I was assuming that the case we're interested in is when the drop occurs after the neutron star and the small object are inside the horizon of the supermassive black hole. Otherwise the question of whether horizons can be nested doesn't arise.

Consider spacetime in the frame of our observer, treat the large BH as perturbation to the field of our small one.

Do you mean just locally, i.e., at some particular event on the idealized worldline of the smaller BH? Or do you mean globally, i.e., all along the idealized worldline of the smaller BH? (I say "idealized worldline" because, if we consider the actual spacetime geometry of the smaller BH, it does not have a worldline the way an ordinary object does, because it doesn't have a center of mass the way an ordinary object does. The singularity at $r = 0$ is spacelike, not timelike. The "idealized worldline" is what we get if we ignore the actual spacetime geometry of the smaller BH and treat it as a test object in the geometry of the larger BH--a more accurate term would be "world tube", since what we are basically doing is taking a tube out of the larger geometry and pretending there's only a test object inside, unless and until we actually try to consider the geometry of the smaller BH.)

Locally, the large BH's geometry is negligible, by hypothesis--we are considering small enough patches of that geometry that the tidal gravity of the large BH is negligible. But that won't tell us whether horizons are nested.

Globally, the perturbation created by the large BH is time-dependent, but also, we can no longer neglect the larger BH's tidal gravity, because we are considering a length of the smaller BH's world tube that is too long to fit in a single local "inertial" frame with respect to the larger BH. So I'm not sure what the outcome of such an analysis would be (but see below).

there are still two different inward speeds, corresponding to two different directions of light for our observer.

Yes, but the really important criterion is not "inward speed" but the expansion of the congruence of null geodesics. Let me re-state the key point in more technical language.

Consider a 2-sphere which is aligned with the spherical symmetry in some spherically symmetric spacetime. (That is, the 2-sphere is spanned by orbits of the 3 spacelike Killing vector fields associated with the spherical symmetry of the spacetime.) There will be two sets of radial null geodesics that can be emitted from this 2-sphere (these are often called "null normals" in the literature), the "outgoing" set and the "ingoing" set.

In ordinary flat spacetime, the outgoing set of null geodesics always has positive expansion, and the ingoing set always has negative expansion. (The expansion, or more precisely the expansion scalar, is one of the standard components of the kinematic decomposition, as described for example here.) In a black hole spacetime, the ingoing set still always has negative expansion, but the expansion of the outgoing set is only positive outside the apparent horizon. At the apparent horizon, the expansion of the outgoing set becomes zero; and inside the apparent horizon, it is negative. (This is what I was describing colloquially as "outgoing light moving inward".)

Describing things this way does open up a possibility I hadn't considered for how apparent horizons could be nested. The expansion of a given congruence of null geodesics is an invariant; but which particular congruence of null geodesics are the "radially outgoing null normals" that are used to define an apparent horizon can be observer-dependent. So a 2-surface that is an apparent horizon with respect to, say, the free-falling observer after he has dropped the small object into the neutron star (but not with respect to an observer just outside the larger BH), could, in principle, be nested inside a 2-surface that is an apparent horizon for an observer just outside the larger BH (but not for the observer free-falling next to the neutron star/smaller BH).

I'm not sure if the math would actually show this on analysis; but since apparent horizons can be observer-dependent, I do agree that the scenario I've just described is possible in principle, and is more or less what emerges from heuristically considering the small BH as being within a local "inertial" frame with respect to the large BH when it forms (i.e., when the small object is dropped into the neutron star). In other words, apparent horizons with respect to the same observer can't be nested (or at least, not in the simple way we've been considering); but apparent horizons with respect to different observers can (each horizon is only a horizon with respect to one observer, not the other).

(Note, btw, that none of the above applies to event horizons; those are not observer-dependent and can't be nested, for the reasons I've already given.)

 

[mfb]

Do you mean just locally, i.e., at some particular event on the idealized worldline of the smaller BH? Or do you mean globally, i.e., all along the idealized worldline of the smaller BH?

Locally, at the world line of the observer close to the small BH-like object.

Spoiler: irrelevant rant

We should call all black holes BH-like objects until we actually fall into one, in the same way the Higgs boson has been called Higgs-like boson way too long.

I think it becomes a matter of definition. Can we say "it is a black hole" within a finite time, no matter what happens at some point in the future? If not, then we are screwed because it will eventually fall into the singularity of the larger BH, or evaporate. If yes, then it has all properties we can measure of a black hole within a finite time.