Gravity is a weak force, which makes it extremely difficult to do actual experiments (or perform astronomical observations) that would give us any detailed, up-close-and-personal data about the behavior of quantum gravity. We should be thankful, therefore, that we’ve been able to learn as much as we have about quantum gravity (and we do know some things) just by sitting in our chairs and doing thought experiments, constrained only by the basic principles of general relativity and quantum mechanics. Undoubtedly the most prolific thought-experiment laboratories have been black holes. In particular, Hawking’s discovery that black holes radiate and have entropy has driven an enormous amount of research, and some of it has actually been productive! One of the highlights was certainly the calculation in 1996 by Strominger and Vafa, who used some tricks from string theory to actually count the number of quantum states hidden in a black hole, in a way that would have made Boltzmann proud, and come up with an answer that matched Hawking’s formula precisely.
There are still puzzles, however, as you might guess. Foremost among them is “How does the information get out?” An increasing number of physicists believe that the evaporation of black holes conserves information, but they don’t know precisely how the details of the state which created the black hole get preserved and then encoded in the outgoing Hawking radiation.
A lesser-known puzzle, which many people don’t even consider a puzzle, hearkens back to a 1994 paper by Stephen Hawking, Gary Horowitz, and Simon Ross. They were trying to use the particular technique called Euclidean Quantum Gravity (in which you temporarily forget that time is any different than space) to calculate rates at which different things could happen, when the stumbled across a puzzle. They calculated the entropy of black holes with electric charge, and in particular of extremal black holes — configurations where all of the energy really comes from the electric field itself, none from any purported mass that might have fallen into the black hole. And for an extremal black hole, they found an unusual answer: zero! That was a surprise, because it is not what Hawking’s original formula (entropy is proportional to area of the event horizon) should give you for such a situation.
Most people (including, I think, the authors) believe that this result is not trustworthy, and reflects a breakdown of the particular method used, rather than a deep truth about extremal black holes. But in a field where actual data is sparse on the ground, it’s worth keeping puzzles in mind, hoping that some day they will teach you something.
Matt Johnson, Lisa Randall and I just submitted a paper in which we revisit this puzzle. We suggest that maybe it’s not just a simple breakdown of the methods of Euclidean quantum gravity, but perhaps something interesting is going on.
Extremal limits and black hole entropy
Authors: Sean M. Carroll, Matthew C. Johnson, Lisa RandallAbstract: Taking the extremal limit of a non-extremal Reissner-Nordström black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate — while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of $AdS_2times S^2$. In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordström solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.
Let’s unpack this a little bit. A fascinating property of a charged (“Reissner-Nordström”) black hole is that it has more than one event horizon. In an ordinary uncharged (“Schwarzschild”) black hole, there is a single horizon, corresponding to the point of no return — once you pass the event horizon, you can never escape back to the rest of the world. Inside, there is a singularity, and you are forced to crunch into the singularity in a finite time. In the charged black hole, what we call the outer event horizon (r+ on the diagram) is the point of no return. But in between the outer event horizon and the singularity is an inner event horizon (r– on the diagram). It is, again, a point of no return — once you cross the inner horizon, you can’t get back out — but you are not forced to hit the singularity in the middle. Inside the inner horizon, you can avoid the singularity if you wish. On the diagram, we’ve portrayed this by arrows, indicating the direction of “moving forward in time.” Inside the outer horizon, moving forward in time is moving toward the inner horizon, and can’t be helped; but outside the black hole, and inside the inner horizon, moving forward in time looks pretty conventional, and you’re not forced anywhere.
An uncharged black hole has no inner horizon; it should come as no surprise, then, that as you increase the charge (keeping the mass of the black hole fixed), the two horizons at r+ and r– come together. In an extremal black hole, where all of the energy comes from the electric field itself, the two horizons coincide: we have
r+ = r– (extremality).
Everyone has known that forever. But Matt, Lisa and I stumbled across an interesting miracle of curved spacetime, which I think some people recognized but certainly isn’t widely appreciated: as you creep up on extremality, increasing the charge of the black hole while keeping its mass fixed, and therefore driving r– ever closer to r+, the spacetime volume of the region in between them does not go to zero. It approaches some finite size and stays there.
So we have a region of spacetime of fixed volume, which doesn’t shrink to zero as you increase the charge, but which suddenly disappears entirely when you hit exactly the extremal value. In other words, the limit is discontinuous.
Among ourselves, we referred to this region as “Whoville.” (That didn’t make it into the paper — another reason why blogs are better.) The spacetime geometry of this region looks like the product of a piece of two-dimensional anti-de Sitter spacetime and a two-dimensional sphere.
And … so? Well, mostly we just wanted to point to the interesting feature of the discontinuous limit. But it’s difficult to resist connecting this puzzle with the puzzle of the vanishing entropy. Hawking, Horowitz and Ross found that the entropy of a charged black hole approached a smooth limit as the charge was increased, but discontinuously went to zero exactly at extremality. We found that the volume of spacetime between the two horizons approached a smooth limit as the charge was increased, but discontinuously disappeared exactly at extremality. It certainly suggests a new angle on the behavior of the entropy: maybe HHR were right after all, and the entropy of a precisely extremal black hole really is zero — because the entropy that should be there has escaped into Whoville, this new anti-de Sitter spacetime that is a different part of the discontinuous limit.
Maybe; I wouldn’t stake my life on it at this point, but it’s certainly a viable possibility worth taking seriously. The best argument against it is the kind of microstate counting from string theory pioneered by Strominger and Vafa — they were always looking precisely at the extremal case, and found a non-zero entropy. On the other hand, that technique did always have a step in which you equated the states at very weak coupling (where you could do the counting, but there was no black hole) to those at strong coupling (where you couldn’t directly count, but there is an extremal black hole). It’s long been recognized that one could worry about phase transitions as the coupling was varied — but it got the right answer, so why worry too much? Perhaps the states disappear into Whoville, rather than really describing an extremal black hole.
In the meantime, it’s been fun to think about these things. Despite having written a textbook on general relativity, I had never written a paper directly about black hole physics. Always good to learn new things, and I look forward to learning more.
Quick question: if the limit of approaching extremality is discontinuous, doesn’t that mean that the exactly extremal space, the one with zero entropy, is unstable to infinitesimal perturbations, and therefore unphysical?
Well, there should be an exact quantum state corresponding to the extremal black hole, since the mass and charge are conserved quantities one can evaluate at infinity. If the entropy is zero, that just means the state is unique. (Note also that there are very good reasons why you can’t make an extremal black hole classically, which might have something to do with all this.)
On the other hand, the local geometry should surely fluctuate in a real theory of quantum gravity. So the semi-classical kinds of words we are using might just not apply to this situation. That might be an escape hatch, I’m willing to grant.
Nobody knows how derive the BH geometry from some sort of coarse graining of the exact quantum microstates, so we are at the level of gut feelings. Mine is that any statement that is unstable to small perturbations is probably an artifact of using the strict classical limit, and is not going to survive in the quantum theory (that includes e.g the inner horizon).
“Nobody knows how derive the BH geometry from some sort of coarse graining of the exact quantum microstates, so we are at the level of gut feelings.”
Dear Moshe
Your statement isn`t correct! String theorists know how to do that!
For example:
http://arxiv.org/abs/0811.0263
Yeah, good point, that is definitely a step in the right direction. This may even be enough to resolve the current issue for extremal BH, provided their techniques can apply to BH with non-zero horizon area.
Moshe:
Are you implying that the actual existence of the inner horizon (and the whole “able to avoid the singularity” region within it) would be incompatible with a correct QTG? What particular aspect of the “unstable to small perturbations” rule that you wish to invoke applies?
Of course that whole inner region qualifies as a boojum for me anyway, if the singularity is naked, though I don’t know if that is true.
Sean:
I am curious about what the existence of the electric field from the charge implies for the characteristic of the Hawking radiation? Does a positively charged black hole preferentially emit positive particles (e.g. positrons?). Because this sort of emission probably isn’t efficient until you get kT = mc^2 for the horizon temperature, this would imply that as the black hole evaporates it would tend to become more extremal (Q/M increases), though I wouldn’t have a clue of the long term track of this.
If the inner horizon is real, then should it also produce Hawking radiation? Are there implications for the entropy that is associated with this emission, particularly the part that terminates in the singularity?
The old “black holes have no hair” theorems were based on the finite range of the interactions, does at some point with a sufficiently high temperatures could you have a weak charge on a black hole? (I’m not sure how the reunification phase transition is driven by the horizon.)
Physically, of course, I would expect it to be very difficult to create extremal charge black holes. But we have some evidence that nearly extremal Kerr holes (angular momentum) might exist (and even be “common”. Does the Kerr solution have some equally juicy behavior?
Does any of this have observable consequences?
Ras, anyone is entitled to their gut feeling. My gut feeling is that the inner horizon and all its features are an artifact of using classical GR with no matter, exactly empty space. Any infinitesimal change in these assumptions (for example looking at the leading order back-reaction, or looking at BH formed by collapse) will discontinuously change the situation to the more generic, aka “physical” one.
I agree with Moshe 🙂
Geez, just to make sure- what I wrote above, which is not very deep, has nothing to do with that paper.
Is there similar weirdness in extremal limits of all the black brane geometries?
The story doesn’t depend on the number of dimensions, of either the black hole/brane or the background. It will go through as long as we’re just talking about Einstein’s equation with a gauge field.
To repeat Jeff’s question:
Does any of this have observable consequences?
So far as I know, nothing in string theory has ever been confirmed by experiment or observation. From that POV, statements such as Someone’s ‘string theorists know how to do that’, I tend to regard as coming from people with a high regard for mathematics. I don’t have a problem with that, per se, particularly as I subscribe to the idea of beauty being indicative of truth. But in the final analysis, mathematics is not science.
Is this about laying groundwork for something that might have an observable consequence? A suggestion that astronomers might want to ponder, as to what an observable consequence might consist of?
I’m new here, a layman, undoubtedly missing a lot of context, and more or less at sea. Any pointers would be much appreciated.
This is a question not directly related to the post, but its one that I have been thinking about for a while, without any satisfactory answer that I can get from my layman’s knowledge. So, here goes:
As measured by a distant observer, an infalling object takes an infinite amount of time to cross the event horizon of a black hole. How, then, can the mass of a black hole, and the radius of its event horizon ever increase (again, as measured by a distant observer)?
Related to this is the “Black hole information paradox”. If, for a distant observer, an information-containing object is never actually seen to cross the event horizon, why do we say that the information has been “lost”? (Just what does “distant” mean here, anyway? Any distance outside the event horizon? Or is there a minimum distance which qualifies as “distant”?)
I’m sure someone must have thought about these questions by now. So, what am I missing?
A couple of ignorant questions:
1. Is there anywhere that space is an empty enough plasma that a stellar mass black hole can gain a charge due to the physical separation between protons and electrons?
2. What happens to an (positive) extremal black hole when it swallows a proton?
Greg, I see this disussion as being akin to Einstein’s question “what would it look like if you could catch up to a lightwave?”. It’s a thought experiment. Although catching up with a light wave is not feasible (and we know now, not possible in principle) the consideration of the question led to great insights.
I can smell some very interesting physics lurking around somewhere here…
Nothing to add to the discussion of physics, but can I just say how much I love ‘in a field where actual data is sparse on the ground, it’s worth keeping puzzles in mind, hoping that some day they will teach you something’. In fact, with your permission Sean, I’d like to add it to the scrapbook of useful and interesting advice I sometimes pass on to my students (I’m teach philosophy in the UK).
(Doh: ‘I teach’ not ‘I’m teach’)
Warning naive question ahead…
Is the implication here that gravity seems to be functioning as an entropy “sink” here in some as yet to be identified/quantified way or am I completely missing the point?
Thanks
e.
Amit:
A better way to think about things falling into a black hole is to note that if you drop something from anywhere outside the hole, after a certain finite amount of time it becomes physically impossible to chase after it and catch it before it crosses the event horizon (or even to shine a beam of light towards the falling object so the photons will reach it before it crosses the horizon).
What you see of an infalling object (albeit exponentially dimmed, so you don’t really get to see it), and what you can calculate if you use certain natural but potentially misleading coordinates, might suggest that the object “hovers forever” at the horizon. But if you can’t catch something, even at the speed of light, it’s gone.
The standard way of defining the mass of a black hole is to time the orbits of distant objects. You certainly won’t have to wait forever for the mass of an object dropped into the hole to show up in the total mass by that definition. And, after any gravitational waves from the infalling object have dissipated, the whole spacetime geometry outside the hole very rapidly comes to reflect the hole’s new mass.
Just a few quick points:
* Gravity has lots of observable consequences. So does quantum mechanics. Therefore, it would be nice to understand how they work together, and this paper was one of the many small steps in that direction. If you don’t think that’s an especially interesting thing to spend time thinking about, that’s fine; there’s plenty of good physics out there for everyone.
* Charged black holes are not going to occur in nature, it’s too easy to cancel the charge by absorbing some ions. Even if you had an extremal black hole, it wouldn’t absorb any more equal charge, because the repulsion would be too strong.
* Eliot, I’m not sure that it’s right to think of gravity as an entropy sink. More that the extremal BH geometry doesn’t include all the spacetime you might naively have thought, just by thinking of it as a limit of non-extremal black holes, and therefore the entropy might not be continuous.
* Sam C, you are more than welcome to pass on that quote.
@Greg: A related question, if I may. If I am a distant observer and my “natural but potential misleading coordinates” lead me to conclude that, from the POV of my frame of reference, infalling objects reach but never quite pass the event horizon…
…isn’t it completely unsurprising then that the entropy of the black hole should be proportional to the area of the event horizon since, from my frame of reference, that’s where all the information I toss at the black hole ends up?
Is this a useful way of thinking about the entropy of black holes… or is it misleadingly simplistic?
To my untrained eye the charged black hole and the spinning black hole have some superficial similarities, in that there are two horizons (besides the surface of the ergosphere) and perhaps a similar flipping of the time/space coordinates when crossing the inner horizon. There’s a space between these horizons that shrinks to zero depending on the mass:angular momentum, which reportedly could yield a naked singularity if you believe GR to that point. That strikes me as even more weird than Whoville, though I’ve no idea if there are any entropy issues associated with this beyond the obscene consequences of exposing a singularity to the law-abiding universe. Are there similar discontinuities in this situation (I suppose this would involve an analogous “extremal” verson of the Kerr metric)?
CarlZ:
“completely unsurprising” sounds like a bit of an overstatement to me. “is it misleadingly simplistic?” It is if you think it follows from any statement that’s only true in some coordinate systems, as opposed to a coordinate-invariant statement of actual physics. The statement “infalling objects never pass the horizon” is only true in some coordinate systems.
It is worth stressing that in GR you can use any coordinate system you like, and the Schwarzschild coordinates (in which t goes to infinity before a falling object passes through the horizon) are, very clearly, pathological on the horizon. What’s “natural” about the Schwarzschild coordinates is only that (a) far from the black hole, they merge nicely into ordinary flat spacetime polar coordinates, and (b) adding a constant to the t coordinate of every point in spacetime is a symmetry of the spacetime (and even that’s not quite true for real black holes). But they are terrible for making sense of infalling objects. There are other coordinate systems that do a much better job of that.
The surfaces of constant Schwarzschild t that fail to pass through the horizon aren’t really telling you anything “from the POV of my reference frame”. A reference frame is something local; every observer can pick a local reference frame in which they are at rest, but that doesn’t give them any special, “correct” way to put coordinates on distant events in curved spacetime. There’s nothing at all that compels someone far from a black hole to use a time coordinate in which infalling objects never pass through the horizon.
A priori of figuring out where does entropy go, is it totally rigorously defined? I mean, I have some material and maybe radioactive (so what the atoms do in statistical mechanics is not all there is to it) is there really, a specific value of “entropy” for that? How could such a rigorous definition be made, and unlike energy or momentum we can’t (?) “put it into” something to measure in a simple way, like using final mass to show change in mass-energy etc. And especially about radioactivity, I mean really – I can have some stuff lying around very cold and lowest “entropy” and if it can radioactively decay, it can change and turn into other stuff etc.