Hyper-intelligent fish and black hole thermodynamics

Bill Unruh's recent collection on black hole analogues begins,

Deep beneath the great encircling seas of the Discworld lived a species of hyper-intelligent fish. (Unruh 2007, p.1)

Unusual, but inspiring: Unruh compares Hawking radiation -- the thermal heat bath emitted by black holes -- to a scenario he imagines in Terry Pratchett's Discworld. Pratchett's world is basically a big dish, with water flowing over the edges.

On Unruh's take, the dish-water is filled with little physicist fish, who are trying to determine the laws of physics. The fish are blind, but use sound waves to interpret their environment. And they are mostly successful. However, as water falls off the edge of the world, it reaches speeds faster than the speed of sound. Events beyond this "sound horizon" are thus inaccessible to the fish in the ocean.

One day, a graduate-student-fish goes flying off the edge while the professor-fish observes. (Professor Unruh apparently expects a lot of his students.) The graduate student yells "Help," while falling off. Then he plunges to his doom. But, from the professor's perspective, the sound of the graduate student's scream persists forever, getting ever more bass-shifted, as the student approaches the horizon.

The point is, the unlucky graduate-student-fish is directly analogous to an astronaut falling into a black hole. From the astronaut's perspective, nothing special happens as she crosses the event horizon. But from an outside observer's perspective, the astronaut appears to be forever approaching (but never crossing) the event horizon, and the light she emits getts ever more red-shifted.

Of course, the astronaut will get ripped to shreds by tidal forces, while the fish will not.

And so the "black hole analogue" debate begins. Black holes are widely believed to have a number of thermal properties -- for example, black holes have a temperature proportional to their surface gravity. Analogously, soundless "dumb-holes" (as Unruh calls them) in water can be shown to have interesting thermal properties as well. And -- tantalizingly -- it appears possible to carry out experiments that would actually test the properties of "dumb-holes," even though black holes remain outside our reach.

But does evidence for a sound-based analogue somehow provide us evidence about a real black hole?

I see no plausible way that it can. Although a black hole is mathematically similar to a "dumb hole," it is not the same thing. And history has something to teach us here: gas and fluid vortices are "mathematically similar" to Descartes' aether vortices. But experiments with the former do not provide evidence for the latter. After all, aether vortices don't exist! So, in spite of some interesting recent experiments (see here), we still don't have any new evidence that black holes have thermal properties.

Nevertheless, there might be one thing that sound-based experiments can still teach us about black holes, according to Unruh:

such successful experiments would greatly increase the confidence in the approximation which were being made in both the gravitational and the analogue situations. ... Certainly the suggestions from the sonic case are that Planckian physics is irrelevant to black hole evaporation, and that the radiation emitted by a black hole is due to low energy processes, processes on the length scale set by the black hole, and not by quantum gravity. (Unruh 2007, p.3.)

This to me seems very plausible: an analogy can tell us whether or not scale is relevant to the effect. According to Unruh, sound-based experiments are really teaching us that black hole thermodynamics is about essentially macroscopic effects. So, our prediction of thermal effects like Hawking radiation won't change when a new theory of quantum gravity comes along, and modifies our picture of the (microscopic, high-energy) Planck scale.

It's a bold and intriguing suggestion, but I'll wait for the iron hand of history to decide.
(If you have a Springer subscription, you can see a version of Unruh's article here.)

No, there are STILL no deadly mini-black holes!

Kentucky, who runs a fantastic blog over at arxivblog.org, has pointed out a new calculation about black hole creation here on earth. The paper (arxiv) suggests that mini-black holes might be created at CERN, and actually be long-lasting, existing for as long as a minute. Kentucky is worried about all of our well-beings.

The press for CERN is of course nice. But as I've argued before, there will be no deadly black holes created at CERN.

The theories that predict mini black holes are highly -- I repeat, highly -- speculative. For example, in the paper Kentucky mentions, the authors adopt a Randall-Sundrum (RS) brane world model. In these theories, gravitational interactions take place in 5-dimensional (4+1) anti de Sitter spacetime, in which there is a 4-dimensional embedded hypersurface (a brane) in which the particle interactions of the Standard Model occur.

In classical general relativity, typical gravitational collapse into a black hole happens for bodies with greater than 3 solar masses of mass-energy. That kind of energy would kind of blow the power-grid here on Earth. However, in the brane world model, the extra spatial dimension for gravity has the effect of vastly reducing the amount of mass-energy in our (4-dimensional brane) world needed to create a black hole -- by screwing with that extra dimension, you can still get the necessary curvature conditions. The five-dimensional black hole is then expected to accrete mass until a 4-dimensional black hole is created in our world.

Now, 4-dimensional mini black holes quickly evaporate because of Hawking radiation. So if you believe the RS brane world model, then you're wondering: will the growth of the black hole in 5-dimensions outrun the evaporation of the black hole in 4-dimensions? According to the paper Kentucky discusses, the answer is: no! If black holes appear, they will still evaporate more quickly than they grow.

But here's the real reason why you shouldn't be worried: The RS brane world theory is still a fairy tale, not a confirmed theory!

There are plenty of competing theories of quantum gravity out there. One of them might even someday be shown to be the correct one, especially once the Large Hadron Collider gets up and running. But for now, there is simply no evidence confirming the possible existence of this 5th spatial dimension. So there is no empirical evidence to support the possibility of black hole creation at CERN.

Visualize a Black Hole at the Event Horizon

This cute little animation shows the causal structure around a classical black hole in an interesting way. The perspective is that of a body orbiting just outside the event horizon:

Animation credit: Robert Nemiroff.
Nemiroff's paper discussing this kind of visualization is available on ArXiv.org.

For more animations of spacetimes with screwy causal structures, visit Virtual Trips to Black Holes & Neutron Stars.

Solving the Black Hole Information "Paradox"

In January, Abhay Ashtekar posted a short preprint in which he (along with two collaborators) proposed a new solution to the so-called black hole information paradox in 1+1 dimensions. Their essential trick is to propose a quantum-gravity inspired framework in which there is no black hole singularity, which leads to no loss in quantum information. This paper received a lot of superficial attention in the media last week; here's a little background on what's going on.

Background: Black Hole Entropy.The theory of black hole entropy is a well accepted combination of general relativity and quantum theory, which exploits the following close analogy between black hole physics and thermodynamics:

• black hole mass :: energy of a thermodynamic system;


• surface area of the event horizon :: entropy;


• surface gravity k :: temperature.

The last analogy is more than that: a black hole's surface gravity is literally its temperature. A black hole radiates energy exactly like a black body, with temperature k/2π, where k is the surface gravity of the black hole. This effect, called Hawking radiation (which I have discussed before), entails that eventually, all black holes will radiate away their mass, until they become boring everyday objects.

The Problem. The problem that many have with this process is that, according to a semi-classical analysis first argued by Stephen Hawking, it entails an unusual loss of quantum coherence. Here's a very informal sketch of how it happens.

Consider two particles that are initially correlated. One particle enters the event horizon of the black hole, while the other remains outside. Informally, it appears that half of the initially correlated state will disappear into the black hole singularity. So after the black hole has completely evaporated, we will have "lost" all information about the correlation. The result: the correlation is broken, and we are left with a mixed state.

In ordinary quantum mechanics, there is no mechanism for a state to evolve from pure to mixed in this way. As a result, many take the appearance of this phenomenon in black hole physics to be a fundamental problem for quantum gravity. The phenomenon is even sometimes referred to as a "paradox," although this is pretty inappropriate, since it doesn't seem to give rise to any contradiction. After all, the analysis doesn't actually use ordinary quantum mechanics, but rather quantum field theory.

In quantum field theory, the evolution of a state from pure to mixed is a pretty ordinary phenomenon. It certainly isn't unique to the highly curved spacetimes around black holes; in fact, curved spacetimes aren't even required to produce the effect! For example, Wald (1994, chapter 7.3) shows how this phenomenon can occur in (flat) Minkowski spacetime. Here's the trick.

Consider a massless scalar field that is spread out across all of space (i.e., on a Cauchy spacelike hypersurface); call this the initial (pure) state. Suppose this field evolves into a hyperboloid, or any other non-Cauchy hypersurface; call this the final state (see figure below). Then instead of radiation falling into a black hole, there is radiation propagating out to infinity. As a result, the correlation of the field on the interior domain of dependence of the hyperboloid is only "correlated" with radiation at infinity, and so is really in a mixed state.

Wald's Construction.
This is the same "paradox" of quantum information, but in flat spacetime: a pure quantum state evolves into a mixed state. It's not a paradox that is somehow unique to strong gravitational systems. It is a very general feature of quantum field theory.

If one simply accepts this fact, then it is far from clear that the Black Hole Information "Paradox" is much of a problem at all.

Solving the Problem Anyway. If it's not clear that there's a problem, then why are people proposing so many solutions? Well, perhaps the rhetoric of solving a problem is misplaced here. Perhaps a better characterization would be that most physicists are trying to extend a theory. In both black hole evaporation and in the hyperboloid field evolution described above, the essence of the phenomenon is this:

the final state of some quantum field provides an "incomplete" description of the original field.

So any effort to solve the problem is really an effort to give a more complete description of the field. And much of the physics community now agrees that such a description should be available.

Now that we've got a better grip on what we're actually "solving" here, we can ask: what have Ashtekar and his collaborators brought to this discussion? The solution of Ashtekar et al. draws on ideas from Loop Quantum Gravity (LQG), one of the major competitors of string theory. LCG has recently received attention for its apparent ability to resolve certain kinds of singularities (for example, LCG can resolve the big bang singularity).

What Ashtekar et al. have shown is that, given a suitable quantization of spacetime, black hole singularities can also be resolved. In this framework, no part of the original field "disappears" into a singularity because there is no singularity. Therefore, no information is lost, and a pure state remains a pure state.

As discussed above, this doesn't mean that quantum information loss can't happen; it only suggests a framework in which it doesn't happen as a result of black hole evaporation. Whether or not any such framework can prevent quantum information loss altogether remains to be seen.

LHC Black Holes: Why I'm Not Holding My Breath

The attention that these two nut-jobs are receiving is a bit discouraging.

Some people do hope to see mini-black-holes at CERN, it's true. Some calculate that we will see thousands. But: (1) if you have any empiricist scruples, then you won't believe in these mini-black-holes at CERN; alternatively, (2) if you have no such empiricist scruples, then you'll agree that all mini-black-holes at CERN are short-lived and harmless. Here's why:

Let's begin with (2), and suppose that you're not too hardcore about your empiricism. Consider a Schwarzschild blackhole (spherically symmetric and non-rotating, simplified idealization of what is expected at CERN) with mass M. Then dM/dt = -K/M^2, where K is a (very large) constant (See Hobson et al, pp. 277). Let M get very, very small, since we are dealing with particle collisions and not collapsing super-structures. Then dM/dt will become an enormous negative number. In other words, any emergent mini-black-hole will quickly decay into a boring everyday particle. These black holes are harmless and short-lived!

This effect is called Hawking radiation. But should we really believe it will happen? It hasn't ever been observed. However, the result is far from speculative. It is derived from well-verified results of basic quantum theory and of general relativity. From GR, we need only the causal structure of black holes (which is now well accepted, and if we're wrong about it, then there are no black holes anyway). And from QT we need little more than quantum fluctuations (consisting of particle/anti-particle pairs), which we have good empirical reason to believe in. So this isn't one of those weird fringe cases where "quantum theory and gravity don't mix." As things currently stand, there are great betting odds in favor of Hawking radiation.


Image Credit: Universe Review (2008)

But now let's suppose that you're a hardcore empiricist and you still don't buy it. If that's the case, then you don't have to worry about mini-black-holes at CERN in the first place, as there is absolutely no empirical reason to believe they will appear.

Black holes appear when a sufficiently large mass-energy to be crammed into a sufficiently small radius, which in our example is called the Schwarzschild radius (R_s) of that mass. This is not expected to happen at CERN according to any well-confirmed quantum theory, for reasons that have to do with uncertainty, and our consequent inability to squish that much mass-energy into lengths of the order of a very small R_s.

However, some string theorists think that our four dimensions are just one surface of a many-dimensional world that we apparently can't access. One consequence they derive is a much larger value for R_s for a given mass in these situations (from what I understand, they think it gets stretched out into these extra dimensions). This is the reason people have recently decided to hype up the hope that mini-black-holes might appear at CERN -- string theory says there is a larger R_s, so it's easier to cram sufficient mass into the region. (This idea was sketched in a CERN press release a few years back.) But of course, there is zero empirical evidence for this (and all) string theory. So your empiricist scruples set you free here -- mini-black-holes at CERN are little more than a fancy speculation. Maybe there's monsters in the closet too, but I'm not holding my breath.

Whew. Do you feel liberated? I feel liberated.

For more information, download the expert safety reports at the CERN website.