Ice water on closed timelike curves

Today, let's kick off our shoes and relax our standards for what counts as a reasonable spacetime. Let's talk about spacetimes allowing for a certain kind of time travel, called a closed timelike curve. Being stuck on one would be like being in Nietzsche's "eternally recurring" world -- or more recently, like Bill Murray in the movie Groundhog Day.

Although worlds containing closed timelike curves arise from classical general relativity, one can worry about whether such worlds obey the laws of statistical physics. For example, what would happen to a glass of ice water on a closed timelike curve?

The state of the ice water would have to be cyclic. So, if the ice were able to melt at all, then it would also have to "unmelt," and coagulate as ice again when it came back around. Apparently, entropy in our time-traveling ice water has to regularly and dramatically decrease. But that's supposed to be impossible.

Does that make closed timelike curves incompatible with the principles of statistical physics? Not necessarily. Boltzmann would argue that the reason the ice should always melt is that, among all the possible trajectories that the particles in the ice water might travel, those in which the ice melts form the overwhelming majority.

So, assuming each possible trajectory is equally likely, it follows that the ice is overwhelmingly likely to melt.

But suppose we restrict what counts as a "possible trajectory" for a particle in our time-traveling ice water. After all, we're on a closed timelike curve. The possible trajectories are different, because they are required to be cyclic -- every particle has to end up back where it started. This constraint guarantees that, if the ice melts at all, it always also unmelts. Boltzmann's counting argument then just determines what most commonly happens in between, among these trajectories.

Of course, everything has to happen the same way every time around an individual closed timelike curve. So, "most commonly" would have to be made sense of in terms of an array closed timelike curves, each with a glass of ice water on it. But apparently, no fundamental principles are violated. And so statistical physics might yet live in harmony with closed timelike curves.

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.)

Why does matter follow geodesics?

In the very first lessons on General Relativity, we learn that free particles follow geodesics -- the equivalent of straight lines in curved spacetimes. Why? Well, it's easy to show that point particles must follow geodesics, if Einstein's Field Equations are respected. But if you're like me, you might have had this lingering suspicion:

Real matter is not made of point particles. That might have lead you to ask: why does real matter follow geodesics?

Here's a very interesting answer, provided by Geroch and Jang (1975):

Theorem: Let (M, g_ab) be a space-time. Let γ be a worldline satisfying the following condition: For any neighborhood U of γ, there exists a nonzero, symmetric, conserved tensor field T^ab that satisfies the strong energy condition, and whose support is in U. Then γ is a timelike geodesic.

The idea is this: if there's a any kind of field of matter following a worldline through spacetime, which a) behaves like the matter we're used to, and b) is small compared to the curvature near the worldline, then that matter follows a geodesic.

Now, another lingering question: is there anything about this matter field that guaranteees the geodesic is unique?

Two Tough Cases of Underdetermination

John Earman once pointed out two tough cases of underdetermination, neither of which arises in a silly algorithmic way. Today, I'd like to argue for an important difference between these two examples. Ultimately, I suspect the first is more deeply intractable than the second.

Earman's first example is this.

I claim that there do exist examples of rival empirically indistinguishable theories that posit interestingly different theoretical structures. For instance, TN [Newtonian Theory] (sans absolute space) can be opposed by a theory which eschews gravitational force in favor of a non-flat affine connection and which predicts exactly the same particle orbits as TN for gravitationally interacting particles (Earman 1993, 31).

In short: Newtonian gravitation can be described as particles interacting via forces on a flat spacetime -- or as particles in freefall in curved spacetime, as in Cartan's 'geometricized' Newtonian gravitation. (See Malament (1986) for a discussion of the latter. This underdetermination has been beautifully and rigorously argued for by Jonathan Bain -- PDF.)

Earman's second example is the existence of observationally indistinguishable spacetimes:

Relativity theory tells us that the data available to an observer through such interactions [as observation] are restricted to events that are swept out by the observer's past light cone.... As a result, even idealized observers who live forever may be unable to empirically distinguish hypotheses about global topological features of some of the cosmological models allowed by Einstein's field equations for gravitation (ibid).

Indeed, my distinguished colleague John Manchak has greatly generalized this result, and shown that all spacetimes have an indistinguishable counterpart that preserves all the local properties of spacetime (Philsci-Archive).

What's the difference between Earman's two examples? In short, it is that the first is about underdetermination of theories, while the second is about underdetermination of models of a particular theory, by available empirical evidence.

What's the significance of that? The first example strikes me as a completely convincing case of underdetermination (especially on Bain's treatment). But note: whether or not the second kind of underdetermination obtains depends on how one formulates a theory. For that reason, this underdetermination might often be avoided by a well-motivated reformulation.

For example: there is a particular class of models of Einstein's Field Equations (EFE) that can be described. There is a smaller class of models when one also requires that the Energy Conditions (EC) be satisfied. Any causal condition (CC) will place further restrictions on the class of models. So, we have multiple formulations of general relativity:

• EFE
• EFE + EC
• EFE + CC
• EFE + EC + CC

all of which correspond to different classes of models. So, when one argues, model M and model N are observationally indistinguishable models of our world, this underdetermination might be avoided by moving to a more restricted formulation of General Relativity. And, unfortunately, the issue of which formulation of general relativity is the correct one is deeply contentious.

Of course, there are some cases of observational indistinguishability which are can obtain even in EFE + EC + CC -- for example, David Malament (PDF) describes observationally indistinguishable counterparts to de Sitter spacetime. But one wonders if this might not lead us to a suspicous game: you show me an example of observational indistinguishability, and I try to rule it out by further restricting the class of models of General Relativity.

At any rate, a claim about underdetermination at the level of theories (such as in Earman's first example) does not seem to suffer from this kind of regress.

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.

Map of the Cosmic Acceleration Literature

Ten years ago, the physics community came to agree that the expansion of the Universe is experiencing a positive acceleration. The experts still disagree on why. Everything but the kitchen sink has been proposed (I've mentioned this twice before), but there is a paucity of experimental evidence to favor one proposal over another.

This situation strikes me as a gold-mine for philosophers physics. In particular, I would hope that we could learn something interesting about what kind of reasoning is allowed in such an empirically starved research program. In particular, what I'd like to know is:

What argumentative moves are licit in response to the cosmic acceleration problem?

As a first step toward figuring this out, I made a map of the most common arguments being made. To see my map, click the image on the left (or download the PDF). This map is inspired by a (much smaller) such diagram given by Sean Carroll (2003). Suggestions are more than welcome!

Rotating Discs in GR: Part I

Introduction. Craig Callender (2001) and Jeremy Butterfield (2004) have recently suggested that the Rotating Disc Argument (RDA) fails in General Relativity. I argue that this conclusion is wrong: two recharged versions of the RDA do work in GR. In this post (Part I), we'll review the necessary background material (without GR):

1. The Original Rotating Disc Argument (RDA)
2. Butterfield's Sophisticated RDA.

In the next post (Part II), I'll explain the Callender-Butterfield complaint that the RDA fails in General Relativity, and then argue for two modifications of the RDA do work in GR.

1. The Original Rotating Disc Argument (RDA). The RDA is supposed to be a counterexample to one view about what an object's properties consist in. Fundamental objects can have intrinsic properties (such as the intrinsic angular momentum of an electron), as well as spatial location properties (described however you like). And there is a philosophical view, often called "Humean Supervenience" (HS), which says: those properties are all there are. Every property of every object is made up of fundamental objects (such as electrons), and so the properties of every object are just combinations of the intrinsic properties and spatial locations of its fundamental parts

The putative counterexample to this view: imagine two perfectly homogeneous, continuous discs; one is rotating and the other is not:


The idea is that these discs have different properties, but they they can't be distinguished by the intrinsic and spatial properties of their parts. Here's how this is supposed to work.

First, the continuous homogeneity is supposed to guarantee that you can't pick out the history of any individual point on the disc. So you can't follow the individual worldlines of the points to tell if they are all vertical (motionless through time), or spinning around like a cork-screw.

Second, you are supposed to ignore every other physical effect that angular momentum incurs on the parts of the disc. Then the claim is: these two cases can't be distinguished by the spatial locations and intrinsic properties of their parts, and "Humean Supervenience" is defeated.

2. A More Sophisticated Version. As Callender rightly points out: that second step was pretty bogus. Imagine a spinning bucket. What happens to the surface of the water inside? It becomes convex. So the spatial locations of the points on the rotating disc do distinguish it from the static one. Furthermore, phenomena like this always indicate whether or not an object is spinning in the real world. So if you imagine them away, you really don't have any reason to think the two discs are different.

However, Butterfield suggests a modification of the RDA that avoids this complaint. Instead of a rotating and non-rotating disc, consider these two cases:
In Case 1, two homogeneous discs rotate clockwise. In case 2, one rotates clockwise and the other counter-clockwise. Butterfield's more sophisticated RDA says: Cases 1 and 2 can't be distinguished by the intrinsic properties and spatial locations of the parts of the discs. The Newtonian effects of rotation equally appear on all four discs. But there is clearly a difference between the two cases. So Humean Supervenience is again defeated.

The point of all this has been to get clear on the RDA, not to advocate it. In fact, Butterfield (2004) has given a very original rebuttle of the RDA, by revising the Humean Supervenience claim in a way that allows him to sneak in velocity as an "intrinsic property." I will not be discussing this idea here. Instead, I'd like to argue against a claim that both Butterfield and Callender take for granted: that the RDA fails in GR.

This argument is to be continued on Wednesday.

Get Started Learning General Relativity Online

General Relativity is the theory of gravitation introduced by Einstein in 1915, and developed throughout the 20th century. And you've decided you want to learn it. But why spend hundreds of dollars on textbooks? As I've shown before, it's easy to learn things on the cheap, without sacrificing quality!

In this tutorial, I've compiled a list of introductory material on the physics and philosophy of general relativity, all of which is available for free online. If you know of any online resources that are not on this list, your comments are more than welcome!

Contents:

• Non-mathematical introductions to general relativity


• Mathematical introductions to general relativity


• Philosophy of general relativity


• Other general relativity references.

Non-mathematical introductions to general relativity.

1. Einstein for Everyone, by John D. Norton. This is a complete introductory text. Many pictures and animations illustrating the central features of the theory. Emphasizes philosophical perspectives, where relevant. Available for free online.


2. Introduction to General Relativity on Wikipedia. A quick overview of a few of the essential features of GR, on everyone's favorite non-scholarly resource.

Mathematical introductions to general relativity.

1. Oz and the Wizard, by John Baez. This is very entertaining introduction to general relativity in the form of a dialogue. It also contains a fantastic dictionary of common terms in GR.


2. David Malament's lecture notes on GR (PDF) are also a pleasure to read. They are really a textbook introduction to GR, with an emphasis on mathematical and philosophical perspectives.


3. Lecture notes on general relativity, by Sean M. Carroll. These are the course lectures for an MIT graduate course in general relativity, and have since been turned into a book. Also try the 24-page "no-nonsense" version of these notes (PDF).


4. Introduction to Differential Geometry and General Relativity, by Stefan Waner. A beautifully arranged collection of lecture notes on differential geometry. Approach is highly mathematical, taking the reader from basic point-set topology all the way to Einstein's field equations.


5. Tensors and Relativity, by Peter Dunsby. A first course in general relativity, beginning with special relativity. Includes an "assignments" section with hints. The layout is ugly and cumbersome, but the content is good.


6. Introduction to General Relativity (PDF), by Gerald 't Hooft. These lecture notes have since been turned into a book with Wei Chen.


7. Modern Relativity, by David Waite. A first course on general relativity, which assumes good familiarity with special relativity. Includes exercises.

Philosophy of general relativity.

1. Here are some graduate courses in the Philosophy of General Relativity (with tons of resources), taught by some of the leaders in the field: John Earman and John Norton, David Malament, Brad Skow, Jonathan Bain, and Craig Callendar


2. John Norton's Introduction to the Philosophy of Space and Time gives Norton's characteristically clear (and opinionated) view of the field. Most of Norton's papers on the philosophy of general relativity are available here.


3. David Malament's Classical General Relativity describes the mathematical structure of the theory, and introduces a few of the philosophical problems associated with it. Most of Malament's papers on the philosophy of general relativity are available here.


4. The Philsci Archive's Relativity category contains most of the latest preprints in the philosophy of general relativity. (I have written some tips for using this resource.)


5. John Earman v. Tim Maudlin: Two philosophy of physics giants discuss what general relativity teaches us about the nature of time. (Not really introductory, but too entertaining not to mention).

Other general relativity resources.

1. Black Holes, by Paul Townsend. From his Cambridge course on the black hole physics developed in the 60's and 70's.


2. The Relativity Bookshelf at U. Toronto is brief but informative.


3. Arxiv.org's general relativity category, on the other hand, is like drinking from a fire-hose. Hundreds of articles on general relativity appear every week.

That's all the free general relativity for today. Enjoy!

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.

Observationally Indistinguishable Dark-Energy Theories

Fundamentally different theories of dark energy may be underdetermined by observation, according to a recent preprint by Sanil Unnikrishnan.

Background. Dark energy was first proposed as a solution to the problem of cosmic acceleration. That is, if you propose the existence of an exotic form of matter (generically labeled "dark energy"), you can account for the observed FRW-expansion, as well as for the late-time ISW effect. However, there is now a whole zoo of different ideas about what dark energy is and what it's like: quintessence, k-essence, "phantom" energy, or even tachyons -- objects travelling at super-luminal speeds. (Here's a nice review of some of the options, by Sean Carroll.)

The Result. Whatever the nature of dark energy, the only observational access we currently have to it is through its effect on cosmic acceleration. This effect is determined by (global) parameters: the ratio of pressure to density of dark energy, and the ratio of pressure-fluctuations to density-fluctuations.

But what if there were two distinct models of dark energy that gave the same values for these parameters? Then those two models would be indistinguishable by observation. This is the situation that Unnikrishnan argues that we are in. In short: suppose dark energy is modeled as a scalar field (as it is in most of the above proposals). Then any values for these two parameters that you might measure can be derived from distinct scalar field Lagrangians -- different models of dark energy.

Some Implications. This would appear to lead to an epistemological quandary. Dark energy is being proposed as a fundamental feature of our universe. So it would be very fishy indeed if the fundamental nature of dark energy were in principle unknowable.

However, the underdetermination isn't quite as bad as that. If dark energy is what the physics community ends up accepting, then it's unlikely that our current observational access to dark energy is all we'll ever get. For example, if we ever succeeded in making any kind of local observation of dark energy in the lab or solar system, then we'd have new dark energy parameters to measure, which would not be underdetermined by this trick.

Furthermore, there are plenty of excuses for cosmic acceleration that don't model dark energy as a scalar field -- for example, those in which a cosmological constant is added to the field equations, and those which model it as the result of early-universe inhomogeneities.

Still, one can't help but feel a bit uneasy about the whole dark energy program. Historically, it looks a bit too much like Descartes' vortices -- and results like this one don't make us feel any better.

Is the cosmological metric about to flip Euclidean?

The 'wildest excuse for cosmic acceleration' prize so far should go, in my view, to these guys. They argue that the metric form of spacetime is about to flip from (-+++) to (++++), which they say produces the appearance of an accelerated expansion. Here's a little background on why they're suggesting this.

Cosmic Acceleration: a very brief background. Around 1998, the majority of physicists accepted that expansion of the Universe is accelerating. Since then, all bets are off as to the large scale structure of the spacetime. The experts simply don't agree as to what kind of Universe we live in, and it sometimes seems that the wilder the proposal, the better. This makes peering over the fence at the zoo of cosmological theories rather entertaining.

There are a lot of competing theories out there -- that exotic matter causing negative vacuum pressure is spread throughout spacetime (this is the "dark energy" that you hear so much hype about), that backreactions due to the inhomogeneities of the early Universe gave the expansion an extra kick, or that the speed of light is not equal to c on cosmic scales (so that we're not interpreting our data correctly).

But a changing metric signature? As it turns out, this idea isn't totally unheard of; as of today there seem to be around 50 papers on arXiv dealing with models of quantum gravity in which the signature of spacetime changes. It just wasn't clear until recently that this could be used to account for anything to do with the accelerated expansion.

(Wait a minute -- so why didn't a treatment of changing signature appear in my general relativity textbook? Well, the metric signature doesn't change in traditional general relativity. Most derivations of the Einstein Field Equations assume non-degeneracy of the metric, which is sufficient to fix the metric form for all of time. So the approach to cosmic acceleration here falls under the category of -- *gasp* -- changing the field equations.)