Prior to Einstein, physicists believed that light waves, like water waves, were ripples in a medium: instead of the ocean, they posited the existence of the luminiferious aether, some form of substance which supported the propagation of electromagnetic waves. If that idea had been true, one would imagine there would be a unique frame of reference in which the aether was at rest, while it was moving in other frames; consequently, the speed of light would depend on one's motion through the aether. This idea was basically scotched by the Michelson-Morley experiment, which showed that the speed of light was unaffected by the motion of the Earth around the Sun. The idea was eventually superseded by special relativity, although (as with most interesting ideas) some adherents gave up only reluctantly. Indeed, if you had asked Hendrik Antoon Lorentz himself about the meaning of the famous Lorentz transformations he invented, he would not have said "they relate physical quantities measured in different inertial frames"; he would have said "they relate quantities as measured in some moving reference frame to their true values in the rest frame of the aether." We know a lot more about field theory as well as about relativity these days, so we don't need to invoke a concept like the aether to explain the propagation of light, and the idea that there is no special preferred frame of rest has been experimentally tested to exquisite precision. But precision, even when exquisite, is never absolute, and important discoveries are often lurking in the margins. So it's interesting to contemplate the possibility that there really is some kind of field in the universe that defines an absolute standard of rest, within the modern context of low-energy effective field theories. Instead of a light-carrying medium, we are interested in the possibility of a Lorentz-violating vector field -- some four-dimensional vector that has a fixed non-zero length and points in some direction at every event in spacetime. But the name "aether" is too good to abandon, so we've re-purposed it for modern use. A lot of work has gone into exploring the possible consequences and experimental constraints on the idea of an aether field pervading the universe (see reviews by Ted Jacobson or David Mattingly, or Alan Kostelecky's web page). But the ideas are still relatively new, and there are still questions about whether such models are fundamentally well-defined. Tim Dulaney, Moira Gresham, Heywood Tam and I have been thinking about these issues for a while, and we've just come out with two papers presenting what we've worked out. Here is the first one:

Instabilities in the AetherAuthors: Sean M. Carroll, Timothy R. Dulaney, Moira I. Gresham, Heywood Tam Abstract: We investigate the stability of theories in which Lorentz invariance is spontaneously broken by fixed-norm vector "aether" fields. Models with generic kinetic terms are plagued either by ghosts or by tachyons, and are therefore physically unacceptable. There are precisely three kinetic terms that are not manifestly unstable: a sigma model $(partial_mu A_nu)^2$, the Maxwell Lagrangian $F_{munu}F^{munu}$, and a scalar Lagrangian $(partial_mu A^mu)^2$. The timelike sigma-model case is well-defined and stable when the vector norm is fixed by a constraint; however, when it is determined by minimizing a potential there is necessarily a tachyonic ghost, and therefore an instability. In the Maxwell and scalar cases, the Hamiltonian is unbounded below, but at the level of perturbation theory there are fewer degrees of freedom and the models are stable. However, in these two theories there are obstacles to smooth evolution for certain choices of initial data.

As the title says, here we're investigating whether aether theories are stable. That is, when you have the vector field in what you think should be the "vacuum" state, with all of the vectors aligned and nothing jiggling around, can a small perturbation lead to some sort of runaway growth, or would it just oscillate peacefully? If you do get runaway behavior, the theory is unstable, which is bad news for thinking of the theory as a sensible starting point for experimental tests. This is one of the first questions you should ask about any theory, and it's been investigated quite a bit in the case of aether. But there is a subtlety: because you have violated Lorentz invariance, it's not enough to check stability in the aether rest frame, you need to do it in every frame. (A perturbation caused by a source moving rapidly in a rocket ship is still a legitimate perturbation.) What we found was that almost all aether theories are unstable in some frame or another. There are just three exceptions, which we called the "sigma model" theory, the "Maxwell" theory, and the "scalar" theory. You might ask, what is this talk about "theories"? Why is there more than one theory? For a vector field, it turns out that there are a number of different quantities you can define (three, to be precise) that might play the role of a "kinetic energy." So we study a three-dimensional parameter space of theories, corresponding to any possible mixture of those three quantities. The three theories we pick out as stable are simply three specific mixtures of the different kinds of kinetic energy. The Maxwell theory is very similar to ordinary electromagnetism, while the scalar theory more closely resembles a scalar field than a vector field. The other theory is actually our favorite, as both the Maxwell and scalar cases seem to have potential lurking pathologies that we can't completely get rid of (although the situation is a bit murky). So we wrote a shorter paper examining the empirical behavior and constraints on that model:

Sigma-Model AetherAuthors: Sean M. Carroll, Timothy R. Dulaney, Moira I. Gresham, Heywood Tam Abstract: Theories of low-energy Lorentz violation by a fixed-norm "aether" vector field with two-derivative kinetic terms have a globally bounded Hamiltonian and are perturbatively stable only if the vector is timelike and the kinetic term in the action takes the form of a sigma model. Here we investigate the phenomenological properties of this theory. We first consider the propagation of modes in the presence of gravity, and show that there is a unique choice of curvature coupling that leads to a theory without superluminal modes. Experimental constraints on this theory come from a number of sources, and we examine bounds in a two-dimensional parameter space. We then consider the cosmological evolution of the aether, arguing that the vector will naturally evolve to be orthogonal to constant-density hypersurfaces in a Friedmann-Robertson-Walker cosmology. Finally, we examine cosmological evolution in the presence of an extra compact dimension of space, concluding that a vector can maintain a constant projection along the extra dimension in an expanding universe only when the expansion is exponential.

Even this theory, as interesting as it is, is plagued by a problem. In the spirit of low-energy phenomenology, we basically fix the length of the vector field by hand. But we recognize that in a more complete description, there is probably some potential energy that gets minimized when the vector takes on that value. But if you allow for any variation whatsoever in the length of the vector, you are immediately confronted with a dramatic instability once more. So, to be honest, there are no aether theories that we can guarantee are perfectly well-behaved, even as low-energy effective theories. (All the problems we identify exist at arbitrarily low energies, and don't rely on the short-distance behavior of the models.) The three theories to which we gave names are problematic but not manifestly unstable, so it will be worth further investigation to see if they can be patched up and made respectable.