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 The idea was eventually superseded by special relativity, although (as with most interesting ideas) some adherents gave up only reluctantly.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 Aether
Authors: Sean M. Carroll, Timothy R. Dulaney, Moira I. Gresham, Heywood TamAbstract: 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 Aether
Authors: Sean M. Carroll, Timothy R. Dulaney, Moira I. Gresham, Heywood TamAbstract: 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.
1. What observation (or intuition) prompts one to postulate such a field? 2. I gather that the Higgs field is not, and has never been, thought of as a neo-aether. The Higgs’ popular descriptions do not make that distinction obvious. How does it differ?
Pat Dennis
I was surprised when I learned that the Milky Way has a measurable velocity wrt the CMB radiation. Wouldn’t that make the CMB frame a common reference frame that any observer in any galaxy can reference motion to; i.e. a kind of absolute frame of reference? Two observers may not agree on who is moving or what a length is or time interval is, but if they reference the length and time in the CMB frame, wouldn’t they each get the same result?
Has anything like a two-scalar field been tried? Two-scalar in the sense of ‘electro’ and ‘magnetism’, so that in a case of absolute rest the ratios of their strengths are minimized (maybe non-zero ‘E^*’ and zero’ M^*’) and any relative motion results in a change of ratios. I’m an algebraist, so I wouldn’t begin to know how to manage something like that. Or if it’s even possible. The differing parts of the EM field is just a vector projection, iirc. Hard to do a projection on a scalar potential.
I was also curious as to motivation: is it dark energy? Thx.
For the most part, the motivation is not to try to solve some pre-existing problem, like dark matter or dark energy. (Although Jacob Bekenstein’s relativistic version of MOND, which is meant to replace dark matter, includes an aether component.) It’s to provide a framework in which to conduct precision experimental tests. It’s one thing to say “let’s test how well Lorentz invariance is preserved in the real world,” but to actually go out and do it you need to have a model of how Lorentz invariance might be broken in the real world. That’s what these theories try to provide. The reason that’s an interesting strategy is because tests of symmetries can indeed be extremely precise; in some aether models, the experimental constraints are already above the Planck scale, which is not something you can achieve in conventional models.
The Higgs, or scalars more generally, just don’t do the trick, as a scalar with a nonzero expectation value is still Lorentz-invariant — you can’t measure your speed with respect to it.
The CMB provides an approximate rest frame for the universe, but only in the same sense as the Earth provides a preferred notion of “up” and “down.” What we’re looking for here is symmetry breaking in the vacuum, not just because there is some “stuff” hanging around with respect to which we can measure our speed.
Hi Sean,
I am trying to figure out the difference between the stuff we’ve done and these papers. In my thesis, I worked out the constraints on the (b1,b2,b3) (equivalent to your (alpha,1) parameter set with b1=-b3), using stability and boundedness of Hamiltonian of the spin-0,1,2 components of the vector-tensor spectra.
In particular, I showed that we need b2>0, which I think is equivalent to saying that alpha=-1 (since we are allowed to rescale it as your set of parameters is one less than the old one). But here, you are saying that (?) there is only 1 unique choice *only* if we impose the additional condition that we want the vector to be not superluminal in all frames, which you showed by considering the space of all possible lorentz boosts of the vector. My question is that : the calculation that Ted+David did (and we did in the old papers for dS space) is “manifestly lorentz invariant”, i.e. it’s written with well behaved lorentz indices. The constraints that was derived by us seems, as far as I can understand, lorentz invariant. So I am not sure what is the additional ingredients you have put in??
(Sorry if I am rambling a little bit!)
Hi Eugene– I believe that what you are calling “manifestly Lorentz invariant” is, we are arguing, not enough. (I thought it was enough at the time, but not any more.) In particular, when one calculates a Hamiltonian, or more importantly when one considers the evolution of initially small plane-wave perturbations defined on some constant-time hypersurface, one needs to choose a frame. And of course it’s natural to choose the aether rest frame. But we are pointing out that there are unstable modes that only look perturbative in highly boosted frames, so that kind of search would never find them.
Clearly, “quintessence” would have been (because it was) the proper term for this. Alas, it has been assigned to another thing. But that other thing is a DE scalar field. So perhaps “sextessence” would be appropriate for this.
See also: http://arxiv.org/abs/astro-ph/0703783 “Natural Dark Energy”, which proposes to squat on ‘sextessence’, ‘septessence’, etc.
Dear Sean,
What led your group to decide to present this research as two related papers, instead of one big paper? Can that choice to provide hints on the age-old lumping vs splitting debate for unrelated areas of academia?
Originally it was one long paper, but it eventually became clear that there were two different sets of results: one about the instabilities of aether theories generally, the other about the phenomenological consequences of one specific model. It’s very easy to imagine a reader being interested in one thing but not the other, so splitting into two papers was a no-brainer. Whatever makes the reader’s life easiest should determine that kind of thing.
Well, maybe this is suitable: a preferred direction in space doesn’t equate to a preferred reference frame per velocity – maybe some further clarification would help for middle brows here as to what the implications are of this field.
A Lorentz violation would seem to imply something very strange about the Higgs field. It seems to me that if you have a wave equation then in general a Klein-Gordon or similar wave equation is going to give you something of the form
k_ak^a = m^2 + A(k_5)^2, a = 1, 2, 3, 4,
where the k_5 is the bit that propagates into some additional dimension which involves the ether. I am thinking that the Lorentz invariance is violated because this additional dimension under a symmetry breaking acts as a mass-like term. This might then be an ether if k_5 = k_5(k_a) which gives a preferred coordinate condition in the universe or spacetime. The A term above might be some term from a Higgs type of field.
If this sort of thing happened to my mind there is some preferred frame for the Higgs field to give mass to particles and it would appear to give a preferred direction for the propagation of light as well. I have not thought enough about this, but this would seem to imply a different type of Higgs field than what we are familiar with.
Lawrence B. Crowell
I move that we start referring to Dark Energy as ‘it’. The Universe has ‘it’. Just like an ‘it’ girl.
Here, have a coupla ligatures: Æ & æ.
Chewing… Thanks, Sili, those were tasty.
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Since c is a constant, and has come to be our way of measuring time…and the Planck length is the smallest observable space and, therefore, our base for measuring distance, how is our system of measuring (d/t) not fundamentally scale-invariant?
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