# Aether Compactification

Even in an election year, physics marches on. Physics is forever.

In this case it’s a fun little paper by Heywood Tam (a grad student here at Caltech) and me, arXiv:0802.0521:

We propose a new way to hide large extra dimensions without invoking branes, based on Lorentz-violating tensor fields with expectation values along the extra directions. We investigate the case of a single vector “aether” field on a compact circle. In such a background, interactions of other fields with the aether can lead to modified dispersion relations, increasing the mass of the Kaluza-Klein excitations. The mass scale characterizing each Kaluza-Klein tower can be chosen independently for each species of scalar, fermion, or gauge boson. No small-scale deviations from the inverse square law for gravity are predicted, although light graviton modes may exist.

This harkens back to the idea of a vector field that violates Lorentz invariance (which Ted Jacobson and friends have dubbed “aether,” appropriately enough), and in particular a vector that picks out a preferred direction in space. I explored this possibility last year in a paper with Lotty Ackerman and Mark Wise, and Mark recently wrote a followup with Tim Dulaney and Moira Gresham. (Our paper was detailed in the “anatomy of a paper” series, 1 2 3.)

There is an obvious problem with the notion of a vector field that violates rotational invariance by picking out a preferred direction through all of space — we don’t see any evidence for it! Physics as we have thus far experienced it seems pretty darn rotationally invariant. In my paper with Lotty and Mark, we sidestepped this issue by imagining that the vector was important in the early universe, and subsequently decayed away.

But there’s another way to sidestep the issue, pretty obvious in retrospect: have the vector point in a direction we don’t see! Extra dimensions are of course a popular theoretical construct, and once you make that leap you can ask what would happen if an unseen extra dimension contained a constant vector field. That would leave good old four-dimensional Lorentz invariance completely unbothered, so it’s not immediately constrained by any well-known experimental bounds.

So, beyond being fun and not ruled out, is it good for anything? The answer is: quite possibly. Heywood and I calculated what the influence of such a vector would be on other fields that propagated in a single extra dimension. In good old-fashioned Kaluza-Klein theory, momentum in the extra dimension can only take on discrete values (it’s quantized, in other words), and each kind of field breaks into an infinite “tower” of particles of different masses. The separation between different mass levels is just the inverse of the size of the extra dimension in natural units. What’s that? You insist upon seeing the equation? Okay, if the original mass of the field is *m* and the size of the extra dimension is *R*, we have a series of masses indexed by *n*:

$latex displaystyle m_n^2 = m^2 + left(frac{hbar n}{cR}right)^2,.$

Here, $latex hbar$ is Planck’s constant, *c* is the speed of light, and *n* is just a whole number that can be anything from 0 to infinity. So the effect of the compact fifth dimension is to give us an infinite set of four-dimensional particles, indexed by *n*, each with a different mass. Not a very big mass, unless the extra dimension is pretty small; separating the levels by about 1 electron volt requires a dimension that is about 1 micrometer across. We would certainly have noticed all those new particles unless the extra dimensions were considerably smaller than a Fermi (10^{-15} meters).

The interesting thing that Heywood and I discovered is that the effect of an aether field pointing in the extra dimension is to boost all of the mass levels in the Kaluza-Klein tower. There is a new set of coupling constants, α_{i} for every kind of particle *i*, that tells us how strongly that particle interacts with the aether. The mass formula is modified to read

$latex displaystyle m_n^2 = m^2 + left(alpha_ifrac{hbar n}{cR}right)^2,.$

So if α_{i} is huge, you could have a huge mass splitting even with an extra dimension that was pretty large. This gives a new way to hide extra dimensions — not just make them invisibly small (the old-school Kaluza-Klein method) or confine us to some thin brane (the new-school ’90s style), but to boost the effective masses associated with momentum in the new direction. And there is an obvious experimental test, if you were to find all of these new particles: unlike plain vanilla compactification, where the towers associated with each kind of field have the same mass splittings, here the splittings could be completely different for every kind of particle, just by choosing different α_{i}‘s.

To be fair, this idea does not by itself suggest any *reason* why the extra dimensions should be large. To allow for a millimeter-sized dimension, the coupling α_{i} has to be at least 10^{15}, which any particle physicist will tell you is an unnaturally big number. But the aether at least allows for the possibility, which I think is worth exploring. Who knows, some clever young graduate student out there might figure out how to use this idea to solve the hierarchy problem and the cosmological constant problem, then we would discover aetherized extra dimensions at the LHC, and everyone would become famous.

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