Aether Compactification

By Sean Carroll | February 6, 2008 12:22 pm

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, αi 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 αi 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 αi‘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 αi has to be at least 1015, 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.

  • B

    Hi Sean:
    This is interesting, I will have a look at the paper (colloquium coming up…). A question – sorry if that sounds funny, but I do not mean it sarcastic in any way: do you really believe this is how nature could work? Best,


  • Sean

    I do, yeah. I would give better than 50-50 odds on the existence of extra dimensions, and they have to be hidden somehow, so we should be open-minded about what might be going on. I think the percentage chance that this idea describes nature is pretty small, but it’s not completely negligible, and the payoff if it’s true is pretty large.

  • B

    thanks. In fact, I too believe in extra dimensions, though I don’t like any of the ways to ‘hide’ them that are presently around. I find it rather unlikely though the LHC will help very much in this regard.

  • George Musser

    Why does the vector field point in the extra dimension only, with no component in 3+1 space?

  • Sean

    Just because we said so. It may or may not be related to why some dimensions are compact and some non-compact.

  • Moshe

    Naive question, apologies if this is in the paper. Having a small extra dimension rotate sounds very energetic, and energy gravitates, doesn’t that overclose the universe?

  • Sean

    Sorry, nothing is rotating; just a fixed vector pointing in the extra dimension (compactified on a circle). In fact, one of the very nice things about these fields is that their energy-momentum tensor vanishes in a flat spacetime background, which this is — so there is at least a solution which is just 5-d Minkowski space with one compact direction. Ultimately you’d like to stabilize it, just as in any KK model.

  • Moshe

    OK, got confused, it is not one of the KK modes that gets turned on, sorry.

  • Josh


    Is this qualitatively different from flux compactifications in supergravity/string theory? There you wrap form fields around cycles in some deformed Calabi-Yau type manifold. From the lower-dimensional POV, one is generating potentials for what would otherwise be massless fields(the moduli). What you’re describing sounds pretty similar, but maybe I am missing something.

  • Sean

    Josh, it’s very different, actually. The role of fluxes is to stabilize the moduli, like you say, but in the end you still need a very small compactification manifold so as not to get light KK states. We have a large compactification manifold (potentially), do not have any particular insight on stabilizing the moduli, and the spacings of the KK towers would all be different in our case (whereas they wouldn’t be in the flux case).

  • Josh

    Thanks, Sean. I took a closer look at your paper too, and I see what my confusion was. In fact, it looks like your configuration for the aether field is closer to a Wilson loop around the circle. But the effect is different since the aether isn’t a gauge field.

  • dusty59

    More naiveté- but does not time represent a asymmetric vector?

  • Metal

    Sean, I haven’t read the paper yet – just the post. I wonder if it is possible to use this idea to explain why the extra dimensions are compactified in the first place.

  • Maybe

    Extra dimensions are about as likely or unlikely as supersymmetry. So dream on.

  • David Nataf

    I have a question about esoteric mathematical physics,
    Why is the HEP-th community open to all sorts of exotic spatial dimensions, but never considers the possibility of additional dimensions of time?
    Is one more common sense than the other?
    Why not add super-time?

  • David Nataf

    For some reason I thought Sean Carroll was at the University of Chicago?

  • Sean

    David, they do think about multiple time dimensions, but it’s generally harder to make that idea work. And I was at Chicago, now I’m at Caltech — you have to keep up!

    Metal, I don’t see any obvious connection, except that there may be a mechanism that makes a vector line up preferentially along compact directions when it has a choice between compact and non-compact ones. But I don’t know of any such mechanism.

    dusty59, time is not really a vector. And, as far as we know, ordinary physics doesn’t pick out a preferred reference frame.

  • Moshe

    In addition to what Sean said, it is really difficult to make sense of physics with more than one time direction, it is not clear what you mean by very basic notions such as time evolution, quantum mechanical probabilities, configuration or phase space etc. Despite the exotic sounding “extra dimensions`, they represent no conceptual change, every physicist would know immediately what to calculate and how to interpret the results.

    (I think Bars calling his program two-time physics is a little tongue in cheek, certainly there is no additional physical time direction there, one of the time directions is always gauge fixed. Allowing thisd extra redundancy is useful mathematical trick though.)

  • Neil B.

    So why does the universe need to hide other large dimensions anyway, or does it? Many say, there is no inherent physical need for space to have three large dimensions (need, to avoid contradiction issues, not about anthropic suitability.) But can it seriously be said that no quirky problems of physical consistency would arise (assuming a logical extrapolation or analogy to our laws, I presume – is that unambiguous however?) in universes with other than 3 + 1 dimensions? Think about for example trying to apply dimensional analysis (of the MLT type, so as not to confuse with “number of space dimensions”) to the analogous expressions for radiation reaction, radiation power, etc. I have fiddled with that and it doesn’t seem to work right. Some have noted there need to be special constants in other kinds of spaces, even for banal things like electromagnetic radiation, which make those spaces appear “contrived” (i.e., they need special propping up “by hand” to avoid logical loose ends perhaps. Well, ours is contrived too in some ways I suppose.)

  • onymous

    the coupling αi has to be at least 1015, which any particle physicist will tell you is an unnaturally big number

    In particular, why doesn’t one run into intractable nonperturbative physics whenever this coupling is much greater than 1?

  • sammyo

    Woo, ‘Jumpers’ in real life soon! 😉

  • Sean

    onymous, that number doesn’t appear directly in the action; it’s a ratio of two different mass scales. In particular, the coupling of excitations of the vector to other fields is actually suppressed by a (potentially) large number. But it’s certainly a good question why such a number would appear.

  • George Musser

    Sean, wasn’t one of Ted Jacobson’s concerns that a Lorentz-violating field could permit multiple event horizons and violations of the Second Law?

  • onymous

    onymous, that number doesn’t appear directly in the action; it’s a ratio of two different mass scales.

    But if you have physics that generates the operator coupling your Lorentz-violating vector to the kinetic term of your fields, wouldn’t you tend to expect to also generate operators coupling it to other terms? It just seems a little unnatural. And if it appears in front of other terms, it really can play the role of a large coupling.

  • Haelfix

    It would be nontrivial to figure out a mechanism for fixing one direction like that so exactly, at least so long as we don’t relegate the dynamics to the very early universe.

    The other issue I think at first glance is that while its true the zero modes might spread out and average out over space in the infinite volume limit, it probably need not be true for V finite. Also, it would be nice to to reconstruct the solution for a nontrivial metric with compact 3D spacial topology (say a closed FRW universe).

  • Odani the Multi-talented Undergrad

    Caltech? Not UW-Madison? You sure of that?

    Oh. These genetics posts did seem kind of weird. I thought it was just me–turns out that you weren’t really you.

  • spayced

    I hate this crap. When we start getting into scientists saying “I believe…” we have a problem. This is all great fun math gymnastics, but means nothing practicality wise as it is based on amazing calculations not observation, evidence, or anything resembling proof.

    Scientists used to look at something in nature that we had not understood, like light for instance. They would come up with ideas about how it would work, then go to observation to confirm and calculations for exactness. Now we go from calculations to theories and back again, veering so far from observable reality and the other steps, primarily confirmation before continuing, the result is nonsense. I know all of you mean well (I’m not saying all of these people are wrong) it’s just the process of coming up with raw theory is not science.

  • Lawrence B. Crowell

    Maybe this is a way of deriving the axion particle. A compactification [itex]nhbar/Rc[/itex] for R large might create just the right particle mass for the CP-preserving field with QCD. In E_8 the two exceptional groups F_4 and G_2 or

    e_8~=~f_4~+~g_2~+~26times 7

    where f_4 contains the so(3,1) for gravity and the so(4) for weak interactions with so(4) = su(2) + su(2) for the left and right handed parts. CP violation leaves the left su(2), but we have no analogous handedness breaking in QCD. Then to balance everything out maybe large scale compactification in an S-duality sense leaves generates a tiny mass field —- the axion, and there is a dual field which compactifies on a tiny scale ~ near L_p. This would “balance” the role of CP violation in the gravitation plus weak F_4 sector with the hypercharge plus QCD G_2 sector.

    Lawrence B. Crowell

  • Rhys

    Sean: “I would give better than 50-50 odds on the existence of extra dimensions…”

    I would probably take up that bet if it was possible for it to be settled in my favour! Why do you have such great confidence in the existence of extra dimensions?

  • CMT


    You do realize that most scientists do exactly what you want them to do? And in fact, so do most physicists?

  • Sean

    Rhys, I don’t think that “better than 50-50 odds” translates easily into “such great confidence.” But I think that some version of string theory is probably correct, and string theory requires extra dimensions.

  • Moshe

    Regarding extra dimensions. the existing paradigm in fundamental physics is that things becomes simpler, more symmetric and more unique at short distances, and complexity arises in the process of symmetry breaking. If this continues we expect new degrees of freedom at shorter distances and new symmetries to relate them. Extra dimension is just a way of organizing those new degrees of freedom (as harmonics in some space) which works for a large class of models. The new symmetries are then rotations in those extra dimensions. Having new degrees of freedom without new symmetries I find less motivated, but it is a matter of taste.

  • Blake Stacey

    Why am I suddenly getting a Hunter S. Thompson voice in my head?

    “There is nothing in the world more helpless and compactified than a man in the depths of an aether binge. . . .”

  • Neil B.

    OK, what about “quintessence”? Any way to work that in, how does it relate to dark energy in general etc? I get the impression, Q isn’t just a mere synonym for DE in general. I wish I’d thought to ask earlier, but anyone LMK if have good scoop.

  • Lawrence B. Crowell

    This compactification scheme predicts the existence of light particles. If this is an unknown particle then maybe the axion is a candidate. Tight compactification, R very small in the above formula, would correspond to a massive particle — maybe dark matter.

    Quintessence and dark energy may involve a phase of the vacuum state, or at least this is what I suspect. In fact dark matter might be local changes in this phase. So called phantom energy might also be such a phase, though its amplitude or probability in the universe I think is near zero.

    Lawrence B. Crowell

  • Jonathan Vos Post

    Re: Blake Stacey, #33:

    More Pynchon than Thompson, though your joke was nostalgic…:

    “Against the Day”,
    Page 566

    In a dream…

    This passage, describing Kit’s dream of Umeki and the message it conveys, pulls together many of the main themes of Against the Day, tying things together in a way that Pynchon seldom does, almost as if he’s providing a rather large piece of the puzzle to help the reader understand the novel:

    “Deep among the equations describing the behavor of light, field equations, Vector and Quaternion equations, lies a set of directions, an intinerary, a map to a hidden space. Double refraction appears again and again as a key element, permitting a view into a Creation set just to the side of this one, so close as to overlap, where the membrane between the worlds, in many places, has become too frail, too permeable, for safety…. Within the mirror, with the scalar term, within the daylit and obvious and taken-for-granted has always lain, as if in wait, the dark intinerary, the corrupted pilgrim’s guide, the nameless Station before the first, in the lightless uncreated, where salvation does not yet exist.”

    See references in Against the day also at:

    55; 58; 132-133; 140; 306; 320; “sounds like light” 426; 458; 557; 565-66; 595; aka Akasa, 613; 620; aka Luminiferous aether ; Wikipedia article on Luminferous aether; 1911 Encyclopedia Britannica

    “Quaternion-ray weapons” 445; “intelligence of a Quaternionic Weapon, a means to unloose upon the world energies hitherto unimagined — hidden … ‘innocently,’ inside the w term.” 542; “Unfamiliar with the Tesla system and alarmed by the strengths of the electric and magnetic fields, de Decker’s people naturally conflated this with those recent rumors of a Quaternion weapon…” 549; “as if this mysterious Q-weapon were a common firearm and he hoping the seller would allow him a few courtesy shots [emphasis added]” 557; “alive in Woevre’s hands” 563; “Kit found himself alone with the enigmatic object, back inside the leather case. He slung it nonchalantly by its strap over one shoulder” 564; technical details, 564-565; 784;


    130; In mathematics, quaternions are a non-commutative extension of complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. At first, quaternions were regarded as pathological, because they disobeyed the commutative law ab = ba. Although they have been superseded in most applications by vectors, they still find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations. James Clerk Maxwell first published his famous theory describing electricity and magnetism as a set of twenty equations, but he was later able to reformulate it as four equations using quaternions. Heaviside translated these into four vector equations, the form typically taught in basic physics classes today. Heaviside’s vector version is compatible with Einstein’s special relativity, but the quaternion form is not; 131; 156; 511; 525; Wars, 526, 548; 533-34; 538-39; Quaternionic Weapon, 542; 557; 564; 590; Wikipedia entry; Quaternions at MathWorld; Hamiltonian quaternions at PlanetMath; Finding the site of Hamilton’s inspiration (by mathematical physicist John Baez); Conspiracy-theory takes on mathematical history [1] [2] (by Tom Bearden, promotor of a dubious free energy machine); Primer on quaternions and their history


    156; In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms. Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, science, and engineering. 158; 165; vectorists, 534; and Monotony, 560; “in five dimensions” 675

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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