Making Extra Dimensions Disappear

One of the big questions for people who believe in extra dimensions is: Why don’t we see them? Sure, we have methods for hiding them, usually by making them really tiny, but then we need to ask: Why are they tiny?

Matt Johnson, Lisa Randall and I just came out with a paper that takes a partial stab at this question: Dynamical Compactification from de Sitter Space. (And a similar-sounding paper came out the same day from Jose Blanco-Pillado, Delia Schwartz-Perlov, and Alex Vilenkin.) It’s an intriguing idea, if I do say so myself: starting with nothing more complicated than a higher-dimensional spacetime with a positive vacuum energy and an electromagnetic field (or a higher-dimensional generalization thereof), you will automatically get quantum fluctuations into lower-dimensional spacetimes! If we really believe in extra dimensions, we need to understand how regions with different effective dimensionalities are cosmologically related, and this is a step in that direction.

Normally I’d blog all about it, but on this occasion we’re outsourcing to a guest blogger. My collaborator Matt Johnson is a postdoc at Caltech, and before that was a grad student at UC Santa Cruz, where he worked with Anthony Aguirre — a previous guest-blogger of ours! We like to keep things in the family.

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Extra dimensions. Sounds preposterous at first. Well, perhaps more accurately, it sounds preposterous to most people who don’t do high-energy theory. But, really I assure you, there are many well-motivated reasons why us wacky theorists like to ponder the existence of extra dimensions.

For one, as shown long ago by Kaluza and Klein, it is possible to get Maxwell’s equations of electromagnetism in four dimensions by taking 5 dimensional General Relativity and wrapping one of the spatial dimensions up in a circle too small to see. The smaller the circle is, the harder it is to move in this “other direction,” and so there is no danger in getting lost on the way home. In this way, Maxwell’s equations have an elegant geometrical origin and gravity and electricity & magnatism are combined into one force (5 dimensional gravity).

Another strong motivation comes from string theory, which is only a consistent quantum theory of gravity if there are 10 or 11 dimensions in total. Again, since we don’t see them, it is necessary to hide the existence of the extra dimensions. Inspired by the fact that it was possible to hide one extra dimension by wrapping it up in a circle, generally the extra 6 or 7 dimensions are thought to be “compactified” into a very small compact geometry like a sphere or a torus.

At this point, the five-year-old in the audience is insistently asking, “If you have all these extra dimensions, and you are telling me that they are wrapped up into this tiny ball, how did they get wrapped up in the first place? Why are the four dimensions we see so large, and the others so small?”

After nearly a century of thinking about the existence of extra dimensions, there are surprisingly few plausible answers to this very simple question. One of the few answers was proposed by Brandenberger and Vafa. They studied the thermodynamics of strings in a torus-shaped hot early-universe, and found that miraculously it is favorable for only four of the dimensions to become large. Pretty nice, if the universe is a torus and all the dimensions started out small and compact. But, it would be nice to have some alternatives in case this turns out not to be viable.

Sean Carroll, Lisa Randall, and I recently wrote a paper that revisits the five-year-old’s question. We wanted to start with the very simplest model that has extra dimensions and solutions in which some of them can be compactified. A minimal set of ingredients needed to accomplish this includes 1) D-dimensional gravity, 2) a positive D-dimensional cosmological constant, and 3) a (D-4)-form gauge field (think E&M, but with more indices). This theory has long been known to have solutions where 4 of the dimensions are non-compact and (D-4) of them correspond to directions on a sphere, whose size is stabilized by the energetics of curvature and a background Electric or Magnetic field.

More interestingly, we showed that some of the spacetimes that are solutions to this theory contain a four-dimensional universe that lives behind the event horizon of an extended object, a “p-brane” or “black brane,” that is embedded in a background D-dimensional spacetime. Moreover, there are mechanisms that dynamically give rise to such objects, thanks to the magic of quantum mechanics, and this leads to an explanation for why some number of extra dimensions became compact!

Sounds complicated, but you can actually go a long way towards understanding what we did by considering plain-old four dimensional black holes. If you are falling into a black hole, eventually you cross the event horizon, a spherical surface that denotes the point-of-no-return: once you get in, you can’t get out. Sadly for you, everyone knows that lurking in the interior of the black hole is a singularity, and your fate will be spaghettification or worse.

What if there was no singularity? By this, I don’t necessarily mean some quantum gravity resolution, but rather, what if the geometry was such that the radius of spheres became frozen at some value close to the value on the event horizon. In this case, you don’t die, but you are still suck inside of the black hole due to the event horizon. You also notice that It is not very spacious in there, because the black hole is of finite size. Effectively, two of the original four dimensions have become compactified on a sphere. Therefore, falling into this made-up black hole, you would experience two of the four dimensions compactifying!

Upping the number of dimensions to D, it is possible to find black branes that do exactly this – when you cross an event horizon, you enter a region in which some number of the extra dimensions have become compact. The number of compact dimensions is determined by the symmetry of the event horizon. If the event horizon has (D-4)-dimensional spherical symmetry, then the region inside of the black brane is effectively four dimensional. Further, the 4 dimensional region has a natural “slicing” into space and time, that yields a cosmology. The big-bang in this 4-dimensional cosmology corresponds to the event horizon of the black brane.

These black branes can be embedded in a D-dimensional de Sitter space, which has some very interesting properties itself. Most relevant for us is that energy conservation does not work in de Sitter space, since it has a finite temperature. This means that every once in a while, a fluctuation will occur that produces one of the black brane solutions, and therefore a 4-dimensional universe. This is a mechanism of dynamical compactification.

The devil is now in the details. First off, the brane needs to be charged under the gauge field for there to be one of these interesting solutions. For each value that the charge can take, a slightly different lower-dimensional universe will be produced, and if there are different types of gauge fields, then universes with different numbers of dimensions will be produced as well. There is a “landscape” of many possible lower-dimensional vacua, just like in string theory. Dynamical compactification will be happening all over the place in the D-dimensional de Sitter space, realizing all of these possibilities in different spacetime regions, and yielding what is referred to as a “multiverse.”

The zeroth order question is if any of these universes look like ours. This involves having an early time epoch of inflation and late-time evolution towards a universe dominated by a small cosmological constant. We found that the answer can be yes on both counts for the models we studied.

A more ambitious goal is then to ask if there is any reason that picks out the universe we observe. This is of course a scary path to walk down, fraught with numerous incarnations of the Anthropic monster, but is a necessary demon to face in a theory where the fundamental parameters vary in different spatiotemporal regions of the multiverse. Although we did not explore this question in depth, there are some suggestive hints. First of all, it is easier to get into a vacuum with a small positive cosmological constant than it is to get out. It is also easier to produce universes with a smaller value for the late-time cosmological constant, although it is much more likely to get a negative vacuum energy solution, which undergoes a crunch. Depending on the total number of dimensions, we also found some cases where the rates to produce four dimensional universes was highest. These questions will be interesting to address in the future.

So, next time you go falling into an event horizon, you can hold out the hope that you will simply be banished to a lower-dimensional universe!