The Shape of Gravity

The Shape of Gravity

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Today’s paper is part of a special series in anticipation of The Science & Ultimate Reality Symposium in Princeton, a symposium in honor of the 90th year of John Archibald Wheeler, a great physicist and teacher of physicists.



More than 80 years ago the German mathematician Theordor Kaluza spotted a curious property about Einstein’s general theory of relativity. The field equations of this theory describe how spacetime is warped by matter, which is generally accepted as the most satisfactory account of gravitation. Kaluza noticed that if Einstein’s equations are written down for a universe in which space has four dimensions rather than three, then not only is gravity correctly described, but electromagnetism too. In other words, if the world were really five-dimensional rather than four (adding in time as well) then both electromagnetism and gravitation would have a common geometrical basis. By all accounts Einstein wasn’t very impressed with the idea.

Clever though Kaluza’s theory was, it had a major drawback. Where is the fifth dimension? Why don’t we see it? A possible answer was provided by Oskar Klein. Imagine viewing a hosepipe from afar; it would look like a wiggly line. On closer inspection, however, the line would be revealed as a tube, and what was apparently a point on the line would turn out to be a little circle going around the tube. In the same way, what we might take to be a structureless point in three-dimensional space might in actuality be a little circle going around a fourth space dimension. So the reason we don’t see the extra space dimension could be because it is rolled up to a tiny size (a configuration known to physicists as ‘compactification’). Klein computed the circumference to be about 20 powers of 10 smaller than anatomic nucleus.

The idea can be generalized. Perhaps there are two, three, four, extra space dimensions folded up out of sight in this manner? Maybe the nuclear forces could be incorporated this way into a Kaluza-Klein theory, thus reducing all the forces of nature to pure geometry? Such theories were developed in earnest in the 1980s. By the time string theory came along, the assumption of extra dimensions seemed natural. A popular string model, for example, has 26 dimensions in total.

But rolling dimensions up is only one way to hide them. Another is to suppose that although real space might have four dimensions, we are trapped in three of them, just as a two-dimensional being is trapped in a surface in Edwin Abbott’s famous (but outrageously sexist) Flatland fable. The confining entity in the case of three-dimensional space embedded inside four dimensions is called a ‘brane’ (after membrane). We could be trapped in a three-brane if the forces that control normal matter, and the photons whereby we see other matter, are confined by a sort of potential well. So in normal circumstances we would not be able to see out into the enveloping higher dimension. But it would be there alright, and might affect the physical within our confining brane, for example, by modifying gravity on a small scale. We can even imagine that collisions between neighboring branes might occur, creating big bangs. Braney researchers have explored many such speculative ideas

Lisa Randall is a high-energy physicist from Harvard who hopes we will be able to detect the fifth dimension at work through subtle experiments. If she is right, then in cosmology what you get might be much more than what you see.

—Paul Davies



When thinking about the outstanding issues in cosmology, it is a good idea to separate the late-time from the early-time issues. It is fairly clear that at late times standard FRW evolution applies because of the observations of the CMBR, the abundance of the elements as predicted by big-bang nucleosynthesis, and the Hubble expansion. However, these probe only late-time/low-energy cosmology back to when the temperature of the universe was of order an MeV. The evolution of the early universe is much less definitive. Early universe cosmology could differ substantially from the conventional picture. Many of the open questions in cosmology center around the range of possibilities for this early universe evolution. Some, such as inflation, are motivated by specific flaws in the conventional picture (in themselves important questions in cosmology). Some, such as extra dimensions, are interesting in that they permit substantially different early universe evolution while nonetheless conforming at late times to what has been observed. New insights into theories with extra dimensions have the potential to address other outstanding issues.

Before discussing any specific theory, we list some of the major problems that cosmologists face. There are the horizon, flatness, and homogeneity problems that might be addressed by inflation, which itself raises questions about its implementation. There are questions related to problems raised by gravity as measured on long distance scales, namely the dark matter and dark energy problems. There is the question of why the world appears to be four-dimensional. There is the black hole information paradox, and questions about the holographic nature of gravitational systems and possible evidence for nonlocality. And of course there is the long-standing dilemna of the cosmological constant. Having listed some problems, we now list possible particle physics or gravity systems and which problems they might help address.

As with the standard model of particle physics, which agrees with all existing low-energy data but leaves many naturalness problems unresolved, the standard theory of late-time cosmology leaves open several naturalness problems of at least as big proportion. These are successfully addressed by inflation. However, we have yet to find a fully satisfactory inflationary model, that is one that does not require some unnatural assumption or parameter choice. Moreover, the existence of theories of inflation, which rely on a time period of large nonvanishing cosmological constant, cannot necessarily be decoupled from the ultimate resolution of the cosmological constant problem.

Many other intriguing questions have evolved around the issue of the dimensionality of space. Ultimately, we would like to address the question of why our universe appears to be four-dimensional. There is the associated question of whether the ultimate theory is four-dimensional, or only appears so in cosmology and particle physics that has been observed. Given that they might exist, it is important to examine the role they might play in addressing questions in particle physics or cosmology. Within this realm, there exist many potential directions, including discovering interesting aspects of gravity theories, exploring new and different time-dependent solutions, and possibly gaining insight into the fundamental nature of quantum gravity. By interesting aspects of gravity theories, I refer to the many new things we have discovered about gravitational theories within only the last few years. For example, the fact that compactification of additional dimensions is not essential, that the graviton can have a mass in AdS space, and the fact that four-dimensional gravity can be a local phenomenon are all new developments. The last point means that local physics can be independent of the space far away. Have we been assuming too much by assuming all of the universe evolves four-dimensionally?

Another property of note is that the cosmological constant problem is completely revamped in the context of brane-world physics. The problem is no longer why there is no vacuum energy, but instead why there is a precise relation between brane energy and that of the surrounding bulk spacetime. By insight into the fundamental nature of gravity, I refer to the fact that with explicit new solutions, we can explore questions about holography, for example, which have precise and specific implications in particular theories. These allow tests of holographic conjectures in regulated versions of the theories. By exploring features of known holographic examples, we might also learn features that can be extrapolated to gravitational theories in general.

It is likely there are many more unanticipated phenomena yet to be discovered, and that some major problems remaining in cosmology might thereby be addressed.