Now, here is the really interesting part: Some of the effects predicted by the theory appear to be in conflict with one of the principles of Einstein's special theory of relativity, the theory that says that the speed of light is a universal constant. It's the same for all photons, and it is independent of the motion of the sender or observer.

How is this possible, if that theory is itself based on the principles of relativity? The principle of the constancy of the speed of light is part of special relativity, but we quantized Einstein's general theory of relativity. Because Einstein's special theory is only a kind of approximation to his general theory, we can implement the principles of the latter but find modifications to the former. And this is what seems to be happening!

So Gambini, Pullin, and others calculated how light travels in a quantum geometry and found that the theory predicts that the speed of light has a small dependence on energy. Photons of higher energy travel slightly slower than low-energy photons. The effect is very small, but it amplifies over time. Two photons produced by a gamma-ray burst 10 billion years ago, one redder and one bluer, should arrive on Earth at slightly different times. The time delay predicted by the theory is large enough to be detectable by a new gamma-ray observatory called GLAST (for Gamma-ray Large Area Space Telescope), which is scheduled for launch into orbit in 2006. We very much look forward to the announcement of the results, as they will be testing a prediction of a quantum theory of gravity.

A very exciting question we are now wrestling with is, How drastically shall we be forced to modify Einstein's special theory of relativity if the predicted effect is observed? The most severe possibility is that the principle of relativity simply fails. The principle of relativity basically means that velocity is relative and there is no absolute meaning to being at rest. To contradict this would mean that after all there is a preferred notion of rest in the universe. This, in turn, would mean that velocity and speed are absolute quantities. It would reverse 400 years of physics and take us back before Galileo enunciated the principle that velocity is relative. While the principle may have been approximately true, we have been confronting the frightening possibility that the principle fails when quantum gravity effects are taken into account.

Recently, people have understood that this possibility appears to be ruled out by experiments that have already been done: that is, if the principle of relativity fails when quantum gravity effects are taken into account, effects would already have been seen in certain very delicate measurements involving atomic clocks and in certain astrophysical processes involving supernova remnants. These effects are not seen, so this drastic possibility seems less likely. So a hypothesis about the structure of space and time on scales twenty orders of magnitude smaller than an atomic nucleus has been ruled out by experiment!

But there is another possibility. This is that the principle of relativity is preserved, but Einstein's special theory of relativity requires modification so as to allow photons to have a speed that depends on energy. The most shocking thing I have learned in the last year is that this is a real possibility. A photon can have an energy-dependent speed without violating the principle of relativity! This was understood a few years ago by Amelino Camelia. I got involved in this issue through work I did with João Magueijo, a very talented young cosmologist at Imperial College, London. During the two years I spent working there, João kept coming to me and bugging me with this problem. His reason for asking was that he had realized that if the speed of light could change according to conditions—for example, when the universe was very hot and dense—you might get an alternative cosmological theory. He and Andreas Albrecht (and before them John Moffat) had found that if the speed of light was higher in the early universe, you get an alternative to inflationary cosmology that explains everything inflation does, without some of the baggage.

These ideas all seemed crazy to me, and for a long time I didn't get it. I was sure it was wrong! But João kept bugging me and slowly I realized that they had a point. We have since written several papers together showing how Einstein's postulates may be modified to give a new version of special relativity in which the speed of light can depend on energy.

Meanwhile, in the last few years there have been some important new results concerning loop quantum gravity. One is that the entropy of a black hole can be computed, and it comes out exactly right. Jacob Bekenstein found in his PhD thesis in 1971 that every black hole must have an entropy proportional to the area of its horizon, the surface beyond which light cannot escape. Stephen Hawking then refined this by showing that the constant of proportionality must be, in units in which area is measured by the Planck length squared, exactly one quarter. A challenge for all quantum theories of gravity since then has been to reproduce this result. Moreover, entropy is supposed to correspond to a measure of information: It counts how many bits of information may be missing in a particular observation. So if a black hole has entropy, one has to answer the question, What is the information that the entropy of a black hole counts?

Loop quantum gravity answers these questions by giving a detailed description of the microscopic structure of the horizon of a black hole. This is based on the atomic description of spatial geometry, which implies that the area of a black hole horizon is quantized—just as space is, it is made up of discrete units. It turns out that a horizon can have, for each quantized unit of area, a finite number of states. Counting them, we get exactly Bekenstein's result, with the one quarter.

This is a very recent result. When we first did this kind of calculation, in the mid-1990s, we got the entropy right up to an overall constant. A few months ago, in a brilliant paper, Olaf Dreyer, a postdoc at the Perimeter Institute, found a very simple and original argument that fixes that constant, using a completely classical property of black holes. He uses an old argument of Neils Bohr called the correspondence principle, which tells us how to tie together classical and quantum descriptions of the same system. Once the constant is fixed, it gives the right entropy for all black holes.

Another big development of loop quantum gravity is that we now know how to describe not only space but spacetime—including causality, light cones, and so on—in loop quantum gravity. Spacetime also turns out to be discrete, described by a structure called a spin foam. Recently there have been important results showing that dynamical calculations in spin-foam models come out finite. Together these two results strongly suggest that loop quantum gravity is giving us sensible answers to questions about the nature of space and time on the shortest scales.

Let me now say something about string theory, which is the other approach to quantum gravity that has been well studied.

String theory is a very beautiful subject. It attempts to unify gravity with the other forces by postulating that all particles and forces arise from the vibrations of extended objects. These include one-dimensional objects (hence the name "strings"), but there are also higher-dimensional extended objects that go by the name of "branes" (for generalizations of membranes). String theory comes from the observation that all the quanta that carry the known forces, and all the known particles, can be found among the vibrations of these extended objects.

String theory is not a complete quantum theory of gravity, for reasons I'll come to in a minute, but it does work to a certain extent. It gives, to a certain order of approximation, sensible predictions for some quantum gravity effects. These include the scattering of gravitons (quanta of gravity analogous to photons) with other particles. For certain very limited kinds of black holes (actually, not real black holes but systems with properties similar to certain special black holes), it gives predictions that agree with the results of Bekenstein and Hawking. And it does succeed in unifying gravity with the other forces.

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