Warped Passages - Lisa Randall [198]
This rescaling might seem arbitrary, but it’s not. It is subtle, however, so let’s first consider an analogous situation. Suppose that we were to measure time in terms of how long it takes to travel 100 km by train. I will call these units of time TT (train time) units. This is a fine measure of time, except that your determination of time would depend on where you are traveling: are the trains fast there or are they slow? For example, suppose that a movie lasted two hours. If an American train took an hour to travel 100 km, an American viewer would cover 200 km over the course of that movie and say that the movie lasted 2 TTs. A French viewer riding the TGV, on the other hand, would think that the movie lasted 6 TTs, because express trains in France travel about three times faster and the French viewer would need to watch their DVD during a 600-km-long train ride to see how it ends. Because the French viewer’s train covers 100 km in 20 minutes, whereas an American’s train covers the same distance in an hour, you need to rescale train time if Americans and French are to share common units and agree on the TT length of the movie. To convert from French to American time, you would have to rescale the French train time by a factor of three.
Similarly, on the Weakbrane, where the graviton interaction is far smaller than on the Gravitybrane, the units for the scale used to measure energy must be rescaled to take account of gravity’s weakness. At the Weakbrane, the rescaling is by an enormous amount, 1016, ten million billion. What this means is that whereas on the Gravitybrane all fundamental masses are expected to be MPl (the Planck scale mass), on the Weakbrane they are expected to be only about 1,000 GeV, a factor of 1016 smaller. Masses of new particles that live on the Weakbrane might be somewhat larger, perhaps 3,000 or 5,000 GeV, but they shouldn’t be much larger than that since all masses have been rescaled enormously.
The hierarchy problem arises when all masses get raised to the largest mass around. If that mass is the Planck scale mass, all masses are expected to be about as big as the Planck scale mass. But owing to the rescaling, if you originally thought that the Planck scale mass was the expected mass for everything on the Gravitybrane, then on the Weakbrane you would conclude that a TeV, sixteen orders of magnitude smaller, is the expected mass.* This means that the mass of the Higgs particle is not at all disturbing: a mass of about a TeV—ten million billion times smaller than the Planck scale mass—is expected, even though gravity is weak. The rescaling, which is essential in this interpretation, solves the hierarchy problem.
By the same reasoning, all new objects on the Weakbrane, including strings, should have mass of about a TeV. This tells us that this model could have dramatic experimental consequences. On the Weakbrane, the extra particles associated with strings would be very much lighter than those on the Gravitybrane—or in a four-dimensional world, for that matter. The Weakbrane presents a fabulous scenario from the perspective of discovering extra dimensions. If this idea is correct, then low-mass particles from extra dimensions should be near at hand. TeV-mass particles would abound on the Weakbrane.
Everything on the Weakbrane is expected to be lighter than the Planck scale mass by a factor of 1016. And according to quantum mechanics, smaller mass means larger size. Athena’s shadow would grow as she went from the Gravitybrane