Warped Passages - Lisa Randall [197]
In this section, we’ll see that gravity’s weakness on the Weakbrane can also be explained as a consequence of objects getting bigger and lighter as you move away from the Gravitybrane and approach the Weakbrane. Were Athena to move from the Gravitybrane to the Weakbrane (as she will in the story in the next chapter), she would see her shadow on the Gravitybrane increase in size as she moved away. And the amount her shadow would increase is enormous—it would grow by sixteen orders of magnitude!
We will also see that in this geometry, heavy and light particles can peacefully coexist. Even when there are Planck-scale-mass particles on one of the two branes, there are only weak-scale-mass particles on the other. Therefore, there is no longer a hierarchy problem.
To understand how this works, suppose that, like most people (at least those who haven’t read this book), you were completely ignorant of the fifth dimension—which is, after all, invisible. Untroubled in the belief that you live in four dimensions, you would know only about four-dimensional gravity, which you would believe was communicated by a conventional four-dimensional graviton. In the four-dimensional effective theory that describes what you see, there would be only one gravitational force, and there could therefore be only a single type of four-dimensional graviton. All particles would interact with that single type of graviton. But that graviton wouldn’t contain any information about the location of a particle in the original, higher-dimensional theory.
This reasoning makes it look as though all graviton interactions should be the same—that is, independent of where in the fifth dimension an object originated. After all, you wouldn’t know that the object originated in the fifth dimension, or even that there was a fifth dimension. Newton’s constant of gravitation, which determines the graviton’s interaction strength, would be the single quantity that determines the strength of all four-dimensional gravitational interactions. But in the previous section, we saw that gravity’s interactions are weaker as you moved from the Gravitybrane towards the Weakbrane. This leaves the question, how can gravity’s strength encompass information about an object’s fifth-dimensional location?
The resolution to the apparent paradox hinges on the fact that gravitational attraction is also proportional to mass, and mass at different points along the fifth dimension can and must be different. The only way to reproduce the weakened graviton interaction on each successive slice along the fifth dimension is to measure mass differently on each four-dimensional slice.
One of the many remarkable properties of warped spacetime is that as you move from the Gravitybrane to the Weakbrane, energies and momenta shrink. The shrinking energies and momenta (and consistency with quantum mechanics and special relativity) also tell us that distance and time must expand (as shown in Figure 82). In the geometry I am describing, size, time, mass, and energy all depend on location. Four-dimensional sizes and masses inherit values that depend on their original five-dimensional positions. Physics looks four-dimensional. But the ruler with which length is measured, or the scale with which mass is measured, depends on the original five-dimensional location. Residents of the Gravitybrane or the Weakbrane both see four-dimensional physics, but they would measure different sizes and expect different masses.
Figure 82. Sizes increase (and masses and energies decrease) as one moves from the Gravitybrane to the Weakbrane.
The gravitational attraction of masses of particles that originate further away from the Gravitybrane in the original five-dimensional theory is smaller in the four-dimensional effective theory