Warped Passages - Lisa Randall [194]
A curved space with flat slices is pictured in Figure 79. It is a filled-in funnel. We could slice the funnel into flat sheets with a cleaver, but the surface of the funnel is clearly curved. This is similar in some respects to the curved five-dimensional spacetime we’re considering. But the analogy isn’t perfect, because the boundary of the funnel, the funnel’s surface, is the only place where it’s curved, whereas in the warped spacetime the curvature is everywhere. This curvature would be reflected in an overall rescaling of the measuring rod of space and the clock speed for time, which would be different at each point in the fifth dimension.36
Figure 79. A filled-in funnel consists of flat slices glued together.
A simpler way of illustrating the curvature of warped spacetime is through the shape of the graviton’s probability function. The graviton is the particle that communicates the gravitational force, and its probability function tells us the likelihood of finding the graviton at any fixed position in space. The strength of gravity is reflected in this function: the larger its value, the stronger the graviton’s interactions at that particular point, and the stronger the force of gravity.
For flat spacetime, the graviton would be equally likely to be found anywhere. The probability function for a graviton in flat spacetime would therefore be constant. But for curved spacetime, as in the warped geometry we are considering, this would no longer be the case. Curvature tells us about the shape of gravity. When spacetime is curved, the value of the graviton’s probability function is different at different locations in spacetime.
Because each slice of spacetime is completely flat in our warped geometry, the graviton’s probability function doesn’t vary along the three standard spatial dimensions—it changes only along the fifth dimension.* In other words, even though the graviton’s probability function has different values at different places along the fifth dimension, so long as two points are equally far along the fifth dimension, the value will be the same. This tells us that the graviton’s probability function depends only on position in the fifth dimension. Nonetheless, it completely characterizes the warped spacetime’s curvature. And because that function varies only with a single coordinate, that of the fifth dimension, it is simple to plot.
This graviton’s probability function along the fifth dimension is shown in Figure 80. It decreases exponentially quickly, which is to say extraordinarily rapidly, as one leaves the first brane, which we will call the Gravitybrane, and heads towards the second brane, which we’ll call the Weakbrane. The Gravitybrane and the Weakbrane are different because the first carries positive energy, while the second carries negative energy. And this energy assignment makes the graviton’s probability function much bigger in the vicinity of the Gravitybrane.
Figure 80. The graviton’s probability function falls off exponentially as it moves away from the Gravitybrane and towards the Weakbrane.
The effect of the plummeting probability function is that the graviton, the physical particle whose exchange generates gravitational attraction, has very little chance to be found near the Weakbrane. The graviton’s interactions on the Weakbrane are therefore highly suppressed.
The strength of gravity depends so strongly on position along the fifth dimension that the strengths of gravity experienced on the two branes, which border the opposite ends of this warped five-dimensional world, are extraordinarily different. Gravity is strong on the first brane, where gravity is localized, but feeble on the second, where the Standard Model resides. Because the graviton’s probability function is