Warped Passages - Lisa Randall [193]
The large extra-dimensional theories of the previous chapter took advantage of the fact that branes could trap particles and forces, but neglected the energy that the branes themselves could carry. Raman and I weren’t sure that this was always a good assumption, since a central component of Einstein’s theory of general relativity is that energy induces a gravitational field, which means that when branes carry energy, they should curve space and time. In a universe with only a single extra dimension, which was what we intended to study, it was not at all clear that one could neglect brane and bulk energy: the gravitational effects of the brane don’t dissipate very rapidly, so one would expect distortions of spacetime, even far away from the branes.
We wanted to know how spacetime would curve in the presence of two energetic branes that bounded the extra dimension of space. Raman and I solved Einstein’s gravity equations for this two-brane setup, assuming that there was energy both in the bulk and on the branes. We discovered that such energy was indeed very important—the resulting spacetime was dramatically curved.
In some cases, curved spaces are easy to picture. The surface of a sphere, for example, is two-dimensional—you need only latitude and longitude to know your location—but it is nonetheless clearly curved. However, many curved spaces are more difficult to draw because they can’t readily be represented in three-dimensional space. The particular warped spacetime that we will now consider is such an example. It is part of a spacetime known as anti de Sitter space. Anti de Sitter space has negative curvature, more like a Pringles potato chip than a sphere. The name comes from the Dutch mathematician and cosmologist Willem de Sitter, who studied a space with positive curvature that is now called de Sitter space. Although we don’t need the name here, we’ll refer to it later on when we connect this theory to a theory of anti de Sitter space that string theorists had been studying.
Although we’ll soon explore the interesting way in which the five-dimensional spacetime is curved, let’s first focus for a moment on the two branes at the edges of the fifth dimension. These two boundary branes are completely flat. If you were on the brane at either boundary, you would be stuck on a three-plus-one-dimensional world (three dimensions of space and one of time),* which would extend infinitely far in the three spatial dimensions and look like flat spacetime, with no peculiar gravitational effects.
Furthermore, the curved spacetime has the special property that were you to restrict yourself to any single slice along the fifth dimension—not just the branes at the ends—you would find that this slice is completely flat. That is, although there aren’t branes anywhere in the fifth dimension except at the ends, the geometry of the three-plus-one-dimensional surfaces that you get by restricting yourself to any single five-dimensional point looks flat: it has the same shape as the large flat branes at the boundaries. If you think of the boundary branes as the heels of a loaf of bread, the flat, parallel four-dimensional regions at any location along the fifth dimension of spacetime are like the flat slices of bread from the interior of the loaf.
But the five-dimensional spacetime we are considering is nonetheless curved. That is reflected in the way the four-dimensional flat spacetime slices are glued together along the fifth dimension. I first spoke about this geometry at the Kavli Institute for Theoretical Physics in Santa Barbara, where the string theorist Tom Banks informed me that, technically speaking, the five-dimensional geometry Raman and I found is warped. Although many curved spacetimes are colloquially called warped, the technical term refers to geometries in which each slice is flat,† but they are put together with an overall warp factor. The warp factor is a function that changes the