Randall-Sundrum model
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In physics, Randall-Sundrum models imagine that the real world is a higher-dimensional Universe described by warped geometry. More concretely, our Universe is a five-dimensional Anti de Sitter space and the elementary particles except for the graviton are localized on a 3+1-dimensional brane or branes.
The models were proposed in 1999 by Lisa Randall and Raman Sundrum while studying technicolor models.
There are two popular models. The first, called RS1, has a finite size for the extra dimension with two branes, one at each end. The second, RS2, is similar to the first, but one brane has been placed infinitely far away, so that there is only one brane left in the model.
The RS1 model
The RS1 model attempts to address the hierarchy problem. The warping of the extra dimension is analogous to the warping of spacetime in the vicinity of a massive object, such as a black hole. This warping, or red-shifting, generates a large ratio of energy scales so that the natural energy scale at one end of the extra dimension is much larger than at the other end.
- <math>ds^2={1\over k^2 y^2}(dy^2+\eta_{\mu\nu}dx^\mu dx^\nu)<math>
where k is some constant and η has -+++ metric signature. This space has boundaries at y=1/k and y=1/Wk, with <math>0\le {1 \over k} \le {1 \over Wk}<math> where k is around the Planck scale and W is the warp factor and Wk is around a TeV. The boundary at y=1/k is called the Planck brane and the boundary at y=1/Wk is called the TeV brane. The particles of the standard model reside on the TeV brane. The distance between both branes is only -ln(W)/k, though.
In another coordinate system,
- <math>\phi\equiv -{\pi \ln(ky)\over \ln(W)}<math>
so that
- <math>0\le \phi \le \pi<math>
and
- <math>ds^2=\left ({\ln(W)\over \pi k}\right )^2 d\phi^2+e^{2\ln(W)\phi\over \pi}\eta_{\mu\nu}dx^\mu dx^\nu<math>
The RS2 model
The RS2 model uses the same geometry as RS1, but there is no TeV brane. The particles of the standard model are presumed to be on the Planck brane. This model was originally of interest because it represented an infinite 5-dimensional model which, in many respects, behaved as a 4-dimensional model. This setup may also be of interest for studies of the AdS/CFT conjecture.