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In physics, Randall–Sundrum models (also called 5-dimensional warped geometry theory) imagines 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 because they were dissatisfied with the universal extra-dimensional models then in vogue. Such models require two fine tunings; one for the value of the bulk cosmological constant and the other for the brane tensions. Later, while studying RS models in the context of the anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence, they showed how it can be dual to 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.

Overview

The model is a braneworld theory developed while trying to solve the hierarchy problem of the Standard Model. It involves a finite five-dimensional bulk that is extremely warped and contains two branes: the Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane") and the Tevbrane (our home with the Standard Model particles; also called "Weakbrane"). In this model, the two branes are separated in the not-necessarily large fifth dimension by approximately 16 units (the units based on the brane and bulk energies). The Planckbrane has positive brane energy, and the Tevbrane has negative brane energy. These energies are the cause of the extremely warped spacetime.
Graviton probability function

In this warped spacetime that is only warped along the fifth dimension, the graviton's probability function is extremely high at the Planckbrane, but it drops exponentially as it moves closer towards the Tevbrane. In this, gravity would be much weaker on the Tevbrane than on the Planckbrane.
Growing, shrinking, and changing weight: solving the hierarchy problem

Also in this model, any object moving from the Planckbrane to the Tevbrane in the bulk would be growing, becoming lighter, and moving more slowly through time. Distance and time expand near the Tevbrane, and mass and energy shrink near it. This creates an alternate explanation of gravity's weakness in the Tevbrane: everything is lighter. An interesting part of this is that the hierarchy problem is automatically solved. The main scale on the Planckbrane would be the Planck scale. However, the change by 16 units causes the scales to change by 16 orders of magnitude. On the Planckbrane, strings would be 10−33 cm in size, but on the Tevbrane they'd be 10−17 cm. In fact, this makes the mass scale for the Weakbrane based on about a TeV. Therefore, you no longer have the strange range of masses and energies. If this is the case, we should see evidence when the Large Hadron Collider begins working.

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.

\( \mathrm{d}s^2={1\over k^2 y^2}(\mathrm{d}y^2+\eta_{\mu\nu}\,\mathrm{d}x^\mu\, \mathrm{d}x^\nu) \)

where k is some constant and η has "-+++" metric signature. This space has boundaries at y = 1/k and y = 1/Wk, with 0\le {1 \over k} \le {1 \over Wk} 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,

\( \varphi\ \stackrel{\mathrm{def}}{=}\ -{\pi \ln(ky)\over \ln(W)}, \)

so that

\( 0\le \varphi \le \pi \)

and

\( \mathrm{d}s^2=\left ({\ln(W)\over \pi k}\right )^2\, \mathrm{d}\varphi^2+e^{2\ln(W)\varphi\over \pi}\eta_{\mu\nu}\,\mathrm{d}x^\mu\, \mathrm{d}x^\nu. \)

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.

Prior models

In 1998/99 Merab Gogberashvili published on arXiv a number of articles on a very similar theme.[1][2][3] In those papers he showed that if the Universe is considered as a thin shell (a mathematical synonym for "brane") expanding in 5-dimensional space then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant and Universe thickness, and thus to solve the hierarchy problem. It was also shown that four-dimensionality of the Universe is the result of stability requirement since the extra component of the Einstein field equations giving the localized solution for matter fields coincides with the one of the conditions of stability.
See also

Kaluza–Klein theory
ADD model
DGP model

References

M. Gogberashvili, Hierarchy problem in the shell universe model, arXiv:hep-ph/9812296.
M. Gogberashvili, Our world as an expanding shell, arXiv:hep-ph/9812365.

M. Gogberashvili, Four dimensionality in noncompact Kaluza-Klein model, arXiv:hep-ph/9904383.