Editing Randall–Sundrum model (section)

==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>\mathrm{d}s^2 = \frac{1}{k^2 y^2}(\mathrm{d}y^2 + \eta_{\mu\nu}\,\mathrm{d}x^\mu\,\mathrm{d}x^\nu),</math>

where ''k'' is some constant, and η has "−+++" [[metric signature]]. This space has [[Boundary (topology)|boundaries]] at ''y'' = 1/''k'' and ''y'' = 1/(''Wk''), with <math>0 \le 1/k \le 1/(Wk)</math>, where ''k'' is around the [[Planck scale]], ''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>\varphi\ \stackrel{\mathrm{def}}{=}\ -\frac{\pi \ln(ky)}{\ln(W)},</math>

so that

:<math>0 \le \varphi \le \pi,</math>

and

:<math>\mathrm{d}s^2 = \left(\frac{\ln(W)}{\pi k}\right)^2\, \mathrm{d}\varphi^2 + e^\frac{2\ln(W)\varphi}{\pi} \eta_{\mu\nu}\,\mathrm{d}x^\mu\, \mathrm{d}x^\nu.</math>