The RS models and the subsequent generalization from a Minkowski brane to a Friedmann–Robertson–Walker (FRW) brane [38, 260, 216, 226, 183, 330, 209, 145, 154] were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the Z2-symmetric brane2. A broader perspective, with useful insights into the inter-play between 4D and 5D effects, can be obtained via the covariant Shiromizu–Maeda–Sasaki approach , in which the brane and bulk metrics remain general. The basic idea is to use the Gauss–Codazzi equations to project the 5D curvature along the brane. (The general formalism for relating the geometries of a spacetime and of hypersurfaces within that spacetime is given in .)
The 5D field equations determine the 5D curvature tensor; in the bulk, they are
Let be a Gaussian normal coordinate orthogonal to the brane (which is at without loss of generality), so that , with being the unit normal. The 5D metric in terms of the induced metric on surfaces is locally given by
Some other useful projections of the 5D curvature are:
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