3 Covariant Approach to Brane-World Geometry and Dynamics

The RS models and the subsequent generalization from a Minkowski brane to a Friedmann–Robertson–Walker (FRW) brane [38Jump To The Next Citation Point, 260, 216Jump To The Next Citation Point, 226, 183, 330, 209, 145, 154Jump To The Next Citation Point] 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 [388Jump To The Next Citation Point], 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 [423].)

The 5D field equations determine the 5D curvature tensor; in the bulk, they are

(5)GAB = − Λ5(5)gAB + κ25(5)TAB, (50 )
where (5) TAB represents any 5D energy-momentum of the gravitational sector (e.g., dilaton and moduli scalar fields, form fields).

Let y be a Gaussian normal coordinate orthogonal to the brane (which is at y = 0 without loss of generality), so that n dXA = dy A, with nA being the unit normal. The 5D metric in terms of the induced metric on {y = const.} surfaces is locally given by

(5)g = g + n n , (5)ds2 = g (xα,y)dx μdxν + dy2. (51 ) AB AB A B μν
The extrinsic curvature of {y = const.} surfaces describes the embedding of these surfaces. It can be defined via the Lie derivative or via the covariant derivative:
KAB = 1£ngAB = gAC (5)∇C nB, (52 ) 2
so that
K [AB] = 0 = KABnB, (53 )
where square brackets denote anti-symmetrization. The Gauss equation gives the 4D curvature tensor in terms of the projection of the 5D curvature, with extrinsic curvature corrections:
R = (5)R g Eg Fg Gg H + 2K K , (54 ) ABCD EF GH A B C D A[C D]B
and the Codazzi equation determines the change of KAB along {y = const.} via
∇BKBA − ∇AK = (5)RBC gABnC , (55 )
where K = KA A.

Some other useful projections of the 5D curvature are:

(5) E F G H REF GH gA gB gC n = 2∇ [AKB ]C, (56 ) (5)REF GH gAEnF gBGnH = − £nKAB + KAC KC B, (57 ) (5) C D C RCDgA gB = RAB − £nKAB − KKAB + 2KAC K B. (58 )
The 5D curvature tensor has Weyl (tracefree) and Ricci parts:
(5) (5) 2((5) (5) (5) (5) ) 1(5) (5) (5) RABCD = CACBD + 3 gA[C RD ]B − gB[C RD ]A − 6 gA [C gD]B R. (59 )

 3.1 Field equations on the brane
 3.2 5-dimensional equations and the initial-value problem
 3.3 The brane viewpoint: A 1 + 3-covariant analysis
 3.4 Conservation equations
 3.5 Propagation and constraint equations on the brane

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