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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 [27Jump To The Next Citation Point18115516212824314999104Jump To The Next Citation Point] were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the Z 2-symmetric brane. 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 [291Jump 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 [315].)

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

(5) (5) 2(5) GAB = - /\5 gAB + k 5 TAB, (44)
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 nAdXA = dy, 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 (xa, y)dxmdxn + dy2. (45) AB AB A B mn
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, (46) 2
so that
K[AB] = 0 = KABnB, (47)
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 F g Gg H + 2K K , (48) ABCD EFGH 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 , (49)
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 , (50) (5)REF GH gAEnF gBGnH = - £ nKAB + KAC KC B, (51) (5) C D C RCD gA gB = RAB - £ nKAB - KKAB + 2KAC K B. (52)
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. (53)

 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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