

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 [27
, 181, 155, 162, 128, 243, 149, 99, 104
] were derived as
solutions in particular coordinates of the 5D Einstein equations,
together with the junction conditions at the
-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 [291
], 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
where
represents any 5D energy-momentum of
the gravitational sector (e.g., dilaton and moduli scalar fields,
form fields).
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
The extrinsic curvature of
surfaces
describes the embedding of these surfaces. It can be defined via
the Lie derivative or via the covariant derivative:
so that
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:
and the Codazzi equation determines the change of
along
via
where
.
Some other useful projections of the 5D curvature
are:
The 5D curvature tensor has Weyl (tracefree) and Ricci parts:

