

3.2 5-dimensional equations and
the initial-value problem
The effective field equations are not a closed system. One needs to
supplement them by 5D equations governing
, which are obtained from the 5D Einstein and Bianchi
equations. This leads to the coupled system [281
]
where the “magnetic” part of the bulk Weyl tensor, counterpart to
the “electric” part
, is
These equations are to be solved subject to the boundary conditions
at the brane,
where
denotes
.
The above equations have been used to develop a
covariant analysis of the weak field [281
]. They can also be
used to develop a Taylor expansion of the metric about the brane.
In Gaussian normal coordinates, Equation (45), we have
. Then we find
In a non-covariant approach based on a specific form of the bulk
metric in particular coordinates, the 5D Bianchi identities would
be avoided and the equivalent problem would be one of solving the
5D field equations, subject to suitable 5D initial conditions and
to the boundary conditions Equation (62) on the metric. The
advantage of the covariant splitting of the field equations and
Bianchi identities along and normal to the brane is the clear
insight that it gives into the interplay between the 4D and 5D
gravitational fields. The disadvantage is that the splitting is not
well suited to dynamical evolution of the equations. Evolution off
the timelike brane in the spacelike normal direction does not in
general constitute a well-defined initial value problem [8]. One needs to specify
initial data on a 4D spacelike (or null) surface, with boundary
conditions at the brane(s) ensuring a consistent
evolution [242, 147]. Clearly the
evolution of the observed universe is dependent upon initial
conditions which are inaccessible to brane-bound observers; this is
simply another aspect of the fact that the brane dynamics is not
determined by 4D but by 5D equations. The initial conditions on a
4D surface could arise from models for creation of the 5D
universe [104
, 178, 6, 31, 284], from dynamical
attractor behaviour [247] or from suitable
conditions (such as no incoming gravitational radiation) at the
past Cauchy horizon if the bulk is asymptotically AdS.

