### 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 [147, 242]. 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 [6, 31, 105, 178, 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.