On a Friedmann brane, we get
whereThe absence of bulk source terms in the
conservation equations is a consequence of having as the only 5D source in the bulk. For example, if
there is a bulk scalar field, then there is energy-momentum
exchange between the brane and bulk (in addition to the
gravitational interaction) [225
, 16
, 236
, 97
, 194
, 98
, 35
].
Equation (73) may be called the
“nonlocal conservation equation”. Projecting along
gives the nonlocal energy conservation equation,
which is a propagation equation for
. In the general,
nonlinear case, this gives
In particular cases, the Weyl anisotropic stress
may drop out of the nonlocal conservation equations,
i.e., when we can neglect
,
, and
. This is the case when we
consider linearized perturbations about an FRW background (which
remove the first and last of these terms) and further when we can
neglect gradient terms on large scales (which removes the second
term). This case is discussed in Section 6. But in general, and especially
in astrophysical contexts, the
terms cannot be
neglected. Even when we can neglect these terms,
arises in the field equations on the brane.
All of the matter source terms on the right of
these two equations, except for the first term on the right of
Equation (112), are imperfect fluid
terms, and most of these terms are quadratic in the imperfect
quantities
and
. For a single
perfect fluid or scalar field, only the
term on the
right of Equation (112
) survives, but in
realistic cosmological and astrophysical models, further terms will
survive. For example, terms linear in
will carry the
photon quadrupole in cosmology or the shear viscous stress in
stellar models. If there are two fluids (even if both fluids are
perfect), then there will be a relative velocity
generating a momentum density
, which will serve to source nonlocal effects.
In general, the 4 independent equations in
Equations (111) and (112
) constrain 4 of the 9
independent components of
on the brane. What is
missing is an evolution equation for
, which has up to
5 independent components. These 5 degrees of freedom correspond to
the 5 polarizations of the 5D graviton. Thus in general, the
projection of the 5-dimensional field equations onto the brane does
not lead to a closed system, as expected, since there are bulk
degrees of freedom whose impact on the brane cannot be predicted by
brane observers. The KK anisotropic stress
encodes the nonlocality.
In special cases the missing equation does not
matter. For example, if by symmetry, as in the case
of an FRW brane, then the evolution of
is determined by
Equations (111
) and (112
). If the brane is
stationary (with Killing vector parallel to
), then evolution equations are not needed for
, although in general
will still be
undetermined. However, small perturbations of these special cases
will immediately restore the problem of missing information.
If the matter on the brane has a perfect-fluid or
scalar-field energy-momentum tensor, the local conservation
equations (105) and (106
) reduce to
A simple example of the latter point is the FRW
case: Equation (116) is trivially
satisfied, while Equation (115
) becomes
If , then the field equations on
the brane form a closed system. Thus for perfect fluid branes with
homogeneous density and
, the brane field equations
form a consistent closed system. However, this is unstable to
perturbations, and there is also no guarantee that the resulting
brane metric can be embedded in a regular bulk.
It also follows as a corollary that inhomogeneous
density requires nonzero :