

3.5 Propagation and
constraint equations on the brane
The propagation equations for the local and nonlocal energy density
and momentum density are supplemented by further
-covariant propagation and constraint equations for
the kinematic quantities
,
,
,
, and for the free gravitational field
on the brane. The kinematic quantities govern the relative motion
of neighbouring fundamental world-lines. The free gravitational
field on the brane is given by the brane Weyl tensor
. This splits into the gravito-electric and
gravito-magnetic fields on the brane:
where
is not to be confused with
. The Ricci identity for
and the Bianchi identities
produce the fundamental evolution and constraint equations
governing the above covariant quantities. The field equations are
incorporated via the algebraic replacement of the Ricci tensor
by the effective total energy-momentum tensor,
according to Equation (63). The brane equations
are derived directly from the standard general relativity versions
by simply replacing the energy-momentum tensor terms
by
. For a general fluid source,
the equations are given in [218
]. In the case of a
single perfect fluid or minimally-coupled scalar field, the
equations reduce to the following nonlinear equations:
- Generalized Raychaudhuri equation (expansion
propagation):
- Vorticity propagation:
- Shear propagation:
- Gravito-electric propagation (Maxwell-Weyl
E-dot equation):
- Gravito-magnetic propagation (Maxwell-Weyl
H-dot equation):
- Vorticity constraint:
- Shear constraint:
- Gravito-magnetic constraint:
- Gravito-electric divergence (Maxwell-Weyl div-E
equation):
- Gravito-magnetic divergence (Maxwell-Weyl div-H
equation):
- Gauss-Codazzi equations on the brane (with
):
where
is the Ricci tensor for 3-surfaces
orthogonal to
on the brane, and
.
The standard 4D general relativity results are
regained when
and
, which sets all right hand sides to zero in
Equations (123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134). Together with
Equations (113, 114, 115, 116), these equations
govern the dynamics of the matter and gravitational fields on the
brane, incorporating both the local, high-energy (quadratic
energy-momentum) and nonlocal, KK (projected 5D Weyl) effects from
the bulk. High-energy terms are proportional to
, and are significant only when
. The KK terms contain
,
, and
, with the latter two
quantities introducing imperfect fluid effects, even when the
matter has perfect fluid form.
Bulk effects give rise to important new driving
and source terms in the propagation and constraint equations. The
vorticity propagation and constraint, and the gravito-magnetic
constraint have no direct bulk effects, but all other equations do.
High-energy and KK energy density terms are driving terms in the
propagation of the expansion
. The spatial gradients of
these terms provide sources for the gravito-electric field
. The KK anisotropic stress is a driving term in the
propagation of shear
and the
gravito-electric/gravito-magnetic fields,
and
respectively, and the KK momentum
density is a source for shear and the gravito-magnetic field. The
4D Maxwell-Weyl equations show in detail the contribution to the 4D
gravito-electromagnetic field on the brane, i.e.,
, from the 5D Weyl field in the bulk.
An interesting example of how high-energy effects
can modify general relativistic dynamics arises in the analysis of
isotropization of Bianchi spacetimes. For a Binachi type I
brane, Equation (134) becomes [221
]
if we neglect the dark radiation, where
and
are the average scale factor and expansion rate, and
is the shear constant. In general relativity, the
shear term dominates as
, but in the brane-world, the
high-energy
term will dominate if
, so that the matter-dominated early universe is
isotropic [221
, 48
, 47
, 308, 280, 18, 64]. This is illustrated in
Figure 4.
Note that this conclusion is sensitive to the
assumption that
, which by Equation (115) implies the
restriction
Relaxing this assumption can lead to non-isotropizing
solutions [1, 65, 46].
The system of propagation and constraint
equations, i.e., Equations (113, 114, 115, 116) and (123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134), is exact and
nonlinear, applicable to both cosmological and astrophysical
modelling, including strong-gravity effects. In general the system
of equations is not closed: There is no evolution equation for the
KK anisotropic stress
.

