

6.5 Vector perturbations
The vorticity propagation equation on the brane is the same as in
general relativity,
Taking the
of the conservation equation (106) (for the case of a
perfect fluid,
), and using the identity in
Equation (249), one obtains
as in general relativity, so that Equation (323) becomes
which expresses the conservation of angular momentum. In general
relativity, vector perturbations vanish when the vorticity is zero.
By contrast, in brane-world cosmology, bulk KK effects can source
vector perturbations even in the absence of vorticity [219
]. This can be seen
via the divergence equation for the magnetic part
of the 4D Weyl tensor on the brane,
where
. Even when
, there is a source for gravimagnetic terms on the
brane from the KK quantity
.
We define covariant dimensionless vector
perturbation quantities for the vorticity and the KK gravi-vector
term:
On large scales, we can find a closed system for these vector
perturbations on the brane [219
]:
Thus we can solve for
and
on super-Hubble scales, as for density
perturbations. Vorticity in the brane matter is a source for the KK
vector perturbation
on large scales. Vorticity decays
unless the matter is ultra-relativistic or stiffer (
), and this source term typically provides a decaying
mode. There is another pure KK mode, independent of vorticity, but
this mode decays like vorticity. For
,
the solutions are
where
.
Inflation will redshift away the vorticity and
the KK mode. Indeed, the massive KK vector modes are not excited
during slow-roll inflation [40, 269].

