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 . 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
On large scales, we can find a closed system for these vector perturbations on the brane :
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
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].