### 6.1 1 + 3-covariant perturbation equations on the brane

In the 1+3-covariant approach [303, 282, 305], perturbative quantities are projected vectors,
, and projected symmetric tracefree tensors, , which are gauge-invariant since
they vanish in the background. These are decomposed into (3D) scalar, vector, and tensor modes as
where and an overbar denotes a (3D) transverse quantity,
In a local inertial frame comoving with , i.e., , all time components may be set to zero:
, , .
Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar
potentials, i.e.,

Scalar perturbative quantities are formed from the potentials via the (3D) Laplacian, e.g.,
. Purely vector perturbations are characterized by
where is the vorticity, and purely tensor by
The KK energy density produces a scalar mode (which is present even if in the
background). The KK momentum density carries scalar and vector modes, and the KK anisotropic stress
carries scalar, vector, and tensor modes:

Linearizing the conservation equations for a single adiabatic fluid, and the nonlocal conservation equations,
we obtain
Linearizing the remaining propagation and constraint equations leads to
Equations (259), (261), and (263) do not provide gauge-invariant equations for perturbed quantities, but
their spatial gradients do.
These equations are the basis for a 1+3-covariant analysis of cosmological perturbations from the brane
observer’s viewpoint, following the approach developed in 4D general relativity (for a review, see [137]).
The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not
a closed system of equations until is determined by a 5D analysis of the bulk perturbations.
An extension of the 1+3-covariant perturbation formalism to 1+4 dimensions would require a
decomposition of the 5D geometric quantities along a timelike extension into the bulk of the brane
4-velocity field , and this remains to be done. The 1+3-covariant perturbation formalism is
incomplete until such a 5D extension is performed. The metric-based approach does not have this
drawback.