Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar potentials, i.e.,
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:
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 ). 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.
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