Scalar–vector–tensor decomposition

In cosmological perturbation theory, the scalar-vector-tensor decomposition states that the evolution equations for the most general linearized perturbations of the Friedmann-Robertson-Walker metric can be decomposed into four scalars (two of which can be made to vanish by gauge choice), a divergence-free spatial vector field (that is, with a spatial index running from 1 to 3), and a traceless, symmetric spatial tensor field with vanishing doubly and singly longitudinal components. This means that the tensor field $S_{ij}$ must not have any component that can be written

$S_{ij}^{||}=(\nabla _{i}\nabla _{j}-{\frac {1}{3}}\delta _{ij}\nabla ^{2})\phi$,

(where $\phi$ is a scalar, $\nabla _{i}$ is a spatial derivative, $\delta _{ij}$ is the Kronecker delta) or

$S_{ij}^{\perp }=\nabla _{i}S_{j}^{\perp }+\nabla _{j}S_{i}^{\perp }$,

where $S_{i}^{\perp }$ is a spatial vector with $\delta ^{ij}\nabla _{i}S_{j}^{\perp }=0$. This leaves only two independent components of $S_{ij}$, corresponding to the two polarizations of gravitational waves. The advantage of this formulation is that the scalar, vector and tensor evolution equations are decoupled. In representation theory, this corresponds to decomposing perturbations under the group of spatial rotations. Two scalar components can further be eliminated by gauge transformations, and the vector components are generally ignored, as there are few known physical processes in which they can be generated. As indicated above, the tensor components correspond to gravitational waves.

Reference

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