Scalar-vector-tensor decomposition


In cosmological perturbation theory, the scalar-vector-tensor decomposition is a decomposition of the most general linearized s of the Friedmann–Lemaître–Robertson–Walker metric into components according to their transformations under spatial rotations. It was first discovered by E. M. Lifshitz in 1946. It follows from Helmholtz's Theorem The general metric perturbation has ten degrees of freedom. The decomposition states that the evolution equations for the most general linearized perturbations of the Friedmann–Lemaître–Robertson–Walker metric can be decomposed into four scalars, two divergence-free spatial vector fields, and a traceless, symmetric spatial tensor field with vanishing doubly and singly longitudinal components. The vector and tensor fields each have two independent components, so this decomposition encodes all ten degrees of freedom in the general metric perturbation. Using gauge invariance four of these components may be set to zero.
If the perturbed metric where is the perturbation, then the decomposition is as follows,
where the Latin indices i and j run over spatial components. The tensor field is traceless under the spatial part of the background metric . The spatial vector and tensor undergo further decomposition. The vector is written
where and . The notation is used because in Fourier space, these equations indicate that the vector points parallel and perpendicular to the direction of the wavevector, respectively. The parallel component can be expressed as the gradient of a scalar,. Thus can be written as a combination of a scalar and a divergenceless, two-component vector.
Finally, an analogous decomposition can be performed on the traceless tensor field. It can be written
where
where is a scalar, and
where is a divergenceless spatial vector. This leaves only two independent components of, 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 and one vector component can further be eliminated by gauge transformations. However, 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. The tensor is gauge invariant: it does not change under infinitesimal coordinate transformations.