Scalar–vector–tensor decomposition: Difference between revisions

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In [[cosmological perturbation theory]], the '''scalar-vector-tensor decomposition''' is a decomposition of the most general linearized [[wiktionary:perturbation|perturbation]]s of the [[Friedmann-Robertson-Walker metric]] into components according to their transformations under spatial rotations. It was first discovered by [[E. M. Lifshitz]] in 1946. The general metric perturbation has ten degrees of freedom. The decomposition states that the evolution equations for the most general linearized [[wiktionary:perturbation|perturbation]]sperturbations of the [[Friedmann-Robertson-Walker metric]] can be decomposed into four scalars, two [[divergence|divergence-free]] spatial [[vector field]]s (that is, with a ''spatial'' index running from 1 to 3), and a [[trace (linear algebra)|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 (two scalars and a vector field) may be set to zero.

If the perturbed metric <math>g'_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}</math> where <math>h_{\mu\nu}</math> is the perturbation, then the decomposition is as follows,