Incosmological perturbation theory, thescalar–vector–tensor decomposition is a decomposition of the most general linearizedperturbations of theFriedmann–Lemaître–Robertson–Walker metric into components according to their transformations under spatial rotations. It was first discovered byE. M. Lifshitz in 1946. It follows from Helmholtz's Theorem (seeHelmholtz decomposition.) The general metric perturbation has ten degrees of freedom. The decomposition states that the evolution equations for the most general linearized perturbations of theFriedmann–Lemaître–Robertson–Walker metric can be decomposed into four scalars, twodivergence-free spatialvector fields (that is, with aspatial index running from 1 to 3), and atraceless, symmetric spatialtensor 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${\displaystyle g_{\mu \nu }^{\prime }=g_{\mu \nu }+h_{\mu \nu }}$ where${\displaystyle h_{\mu \nu }}$ is the perturbation, then the decomposition is as follows,$${\displaystyle h_{00}=-2\psi }$$$${\displaystyle h_{0i}=w_i}$$$${\displaystyle h_{ij}=2(\varphi g_{ij}+S_{ij})}$$where the Latin indicesi andj run over spatial components (1,...,3). The tensor field${\displaystyle S_{ij}}$ is traceless under the spatial part of the background metric${\displaystyle g_{ij}}$ (i.e.${\displaystyle g^{ij}{S}_{ij}=0}$). The spatial vector${\displaystyle w_i}$ and tensor${\displaystyle S_{ij}}$ undergo further decomposition. The vector is written$${\displaystyle w_i=w^{||}{}_i+w^{\perp }{}_i,}$$where${\displaystyle \mathrm{\nabla }\times {\mathbf{w}}^{||}=\mathbf{0}}$ and${\displaystyle \mathrm{\nabla }\cdot {\mathbf{w}}^{\perp }=0}$ (${\displaystyle {\mathrm{\nabla }}_i}$ is thecovariant derivative defined with respect to the spatial metric${\displaystyle g_{ij}}$). The notation is used because inFourier 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,${\displaystyle w^{||}{}_i={\mathrm{\nabla }}_{i}A}$. Thus${\displaystyle \mathbf{w}}$ 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${\displaystyle S_{ij}}$.^[1] It can be written$${\displaystyle S_{ij}=S^{||}{}_{ij}+S_{ij}^{\perp }+S^T{}_{ij},}$$where$${\displaystyle S^{||}{}_{ij}=({\mathrm{\nabla }}_i{\mathrm{\nabla }}_j-\frac{1}{3}{g}_{ij}{\mathrm{\nabla }}^2)B,}$$where${\displaystyle B}$ is a scalar (the combination of derivatives is set by the condition that${\displaystyle S}$ be traceless), and$${\displaystyle S^{\perp }{}_{ij}={\mathrm{\nabla }}_i{S}^{\perp }{}_j+{\mathrm{\nabla }}_j{S}^{\perp }{}_i,}$$where${\displaystyle S^{\perp }{}_i}$ is a divergenceless spatial vector. This leaves only two independent components of${\displaystyle S^T{}_{ij}}$, corresponding to the twopolarizations ofgravitational waves. (Since the graviton is massless, the two polarizations are orthogonal to the direction of propagation, just like the photon.)

The advantage of this formulation is that the scalar, vector and tensor evolution equations are decoupled. Inrepresentation theory, this corresponds to decomposing perturbations under the group ofspatial rotations. Two scalar components and one vector component can further be eliminated bygauge 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${\displaystyle S^T{}_{ij}}$ is gauge invariant: it does not change under infinitesimal coordinate transformations.

• Helmholtz decomposition

• E. Bertschinger (2001). "Cosmological perturbation theory and structure formation".arXiv:astro-ph/0101009.Bibcode:2001astro.ph..1009B.{{cite journal}}:Cite journal requires|journal= (help)
• E. M. Lifshitz (1946). "On the gravitational stability of the expanding universe".J. Phys. USSR.10: 116.
• Eanna E. Flanagan, Scott A. Hughes (2005). "The basics of gravitational wave theory".New Journal of Physics.7: 204.arXiv:gr-qc/0501041.Bibcode:2005NJPh....7..204F.doi:10.1088/1367-2630/7/1/204.S2CID 9530657.
• E. Poisson, C. M. Will (2014).Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press. p. 257.

• Physical cosmology
• Mathematical methods in general relativity

• CS1 errors: missing periodical