Scalar–vector–tensor decomposition
In cosmological perturbation theory, the scalar–vector–tensor decomposition is a decomposition of the most general linearized perturbations 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 (see Helmholtz decomposition.) 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 (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. 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 where is the perturbation, then the decomposition is as follows,
Finally, an analogous decomposition can be performed on the traceless tensor field .[1] It can be written
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.
See also[edit]
Notes[edit]
- ^ J. M. Stewart (1990). "Perturbations of the Friedmann-Robertson-Walker cosmological models". Classical and Quantum Gravity. 7 (7): 1169–1180. Bibcode:1990CQGra...7.1169S. doi:10.1088/0264-9381/7/7/013. S2CID 250864898.
References[edit]
- E. Bertschinger (2001). "Cosmological perturbation theory and structure formation". arXiv:astro-ph/0101009. Bibcode:2001astro.ph..1009B.
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(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.