Scalar–vector–tensor decomposition: Difference between revisions

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In [[cosmological perturbation theory]], the '''scalar-vector-tensorscalar–vector–tensor decomposition''' statesis thata thedecomposition evolution equations forof the most general linearized [[wiktionary:perturbation|perturbation]]s of the [[Friedmann-Robertson-WalkerFriedmann–Lemaître–Robertson–Walker metric]] caninto becomponents decomposedaccording intoto fourtheir scalarstransformations under spatial rotations. It was first discovered by [[E. M. Lifshitz]] in 1946. It follows from Helmholtz's Theorem (twosee [[Helmholtz decomposition]].) The general metric perturbation has ten degrees of whichfreedom. canThe bedecomposition madestates tothat vanishthe byevolution gaugeequations choice)for the most general linearized perturbations of the [[Friedmann–Lemaître–Robertson–Walker metric]] can be decomposed into four scalars, atwo [[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. ThisThe meansvector thatand tensor fields each have two independent components, so this decomposition encodes all ten degrees of freedom in the tensorgeneral fieldmetric <math>S_{ij}</math>perturbation. mustUsing notgauge haveinvariance anyfour componentof thatthese cancomponents (two scalars and a vector field) may be writtenset to zero.

:<math>S^{||}_{ij}=(\nabla_i\nabla_j-\frac{1}{3}\delta_{ij}\nabla^2)\phi</math>,

(where <math>\phi</math> is a scalar, <math>\nabla_i</math> is a spatial derivative, <math>\delta_{ij}</math> is the [[Kronecker delta]]) or

:<math>S^\perp_{ij}=\nabla_iS_j^\perp+\nabla_jS_i^\perp</math>,

where <math>S^\perp_i</math> is a spatial vector with <math>\delta^{ij}\nabla_iS_j^\perp=0</math>. This leaves only two independent components of <math>S_{ij}</math>, corresponding to the two [[polarization]]s of [[gravitational wave]]s. 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 [[rotation group|spatial rotation]]s. Two scalar components can further be eliminated by [[gauge transformation]]s, 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.

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,

==Reference==

<math display="block">h_{00}=-2\psi</math>

*{{cite journal | author = E. Bertschinger | title = Cosmological perturbation theory and structure formation | url = http://arxiv.org/abs/astro-ph/0101009 | year = 2001}}

<math display="block">h_{0i}=w_i</math>

<math display="block">h_{ij}=2(\phi g_{ij}+S_{ij})</math>

where the Latin indices ''i'' and ''j'' run over spatial components (1,...,3). The tensor field <math>S_{ij}</math> is traceless under the spatial part of the background metric <math>g_{ij}</math> (''i.e.'' <math>g^{ij}S_{ij}=0</math>). The spatial vector <math>w_i</math> and tensor <math>S_{ij}</math> undergo further decomposition. The vector is written

<math display="block">w_i=w^{||}{}_i+w^{\perp}{}_i,</math>

where <math>\nabla\times\mathbf{w}^{||}=\mathbf{0}</math> and <math>\nabla\cdot\mathbf{w}^{\perp}=0</math> (<math>\nabla_i</math> is the [[covariant derivative]] defined with respect to the spatial metric <math>g_{ij}</math>). 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, <math>w^{||}{}_i=\nabla_iA</math>. Thus <math>\mathbf{w}</math> 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 <math>S_{ij}</math>.<ref>{{cite journal | author = J. M. Stewart | title = Perturbations of the Friedmann-Robertson-Walker cosmological models | journal = Classical and Quantum Gravity | volume = 7 | number = 7 | pages = 1169–1180 | url = http://stacks.iop.org/0264-9381/7/i=7/a=013 | year = 1990 | doi=10.1088/0264-9381/7/7/013| bibcode = 1990CQGra...7.1169S | s2cid = 250864898 | url-access = subscription }}</ref> It can be written

[[Category:Cosmology]]

<math display="block">S_{ij}=S^{||}{}_{ij}+S^{\perp}_{ij}+S^T{}_{ij},</math>

[[Category:General relativity]]

where

{{relativity-stub}}

:<math display="block">S^{||}{}_{ij}=(\nabla_i\nabla_j-\frac{1}{3}\delta_g_{ij}\nabla^2)\phiB,</math>,

where <math>B</math> is a scalar (the combination of derivatives is set by the condition that <math>S</math> be traceless), and

<math display="block"> S^\perp{}_{ij} = \nabla_iS^\perp{}_j + \nabla_j S^\perp{}_i,</math>

where <math>S^\perp{}_i</math> is a divergenceless spatial vector. This leaves only two independent components of <math>S^T{}_{ij}</math>, corresponding to the two [[Polarization (waves)|polarizations]] of [[gravitational wave]]s. (Since the graviton is massless, the two polarizations are orthogonal to the direction of propagation, just like the photon.)

where <math>S^\perp_i</math> is a spatial vector with <math>\delta^{ij}\nabla_iS_j^\perp=0</math>. This leaves only two independent components of <math>S_{ij}</math>, corresponding to the two [[polarization]]s of [[gravitational wave]]s. 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 [[rotationRotation group SO(3)|spatial rotation]]s. Two scalar components and one vector component can further be eliminated by [[gauge transformation]]s,. andHowever, 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 <math>S^T{}_{ij}</math> is gauge invariant: it does not change under infinitesimal coordinate transformations.

==See also==

* [[Helmholtz decomposition]]

==Notes==

{{reflist}}

==References==

*{{citeCite journal | author = E. Bertschinger | title = Cosmological perturbation theory and structure formation | urlarxiv = http://arxiv.org/abs/astro-ph/0101009 | year = 2001 | bibcode = 2001astro.ph..1009B }}

*{{cite journal | author = E. M. Lifshitz |title = On the gravitational stability of the expanding universe|journal=J. Phys. USSR | volume=10| pages=116| year=1946}}

*{{cite journal | author = Eanna E. Flanagan, Scott A. Hughes |title = The basics of gravitational wave theory|journal=New Journal of Physics | volume=7| pages=204| year=2005| doi=10.1088/1367-2630/7/1/204 | s2cid=9530657 | arxiv=gr-qc/0501041| bibcode=2005NJPh....7..204F }}

*{{cite book | author = E. Poisson, C. M. Will |title = Gravity: Newtonian, Post-Newtonian, Relativistic|publisher=Cambridge University Press | pages=257 | year=2014}}

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[[Category:CosmologyPhysical cosmology]]

[[Category:GeneralMathematical methods in general relativity]]