Scalar–vector–tensor decomposition: Difference between revisions

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<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

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

where

<math display="block">S^{||}{}_{ij}=(\nabla_i\nabla_j-\frac{1}{3}g_{ij}\nabla^2)B,</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_jSnabla_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.)

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*{{Cite journal | author = E. Bertschinger | title = Cosmological perturbation theory and structure formation | arxiv = 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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