Scalar–vector–tensor decomposition: Difference between revisions

Revision as of 01:16, 27 August 2009

Revision as of 15:35, 21 January 2010

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

:<math>S^\perp{}_{ij}=\nabla_iS^\perp{}_j+\nabla_jS^\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 [[polarizationPolarization (waves)|polarizations]]s of [[gravitational wave]]s. (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. In [[representation theory]], this corresponds to decomposing perturbations under the group of [[rotation group|spatial rotation]]s. Two scalar components and one vector component can further be eliminated by [[gauge transformation]]s. 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 <math>S^T{}_{ij}</math> is gauge invariant: it does not change under infinitesimal coordinate transformations.