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Bel decomposition

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(Redirected from Magnetogravitic tensor)

In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric field and magnetic field. Such a decomposition was partially described by Alphonse Matte in 1953[1] and by Lluis Bel in 1958.[2]

This decomposition is particularly important in general relativity.[citation needed] This is the case of four-dimensional Lorentzian manifolds, for which there are only three pieces with simple properties and individual physical interpretations.

Decomposition of the Riemann tensor

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In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field , not necessarily geodesic or hypersurface orthogonal, consists of three pieces:

  1. the electrogravitic tensor
  2. the magnetogravitic tensor
  3. the topogravitic tensor
    • Can be interpreted as representing the sectional curvatures for the spatial part of a frame field.

Because these are all transverse (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three-dimensional vectors, or as three-by-three real matrices. They are respectively symmetric, traceless, and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows:

  • is the trace of E2 + L2 - 2 B BT,
  • is the trace of B ( E - L ),
  • is the trace of E L - B2.

See also

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References

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  1. ^ Matte, A. (1953), "Sur de nouvelles solutions oscillatoires des equations de la gravitation", Can. J. Math., 5: 1, doi:10.4153/CJM-1953-001-3
  2. ^ Bel, L. (1958), "Définition d'une densité d'énergie et d'un état de radiation totale généralisée", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 246: 3015