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.
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:
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: