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.
[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.
If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows: