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 tensoredit
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:
the electrogravitic tensor
Also known as the tidal tensor. It can be physically interpreted as giving the tidal stresses on small bits of a material object (which may also be acted upon by other physical forces), or the tidal accelerations of a small cloud of test particles in a vacuum solution or electrovacuum solution.
the magnetogravitic tensor
Can be interpreted physically as a specifying possible spin-spin forces on spinning bits of matter, such as spinning test particles.
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:
^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
^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