# Bel decomposition

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 and by Lluis Bel in 1958.

This decomposition is particularly important in general relativity. 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

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

1. the electrogravitic tensor $\displaystyle{ E[\vec{X}]_{ab} = R_{ambn} \, X^m \, X^n }$
2. the magnetogravitic tensor $\displaystyle{ B[\vec{X}]_{ab} = {{}^\star R}_{ambn} \, X^m \, X^n }$
• Can be interpreted physically as a specifying possible spin-spin forces on spinning bits of matter, such as spinning test particles.
3. the topogravitic tensor $\displaystyle{ L[\vec{X}]_{ab} = {{}^\star R^\star}_{ambn} \, X^m \, X^n }$
• 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:

• $\displaystyle{ K_1/4 }$ is the trace of E2 + L2 - 2 B BT,
• $\displaystyle{ -K_2/8 }$ is the trace of B ( E - L ),
• $\displaystyle{ K_3/8 }$ is the trace of E L - B2.