Bel decomposition

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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 ${\displaystyle {\vec {X}}}$, not necessarily geodesic or hypersurface orthogonal, consists of three pieces:

1. the electrogravitic tensor ${\displaystyle E[{\vec {X}}]_{ab}=R_{ambn}\,X^{m}\,X^{n}}$
   • 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.
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 B^T,
• ${\displaystyle -K_{2}/8}$ is the trace of B ( E - L ),
• ${\displaystyle K_{3}/8}$ is the trace of E L - B².

• Bel–Robinson tensor
• Ricci decomposition
• Tidal tensor
• Papapetrou–Dixon equations
• Curvature invariant

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