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

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

 into four pieces. It was introduced in 1959 by the physicist Lluis Bel.

This decomposition is particularly important in general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. This is the four dimensional case of Lorentzian manifolds, for which there are only three pieces with particularly simple properties, which have individual physical interpretations.

See also

• Electrogravitic tensor
  Electrogravitic tensor
  In general relativity, the tidal tensor or gravitoelectric tensor is one of the pieces in the Bel decomposition of the Riemann tensor. It is physically interpreted as giving the tidal stresses on small bits of a material object , or the tidal accelerations of a small cloud of test particles in a...
  
  
• Magnetogravitic tensor
  Magnetogravitic tensor
  In general relativity, the magnetogravitic tensor is one of the three pieces appearing in the Bel decomposition of the Riemann tensor.The magnetogravitic tensor can be interpreted physically as a specifying possible spin-spin forces on spinning bits of matter, such as spinning test particles....
  
  
• Ricci decomposition
  Ricci decomposition
  In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties...
  
  
• Topogravitic tensor
  Topogravitic tensor
  In general relativity, the topogravitic tensor is one of the three pieces of the Bel decomposition of the Riemann tensor.The topogravitic tensor can be interpreted as representing the sectional curvatures for the spatial part of a frame field....