Bel Decomposition - Decomposition of The Riemann Tensor

Decomposition of The Riemann Tensor

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

  1. the electrogravitic tensor
  2. the magnetogravitic tensor
  3. the topogravitic tensor

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:

  • is the trace of E2 + L2 - 2 B BT,
  • is the trace of B ( E - L ),
  • is the trace of E L - B2.

Read more about this topic:  Bel Decomposition