In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field \vec{X}, not necessarily geodesic or hypersurface orthogonal, consists of three pieces: • the
electrogravitic tensor 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. • the
magnetogravitic tensor 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. • the
topogravitic tensor 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: • K_1/4 is the trace of
E2 +
L2 - 2
B BT, • -K_2/8 is the trace of
B (
E -
L ), • K_3/8 is the trace of
E L -
B2. ==See also==