Under that more common convention, the signs of the Ricci tensor and scalar must be changed in the equations below.)
Define the traceless Ricci tensor and then define three (0,4)-tensor fields S, E, and W by The "Ricci decomposition" is the statement As stated, this is vacuous since it is just a reorganization of the definition of W. The importance of the decomposition is in the properties of the three new tensors S, E, and W. Terminological note.
One may check that the Ricci decomposition is orthogonal in the sense that recalling the general definition
This has the consequence, which could be proved directly, that This orthogonality can be represented without indices by together with One can compute the "norm formulas" and the "trace formulas" Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group (Besse 1987, Chapter 1, §G).
Let V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature).
The curvature tensor is skew symmetric in its first and last two entries: and obeys the interchange symmetry for all x,y,z,w ∈ V∗.
The Ricci contraction mapping is given by This associates a symmetric 2-form to an algebraic curvature tensor.
Conversely, given a pair of symmetric 2-forms h and k, the Kulkarni–Nomizu product of h and k produces an algebraic curvature tensor.
In dimension 4, the Weyl module decomposes further into a pair of irreducible factors for the special orthogonal group: the self-dual and antiself-dual parts W+ and W−.
is the stress–energy tensor describing the amount and motion of all matter and all nongravitational field energy and momentum, states that the Ricci tensor—or equivalently, the Einstein tensor—represents that part of the gravitational field which is due to the immediate presence of nongravitational energy and momentum.
Regions of spacetime in which the Weyl tensor vanishes contain no gravitational radiation and are also conformally flat.