However, the Weyl tensor
being that part of the curvature which is not determined locally by the matter distribution, there is no reason why it should disappear in an "empty" model of spacetime.
The Weyl tensor of a Riemannian manifold (M,g) is defined by
x] M, [alpha], [beta] [member of] R: If at x the curvature tensor R is expressed by R = [gamma]/2 E [conjunction] E, g[gamma] [member of] R, then the Weyl tensor C vanishes at x.
denote the Ricci tensor, Weyl tensor, and the scalar curvature of M with respect to semisymmetric connection [?
ab] has to be intrinsic because it is simply proportional to an invariant of the Weyl tensor
These are often stated interms of Petrov classification of the possible symmetries of the Weyl tensor
or the Segre classification of the possible symmetries of the Ricci tensor.
As is common in practice, their description must therefore be attributed to the Weyl tensor alone, as the only remaining geometric object in emptiness (with the cosmological constant neglected).
v] and instead of expressing the field strength f in terms of the Weyl tensor, let us write its components in the following equivalent form:
which, expressed in terms of the Weyl tensor, the Ricci tensor, and the Ricci scalar, is
Introducing the traceless Weyl tensor W, we have the following decomposition theorem:
Now let us recall that in four dimensions, with the help of the Weyl tensor W, we have the decomposition
Introducing the traceless Weyl tensor
C, we have the following decomposition theorem: