Weyl tensor


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Weyl tensor

[′wīl ‚ten·sər]
(relativity)
A tensor with the symmetries of the curvature tensor such that all contractions on its indices vanish; the curvature tensor is decomposable in terms of the metric, the scalar curvature, and the Weyl tensor.
References in periodicals archive ?
pointing out the special radius value [[rho].sub.*] = 2/[B.sub.*], for which only the components [R.sub.1212] = [B.sup.2.sub.*]/[[LAMBDA].sup.4] = [R.sub.3434] are surviving and the Weyl tensor vanishes.
[19] studied the relationship between conformal symmetries and relativistic spheres in astrophysics and exploit the nonvanishing components of the Weyl tensor to classify the conformal symmetries in static spherical space-time.
The Weyl tensor has the special property that it is invariant under conformal changes to the metric.
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.
This is in opposition to an algebraic constraint which may be imposed, for example, by demanding that the eigenvectors of the Weyl tensor have certain preferred alignments.
[F.sub.ab] is an extrinsic quantity but equation (8) indicates that the invariant [F.sub.ab] [sup.*][F.sup.ab] has to be intrinsic because it is simply proportional to an invariant of the Weyl tensor.
The presented cylindrically symmetric spacetime (1) is conformally flat since the Weyl tensor vanishes, that is, [C.sub.[mu][nu][rho][sigma]] = 0, and static.
Since the source of this problem is directly related to the projection [E.sub.[mu][nu]] of the bulk Weyl tensor on the brane, the first logical step to overcome this issue would be to impose the constraint [E.sub.[mu][nu]] = 0 on the brane.
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.
Instead of writing [f.sub.[mu]v] = [[partial derivative].sub.v] [[phi].sub.[mu]], - [partial derivative], [[phi].sub.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:
where C is the Weyl tensor. Note that the generalized Ricci tensor (given by its components [R.sub.[micro]v]) is generally asymmetric.