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.
The
Weyl tensor of a Riemannian manifold (M,g) is defined by
[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.
