21 research outputs found
The Lanczos potential for Weyl-candidate tensors exists only in four dimensions
We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd
does not generally exist for dimensions higher than four. The technique is
simply to assume the existence of such a potential in dimension n, and then
check the integrability conditions for the assumed system of differential
equations; if the integrability conditions yield another non-trivial
differential system for L_abc and W_abcd, then this system's integrability
conditions should be checked; and so on. When we find a non-trivial condition
involving only W_abcd and its derivatives, then clearly Weyl candidate tensors
failing to satisfy that condition cannot be written in terms of a Lanczos
potential L_abc.Comment: 11 pages, LaTeX, Heavily revised April 200
Linear connections with propagating spin-3 field in gravity
We show that Fronsdal's Lagrangian for a free massless spin-3 gauge field in
Minkowski spacetime is contained in a general Yang--Mills-like Lagrangian of
metric-affine gravity (MAG), the gauge theory of the general affine group in
the presence of a metric. Due to the geometric character of MAG, this can best
be seen by using Vasiliev's frame formalism for higher-spin gauge fields in
which the spin-3 frame is identified with the tracefree nonmetricity one-form
associated with the shear generators of GL(n,R). Furthermore, for specific
gravitational gauge models in the framework of full nonlinear MAG, exact
solutions are constructed, featuring propagating massless and massive spin-3
fields.Comment: References added. Minor corrections and clarifications. To be
published in Phys. Rev.
A local potential for the Weyl tensor in all dimensions
In all dimensions and arbitrary signature, we demonstrate the existence of a
new local potential -- a double (2,3)-form -- for the Weyl curvature tensor,
and more generally for all tensors with the symmetry properties of the Weyl
curvature tensor. The classical four-dimensional Lanczos potential for a Weyl
tensor -- a double (2,1)-form -- is proven to be a particular case of the new
potential: its double dual.Comment: 7 pages; Late