The affine geometry of the Lanczos H-tensor formalism

The Lanczos potential for the Weyl curvature tensor: existence, wave equation and algorithms

“…In Bampi & Caviglia (1983), there is some discussion for spaces with dimension n > 4 †; however, this is for a parallel existence problem (which is shown to be equivalent to the existence problem for potentials of Weyl † Bampi & Caviglia (1983) state, for the parallel problem which disregards the cyclic properties of the Weyl & Lanczos tensors respectively, that, under generic conditions, a potential can exist only in spaces with dimension n 6; this has lead to the incorrect statement that the Lanczos potential for a Weyl candidates only in four dimensions), and so we believe that it is still an open question whether there exist potentials with the properties (1.2) and (1.4) which satisfy the n-dimensional generalization of (1.3) for Weyl tensors in spaces with dimension n > 4. Although we shall not deal directly with the results in this paper, we draw attention to other recent related work: Hammon & Norris (1993) consider the Lanczos potential and, in particular, the question of gauge, in a still more general geometric setting; Torres del Castillo (1995) has obtained Lanczos potentials for a large class of spacetimes, including the Kerr metric, by spinor means; Ares de Parga et al (1989) and López-Bonilla et al (1993) have obtained expressions for the Lanczos potentials of some special spacetimes, in the spin coefficient formalism of Newman & Penrose (1962); it has been shown that the Riemann tensor cannot, in general, be written in terms of a Lanczos potential, by Massa & Pagani (1984) (in four dimensions) and by Edgar (1994b)…”

Lanczos Potential and Perfect Fluid Spacetimes

“…Dolan and Kim [14] not only generalized the results of Novello and Velloso but also made a correction in the earlier versions of the Weyl-Lanczos equations appeared in literature [10,44]. The guage conditions in a more geometric setting were studied by Hammon and Norris [28]. For a larger class of spacetimes, using the spinor formalism, the Lanczos potential was obtained by Torris de Castillo [43], while using Newman-Penrose formalism, Lopez Bonilla, Ares de Parga and co-workers [21][22][23][24][25] and Lopez et al [35] obtained the Lanczos potential for various algebraically special spacetimes.…”

Localizing Energy in Fierz-Lanczos theory

“…Here, we shall use the tensor formalism. This means that the field configuration is described by the "Weyl-like" tensor fulfilling identities (9)(10)(11) typical for the Weyl tensor of a metric connection.…”

“…Originally, the particle's "wave function" is described by the totally symmetric, fourth order spintensor. However, there is a one-to-one correspondence between such spin-tensors and tensors W λµνκ satisfying identities (9)(10)(11) (the transformation between the two pictures can, e.g., be found in [5]). Moreover, evolution of a massless particle is governed by the same field equation (12).…”

“…identities (9)(10)(11) are not treated as a consequence of any "metricity" (there is a priori no metric here!) but are a straightforward consequence of the transformation from the spinorial to the tensorial language.…”