Lanczos potential of Weyl field: interpretations and applications

An attempt is made to uncover the physical meaning and significance of the obscure Lanczos tensor field which is regarded as a potential of the Weyl field. Despite being a fundamental building block of any metric theory of gravity, the Lanczos tensor has not been paid proper attention as it deserves. By providing an elucidation on this tensor field through its derivation in some particularly chosen spacetimes, we try to find its adequate interpretation. Though the Lanczos field is traditionally introduced as a gravitational analogue of the electromagnetic 4-potential field, the performed study unearths its another feature – a relativistic analogue of the Newtonian gravitational force field. A new domain of applicability of the Lanczos tensor is introduced which corroborates this new feature of the tensor.

Gödel spacetime and Lanczos potential

We exhibit that, in Gödel geometry, the Lanczos generator can be written algebraically in terms of the Weyl tensor.

The Lanczos potential and Chern–Simons theory

In electromagnetism, the Faraday tensor [Formula: see text] can be constructed from the vector potential [Formula: see text], it is possible to add term to the Lagrangian depending on [Formula: see text] but not its derivatives called Chern–Simons terms. In gravitation, the Weyl tensor [Formula: see text] can be constructed from the Lanczos potential [Formula: see text], I pursue the analogy to see if terms of Chern–Simons form can be added to the Lagrangian. A new tensor [Formula: see text] is introduced which is constructed from the Lanczos potential and is of the same form as that of the Weyl tensor [Formula: see text] expressed in terms of the Lanczos potential except that covariant differentiation is replaced by transvection with a vector [Formula: see text]. The new tensor has associated invariants [Formula: see text] and [Formula: see text], the first of these can be interpreted as a Chern–Simons term for Weyl [Formula: see text] gravity. Both invariants allow various tensors to be constructed and some of their properties are investigated by using exact examples.