Lanczos Potential Research Articles

On the Lanczos Potential

For arbitrary geometries of Petrov types III, N and O we construct the Lanczos potential for the corresponding Weyl tensor. This will provide a technique to construct Lanczos potential in non-vacuum cases.

Hyperbolic reduction of the Weyl–Lanczos equations

We provide a new framework to find the Lanczos potential of the Weyl tensor in terms of two symmetric and spatial fields. We show that given any space time with the Einstein field equations coupled to a general matter source, a Lanczos potential can be identified by solving a symmetric hyperbolic system of evolution equations.

The Lanczos Potential for Bianchi Spacetime

The form and application of the Lanczos potential for Bianchi spacetime are studied. The Lanczos potential is found in some specific cases, then the general case is studied. It leads to two coupled first order partial differential equations which, although they so far have not been solved in general, can be solved for many configurations. The application is to cosmic energetics: in other words to the study of the energy of the gravitational and other fields in the Universe.

Can a tensorial analogue of gravitational force explain away the galactic rotation curves without dark matter?

The dark matter problem is one of the most pressing problems in modern physics. As there is no well-established claim from a direct detection experiment supporting the existence of the illusive dark matter that has been postulated to explain the flat rotation curves of galaxies, and since the whole issue of an alternative theory of gravity remains controversial, it may be worth to reconsider the familiar ground of general relativity (GR) itself for a possible way out. It has recently been discovered that a skew-symmetric rank-three tensor field — the Lanczos tensor field — that generates the Weyl tensor differentially, provides a proper relativistic analogue of the Newtonian gravitational force. By taking account of its conformal invariance, the Lanczos tensor leads to a modified acceleration law which can explain, within the framework of GR itself, the flat rotation curves of galaxies without the need for any dark matter whatsoever.

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.

Relativistic potential of Weyl: a gateway to quantum physics in the presence of gravitation?

The well-noted correspondence between gravitation and electrodynamics emphasizes the importance of the Lanczos tensor—the potential of the Weyl tensor—which is an inherent structural element of any metric theory of gravity formulated in a 4-dimensional pseudo Riemannian spacetime. However, this ingenious discovery has gone largely unnoticed. We elucidate this important quantity by deriving its expressions in some particularly chosen spacetimes and try to find out what it represents actually. We find out that the Lanczos potential tensor does not represent the potential of the gravitational field, as is ascertained by various evidences. Rather, it is enriched with various signatures of quantum character, which provides a novel insight into the heart of a geometric embodiment of gravity.

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.

A derivation of Weyl-Lanczos equations

The Lanczos potential for the Weyl tensor is derived from a quadratic curvature Lagrangian by making use of the exterior algebra of forms and the variational principles with constraints.

Lanczos potential for some non-vacuum spacetimes

The Lanczos potential is a general relativity analogue of potential in electrodynamics. This potential can be obtained by solving Weyl-Lanczos relations. This computation is easy in the case of vacuum spacetimes and/or spacetimes of particular Petrov type, while obtaining the Lanczos potential for arbitrary spacetime is very difficult. In this paper, we have obtained the Lanczos potential for non-vacuum spacetimes: Vaidya spacetime and Van Stockum spacetime (which is not of any Petrov type).

Lanczos Potential for Arbitrary Petrov Type II Spacetimes

The Lanczos potential for an arbitrary Petrov type II spacetime has been obtained and Robinson–Trautman family of metrics has been considered as an example.

Lanczos generator in Gödel geometry

We exhibit three ways to obtain a Lanczos spintensor for the Gödel cosmological model. In particular, a symmetric tensor of rank two is constructed which satisfies the wave equation and generates a Lanczos potential.

On a new approach of deriving the Weyl-Lanczos equations giving rise to a physical interpretation of the Lagrange multiplier q

In this paper we incorporate Lanczos’ six differential gauge conditions and eight algebraic conditions to the tensor form of the Weyl-Lanczos equation and subsequently apply the null tetrad formalism to obtain the Weyl-Lanczos equation in spin coefficient form. This technique is then applied to the Ricci tensor. As a consequence, Lanczos’ Lagrange multiplier \(\ensuremath Q_{\alpha \beta}\) is found to be related to the Ricci tensor --and explicit forms are obtained involving spin coefficients and the eight complex Lanczos coefficients-- while the Lagrange multiplier q can be interpreted as the Ricci scalar. Furthermore, a relationship is established between \(\ensuremath H_{\alpha \beta \gamma }\) , \(\ensuremath Q_{\alpha \beta}\) and q when the Bianchi identities are invoked.

Lanczos Potential and Perfect Fluid Spacetimes

Using method of general observers and the spin coefficient formalism of Newman and Penrose, the Lanczos potential has been studied for perfect fluid spacetimes.

A Solution of Weyl-Lanczos Equations for Arbitrary Petrov Type D Vacuum Spacetimes

A Lanczos potential for an arbitrary Petrov type D vacuum spacetimes, using the compacted spin coefficient formalism (or GHP-formalism), has been obtained; which in turn leads to a solution of Weyl-Lanczos equations.

Petrov Type D Spacetimes and Lanczos Potential

Using the Newman-Penrose formalism, the Lanczos potential for Petrov type D spacetimes has been found. It is seen that the Lanczos scalars can be expressed in terms of the spin-coefficients. The non-uniqueness character of the Lanczos potential has been established and a possible justification to the name “the Lanczos potential” is given

Lanczos Potential for the van Stockung Space-Time

The Lanczos Potential is a theoretical useful tool to find the conformal Weyl cur- vature tensor Cabcd of a given relativistic metric. In this paper we find the Lanczos potential Labc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with spinor methods in order to find this important three covariant index tensor.

The Riemann–Lanczos equations in general relativity and their integrability

The aim of this paper is to examine the Riemann-Lanczos equations and how they can be made integrable. They consist of a system of linear first-order partial differential equations that arise in general relativity, whereby the Riemann curvature tensor is generated by an unknown third-order tensor potential field called the Lanczos tensor. Our approach is based on the theory of jet bundles, where all field variables and all their partial derivatives of all relevant orders are treated as independent variables alongside the local manifold coordinates (xa) on the given space-time manifold M. This approach is adopted in (a) Cartan’s method of exterior differential systems, (b) Vessiot’s dual method using vector field systems, and (c) the Janet–Riquier theory of systems of partial differential equations. All three methods allow for the most general situations under which integrability conditions can be found. They give equivalent results, namely, that involutivity is always achieved at all generic points of the jet manifold M after a finite number of prolongations. Two alternative methods that appear in the general relativity literature to find integrability conditions for the Riemann–Lanczos equations generate new partial differential equations for the Lanczos potential that introduce a source term, which is nonlinear in the components of the Riemann tensor. We show that such sources do not occur when either of method (a), (b), or (c) are used.

The new potential for the Weyl tensor inNdimensions: gauge and wave equation

Although the Lanczos potential (a (2,1) form, L ab c ) for the Weyl tensor does not exist in dimensions greater than four, a new potential (a (2,3) form, P ab cde , which coincides with the double dual of L ab c in four dimensions) has recently been shown to exist in all dimensions n ≥ 4. In this talk we investigate the question of gauge and discuss the structure of the new potential's wave equation which is obtained from the Bianchi identities; identifying the gauge supplies us with a new direct proof of the existence of P ab cde via the Cauchy-Kowaleski theorem, as well as the foundation for more general investigations of the first order symmetric hyperbolic structure.

Lanczos potentials for linearly perturbed FLRW spacetimes

We study the problem of deriving the Lanczos potential and superpotential for linearly perturbed Friedman-Lemaitre-Robertson-Walker (FLRW) spacetimes.