Lanczos Potential Research Articles

Lanczos potential for the Gödel spacetime

AbstractWe employ the local and isometric embedding of Riemannian spaces to construct one Lanczos potential in Gödel geometry.

Lanczos potentials and a definition of gravitational entropy for perturbed Friedman–Lemaitre–Robertson–Walker spacetimes

We give a prescription for constructing a Lanczos potential for a cosmological model which is a purely gravitational perturbation of a Friedman–Lemaitre–Robertson–Walker spacetime. For the radiation equation of state, we find the Lanczos potential explicitly via Fourier transforms. As an application, we follow up a suggestion of Penrose (1979 Singularities and time-asymmetry General Relativity: An Einstein Centenary Survey ed S W Hawking and W Israel (Cambridge: Cambridge University Press)) and propose a definition of gravitational entropy for these cosmologies. With this definition, the gravitational entropy initially is finite if and only if the initial Weyl tensor is finite.

A weighted de Rham operator acting on arbitrary tensor fields and their local potentials

We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r -fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence of 2 r potentials for any tensor field, where r is its form-structure number. By specialising this result to symmetric double forms, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2, 3)-form potential for the Weyl tensor due to its tracelessness. This latter potential is the n -dimensional version of the double dual of the classical four-dimensional (2, 1)-form Lanczos potential. We also introduce a new concept of harmonic tensor fields, and demonstrate that the new weighted de Rham operator has many other desirable properties and, in particular, is the natural operator to use in the Laplace-like equation for the Riemann tensor.

Some general ansätze for the Lanczos potential spinor

The Weyl curvature spinor of every four-dimensional spacetime is derivable from a potential spinor, called the Lanczos potential. In certain special situations the Lanczos spinor has been found explicitly, but there exists no algorithm for finding it in general. Using a modified Newman–Penrose formalism proposed by Wunsch, we study some ansatze as candidates for the Lanczos potential with respect to an arbitrary spin frame. We show under which necessary and sufficient conditions each ansatz will lead to a Lanczos potential of theWeyl spinor. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Classical model of an elementary particle with a Bertotti-Robinson core and extremal black holes

We discuss the question, whether the Reissner-Nordstr\"om (RN) metric can be glued to other solutions of Einstein-Maxwell equations in such a way that (i) the singularity at $r=0$ typical of the RN metric is removed, and (ii) matching is smooth. Such a construction could be viewed as a classical model of an elementary particle balanced by its own forces without support by an external agent. One choice is the Minkowski interior that goes back to the old Vilenkin and Fomin's idea who claimed that in this case the bare deltalike stresses at the horizon vanish if the RN metric is extremal. However, the relevant entity here is the integral of these stresses over the proper distance which is infinite in the extremal case. As a result of the competition of these two factors, the Lanczos tensor does not vanish and the extremal RN cannot be glued to the Minkowski metric smoothly, so the elementary-particle model as an empty ball inside fails. We examine the alternative possibility for the extremal RN metric---gluing to the Bertotti-Robinson (BR) metric. For a surface placed outside the horizon there always exist bare stresses but their amplitude goes to zero as the radius of the shell approaches that of the horizon. This limit realizes the Wheeler idea of ''mass without mass'' and ''charge without charge.'' We generalize the model to the extremal Kerr-Newman metric glued to the rotating analog of the BR metric.

Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second-order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher-dimensional tensor identities to obtain the analogous results for ( m, m)-forms in 2 m dimensions.

A Method for Finding Lanczos Coefficients with the Aid of Symmetries

A set of algorithms involving the Lanczos potential and certain kinematical quantities for some arbitrary space-times, obtained using timelike vector fields, is considered. It is shown that such algorithms appear most naturally in the context of the spin coefficient formalism. Furthermore, explicit solutions are derived for the new spin coefficient algorithms.

Letter: A Solution of the Weyl–Lanczos Equations for the Schwarzschild Space-Time

The spin coefficient form of the Weyl–Lanczos equations is analysed for the Schwarzschild space-time. The solution obtained yields an alternative form of Lanczos coefficients to the one currently known for this particular metric.

Lanczos tensor potential for conformally flat space-times

A sufficient condition is derived for the Lanczos tensor potential such that the Weyl tensor remains conformal. This condition is then applied to the metric of a conformally flat space-time, and the corresponding Lanczos tensor potential is obtained.

The Weyl–Lanczos relations and the four-dimensional Lanczos tensor wave equation and some symmetry generators

We examine symmetry generators for exterior differential systems and for systems of partial differential equations and apply the Cartan theory of exterior differential systems to the Weyl–Lanczos equations and to the Lanczos wave equation in four dimensions. We look at a number of examples of symmetries for the Weyl–Lanczos equations in four dimensions and give examples of isovectors when the solution manifold is the Schwarzschild, Kasner or Gödel space–time. Solutions of the Weyl–Lanczos system are automatically solutions of the Lanczos wave equation. We give examples of symmetry generators for the Lanczos wave equation and find that they are not automatically symmetry generators for the Weyl–Lanczos equations.

The Weyl–Lanczos equations and the Lanczos wave equation in four dimensions as systems in involution

The Weyl–Lanczos equations in four dimensions form a system in involution. We compute its Cartan characters explicitly and use Janet–Riquier theory to confirm the results in the case of all space–times with a diagonal metric tensor and for the plane wave limit of space–times. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet–Riquier theory, we compute its Cartan characters and find that it forms a system in involution. We compare these Cartan characters with those of the Weyl-Lanczos equations. All results hold for the real analytic case.

Letter: The Non-Existence of a Lanczos Potential for the Weyl Curvature Tensor in Dimensions n ≥ 7

In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n ≥ 7.

Lanczos potential for the Gödel cosmological model

Lanczos potential for the Gödel cosmological model

Existence of Lanczos potentialsand superpotentials for theWeyl spinor/tensor

A new and concise proof ofexistence - emphasizing the very naturaland simple structure - is given for theLanczos spinor potential LABCA' of anarbitrary symmetric spinor WABCDdefined by WABCD = 2∇(AA'LBCD)A'; this proof is easily translated into tensorsin such a way that it is valid in four-dimensional spaces of any signature.In particular, this means that the Weyl spinor ΨABCD has Lanczospotentials in all spacetimes, and furthermore that the Weyl tensor hasLanczos potentials on all four-dimensional spaces, irrespective ofsignature. In addition, two superpotentials for WABCD are identified:the first TABCD ( = T(ABC)D) isgiven by LABCA' = ∇A'DTABCD, while the secondHABA'B' ( = H(AB)(A'B')) (which isrestricted to Einstein spacetimes) is givenbyLABCA' = ∇(AB'HBC)A'B'. The superpotential TABCD is used to describe the gaugefreedom in the Lanczos potential.

Local existence of symmetric spinor potentials for symmetric (3,1)-spinors in Einstein space–times

We investigate the possibility of existence of a symmetric potential H ABA′ B′ = H ( AB)( A′ B′) for a symmetric (3,1)-spinor L ABCA′ , e.g., a Lanczos potential of the Weyl spinor, as defined by the equation L ABCA′ =∇ ( A B′ H BC) A′ B′ . We prove that in all Einstein space–times such a symmetric potential H ABA′ B′ exists. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of space–times. A tensor version of this result is also given. We apply similar ideas and results by Illge to Maxwell’s equations in a curved space–time.

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.

Spin coefficients as Lanczos scalars: Underlying spinor relations

It has been conjectured by Lopez-Bonilla and co-workers that there is some linear relationship between the NP spin coefficients and the Lanczos scalars, and examples have been given for a number of different classes of space–times. We show that in each of those examples a Lanczos potential can be defined in a very simple way directly from the spinor dyad. Although some of these examples seem to have no deeper geometric meaning, we emphasize that there are structural links between Lanczos potential and spin coefficients which we highlight in some other examples. In particular we show that the direct identification of Lanczos potentials with spin coefficients is possible for some important classes of space–times while the direct identification of Lanczos potentials with the properly weighted spin coefficients is also possible for several important classes of space–times. In both of these cases we obtain the necessary and sufficient conditions on the spin coefficients for such identifications to be possible, which enables us to test space–times directly.

LETTER: A Note on the Algebraic Symmetries of the Riemann and Lanczos Tensors

The usual algebraic symmetries of the Riemanntensor involve four convenient relations. It is shownthat these can be encompassed in three, two or just onerelation. Similar results are shown for the Lanczos tensor.

The Lanczos potential for vacuum space–times with an Ernst potential

By a particular choice of tetrad and coordinate systems, we find the Lanczos potential for general vacuum stationary axisymmetric space–times and for general vacuum cylindrically symmetric solutions. This result is particularly elegant and unifies the treatment of the Lanczos potential and the Ernst potential for vacuum stationary axially symmetric and cylindrically symmetric space–times. The Lanczos potential for such large classes of vacuum space–times is expressed very simply in terms of spin-coefficients for a null tetrad.

GRAVITATIONAL POTENTIALS AND THE EXISTENCE OF GRAVITATIONAL GREEN'S TENSORS

The non-local part of the gravitational field Cabcd can be generated by the 16-component Lanczos tensor potential Labc. When six gauge conditions are imposed, Labe;e=0, its ten degrees of freedom match those of the Weyl tensor. The Penrose wave equation for Cabcd can be independently derived from that for Labc. The consistency between Labc and Cabcd is also shown by the compatibility of their algebraic classifications. An unexpected insight into the relationship of Labc and Cabcd is found in "Euclidean gravity" which in turn leads to the introduction of a gravitational Green's tensor [Formula: see text] corresponding to the potential Labc.