Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

The Plebanski–Demianski metric, and those that can be obtained from it by taking coordinate transformations in certain limits, include the complete family of space–times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non-expanding type D solutions are also identified. All special cases are derived in a simple and transparent way.

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A New look at the Plebanski-Demianski family of solutions

The Plebanski-Demianski metric, and those that can be obtained from it by taking coordinate transformations in certain limits, include the complete family of space-times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non-expanding type D solutions are also identified. All special cases are derived in a simple and transparent way.

A new look at the Plebanski-Demianski family of solutions

Within nonlinear electrodynamics of Born–Infeld type allowing for the freedom of duality rotations, explicit type D solutions are constructed. The obtained type D solutions, which generalize the charged Taub–NUT (Newman–Unti–Tamburino) metric with λ, exhaust all solutions within the considered class, under the assumption that the real eigenvectors of the electromagnetic field are aligned along the geodesic and shear‐free principal null directions. Various relevant limiting transitions, in particular those of the flat space‐time, are studied in some detail.

Duality rotations and type D solutions to Einstein equations with nonlinear electromagnetic sources

Canonical theories of Lagrangian dynamical systems in physics

We present an exhaustive integration of the type D vacuum and electrovac field equations with cosmological constant admitting a nonsingular aligned Maxwell field satisfying the generalized Goldberg–Sachs theorem. We derive a single expression for the general solution from which one may obtain all particular cases known until now in partial versions. We also investigate in detail the separability properties of the Hamilton–Jacobi equation for the charged particle orbits and of the Klein–Gordon equation for a massive charged spin‐zero test particle and their corresponding Killing tensors.

Exhaustive integration and a single expression for the general solution of the type D vacuum and electrovac field equations with cosmological constant for a nonsingular aligned Maxwell field

It is shown without explicit integration that all Petrov type‐D vacuum solutions with cosmological constant admit at least a two‐parameter, abelian, orthogonally transitive group of local isometries. In the case when the orbits are non‐null the group is invertible and a symmetric null tetrad is shown to exist in which it is manifest that the principal null congruences defined by the type‐D Weyl tensor are indistinguishable. General forms for the metrics are given for both the case of non‐null and null group orbits. It is also demonstrated that there always exists a system of coordinates for these solutions in which the Hamilton–Jacobi equation for the null geodesics is solvable by separation of variables, a fact that explains the existence of a conformal Killing tensor therein. A partial invariant characterization of Kinnersley’s type‐D vacuum solutions is given from which it follows that all his metrics except those of Case III admit a (0,2) Yano–Killing tensor and hence a full (0,2) Killing tensor.

Orthogonal transitivity, invertibility, and null geodesic separability in type D electrovac solutions of Einstein’s field equations with cosmological constant

It is shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2-form and the exterior derivative of a related 2-form are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group with nonsingular orbits

Riemannian-Maxwellian invertible structures in general relativity

A single expression for the general solution of Einstein's vacuum and electrovac field equations with cosmological constant for Petrov type D admitting a non-singular aligned Maxwell field

Robert Debever

Papers

Exhaustive integration and a single expression for the general solution of the type D vacuum and electrovac field equations with cosmological constant for a nonsingular aligned Maxwell field

Orthogonal transitivity, invertibility, and null geodesic separability in type D electrovac solutions of Einstein’s field equations with cosmological constant

Riemannian-Maxwellian invertible structures in general relativity

A single expression for the general solution of Einstein's vacuum and electrovac field equations with cosmological constant for Petrov type D admitting a non-singular aligned Maxwell field

Etude géométrique du tenseur de Riemann-Christoffel des espaces de Riemann à quatre dimensions