Ivor Robinson

Plane‐fronted gravitational and electromagnetic waves in spaces with cosmological constant

Plane‐fronted waves in spaces with nonzero cosmological constant are studied. In particular their complete classification, which depends essentially on the sign of the cosmological constant and that of some second‐order invariant determined by the congruence of null rays, is provided.

On the Bel - Robinson tensor

A completely symmetrical form of the Bel - Robinson (BR) tensor is shown to give rise, in n-dimensional Lorentzian space, to a gravitational density. For n = 4, a physical identification is proposed.

CONFORMAL GEOMETRY OF FLOWS IN N DIMENSIONS

Flows generated by smooth vector fields are considered from the point of view of conformal geometry. A flow is defined to be conformally geodesic if it preserves the distribution of vector spaces orthogonal to the lines of the flow. It is shear‐free if, moreover, it preserves the conformal structure on these vector spaces. Differential equations characterizing such flows are derived for the general case of an n‐dimensional conformal space of arbitrary signature. In the special case of null flows in spacetime, one obtains a refined version of the theorem connecting null solutions of Maxwell’s equations with null flows that are geodesic and shear‐free.

The Complex Vacuum Metric with Minimally Degenerated Conformal Curvature

By applying Plebanski-Hacyan theorem, the canonical forms of the metric are established for all complex Einstein flat with the minimally (one-sided) algebraically degenerate — conformal curvature. Then Einstein equations are integrated. The solution is expressed in the terms of only one fundamental key function which is determined by a differential equation of the second order and with quadratic non-linearity only, this equation being a generalization of the second heavenly equation.

Lightlike contractions on Minkowski space‐time

The lightlike contractions of electromagnetic fields are discussed. In particular Dirac’s delta type behavior on null hyperplanes is established and the lightlike limit of the electromagnetic field of a spinning, charged particle is obtained.

A Plain Man’s Guide to Bivectors, Biquaternions, and the Algebra and Geometry of Lorentz Transformations

We here present an elementary account of the algebra and geometry of bivectors and of the biquaternions they give rise to. We then proceed to show how the latter allow a detailed study of the algebra and geometry of Lorentz transformations.

Electromagnetic and gravitational Hertz potentials

With generality of complex relativity, the classical theory of the electromagnetic Hertz potentials is outlined in terms of spinors and forms. Particularly interesting are D (0,1) and D (1,0) null Hertz potentials. Then, a new spinorial approach to heavens (H spaces) is proposed, which reveals in their structure the presence of the left null gravitational Hertz potential [of the type D (0,2)]. The relevant hints which follow from our results and concern the structure of the most general solutions of the Einstein vacuum equations (type G⊗G), are discussed, in particular on the level of the linearized theory.

On vacuum metrics of type (3, 1)

A technique is described for constructing solutions of Einsteins equations for empty space, in which the Riemann tensor has a triply degenerate principal null direction with twist.

Null strings and complex Einstein-Maxwell fields with cosmological constant

Previous results of two of us (J. P. and I. R.) concerning left-degenerate Einstein-flat complex space-times, and preliminary results concerning the electromagnetic field [1], are here generalized and worked out in some detail for the system of Einstein-Maxwell equations with a cosmological constant. On the assumption that there exists a congruence of totally null surfaces, the system is reduced to a pair of equations for two unknown functions.

Integrable optical geometry

It is shown that a Lorentzian 4-manifold admitting a congruence of optical (null) geodesics without shear and twist defines an optical geometry which is integrable (locally flat) in the sense of the theory of G-structures. The existence of a symmetric linear connection compatible with the optical geometry is another condition equivalent to the integrability of the optical G-structure.