R. P. Kerr

Solutions of the Einstein and Einstein‐Maxwell Equations

Algebraically degenerate solutions of the Einstein and Einstein‐Maxwell equations are studied. Explicit solutions are obtained which contain two arbitrary functions of a complex variable, one function being associated with the gravitational field and the other mainly with the electromagnetic field.

EINSTEIN SPACES WITH SYMMETRY GROUPS.

In this paper, we construct all possible groups of motion (symmetry groups) for empty Einstein spaces admitting a diverging, geodesic, and shear‐free ray congruence. (Minkowski space is excluded throughout the discussion.) It is proved that any such Einstein space cannot admit a symmetry group with dimension greater than four. Although the field equations are not solved completely for spaces with groups of dimension one or two, a generalization of the Kerr spinning‐mass solution is obtained from the 2‐dimensional class. It is shown that all such spaces with 4‐dimensional symmetry groups are well known: Schwarzschild, NUT (Newman, Unti, and Tamborino), and a particular hypersurface orthogonal Kerr‐Schild metric. The only member of these spaces admitting a 3‐dimensional symmetry group is a Petrov Type III hypersurface orthogonal metric.

Some Applications of the Infinitesimal‐Holonomy Group to the Petrov Classification of Einstein Spaces

The classifications of Einstein spaces by Schell and Petrov are combined and certain nonlocal results are obtained. In particular, we show that an Einstein space cannot be type I with a rank four Riemann tensor in a four‐dimensional region. On using the notion of a perfect or imperfect infinitesimal‐holonomy group, we establish the conditions under which an Einstein space possesses a two‐, four‐, or six‐parameter group. We find that two‐ and four‐parameter groups are associated with special cases of type II null and type III, respectively.

Einstein Spaces With Four‐Parameter Holonomy Groups

The metric tensor is constructed for Einstein spaces which are Petrov type III and whose holonomy group is four parametric. Together with the previously known plane fronted wave solutions, this completes the study of all metrics whose holonomy groups are less than six parameter. The Killing vector equations are studied and it is found that the space cannot admit more than two independent motions.

Asymptotic Properties of the Electromagnetic Field

From the integral form of the general solution for the retarded electromagnetic field of a localized charge‐current distribution, the asymptotic field is shown to have the behavior Fμν = Nμν/R + IIIμν/R2 + 2Jμν/R3, where the coefficients satisfy Nμνkν = 0, IIIμνkν = Akμ, and kμkν = 0. The remainder 2Jμν is shown to be bounded by using the second‐mean‐value theorem. Thus the algebraically special character of the asymptotic electromagnetic field is exhibited.

Singularities in the Kerr-Schild metrics

It is shown that the only empty space solution of the type “flat space plus the square of a null vector” whose singularities are confined to a bounded region is the Kerr metric.

Diverging Type-D Metrics

This paper contains an investigation of algebraically special spaces with two commuting Killing vectors. It is shown that the field equations for these spaces can be reduced to two ordinary differential equations, one of which is quasi-linear in one of the variables. The metric is type D iff it possesses a two dimensional, abelian, orthogonally transitive symmetry group. Finally, the type D metrics of Kinnersley are expressed in various coordinates, including those of Plebanski and Demianski.

Einstein spaces and homothetic motions. I

Algebraically special, nonflat vacuum Einstein spaces with an expanding and/or twisting geodesic principal null congruence are considered. These spaces are assumed to possess locally a homothetic symmetry as well as two or more Killing vectors. All metrics of such spaces are determined along with the form of the homothetic Killing vector admitted. All but one of the metrics are twist free. It is proved that two of the NUT metrics do not admit a homothetic motion.

Homogeneous Dust‐Filled Cosmological Solutions

The Einstein field equations with incoherent matter are discussed for the case of homogeneous space‐time, i.e., for metrics allowing a four‐parametric, simply transitive group of motions. It is proved that the only universes satisfying the above are those of Einstein, Godel, and Ozsvath.

On the quasi-static approximation in general relativity

SummaryThe three main methods used for solving the quasi-static field equations are discussed and it is shown that certain theorems have to be proved before it can be said that the physical equations of motion follow from the symmetry of the field around the sources. We have proved these results, and in the process have shown that there are seven and only seven physical equations of motion for each particle. These correspond to the classical equations of energy, motion and angular momentum. Also, we have shown that the quasi-static field equations may be integrated without expanding any particle parameters, and without introducing a stress-energy tensor.RiassuntoSi discutono i tre metodi principali usati per la risoluzione delle equazioni di campo quasi statiche e si mostra che alcuni teoremi devono essere dimostrati prima di poter dire che le equazioni fisiche del moto procedono dalla simmetria del campo attorno alle sorgenti. Abbiamo dimostrato questi risultati e durante lo svolgimento abbiamo mostrato che ci sono sette e soltanto sette equazioni fisiche del moto per ogni particella. Queste corrispondono alle equazioni classiche dell’energia, del moto e della quantità di moto angolare. Abbiamo anche mostrato che le equazioni di campo quasi statiche possono essere integrate senza sviluppare in serie alcnni dei parametri della particella, e senza introdurre un tensore sforzo-energia.