A. Schild

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.

Generalization of a Theorem by Goldberg and Sachs

A connection is established between algebraic degeneracy of the Weyl tensor, the existence of a null geodesic shear‐free congruence, and certain restrictions on the Ricci tensor which are weaker than the gravitational equations for empty space. The result is roughly, with some important qualifications, that any two of these conditions imply the third. These restrictions on the metric are shown to be invariant under conformal transformations.

Geometry in a manifold with projective structure

Parallel transport of line elements, surface elements etc. along geodesics and more general curves in a projectively connected manifold is investigated analytically and in terms of geometrical constructions. Projective curvature is characterized geometrically by a projective analogue of the geodesic deviation equation and by a geometrical construction. The results are interpreted physically as statements about free fall world lines in space-time.

Complex relativity and double KS metrics

SummaryThe basic motivations and the foundamental facts of the theory of the double KS metrics are outlined. The complex metricsnμν+2Pkμkν+2Qlμlνwithkμandlμnull and mutually orthogonal become particularly interesting whenkμandlμare surface forming,i.e. when they determine a congruence of the complex null strings.RiassuntoSi delineano i motivi basilari ed i fatti fondamentali della teoria delle metriche doppie di KS. Le metriche complessenμν+2Pkμkν+2Qlμlνconkμelνnulli e reciprocamente ortogonali diventano particolarmente interessanti quandokμelνsono formatrici della superficie, ciè quando determinano una congruenza di corde nulle complesse.

Fokker Action Principle for Particles with Charge, Spin, and Magnetic Moment

A Fokker action principle is obtained for a system of particles with charge, spin, and magnetic moment, interacting through time‐symmetric electromagnetic fields. Conservation laws are derived for the energy, and the linear and angular momentum of the system. A limiting process is performed, which involves a renormalization of mass, in order that the magnitude of the spin of each particle remain constant.

Equivalence Principle and Red-Shift Measurements

Two questions are discussed. The first asks whether experiments on accelerated systems (e.g., red-shifts produced in rotating disks) can serve to verify the general theory of relativity. The answer is “no.” The second asks to what extent the special theory of relativity and the principle of equivalence determine the well-known effects of the general theory of relativity. It is important to formulate this question very carefully because special relativity and the equivalence principle do not form a consistent theoretical system. If this is done, then the answer is that the equivalence principle leads to the same gravitational red-shift as general relativity but does not lead to specific values for the bending of light rays by a star or for the perihelion rotation of a planetary orbit.

A generalized Reissner-Nordström solution

A new class of solutions to the Einstein-Maxwell equations is presented; it stands in the same relation to the Robinson & Robinson (1969) metrics as the Reissner-Nordström solution to Schwarzschilds.

Electromagnetic Two‐Body Problem for Particles with Spin

A two‐body system is considered for classical particles with charge, spin, and magnetic moment. The particles move in circular orbits, with spins orthogonal to the orbital plane, and they interact through time‐symmetric electromagnetic fields. Rigorous relativistic equations of motion and rigorous expressions for the total energy and angular momentum of the system, including contributions from the field, are obtained.

On Gravitational Theories of Whitehead's Type

An infinity of gravitational theories are obtained which, like Whitehead’s theory, are action-at-a-distance theories. The motions of mass particles and of light rays in the gravitational field of a single heavy particle at rest are the same as in Einstein’s general theory of relativity.

Fokker action principles with self-action. I. General theory: outline of a program for constructing models of elementary particles

Abstract Fokker action principles with self-action are studied. A program is outlined that offers long-odds hope for the construction of classical models of some elementary particles. Part I gives an outline of the general theory, its possibilities and its difficulties. Part II, which will be published shortly, will give applications of the theory to circular motions of a single particle interacting with itself. A specific, but ugly, model that arose in a thusfar unsuccessful attempt at a theory of the electron will be discussed.