States of total pure radiation in general relativity

Summary In this paper we investigate and exhibit space-times which admit states of pure radiation in the sense of Lichnerowicz. In § 1 the notion of special total pure radiation is introduced, and in § 2 we derive the canonical line element for this type of radiation. An additional type of spacetime admitting radiation is considered in § 3. A class of singular integrable electromagnetic fields for the space-times of § 2 are constructed in § 4. The final section is concerned with the radiation condition proposed by Zakharov. Work supported by National Science Foundation Grants GP 6876 and GP 7401.     

A theory of electromagnetic and gravitational fields

Careful calculations using classical field theory show that if a macroscopic ball with uniform surface charge (say, a billiard ball with 1E6 excess electrons) is released near the surface of the earth, it will almost instantaneously accelerate to relativistic speed and blow a hole in the ground. This absurd prediction is just the macroscopic version of the self-force problem for charged particles [1]. Furthermore, if one attempts to develop from electromagnetism a parallel theory for gravitational [2], the result is the same, self-acceleration.

The basis of the new theory is a measure of energy density for any wave equation [3–5]. Given any solution of any four-vector wave equation in spacetime (for example, the potentials (c^-1φA)=(A⁰,A¹,A²,A³) in electromagnetism), one can form the 16th first order partial derivatives of the vector components, with respect to the time and space variables (ct,x) = (x⁰, x¹, x², x³). The sum of the squares of the 16 terms is a natural energy function [6, p. 283] (satisfying a conservation law . Such energy functions are routinely utilized by mathematicians as Lyapunov functions in the theory of stability of waves with boundary conditions. A Lagrangian using this sum leads to a new energy tensor for electromagnetic and gravitational fields, an alternative to that in [7].     

Time-like solutions to the Lorentz force equation in time-dependent electromagnetic and gravitational fields

We find existence and multiplicity results for time-like spatially periodic trajectories of massive particles carrying an electric charge q and subjected to time-dependent gravitational and electromagnetic fields. Such trajectories are obtained by projecting, on the base space-time, time-like geodesics with respect to a suitable Kaluza-Klein metric.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

Exact sets of test uncharged massive spin 3/2 fields in general relativity

A theorem giving the necessary and sufficient conditions for Penrose's exact sets of spinor fields in curved spacetimes is proved. An algorithm for augmenting a system of fields to an exact set and constructing covariant Taylor expansions for the fields of an exact set is proposed. The general approach is applied to test massive spin 3/2 fields. Two possible forms of exact sets are constructed for them on the basis of modified Dirac-Fierz-Pauli equations. The functional arbitrariness in the solutions of the equations for exact sets is determined. In one of the cases, the obtained exact set can be interpreted as a system of fields of spins 3/2 and 1/2 interacting through the gravitational field.State University, Kiev. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 1, pp. 119–133, January, 1995.     

Spinning particles in general relativity

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 456–466, March, 1994.