General Solution of the Complex 4-Eikonal Equation and the "Algebrodynamical" Field Theory

TL;DR

This paper presents a comprehensive algebraic solution to the complex 4-eikonal equation, revealing twistor structures, particle-like singularities, and introducing the algebrodynamical field theory as a new framework.

Contribution

It provides the first explicit algebraic solutions to the complex 4-eikonal equation using twistor methods and introduces the algebrodynamical field theory related to these solutions.

Findings

Existence of twistor and ambitwistor structures for the 4-eikonal equation.

General solutions classified into two classes derived from algebraic twistor functions.

Identification of particle-like singularities with dynamic properties.

Abstract

We explicitly demonstrate the existence of twistor and ambitwistor structure for the 4-dimensional complexified eikonal equation (CEE) and present its general solution consisting of two different classes. For both, every solution can be obtained from a generating twistor function in a purely algebraic way, via the procedure similar to that used in the Kerr theorem for shear-free null congruences. Bounded singularities of eikonal or of its gradient define some particle-like objects with nontrivial characteristics and dynamics. Example of a new static solution to CEE with a ring-like singularity is presented, and general principles of algebraic field theory - algebrodynamics - closely related to CEE are briefly discussed.