On interrelations between Sibgatullin's and Alekseev's approaches to the construction of exact solutions of the Einstein-Maxwell equations

TL;DR

This paper compares Sibgatullin's and Alekseev's methods for constructing exact solutions to the Einstein-Maxwell equations, revealing mathematical inconsistencies in Alekseev's approach and clarifying their interrelations.

Contribution

It demonstrates that Alekseev's integral equations are combinations of Sibgatullin's equations and identifies errors in Alekseev's principal value integrals.

Findings

Alekseev's scheme is mathematically inconsistent due to complex conjugation.

Alekseev's integrals contain an intrinsic unrecognized error.

A double-step algorithm relates solutions from both methods.

Abstract

The integral equations involved in Alekseev's "monodromy transform" technique are shown to be simple combinations of Sibgatullin's integral equations and normalizing conditions. An additional complex conjugation introduced by Alekseev in the integrands makes his scheme mathematically inconsistent; besides, in the electrovac case all Alekseev's principal value integrals contain an intrinsic error which has never been identified before. We also explain how operates a non-trivial double-step algorithm devised by Alekseev for rewriting, by purely algebraic manipulations and in a different (more complicated) parameter set, any particular specialization of the known analytically extended N-soliton electrovac solution obtained in 1995 with the aid of Sibgatullin's method.