Weyl metrics and Wiener-Hopf factorization

TL;DR

This paper rigorously proves that Wiener-Hopf factorization of monodromy matrices can generate solutions to four-dimensional Weyl metrics, providing a new method to construct explicit solutions to Einstein's equations.

Contribution

It offers the first rigorous proof linking Wiener-Hopf factorization of monodromy matrices to Weyl metric solutions, expanding the mathematical tools for solving Einstein's equations.

Findings

Established a rigorous connection between Riemann-Hilbert factorization and Weyl metrics.

Showed that different contour choices yield various solutions from the same monodromy matrix.

Constructed explicit solutions, including new ones, to Einstein's equations using complex analysis.

Abstract

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows us to construct explicit solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results.