Solving gravitational field equations by Wiener–Hopf matrix factorization and beyond

By viewing Einstein’s field equations—reduced to two dimensions—as an integrable system, one can simultaneously obtain exact solutions to both the equations themselves and their associated Lax pair via a canonical Wiener–Hopf factorization of a so-called monodromy matrix. In this article, we review this remarkable interplay between gravitational field equations, integrable systems, Riemann–Hilbert problems and Wiener–Hopf factorization theory, with particular emphasis on developments from the past decade enabled by advances in Wiener–Hopf factorization techniques arising from the study of singular integral equations and Toeplitz operators. Through a variety of concrete examples, we illustrate how Wiener–Hopf factorization yields explicit, exact solutions to the field equations of gravitational theories and how its generalization through a so-called τ-invariance property provides a new solution-generating method. Along the way, we aim to demonstrate the importance of an interdisciplinary approach—grounded in General Relativity, Complex Analysis and Operator Theory—for the study of gravitational field equations.