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Solving gravitational field equations by Wiener-Hopf matrix factorisation, and beyond

M. Cristina Câmara, Gabriel Lopes Cardoso

Abstract

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 factorisation 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 factorisation theory, with particular emphasis on developments from the past decade enabled by advances in Wiener-Hopf factorisation techniques arising from the study of singular integral equations and Toeplitz operators. Through a variety of concrete examples, we illustrate how Wiener-Hopf factorisation yields explicit, exact solutions to the field equations of gravitational theories, and how its generalisation 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.

Solving gravitational field equations by Wiener-Hopf matrix factorisation, and beyond

Abstract

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 factorisation 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 factorisation theory, with particular emphasis on developments from the past decade enabled by advances in Wiener-Hopf factorisation techniques arising from the study of singular integral equations and Toeplitz operators. Through a variety of concrete examples, we illustrate how Wiener-Hopf factorisation yields explicit, exact solutions to the field equations of gravitational theories, and how its generalisation through a so-called -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.
Paper Structure (11 sections, 20 theorems, 146 equations, 2 figures)

This paper contains 11 sections, 20 theorems, 146 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The gravitational field equations as an integrable system
  3. Solving the field equations by canonical Wiener-Hopf factorisation
  4. Canonical Wiener-Hopf factorisation: the question of existence
  5. Beyond Wiener-Hopf factorisation: $\tau$-invariance and solution generation by multiplication
  6. Examples
  7. Deforming the Schwarzschild monodromy matrix
  8. Deforming a static attractor solution
  9. Open questions
  10. Toeplitz operators, RH problems and WH factorisation
  11. Canonical WH factorisation and Ward's factorisation approach

Key Result

Theorem 2.1

Lu:2007jcAniceto:2019rhg Let $\varphi \in {\cal T}$. If, for a given $A = M^{-1} dM$, there exists $X(\tau, \rho, v)$ such that upon substituting $\tau = \varphi$, we have $X \in C^2, X^{-1} \in C^1$ and then $M$ is a solution to fi2d. Note, however, that although the Breitenlohner–Maison linear system laxx is formally a system of linear PDE's for the unknown $X$, with coefficient $A(\rho, v)$, i

Figures (2)

  • Figure 1: $-m < v < m$: four distinct choices of contours.
  • Figure 2: Curve $\mathcal{C}$ in the Weyl coordinates upper half-plane $(\rho >0, v)$ for the values $m= 2, a = 1$. The horizontal axis represents $v \in \mathbb{R}$, while the vertical axis represents $\rho> 0$.

Theorems & Definitions (31)

  • Theorem 2.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Example 3.4
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Proposition 4.5
  • ...and 21 more