Weyl metrics and Wiener-Hopf factorization

M.C. Câmara,M. Rosselló,P. Aniceto,G.L. Cardoso

doi:10.1007/jhep05(2020)124

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.

Highlights

The field equations of gravitational theories in D space-time dimensions are a system of non-linear PDE’s for the space-time metric which are, in general, very difficult to solve
We illustrate the power of this approach by showing that all type A space-time metrics [7, 8], which includes the Schwarzschild solution, can be obtained from a single class of diagonal matrices, and by presenting new solutions to the gravitational field equations that we believe would be difficult to obtain through other methods
We show in this paper that by taking advantage of the possible choices of factorization contours Γ and appropriate changes of coordinates, each monodromy matrix gives rise not to one solution, but to a whole class of exact solutions

Summary

Introduction

The field equations of gravitational theories in D space-time dimensions are a system of non-linear PDE’s for the space-time metric which are, in general, very difficult to solve. We show in this paper that by taking advantage of the possible choices of factorization contours Γ and appropriate changes of coordinates, each monodromy matrix gives rise not to one solution, but to a whole class of exact solutions We illustrate this surprising result by showing that from the canonical Wiener-Hopf factorization of a rational diagonal matrix of a very simple kind that is factorizable, one obtains a wide class of metrics that includes all type A space-time metrics, a cosmological Kasner solution and the Rindler metric, as well as solutions whose metric tensor is continuous but not smooth due to the presence of null hypersurfaces [7, 8, 12, 13]. In appendix C we summarize the class of A-metrics, which here is obtained from the canonical factorization of the monodromy matrix (2.19)

Summary of the main results

Preliminary results

Affine transformations

The Breitenlohner-Maison linear system

Monodromy matrix

Canonical factorization gives a solution to the BM linear system

Meromorphic factorizations: a case study

The Schwarzschild monodromy matrix

Region I

Extending solutions: the interior region of the Schwarzschild solution

Regions A and B

10 Solutions with two Killing horizons