{
  "id": "2404.03373",
  "version": "v1",
  "published": "2024-04-04T11:16:22.000Z",
  "updated": "2024-04-04T11:16:22.000Z",
  "title": "Riemann-Hilbert problems, Toeplitz operators and ergosurfaces",
  "authors": [
    "M. Cristina Câmara",
    "Gabriel Lopes Cardoso"
  ],
  "comment": "23 pages",
  "categories": [
    "math-ph",
    "gr-qc",
    "hep-th",
    "math.AP",
    "math.FA",
    "math.MP"
  ],
  "abstract": "The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix $M$ depending on the Weyl coordinates $\\rho$ and $v$, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the $(\\rho,v)$ plane on which some elements of $M(\\rho,v)$ tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-04T11:16:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "toeplitz operators",
      "riemann-hilbert problems",
      "ergosurfaces",
      "canonical wiener-hopf factorisation determines curves",
      "space-time dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}