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Goursat's Problem and the Holographic Principle

Enrique Alvarez, Jorge Conde, Lorenzo Hernandez

TL;DR

This work investigates holography for Ricci-flat spacetimes with vanishing cosmological constant by recasting bulk-boundary relations as Goursat (characteristic) problems on a null finite boundary.It develops boundary-to-bulk propagators using Riesz potentials for massless fields and generalized Bessel kernels for massive fields within Milne-type setups, and analyzes their behavior under nontrivial gravitational perturbations.Explicit 3D and 4D examples are derived, including boundary actions, conformal scaling properties, and perturbative corrections, together with a discussion of Brown-York quasilocal energy and its connection to conformal anomalies.The results point to a holographic mapping tied to the finite boundary's conformal structure, with potential connections to conformal field theories and matrix-model frameworks in the zero-$\Lambda$ regime.

Abstract

The whole idea of holography as put forward by Gerard 't Hooft assumes that data on a boundary determine physics in the volume. This corresponds to a Dirichlet problem for euclidean signature, or to a Goursat (characteristic) problem in the lorentzian setting. Is this last aspect of the problem that is explored here for Ricci flat spaces with vanishing cosmological constant.

Goursat's Problem and the Holographic Principle

TL;DR

This work investigates holography for Ricci-flat spacetimes with vanishing cosmological constant by recasting bulk-boundary relations as Goursat (characteristic) problems on a null finite boundary.It develops boundary-to-bulk propagators using Riesz potentials for massless fields and generalized Bessel kernels for massive fields within Milne-type setups, and analyzes their behavior under nontrivial gravitational perturbations.Explicit 3D and 4D examples are derived, including boundary actions, conformal scaling properties, and perturbative corrections, together with a discussion of Brown-York quasilocal energy and its connection to conformal anomalies.The results point to a holographic mapping tied to the finite boundary's conformal structure, with potential connections to conformal field theories and matrix-model frameworks in the zero-$\Lambda$ regime.

Abstract

The whole idea of holography as put forward by Gerard 't Hooft assumes that data on a boundary determine physics in the volume. This corresponds to a Dirichlet problem for euclidean signature, or to a Goursat (characteristic) problem in the lorentzian setting. Is this last aspect of the problem that is explored here for Ricci flat spaces with vanishing cosmological constant.

Paper Structure

This paper contains 24 sections, 204 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The simplest example: the Milne Universe
  3. Riesz' potential in the massless case.
  4. Generalities on the characteristic problem
  5. The Generalized Potential of Order $a$.
  6. Stokes'theorem at work
  7. The concept of characteristic propagator
  8. Three and four-dimensional examples
  9. The Characteristic boundary-bulk propagator for Masssive fields.
  10. Scalar Fields in Milne space
  11. Point Splitting
  12. Boundary-boundary Propagators for massless scalar fields
  13. The regularized boundary and the infinite curvature limit
  14. The effect of nontrivial gravitational fields
  15. Canonical coordinates
  16. ...and 9 more sections

Figures (4)

  • Figure 1:
  • Figure 2: Horospheric coordinates only cover half of the whole anti de Sitter space ( namely the portion which is not filled).
  • Figure 3: The fundamental diamond with angular coordinates.
  • Figure :