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QNM orthogonality relations for AdS black holes

Paolo Arnaudo, Javier Carballo, Benjamin Withers

TL;DR

The paper develops orthogonality relations for quasinormal modes of asymptotically AdS black holes by introducing a CPT-modified Klein-Gordon inner product defined on a complex radial contour $\Gamma$ that avoids horizon branch points and links two copies of the dual QFT on a thermal Schwinger-Keldysh contour. By proving that the Hamiltonian satisfies $\mathcal{H}^\dagger = \mathcal{H}$ under this product, it yields orthogonality between eigenfunctions with distinct frequencies via $(\omega_j - \omega_i) \langle v_i, v_j \rangle = 0$, and shows that regular, normalisable eigenfunctions on $\Gamma$ are precisely the QNMs and anti-QNMs. The framework is developed for a broad class of AdS black holes, with explicit BTZ results and numerical verification for Schwarzschild-AdS$_4$, providing a real-time holographic interpretation through the Schwinger-Keldysh contour and a potential basis for late-time QNM expansions. This CPT-contour approach clarifies the spectral structure of dissipative perturbations in AdS backgrounds and suggests extensions to AdS$_2$ and asymptotically flat spacetimes, as well as connections to double-cone wormholes and holographic correlators.

Abstract

We present orthogonality relations for quasinormal modes of a wide class of asymptotically AdS black holes. The definition is obtained from a standard product, modified by a CPT operator and placed on a complex radial contour which avoids branch points of the modes. They are inspired by existing constructions for de Sitter and Kerr spacetimes. The CPT operator is needed to map right eigenfunctions of the Hamiltonian into left eigenfunctions. The radial contour connects two copies of the dual QFT on a thermal Schwinger-Keldysh contour, making contact with real-time holography and the double cone wormhole.

QNM orthogonality relations for AdS black holes

TL;DR

The paper develops orthogonality relations for quasinormal modes of asymptotically AdS black holes by introducing a CPT-modified Klein-Gordon inner product defined on a complex radial contour that avoids horizon branch points and links two copies of the dual QFT on a thermal Schwinger-Keldysh contour. By proving that the Hamiltonian satisfies under this product, it yields orthogonality between eigenfunctions with distinct frequencies via , and shows that regular, normalisable eigenfunctions on are precisely the QNMs and anti-QNMs. The framework is developed for a broad class of AdS black holes, with explicit BTZ results and numerical verification for Schwarzschild-AdS, providing a real-time holographic interpretation through the Schwinger-Keldysh contour and a potential basis for late-time QNM expansions. This CPT-contour approach clarifies the spectral structure of dissipative perturbations in AdS backgrounds and suggests extensions to AdS and asymptotically flat spacetimes, as well as connections to double-cone wormholes and holographic correlators.

Abstract

We present orthogonality relations for quasinormal modes of a wide class of asymptotically AdS black holes. The definition is obtained from a standard product, modified by a CPT operator and placed on a complex radial contour which avoids branch points of the modes. They are inspired by existing constructions for de Sitter and Kerr spacetimes. The CPT operator is needed to map right eigenfunctions of the Hamiltonian into left eigenfunctions. The radial contour connects two copies of the dual QFT on a thermal Schwinger-Keldysh contour, making contact with real-time holography and the double cone wormhole.
Paper Structure (7 sections, 52 equations, 1 figure, 1 table)

This paper contains 7 sections, 52 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Orthogonality for a general class of black holes
  3. Regular, normalisable eigenfunctions of $\mathcal{H}$ on $\Gamma$
  4. Explicit evaluations on QNMs
  5. Discussion
  6. Further details of the BTZ example
  7. $\mathcal{CPT}$ energy norms

Figures (1)

  • Figure 1: Left: The complex radial contour $\Gamma$ used in our QNM orthogonality product. The contour avoids branch point singularities of QNMs and anti-QNMs on the horizon. Right: The contour $\Gamma$ on the maximally extended spacetime, connecting two copies of the QFT through Euclidean time translation by half the thermal circle, $\beta/2$. Here the red line illustrates a singular past horizon on side 1, corresponding to the branch point singularities of QNMs on that side, avoided by the $i\epsilon$ insertions.