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Bouncing geodesics, black hole singularities, and singularities of thermal correlators

Sašo Grozdanov, Samuel Valach, Mile Vrbica

Abstract

Bouncing geodesics have been used as valuable probes of black hole singularities. In the dual boundary theory, the presence of bouncing geodesics is encoded in the analytic structure of correlation functions. Thus, when their existence is related to the presence of a black hole singularity, this presents a practical holographic framework to analyse, diagnose, and classify spacetimes with curvature singularities. To make this intuition precise, we use the Hadamard theory of hyperbolic differential equations to prove that both bulk and boundary retarded propagators diverge whenever two points can be connected by a null geodesic. We clarify why this statement remains valid beyond the geodesic regime (for operators of any dimension) and examine how holographic renormalisation modifies the structure of the dual propagator. We also present a general characterisation of bouncing geodesics and the associated singularities in correlation functions for arbitrary spacetimes. Furthermore, we compare the analytic structure of the correlators in position and momentum space and discuss explicit examples. Finally, we demonstrate the validity and concrete limitations of the bouncing geodesic approach to the study of black hole singularities. In particular, we show an explicit example of a black hole in the self-dual linear axion model, which has a curvature singularity despite the absence of bouncing geodesics.

Bouncing geodesics, black hole singularities, and singularities of thermal correlators

Abstract

Bouncing geodesics have been used as valuable probes of black hole singularities. In the dual boundary theory, the presence of bouncing geodesics is encoded in the analytic structure of correlation functions. Thus, when their existence is related to the presence of a black hole singularity, this presents a practical holographic framework to analyse, diagnose, and classify spacetimes with curvature singularities. To make this intuition precise, we use the Hadamard theory of hyperbolic differential equations to prove that both bulk and boundary retarded propagators diverge whenever two points can be connected by a null geodesic. We clarify why this statement remains valid beyond the geodesic regime (for operators of any dimension) and examine how holographic renormalisation modifies the structure of the dual propagator. We also present a general characterisation of bouncing geodesics and the associated singularities in correlation functions for arbitrary spacetimes. Furthermore, we compare the analytic structure of the correlators in position and momentum space and discuss explicit examples. Finally, we demonstrate the validity and concrete limitations of the bouncing geodesic approach to the study of black hole singularities. In particular, we show an explicit example of a black hole in the self-dual linear axion model, which has a curvature singularity despite the absence of bouncing geodesics.
Paper Structure (30 sections, 2 theorems, 98 equations, 5 figures)

This paper contains 30 sections, 2 theorems, 98 equations, 5 figures.

Table of Contents

  1. Introduction and summary
  2. The Hadamard theorems
  3. Setup
  4. Singularities
  5. Examples
  6. Lightcone singularity:
  7. Bulk-cone singularities:
  8. Bouncing singularities:
  9. Bouncing geodesics: the position space perspective
  10. Definitions and existence
  11. Claim:
  12. Proof:
  13. Examples of absence
  14. The BTZ black hole:
  15. The self-dual linear axion model:
  16. ...and 15 more sections

Key Result

Theorem 1

For all $Y$ in the normal neighbourhood of $X$,A normal neighbourhood of $X$ is a region where every point $X'$ can be connected to $X$ by a unique geodesic that lies entirely within that region. the retarded Green's function $\mathcal{G}(X,Y)$ on any smooth $D$-dimensional spacetime takes the form where $W(X,Y)$, $U(X,Y)$ and $Q(X,Y)$ are smooth functions, $\delta$ is the Dirac distribution and

Figures (5)

  • Figure 1: Examples of null-geodesics (red) leading to singularities in the boundary retarded propagator. From left to right these are: geodesic propagating at the photon sphere of 1-sided AdS-Schwarzschild, geodesic bouncing off the space-like singularity in 2-sided AdS-black brane, geodesic bouncing off the time-like singularity in AdS-Reissner–Nordström black brane.
  • Figure 2: Blackening factors $f(r)$ near $r=0$ for AdS-Schwarzschild (left), AdS-Reissner-Nordström black hole (middle) and BTZ black hole (right). For the first two cases there exists a real geodesic bouncing off the space-like (resp. time-like) curvature singularity.
  • Figure 3: Penrose diagram for the $D=3$ BTZ and the $D=4$ neutral self-dual axion black holes.
  • Figure 4: The singularities in the complex $t$ plane at fixed spatial momentum $\mathbf k$ for the two-sided correlator (left) and the retarded correlator (right). The former is characterised by a strip of analyticity for $-\frac{\beta}{2}<\Im t<\frac{\beta}{2}$.
  • Figure 5: The singularities in the complex $t$ plane in both the BTZ and self-dual linear axion cases at fixed $\mathbf{k}$ (red dots) and at $x=0$ (green circles) for the two-sided correlator (left) and the retarded correlator (right). Only a subset of the singularities survive the Fourier integration over $\mathbf{k}$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2