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Carrollian correlators in black hole perturbation theory

Jiang Long, Zhan-Jia Qu, Hong-Yang Xiao

Abstract

In this note, we clarify the relationship between the two-point Carrollian correlator and massless scattering in black hole background. It turns out that there are two kinds of Carrollian correlators at the null boundaries of each asymptotically flat spacetime. The correlator from $\mathscr I^-$ to $\mathscr I^+$ should be regularized by subtracting the flat space analog, and it is the position space version of the reflection amplitude of massless scattering. On the other hand, the correlator from $\mathscr I^-$ to the future horizon $\mathcal H^+$ is absent in flat space, and it is the position space version of the transmission amplitude. The poles of the Carrollian correlators are governed by the null geodesics from $\mathscr I^-$ to $\mathscr I^+$ or $\mathcal H^+$, and they define two kinds of classical equations in Carrollian space. These equations establish the relationship between the Shapiro time delay and the deflection angle for light rays and should be understood as the dual descriptions of the quasinormal modes (QNMs) and the branch cut of the Green's function. We find that the time delay contains a logarithmic/quadratic behavior for the correlator from $\mathscr I^-$ to $\mathscr I^+/\mathcal H^+$ for small deflection angles. On the other hand, the time delay is always increasing linearly for both correlators when the deflection angle is large.

Carrollian correlators in black hole perturbation theory

Abstract

In this note, we clarify the relationship between the two-point Carrollian correlator and massless scattering in black hole background. It turns out that there are two kinds of Carrollian correlators at the null boundaries of each asymptotically flat spacetime. The correlator from to should be regularized by subtracting the flat space analog, and it is the position space version of the reflection amplitude of massless scattering. On the other hand, the correlator from to the future horizon is absent in flat space, and it is the position space version of the transmission amplitude. The poles of the Carrollian correlators are governed by the null geodesics from to or , and they define two kinds of classical equations in Carrollian space. These equations establish the relationship between the Shapiro time delay and the deflection angle for light rays and should be understood as the dual descriptions of the quasinormal modes (QNMs) and the branch cut of the Green's function. We find that the time delay contains a logarithmic/quadratic behavior for the correlator from to for small deflection angles. On the other hand, the time delay is always increasing linearly for both correlators when the deflection angle is large.
Paper Structure (14 sections, 167 equations, 7 figures)

This paper contains 14 sections, 167 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Carrollian correlator in black hole background
  3. Carrollian correlator in flat spacetime
  4. Carrollian correlator in Schwarzschild black hole background
  5. Approximation of the Carrollian correlator
  6. Late time behaviour.
  7. Eikonal approximation.
  8. Classical equations at null boundaries
  9. From $\mathscr I^-$ to $\mathscr I^+$
  10. From $\mathscr I^-$ to $\mathcal{H}^+$
  11. Conclusion and discussion
  12. Integration
  13. The equation from $\mathscr I^-$ to $\mathscr I^+$
  14. The equation from $\mathscr I^-$ to $\mathcal{H}^+$

Figures (7)

  • Figure 1: Penrose diagram of two-side Schwarzschild balck hole and Carrollian propagators in an asymptotically flat region.
  • Figure 2: An illustration of the boundary conditions for the in, out, up and down modes.
  • Figure 6: Shapiro delay. A light is deflected by the massive object with mass $M$. The distance between $A/B$ and $M$ is $r'/r$ and the turning point to $M$ is approximately equal to the impact parameter $b$.
  • Figure 7: Deflection of light in Schwarzschild spacetime.
  • Figure 8: The equation $u-v'=h(\gamma)$ that connects the deflection angle and the time delay for $b>b_c$. We have set $M=1$ in the diagram.
  • ...and 2 more figures