Caustic echoes from a Schwarzschild black hole

Abstract

We present the first numerical approximation of the scalar Schwarzschild Green function in the time-domain, which reveals several universal features of wave propagation in black hole spacetimes. We demonstrate the trapping of energy near the photon sphere and confirm its exponential decay. The trapped wavefront passes through caustics resulting in echoes that propagate to infinity. The arrival times and the decay rate of these caustic echoes are consistent with propagation along null geodesics and the large limit of quasinormal modes. We show that the four-fold singularity structure of the retarded Green function is due to the well-known action of a Hilbert transform on the trapped wavefront at caustics. A two-fold cycle is obtained for degenerate source-observer configurations along the caustic line, where the energy amplification increases with an inverse power of the scale of the source. Finally, we discuss the tail piece of the solution due to propagation within the light cone, up to and including null infinity, and argue that, even with ideal instruments, only a finite number of echoes can be observed. Putting these pieces together, we provide a heuristic expression that approximates the Green function with a few free parameters. Accurate calculations and approximations of the Green function are the most general way of solving for wave propagation in curved spacetimes and should be useful in a variety of studies such as the computation of the self-force on a particle.

Scalar Green function of the Kerr spacetime

“…[8] in the Schwarzschild case and more generally by Ref. [11], the g o -related terms are turned on not only after T > 0, but also after the previous g e pulse. We notice that this feature is not captured by our method.…”

“…The Green function is expected to be singular on the points connected by null geodesics, and we confirm this for our approximate Green function using numerical investigations. We also recover the fourfold singular structure of the Green function, as seen in other spacetimes [9,13,14], including Schwarzschild [8,11]. This study makes progress towards solving the problem of the evolution of EMRI systems in the Kerr spacetime using the self-force approximation.…”

“…An immediate next step of this work would be to compare our analytical formula with numerical Green functions such as those presented in Ref. [11].…”

“…This fourfold singular structure matches the earlier expectation by Casals et al [9,10], which was proved by using the Hadamard ansatz for the direct part of the Green function. On the other hand, Zenginoğlu and Galley [11] used numerical methods to obtain the time-domain scalar Green function in the Schwarzschild background. They also observed the fourfold singular structure as a result of caustic echoes.…”

High-Frequency Gravitational Waves From Supermassive Black Holes: Prospects for Ligo-Virgo Detections

“…High frequency GWs are scattered by the gravitational field of the SMBH (e.g. Futterman et al 1988) producing GW echos (Dolan & Ottewill 2011;Zenginoglu & Galley 2012). The GW frequency is not effected to leading order in the WKB limit if the frequency of the incoming wave exceeds M −1 .…”