"Waveforms" at the Horizon

We study perturbations induced by a light particle scattering off a Schwarzschild black hole. Exploiting recent results for the wave propagation in this geometry, we derive the fields that this process induces on the horizon to leading order in the post-Minkowskian (PM) regime, when the light probe is far from the black hole. We then use these results to calculate the fluxes of energy and angular momentum that enter the black hole. We consider the effects due to gravitational, electromagnetic and scalar radiation, finding agreement with recent computations of the absorbed energy, while the absorbed angular momentum provides a new PM result.

💡 Research Summary

The paper investigates the perturbations generated when a light particle (mass µ, possibly carrying electric charge q_e and scalar charge q_p) scatters off a Schwarzschild black hole at large impact parameter. By combining two complementary analytical frameworks—post‑Minkowskian (PM) expansion, which is exact in velocities and masses but assumes weak coupling, and black‑hole perturbation theory, which treats the heavy black hole as a fixed background—the authors obtain a unified description of the fields that reach the horizon.

The authors start from the Teukolsky master equation, which governs scalar (s = 0), electromagnetic (s = ±1) and linearized gravitational (s = ±2) perturbations on a Schwarzschild background. After separating variables into spin‑weighted spherical harmonics Y^s_{ℓm}(θ,ϕ) and a Fourier frequency ω, the problem reduces to a radial ordinary differential equation (2.19). The homogeneous version of this equation is a confluent Heun equation with regular singularities at the horizon (r = 2M) and at the origin, and an irregular singularity at infinity.

Two natural bases of solutions are introduced: one that is purely ingoing at the horizon (R_in) and one that is purely outgoing at infinity (R_up). Their Wronskian is shown to be constant, allowing the construction of the full Green’s function as a product of the two bases. The source term is derived from the particle’s stress‑energy tensor (or current for the electromagnetic and scalar cases) and expanded in the same harmonic basis, yielding T_{ℓm}(r).

To make analytic progress, the authors perform a small‑x expansion, where x = 4iMω corresponds to the PM limit Mω → 0. In this regime the Heun functions can be systematically approximated by hypergeometric functions multiplied by polynomials P_0 and bP_0 whose coefficients are determined recursively. An auxiliary monodromy parameter a(u) appears, linking the problem to the Seiberg‑Witten curve of an N = 2 supersymmetric gauge theory; the coefficients u_i in the expansion of u(a) encode the instanton contributions. Crucially, each spherical harmonic ℓ is suppressed by a factor G^{ℓ+1}, so the lowest non‑trivial ℓ (ℓ = 2) already captures the leading‑order PM result for arbitrary velocities.

With the explicit radial solutions in hand, the authors compute the Noether currents associated with energy and angular momentum. By integrating the appropriate components of the stress‑energy tensor over the horizon, they obtain the absorbed power dE/dt and torque dJ/dt for each spin sector. The gravitational case reproduces the previously known PM result for absorbed energy (matching the EFT calculations of Refs.