Gravitational radiation from a photon on the light ring: multipole fluxes and the logarithmic divergence
Abstract
A massless particle on the circular null geodesic of a Schwarzschild or Kerr black hole radiates gravitational waves at all multipoles ℓ with a power that falls off only as 1/ℓ, so that the total instantaneous power is logarithmically divergent. We derive this result analytically from the Zerilli--Regge--Wheeler and Teukolsky equations with a null stress-energy tensor as source, and find that both the power radiated to infinity and the power absorbed by the horizon approach Ėℓ→ g(a) κE2/(M2ℓ), where E is the energy of the particle, M the mass of the black hole and a its spin, with κ=0.063653 and a spin factor g(a), for which we obtain, for any spin, the closed form g=27M2(r0−M)/[r02(r0+3M)] in terms of the radius r0 of the Kerr light ring. The 1/ℓ behavior has two causes: the near-resonance of the photon with the quasinormal modes, which prevents any exponential suppression at large ℓ, and a cancellation in the source that is specific to massless particles, which reduces the power per multipole by a factor ℓ2 with respect to a generic source. At leading order half of the radiation falls into the black hole, and we compute the first correction to this equal split in the non-spinning case. The same analysis describes massive particles on orbits close to the light ring, where the mass turns the 1/ℓ law into an exponential cut-off. We recover the dominant, even-parity term of the geodesic synchrotron radiation formulae for massive particles of Breuer, Ruffini, Tiomno and Vishveshwara (1973), but we find that their odd-parity term is too small by a factor of four. All results are checked against numerical solutions, up to ℓ=12800 in Schwarzschild and ℓ=800 in Kerr.
Related
- Linearised gravitational radiation in de Sitter spacetime at all multipole orders
- Asymptotic Gravitational Radiation of Photon Rockets with Λ: Point, String, and Sheet Sources
- Multipole blackbody radiation shift in Rydberg atoms
- Few-gluon interactions and multipole radiation in high energy nuclear collisions
- A Multipole Technique for the Simulation and Analysis of Gravitational Waveforms from Extreme Mass Ratio Inspirals
BibTeX
@misc{barausse2026gravitational,
title = {{Gravitational radiation from a photon on the light ring: multipole fluxes and the logarithmic divergence}},
author = {Enrico Barausse},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12082},
eprint = {2610.12082},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12082},
}
RIS
TY - UNPB
TI - Gravitational radiation from a photon on the light ring: multipole fluxes and the logarithmic divergence
AU - Enrico Barausse
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12082
UR - https://arxiv.org/abs/2610.12082
AB - A massless particle on the circular null geodesic of a Schwarzschild or Kerr black hole radiates gravitational waves at all multipoles ℓ with a power that falls off only as 1/ℓ, so that the total instantaneous power is logarithmically divergent. We derive this result analytically from the Zerilli--Regge--Wheeler and Teukolsky equations with a null stress-energy tensor as source, and find that both the power radiated to infinity and the power absorbed by the horizon approach Ė_ℓ→ g(a) κE^2/(M^2ℓ), where E is the energy of the particle, M the mass of the black hole and a its spin, with κ=0.063653 and a spin factor g(a), for which we obtain, for any spin, the closed form g=27M^2(r_0−M)/[r_0^2(r_0+3M)] in terms of the radius r_0 of the Kerr light ring. The 1/ℓ behavior has two causes: the near-resonance of the photon with the quasinormal modes, which prevents any exponential suppression at large ℓ, and a cancellation in the source that is specific to massless particles, which reduces the power per multipole by a factor ℓ^2 with respect to a generic source. At leading order half of the radiation falls into the black hole, and we compute the first correction to this equal split in the non-spinning case. The same analysis describes massive particles on orbits close to the light ring, where the mass turns the 1/ℓ law into an exponential cut-off. We recover the dominant, even-parity term of the geodesic synchrotron radiation formulae for massive particles of Breuer, Ruffini, Tiomno and Vishveshwara (1973), but we find that their odd-parity term is too small by a factor of four. All results are checked against numerical solutions, up to ℓ=12800 in Schwarzschild and ℓ=800 in Kerr.
ER -