Gravitational radiation from a photon on the light ring: multipole fluxes and the logarithmic divergence

A massless particle on the circular null geodesic of a Schwarzschild or Kerr black hole radiates gravitational waves at all multipoles $\mathrm{\ell }$ with a power that falls off only as $1\mathrm{/}\mathrm{\ell }$, 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 ${\dot{E}}_{\mathrm{\ell }}\to g(a)\kappa E^2\mathrm{/}(M^2\mathrm{\ell })$, where $E$ is the energy of the particle, $M$ the mass of the black hole and $a$ its spin, with $\kappa =0.063653$ and a spin factor $g(a)$, for which we obtain, for any spin, the closed form $g=27M^2(r_0-M)\mathrm{/}[r_0^2(r_0+3M)]$ in terms of the radius $r_0$ of the Kerr light ring. The $1\mathrm{/}\mathrm{\ell }$ behavior has two causes: the near-resonance of the photon with the quasinormal modes, which prevents any exponential suppression at large $\mathrm{\ell }$, and a cancellation in the source that is specific to massless particles, which reduces the power per multipole by a factor ${\mathrm{\ell }}^2$ 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 $1\mathrm{/}\mathrm{\ell }$ 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 $\mathrm{\ell }=12800$ in Schwarzschild and $\mathrm{\ell }=800$ in Kerr.