Divergences in gravitational-wave emission and absorption from extreme mass ratio binaries.
A perturbative expansion is a powerful technique for calculating gravitational radiation from binary systems in which the "small" body is treated as a point particle of mass mp moving in the gravitational field generated by the large mass M, and only linear terms in the small mass ratio mp/M are kept.
Even when extended to approximately similar mass ratios, this approach typically produces finite answers that are frequently in good agreement with completely nonlinear numerical relativity results. The instantaneous flux emitted by a tiny body as it circles the light ring of a black hole, and the total energy absorbed by the horizon when a small body plunges into a black hole are two scenarios in which the point-particle approximation produces a divergent conclusion.
We can prove that both of these variables diverge by integrating the Teukolsky (or Zerilli/Regge-Wheeler) equations in the frequency and time domains. We discover that these divergences are a result of the point-particle idealization, and we can explain and regularise this behavior by giving the point particle a fixed size. The Event Horizon Telescope, for example, does not use these divergences while photographing black holes.