No-hair theorem for Black Holes in Astrophysical Environments

TL;DR

The paper frames the problem of astrophysical distortions from companions or disks challenging the isolated no-hair paradigm. It employs the Weyl form for static, axially symmetric spacetimes and covariant source integrals to separate the black-hole contribution from external sources and to define induced multipole moments. The key result is analytic: $U_{ ext{ind}}^{(r)}=0$, so the black hole's second Love numbers vanish ($k_r=0$) in full GR, while horizon multipoles can differ from Schwarzschild due to external distortion; total asymptotic multipole moments need not match Schwarzschild because external sources contribute. This extends the no-hair theorem to realistic, distorted black holes and has implications for gravitational-wave modeling and tests of GR in astrophysical settings.

Abstract

According to the no-hair theorem, static black holes are described by a Schwarzschild spacetime provided there are no other sources of the gravitational field. This requirement, however, is in astrophysical realistic scenarios often violated, e.g., if the black hole is part of a binary system or if it is surrounded by an accretion disk. In these cases, the black hole is distorted due to tidal forces. Nonetheless, the subsequent formulation of the no-hair theorem holds: The contribution of the distorted black hole to the multipole moments that describe the gravitational field close to infinity and, thus, all sources is that of a Schwarzschild black hole. It still has no hair. This implies that there is no multipole moment induced in the black hole and that its second Love numbers, which measure some aspects of the distortion, vanish as was already shown in approximations to general relativity. But here we prove this property for astrophysical relevant black holes in full general relativity.