Novel ‘‘no-scalar-hair’’ theorem for black holes

TLDR

The scalar field is minimally coupled to gravity, has non‑negative energy density for all observers, and its Lagrangian need not be quadratic in derivatives. We formulate a new no‑hair theorem for general‑relativistic black holes that rules out any multicomponent scalar field dressing of asymptotically flat, static, spherically symmetric black holes. The proof relies on energy‑momentum conservation and the Einstein equations. The theorem excludes Higgs fields with double or multiple well potentials and also holds in scalar‑tensor gravity.

Abstract

We formulate a new ``no-hair'' theorem for black holes in general relativity which rules out a multicomponent scalar field dressing of any asymptotically flat, static, spherically symmetric black hole. The field is assumed to be minimally coupled to gravity, and to bear a non-negative energy density as seen by any observer, but its field Lagrangian need not be quadratic in the field derivatives. The proof centers on energy-momentum conservation and the Einstein equations. One kind of field ruled out is the Higgs field with a double (or multiple) well potential. The theorem is also proved for scalar-tensor gravity.