Obstructions towards a generalization of no-hair theorems: I. Scalar clouds around Kerr black holes

Understand

We show that the integral method used to prove the no-hair theorem for Black Holes (BH's) in spherically symmetric and static spacetimes within the framework of general relativity with matter composed by a complex-valued scalar-field does not lead to a straightforward conclusion about the absence of hair in the stationary and rotating (axisymmetric) scenario.

• We argue that such a failure can be used to justify in a simple and heuristic way the existence of non-trivial boson clouds or hair found numerically by Herdeiro and Radu [1,2] and analytically by Hod in the test field limit [3-5].
• This is due to the presence of a contribution that is negative when rotation exists which allows for an integral to vanish even when a non-trivial boson hair is present.
• The presence of such a negative contribution that depends on the rotation properties of the BH is perfectly correlated with the eigenvalue problem associated with the boson-field equation.