New no-scalar-hair theorem for black holes

Abstract

A new no-hair theorem is formulated which rules out a very large class of non-minimally coupled finite scalar dressing of an asymptotically flat, static, and spherically symmetric black-hole. The proof is very simple and based in a covariant method for generating solutions for non-minimally coupled scalar fields starting from the minimally coupled case. Such method generalizes the Bekenstein method for conformal coupling and other recent ones. We also discuss the role of the finiteness assumption for the scalar field.

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“…Interestingly, the corresponding proof for a charged, nonminimally coupled scalar field is valid for all ξ. Related theorems for the neutral case and all values of ξ can be found in two papers by Saa [27,28], which however make various restrictions on the value of the scalar field, whilst the paper of Bekenstein and Mayo has stronger results in that the restrictions on the value of the scalar field required to prove the theorems are themselves proved, using energy arguments. It is striking that energy arguments can be applied to this system, as the non-minimal coupling means that the weak energy condition (1), so crucial in the minimally coupled case, is no longer valid.…”

Phases of 4d Scalar–tensor Black Holes Coupled to Born–infeld Nonlinear Electrodynamics

“…According to the noscalar-hair conjecture, the black-hole solutions in the STT coincide with the solutions from GR. No-scalar-hair theorems treating the cases of static, spherically symmetric, asymptotically flat, electrically neutral black holes and charged black holes in the Maxwell electrodynamics have been proved for a large class of scalar-tensor theories [4,5,6]. The scalar field in these cases is constant, and thus trivial, if one demands that the essential singularity at the center of symmetry is hidden in a regular event horizon.…”

Scalar-tensor black holes coupled to Born-Infeld nonlinear electrodynamics

“…In the last two equations (2)g and (2) g are the determinants of the induced metrics on the horizon in the Jordan and in the Einstein frame, respectively. The term in the FL (26) connected with the magnetic charge is also preserved under the conformal transformations.…”

“…The natural question which arises then is whether the scalar field would lead to the existence of other preserved quantities which would allow a distant observer to distinguish between the Schwarzschild black hole and a black hole with a scalar dressing. Saa [2] was able to prove a no-scalar-hair theorem which rules out the existence of static, spherically symmetric, asymptotically flat, neutral black holes with regular, non-trivial scalar field for a large class of scalar tensor theories in which the scalar field is nonminimally coupled to gravity. He applied an explicit, covariant method to generate the exterior solutions for these theories through conformal transformations from the minimally coupled case.…”