Bypassing Bekenstein’s no-scalar-hair theorem without violating the energy conditions
Panagiotis Dorlis, Nick E. Mavromatos, Sotirios-Neilos Vlachos
Abstract
In this work we establish in a rigorous manner, and a model-independent way, the conditions for bypassing Bekenstein's no-scalar-hair theorem for static, spherically symmetric, and asymptotically flat black holes, while maintaining the validity of the energy conditions. Specifically, we argue that an assumption in the theorem, namely the vanishing of the quantity $\mathcal{G}=\mathcal{E}+{{T}^{\ensuremath{\theta}}}_{\ensuremath{\theta}}$, where $\mathcal{E}$ is the energy density and ${{T}^{\ensuremath{\theta}}}_{\ensuremath{\theta}}$ the corresponding component of the energy-momentum tensor of the scalar field theory, can be relaxed. Indeed, if $\mathcal{G}$ is positive, as a consequence of the assumption on the validity of the energy conditions, then scalar hair is potentially allowed in the black hole's exterior, consistently with the gravitational equations and the generic properties of the (nontrivial) energy-momentum tensor. As an explicit example, in which such a behavior is realized, we discuss the well-known model of a ($3+1$)-dimensional Schwarzschild black hole coupled to a spontaneously broken Yang-Mills $SU(2)$ gauge theory interacting with a Higgs scalar. We present a rather novel approach to obtain analytical black-hole solutions for this system (in contrast to the numerical ones in the existing literature) by applying an appropriate perturbative treatment whereby the black-hole configuration is derived as a result of backreaction of Higgs and gauge fields onto an initially fixed flat spacetime. The massive nature of the scalar and gauge fields in this example requires a special treatment because of their asymptotic form, which we discuss in some detail.