The Interior of the Scalar Hairy Black Hole with Inverted Higgs Potential

TL;DR

This paper explores the interior structure of scalar hairy black holes with an inverted Higgs potential, revealing a singularity at the core and absence of a Cauchy horizon, with implications for cosmic censorship.

Contribution

It provides the first detailed numerical analysis of the interior of HBHs with inverted Higgs potentials, showing the divergence of fields and the violation of energy conditions inside.

Findings

Scalar field and metric diverge at the center

No Cauchy horizon is found inside these black holes

Weak energy condition is violated throughout the interior

Abstract

We investigate the interior structure of asymptotically flat hairy black holes (HBHs) arising in the Einstein-Klein-Gordon theory with nonpositive-definite scalar potentials, where nontrivial scalar hair exists at the event horizon. While exterior properties, including shadow imaging for HBHs supported by an inverted Higgs-like potential have been extensively investigated, their interior structure remains largely unexplored. In many gravitational theories, backreaction of classical fields can significantly eliminate the Cauchy horizon, which is known to be highly unstable due to the mass inflation effect, raising important questions regarding the validity of the Strong Cosmic Censorship conjecture. These considerations motivate us to examine the interior structure of HBHs by numerically integrating the field equations inward from the outer horizon. We find that the scalar field and the metric functions increase monotonically inside the horizon and diverge as $r \rightarrow 0$. The Ricci and Kretschmann scalars also diverge at $r=0$, confirming the presence of a genuine curvature singularity. No additional root of the metric function is observed, indicating the absence of a Cauchy horizon in the electrically neutral HBHs considered here. Furthermore, the weak energy condition is violated throughout the interior region, and the degree of violation becomes more pronounced as the scalar field at the horizon increases. These results provide new insight into the global structure of HBHs and their implications for cosmic censorship.