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

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.

Figures (5)

• Figure 1: Two basic properties of HBHs as the function of $\phi_H$: reduced area of horizon $a_H$ (purple) and reduced Hawking temperature $t_H$ (green). Black dot represents the Schwarzschild black hole.
• Figure 2: Properties inside the HBHs $(r<r_H)$ with $r_H=1$ for several values of $\phi_H$: (a) $\phi(r)$; (b) $m(r)$; (c) $\sigma(r)$; (d) The scaled Kretschmann scalar $K/\mu^2$; (e) The scaled Ricci scalar $R/\mu$; (f) The metric component $-g_{tt}$.
• Figure 3: Properties inside the HBHs with $r_H=0.01$ for several values of $\phi_H$: The profiles of functions: (a) $\phi(r)$; (b) $dm(r)/dr$; (c) $d\sigma(r)/dr$; (d) The scaled Ricci and (e) Kretschmann scalars.
• Figure 4: The scaled weak energy condition $(8\pi G/\mu)\rho$ of HBHs with several $\phi_H$ for (a) $r_H=1.0$; (b) $r_H=0.01$.
• Figure 5: The Penrose diagram for HBH with $\phi_H=1$ for $r_H=1$, where its causal structure divided into four regions (I, II, III, IV), $i^{\pm}$ denotes the future $(+)$ and past $(-)$ timelike infinities, $i^0$ denotes the spatial infinity, $\mathcal{I}^\pm$ denotes the future $(+)$ and past $(-)$ null infinities, the red curvy line represents the spacelike singularity at $r=0$.