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.

• The paper explores the interior geometry of electrically neutral, hairy black holes with an inverted Higgs potential.
• Philo field and metric functions diverge at singularity thus showing mass inflation while confirming scalar hair.
• The absence of a Cauchy horizon is consistent with SCCC, highlighting the critical difference between spacelike and null-like singularities.

Overview

This paper investigates the interior structure of asymptotically flat, electrically neutral hairy black holes (HBHs) in the Einstein–Klein–Gordon (EKG) theory with an inverted Higgs-like potential $V(\phi) = -\Lambda \phi^4 + \mu \phi^2$ (2603.08067). This potential is unbounded from below and negative for $|\phi| > \sqrt{\mu/\Lambda}$, which violates the weak energy condition (WEC) and thereby evades the no-hair theorem, allowing nontrivial scalar hair at the event horizon (2603.08067). While the exterior properties of these solutions—including their shadows—had been studied previously, their interior geometry had not been systematically explored. The authors integrate the field equations inward from the event horizon to determine whether a Cauchy horizon forms inside, with direct implications for the Strong Cosmic Censorship Conjecture (SCCC).

Theoretical framework

The model is defined by the EKG action with a minimally coupled real scalar field. The static, spherically symmetric metric Ansatz uses $N(r) = 1 - 2m(r)/r$, where $m(r)$ is the Misner–Sharp mass function. The field equations reduce to three coupled nonlinear ODEs for $m(r)$, $\sigma(r)$, and $\phi(r)$. Regularity of $\sigma$ and $\phi$ at the horizon requires $1 - 8\pi G r_H^2 V(\phi_H) \neq 0$, and asymptotic flatness fixes $|\phi| > \sqrt{\mu/\Lambda}$0, $|\phi| > \sqrt{\mu/\Lambda}$1. After rescaling, only two governing parameters remain: $|\phi| > \sqrt{\mu/\Lambda}$2 and $|\phi| > \sqrt{\mu/\Lambda}$3. The exterior solutions bifurcate from Schwarzschild at $|\phi| > \sqrt{\mu/\Lambda}$4; as $|\phi| > \sqrt{\mu/\Lambda}$5 increases, the reduced horizon area $|\phi| > \sqrt{\mu/\Lambda}$6 decreases monotonically from unity toward zero while the reduced Hawking temperature $|\phi| > \sqrt{\mu/\Lambda}$7 increases monotonically from unity.

Interior structure

The interior solutions are obtained by integrating the same ODEs inward from $|\phi| > \sqrt{\mu/\Lambda}$8 using the identical boundary data. Three principal findings emerge:

• Monotonic divergence: The scalar field $|\phi| > \sqrt{\mu/\Lambda}$9 and both metric functions $N(r) = 1 - 2m(r)/r$0 and $N(r) = 1 - 2m(r)/r$1 increase monotonically inside the horizon and diverge as $N(r) = 1 - 2m(r)/r$2. The growth steepens with increasing $N(r) = 1 - 2m(r)/r$3. The divergence of $N(r) = 1 - 2m(r)/r$4 near $N(r) = 1 - 2m(r)/r$5 indicates mass inflation in the vicinity of the singularity.
• Curvature singularity at $N(r) = 1 - 2m(r)/r$6: Both the Kretschmann scalar $N(r) = 1 - 2m(r)/r$7 and the Ricci scalar $N(r) = 1 - 2m(r)/r$8 diverge as $N(r) = 1 - 2m(r)/r$9. Notably, $m(r)$0 is negative throughout the interior, consistent with its predominantly negative exterior behavior, whereas $m(r)$1 for Schwarzschild.
• Absence of a Cauchy horizon: No additional root of $m(r)$2 is found for $m(r)$3, so no Cauchy horizon exists in these electrically neutral HBHs. This supports the emerging picture that Cauchy horizon absence is generic for hairy black holes, consistent with results in Einstein–Yang–Mills theories, holographic superconductors, and other asymptotically flat HBHs.

A qualitatively distinct feature concerns the causal character of the singularity, probed via $m(r)$4. For moderate hair ($m(r)$5 with $m(r)$6), $m(r)$7 first decreases to a minimum and then increases toward zero as $m(r)$8, indicating a null-like singularity—an inversion of the Schwarzschild-like behavior. For larger hair ($m(r)$9), $m(r)$0 decreases monotonically and diverges, recovering a spacelike singularity. For small horizons ($m(r)$1), the singularity remains spacelike for all $m(r)$2 considered. The Penrose diagram constructed via Kruskal–Szekeres coordinates confirms that the global causal structure resembles maximally extended Schwarzschild: four regions (exterior, interior, parallel exterior, white hole region) with a spacelike (or occasionally null-like) singularity as the inevitable future boundary, and no inner horizon.

Energy condition violation

Since WEC violation enables the existence of these hairy solutions externally, the authors examine whether it persists inside. The scaled energy density $m(r)$3 is negative throughout the interior, decreases monotonically, and diverges as $m(r)$4, tracking the divergence of $m(r)$5. The severity of the violation grows with $m(r)$6 for both $m(r)$7 and $m(r)$8. This behavior contrasts with the four-stage interior dynamics (Einstein–Rosen bridge collapse, Josephson oscillations, Kasner epoch, Kasner inversion) observed in charged holographic superconductor black holes; here the interior evolution is uniformly monotonic.

Limitations and open questions

The paper concedes several restrictions on its conclusions. First, only electrically neutral solutions are treated; whether a Cauchy horizon can form in charged HBHs supported by asymmetric potentials remains open and is explicitly deferred to future work. Second, the analysis is restricted to static configurations, whereas realistic gravitational collapse produces dynamical interiors; prior double-null studies show that Cauchy horizon formation during collapse can differ substantially from static expectations. Third, quantum-gravitational questions—such as how scalar hair modifies the Wheeler–DeWitt equation on the Kantowski–Sachs interior and whether quantum effects soften the classical singularity—are outside the scope of this work, though the classical solutions presented here provide the necessary background for such analyses.

Conclusion

By integrating the field equations inward from the event horizon, this study establishes that electrically neutral HBHs supported by an inverted Higgs potential possess a genuine curvature singularity at $m(r)$9, exhibit mass inflation near the singularity, violate the WEC throughout the interior with severity increasing with horizon hair, and contain no Cauchy horizon. The identification of a parameter-dependent transition between spacelike and null-like singularities via the behavior of $\sigma(r)$0 is the most distinctive structural result. These findings reinforce the view that Cauchy horizon absence is a generic feature of hairy black holes and thereby support the SCCC within this class of solutions, while leaving the charged and dynamical cases as clearly posed open problems.