Probing the Singularity of Scalar-Haired Black Holes with Holographic Complexity

Technical Summary: Probing the Singularity of Scalar-Haired Black Holes with Holographic Complexity

Problem Statement
Recent developments in the AdS/CFT correspondence have introduced "complexity=anything" (CAny) observables, a broad class of bulk quantities that generalize the "complexity=volume" (CV) and "complexity=action" (CA) proposals. While CV and CA observables are known to probe the interior of black holes, their reach is limited: in uncharged black holes, CV surfaces remain at a finite distance from the singularity, while in charged black holes, observables are often blocked by inner horizons. Previous work [10] demonstrated that certain generalized CAny observables in uncharged, hairless black holes can probe arbitrarily close to the singularity. However, it remains unclear how the introduction of scalar hair—which modifies the near-singularity geometry from a vacuum Kasner form to a continuous family of Kasner spacetimes characterized by varying exponents—affects the viability and probing depth of these observables. This work investigates whether specific CAny functionals remain valid holographic duals of complexity in the presence of scalar hair and determines how deeply they can access the Kasner regime near the singularity.

Methodology
The authors analyze two distinct models of scalar-haired black holes in $d+1$ dimensions:

1. Chamblin–Reall Solution: An analytic solution with an exponential scalar potential $V(\phi) \propto e^{\alpha\phi}$. While not asymptotically AdS, it admits a holographic dictionary via dimensional reduction. The parameter $\alpha$ controls the scalar hair and continuously shifts the Kasner exponents near the singularity.
2. Massive Scalar Solution: A numerical solution with a simple mass term potential $V(\phi) = \frac{1}{2}m^2\phi^2 + \Lambda$. These solutions are asymptotically AdS, but the backreaction of the scalar field requires numerical integration. The scalar field value at the horizon ($\phi_h$) is tuned to modify the Kasner exponents.

The study focuses on two specific codimension-one CAny observables, defined by functionals $F_\zeta[\Sigma]$ evaluated on extremal surfaces $\Sigma$:

• The $C_2$-observable: An embedding-independent functional where the integrand includes the square of the Weyl tensor ($C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$), scaled by a coupling $\lambda_{C_2}$.
• The $K$-observable: An embedding-dependent functional where the integrand includes the trace of the extrinsic curvature ($K$), scaled by a coupling $\lambda_K$.

The authors derive the equations of motion for extremal surfaces in these backgrounds, analyzing the effective potentials governing the radial evolution. They investigate the existence of late-time extremal surfaces (those anchored at large boundary times $\tau$) by checking for local maxima in the effective potential behind the horizon. They further calculate the proper time $\tau_\infty$ from the turning point of these surfaces to the singularity and the late-time linear growth rate of the observables, $\pi_\infty$.

Key Contributions and Results

1. Existence of Late-Time Surfaces:

   • $C_2$-observable: In hairless black holes, this observable admits late-time extremal surfaces only within a narrow, finite window of the coupling constant $\lambda_{C_2}$.
     • In the Chamblin–Reall background, increasing the scalar hair parameter $\alpha$ widens this allowed window.
     • In the massive scalar background, increasing the scalar hair (decreasing the normalized energy density $\langle T_{tt} \rangle/T^3$) shrinks the allowed window. For sufficiently strong hair, the window closes entirely, leaving only the pure volume functional ($\lambda_{C_2}=0$) as a viable candidate.
   • $K$-observable: This observable admits late-time extremal surfaces for all values of the coupling $\lambda_K$ in both backgrounds. The presence of scalar hair does not restrict the existence of these surfaces.

2. Probing Depth (Singularity Access):

   • $C_2$-observable: Due to the bounded nature of the allowed coupling range, the turning points of the extremal surfaces are bounded away from the singularity. Consequently, the $C_2$-observable cannot probe the singularity in either scalar-haired background.
   • $K$-observable: The surfaces can approach the singularity arbitrarily closely. In both backgrounds, increasing the strength of the scalar hair (increasing $\alpha$ or decreasing $\langle T_{tt} \rangle/T^3$) pushes the turning points of the extremal surfaces closer to the singularity, particularly for negative values of the coupling $\lambda_K$.

3. Late-Time Growth Rates and Symmetry Breaking:

   • In hairless black holes, the late-time growth rate of the $K$-observable is symmetric under the sign reversal of $\lambda_K$.
   • In scalar-haired backgrounds, this symmetry is broken. Negative couplings (which correspond to surfaces probing deeper into the interior) yield larger growth rates than positive couplings.
   • The magnitude of this asymmetry correlates with the deviation of the Kasner exponents from their vacuum values. The effect is most pronounced in the Chamblin–Reall background, where the Kasner exponents vary significantly with $\alpha$, compared to the massive scalar background where the variation is more constrained.

4. Boundary Contributions:
   The authors explicitly compute boundary contributions for the Chamblin–Reall geometry, noting that they differ from standard asymptotically AdS cases due to the non-trivial asymptotic behavior of the metric functions. They show that for the $K$-observable, the positive branch of the potential may lead to divergent anchoring times if the scalar hair parameter $\alpha$ exceeds a critical value, rendering the negative branch the physically relevant one for late-time analysis.

Significance and Claims
The paper claims that the viability of holographic complexity observables as probes of the black hole interior is highly sensitive to the specific functional form and the presence of scalar hair.

• The $C_2$-observable is shown to be a fragile probe; its ability to exhibit linear late-time growth is easily suppressed by scalar hair in certain models, and it fundamentally fails to reach the singularity.
• The $K$-observable emerges as a robust and tunable probe. It maintains linear growth across all coupling values and, crucially, allows the extremal surfaces to penetrate arbitrarily close to the singularity. The authors suggest that the $K$-functional provides a sensitive mechanism to detect the structure of the near-singularity Kasner regime, as evidenced by the breaking of coupling-sign symmetry and the correlation between growth rates and Kasner exponents.

The work concludes that while the "complexity=anything" framework offers a rich landscape of observables, not all are equally suitable for probing the deep interior of scalar-haired black holes. The $K$-observable stands out as a viable dual for complexity that can access the universal Kasner geometry near the singularity, whereas the $C_2$-observable is limited to the outer interior regions.