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

TL;DR

This work investigates holographic complexity in scalar-haired AdS black holes by analyzing two CAny observables, one based on $C^2$ and one on the mean extrinsic curvature $K$, across analytic Chamblin--Reall and numerically constructed massive-scalar backgrounds. It shows that scalar hair continuously deforms the near-singularity Kasner exponents and modulates how deeply the observables probe the Kasner regime, with the $C^2$-observable generally limited to a finite parameter window while the $K$-observable can probe arbitrarily close to the singularity and exhibits hair-induced asymmetry in the growth rate. These findings suggest that the $K$-functional provides a more versatile and sensitive probe of interior geometry and Kasner data in hairy black holes, offering a path to connect late-time holographic complexity with singularity structure. The results highlight the role of scalar hair in shaping interior probes and motivate further exploration with gauge fields, alternative complexity functionals, and subleading time dynamics to sharpen the link between holographic complexity and near-singularity geometry.

Abstract

It has been shown that the "complexity=anything" observables allow more possibilities to probe the geometry behind the horizon of AdS black holes compared to the volume complexity. For uncharged black holes, these observables access the geometry all the way to the vicinity of the singularity, while for charged black holes, they only probe up to the inner horizon. Under appropriate conditions, the near-singularity geometry takes the universal form of a Kasner spacetime, characterized by the Kasner exponents. By introducing scalar hair, it is possible to continuously vary the Kasner exponents away from their vacuum values. In this work, we study the behavior of two different observables to determine whether they remain viable holographic duals of complexity in the presence of scalar hair. We also investigate how deeply these observables can probe the Kasner regime near the singularity. To this end, we consider two scalar potentials: an exponential potential, which admits analytic solutions, and a pure mass term, which requires numerical analysis.

Figures (11)

• Figure 1: Chamblin--Reall solution for different values of $\alpha$ in $d=3$ spatial dimensions, with $\phi_\text{h}$ fixed according to Equation \ref{['eq:phi_h']}, showing $\phi(z)$ (left) and $\rho(z)$ (right). The vertical dashed line indicates the location of the horizon. Changing $\alpha$ deforms the Kasner exponents governing the near-singularity behavior, while Equation \ref{['eq:phi_h']} ensures that the asymptotic form of the spacetime remains unchanged.
• Figure 2: Left: Choosing $\phi_\text{h}$ as prescribed in Equation \ref{['eq:phi_h']} ensures that the asymptotic metric remains unchanged when $\alpha$ is modified. Right: The Kasner exponents, which characterize the near-singularity geometry and are independent of $\phi_\text{h}$, change monotonically with $\alpha$. Both panels correspond to $d=3$ spatial dimensions.
• Figure 3: Numerical solutions to Equations \ref{['eq:eom']} for a massive scalar field with $d = 3$ and $m^2 = -2$, showing $\phi(z)$ (left) and $\rho(z)$ (right). The solutions are obtained by integrating from the horizon toward both the boundary at $z \to 0$ and the singularity at $z \to \infty$ independently. Owing to regularity, the values of $\phi(z)$ and $\rho(z)$ near the boundary are fixed, eliminating the need for a shooting method. As shown explicitly in Equation \ref{['eq:asymptotic_boundary']}, both $\phi(z)$ and $\rho(z)$ diverge near the singularity. The vertical dashed line indicates the location of the horizon.
• Figure 4: Left: Plot of the normalized boundary energy density, $\langle T_{tt} \rangle / T^3$, as a function of the scalar field's value at the horizon $\phi_\text{h}$ in $d = 3$ dimensions for a pure mass potential with $m^2 = -2$. Since the relation is monotonic, $\phi_\text{h}$ can be traded for the more physical label $\langle T_{tt} \rangle / T^3$ to classify the numerical solutions. The horizontal dashed line indicates the analytic result $\langle T_{tt} \rangle / T^3 = 64\pi^3/27$ for $\phi_\text{h} = 0$. Right: Kasner exponents as a function of $\langle T_{tt} \rangle / T^3$. The dashed lines mark the reference values $p_X = 2/3$, $p_T = -1/3$, and $p_\phi = 0$, corresponding to the solution without a scalar field, the arrows along the lines indicate the direction of flow as $\phi_\text{h}$ is increased.
• Figure 5: Penrose diagram of an eternal AdS black hole. For the planar black hole, each point represents a spatial slice of topology $\mathbb{R}^{d-1}$. The timelike boundaries of the exterior regions correspond to the domains of two independent CFTs. The stationary surfaces $\Sigma$ connect the two boundaries and are conjectured to encode the complexity of the dual CFTs at the timeslice where they are anchored. For the Chamblin--Reall solution, the figure needs to be modified at the boundary as the spacetime does not asymptote to AdS.