Analytical studies on 3D hairy rotating black hole interiors

TL;DR

The paper addresses the interior dynamics of three-dimensional hairy rotating black holes with a super-exponential potential, revealing an infinite cascade of Kasner epochs connected by Kasner inversions and transitions. It develops analytic descriptions for both the Kasner inversion and the Kasner transitions, including explicit forms for the velocity $v(\rho)$ and associated plateaus, and derives three recurrence relations that govern the late-time interior evolution, treated in the continuous $n$-limit. A key finding is that after an inversion the interior approaches a Kasner regime with exponents $p_t\to 1$ and $p_x\to 0$, effectively a Milne$_{1+1}\times S^1$ geometry, but curvature invariants still diverge due to the non-commuting limits with $v\to\infty$, so the singularity remains curvature-type. The results indicate a richer, non-chaotic interior structure in 3D compared to 4D static cases and hold potential implications for holographic descriptions of black hole interiors and quantum gravity phenomenology.

Abstract

We present an analytical study of the interior structure of hairy rotating black holes in three-dimensional Einstein gravity, minimally coupled to a complex scalar field with a sup-exponential potential. The interior dynamics of these black holes are characterized by an infinite sequence of Kasner epochs, separated by inversion and transitions, each of which admits an analytical description. We derive an explicit analytical expression for this infinite sequence of epochs. At late interior times, the geometry evolves into a curvature singularity, despite the local resemblance of each Kasner epoch to a regular Milne universe on a circle. These results reveal an interior structure richer and more complex than that of its 4D static black hole counterparts.

Figures (9)

• Figure 1: Evolution of $v$ as a function of $\rho$. A specific set of initial values is chosen at the horizon. We name the type of left transitions as 'decreasing' transitions since $|v|$ between two neighboring epochs decreases constantly, and the type of right ones as 'increasing' transitions. The inversion separates the two types of transitions. The right plot shows a zoomed-in view of the Kasner inversion region from the left plot.
• Figure 2: Evolution of $v$ for another different choice of initial values at the horizon. There is no inversion, and only 'increasing' transitions exist in the interior.
• Figure 3: Evolution of $N$ with ( left) and without ( right) Kasner inversion. In the left, we have $N_h=-0.506$, $N_K^i=-1.976$, and $N_K^f=-0.506$; while in the right, $N_h=-0.220$ and $N_K^f=-4.544$.
• Figure 4: This figure shows that $H$ is almost a constant $H_n$ within a selection of an arbitrary $n$-th Kasner transition.
• Figure 5: Evolution of $v$ as a function of $\rho$ within two types of Kasner transition. A bounce refers to the curve between two plateaus. The left figure shows a decreasing transition from $v_n=-0.777$ to $v_{n+1}=0.762$, while the right shows a increasing one from $v_n=2.719$ to $v_{n+1}=-2.763$. Solid blue lines represent the analytical results of \ref{['expr-v-T']} and \ref{['expr-T']}. We have $\rho_n=82.656\,,h_n=-0.0106 \,,c_T=-1.99984\,$ for the left figure and $\rho_n=122.458\,,h_n=0.008 \,,c_T=-2.00035\,$ for the right. The red points are the numerical results.