Black hole interiors of homogeneous holographic solids under shear strain

TL;DR

This work investigates AdS black hole interiors under finite shear strain in holographic axion models with broken translations. It proves that shear anisotropy universally removes the inner Cauchy horizon, driving the interior to a spacelike singularity, and reveals a rich interior dynamics sequence: ER bridge collapse for small shear, emergence of a Lifshitz-domain-wall geometry at large shear, and subsequent Kasner-like evolution whose specifics depend on the axion potential. Depending on the potential, the interior can settle into a stable Kasner universe or undergo endless Kasner transitions toward the singularity, with exponential potentials leading to perpetual alternations. The results illuminate how boundary elasticity and translational symmetry breaking imprint intricate spacetime dynamics inside holographic black holes and point to further holographic probes of interior phenomena.

Abstract

We investigate the interior of AdS black holes under finite shear strain in a class of holographic axion models, which are widely used to describe strongly-coupled systems with broken translations. We demonstrate that the shear anisotropy necessarily eliminates the inner Cauchy horizon, such that the deformed black hole approaches a spacelike singularity. The anisotropic effect induced by the axion fields triggers a collapse of the Einstein-Rosen bridge at the would-be Cauchy horizon, accompanied by a rapid change in the anisotropy of the spatial geometry. In addition, for a power-law axion potential, sufficiently large shear deformations give rise to a domain wall solution that includes a Lifshitz like scaling geometry near the boundary as well as a near horizon Kasner epoch with the Kasner exponents determined by the powers of the potential. Finally, we find that the interior dynamics of black holes generally enter an era described by an anisotropic Kasner universe at later interior time. Depending on the form of the potential, they either tend to stable Kasner universes, or exhibit an endless alternation of different Kasner epochs toward the singularity.

Figures (10)

• Figure 1: The nonlinear elastic stress-strain curve for the potential $V(X,Z)=X^2\sqrt{Z}$. For sufficiently large shear deformation $\epsilon$, the nonlinear stress-strain curve follows a scaling law of $\sigma\sim\epsilon^{3M/(M+2N)}$. Here we set $T/m=0.1$.
• Figure 2: The blackening factor $f(u)$ near the horizon and inside the black hole for different values of $\Omega$. The black dashed line represents the isotropic solution of $f(u)$ under zero shear strain $(\epsilon=0)$ . Here we set $V(X,Z)=X^2\sqrt{Z}$, $T/m=0.1$.
• Figure 3: (a): The behavior of the metric component $g_{tt}$ near the would-be inner horizon under different values of $\Omega$. The black dashed curves show semi-analytical results obtained from Eq. \ref{['eq43']}. (b): The behavior of the function $h'(u)$ with respect to the radial coordinate $u/u_h$ inside the black hole. According to Table. \ref{['tab1']}, the maximum value of $h'$ near the would-be inner horizon $u_i$ corresponds to the ratio $c_2/c_1$. We have chosen $V(X,Z)=X^2\sqrt{Z}$ at $T/m=0.1$.
• Figure 4: The ratio $c_2/c_1$ as a function of $\Omega$, extracted via $h'=c_2/c_1$ in Table \ref{['tab1']} near the would-be inner horizon. As $\Omega$ decreases, $c_2/c_1$ increases, resulting in a more dramatic crossover. Red dots and the black line represent the numerical data and the analytical approximation $c_2/c_1\simeq32.8755/\Omega$, respectively, for the potential $V=X^2\sqrt{Z}$ at $T/m=0.1$.
• Figure 5: Behaviors of $|f(u)|$, $\chi(u)$ and $h(u)$ for different values of $\Omega$. The black dashed line in (a) and the color dashed lines in (b) and (c) represent the analytical solution \ref{['eq48']}. We have fixed $V=X^2\sqrt{Z}$ and $T/m=0.1$.