On the time dependence of holographic complexity for charged AdS black holes with scalar hair

Abstract

In the presence of a scalar hair perturbation, the Cauchy horizon of a Reissner-Nordström black hole disappears and is replaced by the rapid collapse of the Einstein-Rosen bridge, which leads to a Kasner singularity [1, 2]. We study the time-dependence of holographic complexity, both for the volume and for the action proposals, in a class of models with hairy black holes. Volume complexity can only probe a portion of the black hole interior that remains far away from the Kasner singularity. We provide numerical evidence that the Lloyd bound is satisfied by the volume complexity rate in all the parameter space that we explored. Action complexity can instead probe a portion of the spacetime closer to the singularity. In particular, the complexity rate diverges at the critical time tc for which the Wheeler-DeWitt patch touches the singularity. After the critical time the action complexity rate approaches a constant. We find that the Kasner exponent does not directly affect the details of the divergence of the complexity rate at t = tc and the late-time behaviour of the complexity. The Lloyd bound is violated by action complexity at finite time, because the complexity rate diverges at t = tc. We find that the Lloyd bound is satisfied by the asymptotic action complexity rate in all the parameter space that we investigated.

Probing inside a charged hairy black hole in massive gravity

“…See also refs. [16][17][18][19][20][21] for further theoretical developments in the no inner-horizon theorem. Moreover, it has been shown that the Null Energy Condition (NEC) necessarily removes the Cauchy horizon of black holes in the presence of the charged hairs [22][23][24].…”

Interior of helical black holes

“…From the holographic point of view, the interesting internal structure is expected to have corresponding descriptions in the dual field theory. Some observables have been explored to study the physics inside the horizon, including the correlation functions [7,14,[28][29][30], entanglement entropy [7,31], and complexity [32][33][34][35] etc. We have shown that in the interior there are oscillation regimes for both neutral and charged helical black holes, and the collapse of the Einstein-Rosen bridge for charged helical black holes.…”

Anisotropic flows into black holes

“…However, we may ask whether a similar story may be told for more complicated black holes, such as those with charge [51] or anisotropy. There has been some work on this; [52] considers such charged black holes with backreaction, admitting an RG flow interpretation. It would be interesting to more concretely interpret their results in the language of holographic RG flow, radial NECs, and a-functions.…”