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Action growth of charged black holes with a single horizon

Rong-Gen Cai, Misao Sasaki, Shao-Jiang Wang

TL;DR

This work examines the complexity=action conjecture for holographic states dual to charged AdS black holes with a single horizon and a spacelike interior singularity. It analyzes charged dilaton and Born-Infeld black holes, including subcases with two horizons and with a single horizon, and also a phantom Maxwell field to probe methodological robustness. The main results show finite late-time action growth that respects the Lloyd bound for the dilaton and single-horizon Born-Infeld cases, while the phantom Maxwell case yields a finite growth that violates the bound; the two calculational formalisms yield the same results in these scenarios. The findings reinforce the CA framework for a broad class of nontrivial interior geometries and highlight the interplay between horizon structure, singularities, and holographic complexity.

Abstract

According to the conjecture "complexity equals action," the complexity of a holographic state is equal to the action of a Wheeler-DeWitt (WDW) patch of black holes in anti-de Sitter space. In this paper we calculate the action growth of charged black holes with a single horizon, paying attention to the contribution from a spacelike singularity inside the horizon. We consider two kinds of such charged black holes: one is a charged dilaton black hole, and the other is a Born-Infeld black hole with $β^2 Q^2<1/4$. In both cases, although an electric charge appears in the black hole solutions, the inner horizon is absent, instead a spacelike singularity appears inside the horizon. We find that the action growth of the WDW patch of the charged black hole is finite and satisfies the Lloyd bound. As a check, we also calculate the action growth of a charged black hole with a phantom Maxwell field. In this case, although the contributions from the bulk integral and the spacelike singularity are individually divergent, these two divergences just cancel each other and a finite action growth is obtained. But in this case, the Lloyd bound is violated as expected.

Action growth of charged black holes with a single horizon

TL;DR

This work examines the complexity=action conjecture for holographic states dual to charged AdS black holes with a single horizon and a spacelike interior singularity. It analyzes charged dilaton and Born-Infeld black holes, including subcases with two horizons and with a single horizon, and also a phantom Maxwell field to probe methodological robustness. The main results show finite late-time action growth that respects the Lloyd bound for the dilaton and single-horizon Born-Infeld cases, while the phantom Maxwell case yields a finite growth that violates the bound; the two calculational formalisms yield the same results in these scenarios. The findings reinforce the CA framework for a broad class of nontrivial interior geometries and highlight the interplay between horizon structure, singularities, and holographic complexity.

Abstract

According to the conjecture "complexity equals action," the complexity of a holographic state is equal to the action of a Wheeler-DeWitt (WDW) patch of black holes in anti-de Sitter space. In this paper we calculate the action growth of charged black holes with a single horizon, paying attention to the contribution from a spacelike singularity inside the horizon. We consider two kinds of such charged black holes: one is a charged dilaton black hole, and the other is a Born-Infeld black hole with . In both cases, although an electric charge appears in the black hole solutions, the inner horizon is absent, instead a spacelike singularity appears inside the horizon. We find that the action growth of the WDW patch of the charged black hole is finite and satisfies the Lloyd bound. As a check, we also calculate the action growth of a charged black hole with a phantom Maxwell field. In this case, although the contributions from the bulk integral and the spacelike singularity are individually divergent, these two divergences just cancel each other and a finite action growth is obtained. But in this case, the Lloyd bound is violated as expected.

Paper Structure

This paper contains 7 sections, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Charged dilaton black hole in AdS space
  3. Born-Infeld black hole in AdS space
  4. The case with two horizons
  5. The case with a single horizon
  6. Charged black hole with phantom Maxwell field
  7. Conclusion and discussion

Figures (1)

  • Figure 1: A WDW patch and its change due to an infinitesimal time shift $\delta t$ at the left boundary, either for a charged dilaton AdS black hole (BH) or for a Born-Infeld (BI) AdS BH with single horizon (left panel), and the same for a BI-AdS BH with two horizons (right panel). For the charged dilaton AdS BH, the light rays from the boundaries terminate on a nonzero spacelike singularity at $r=r_D=2D$, while for the BI-AdS BH with single horizon, they terminate at $r=r_D=0$. For the BI-AdS BH with two horizons the light rays from both boundaries meet without encountering a singularity. Figures are taken from Lehner:2016vdi and slightly modified.