# Action growth of charged black holes with a single horizon

## TL;DR

This paper calculates the action growth of charged black holes with a single horizon in anti-de Sitter space, exploring contributions from spacelike singularities and testing the Lloyd bound across different black hole models.

## Contribution

It provides the first detailed analysis of action growth for charged black holes with a single horizon, including effects of spacelike singularities and phantom fields.

## Key findings

- Action growth is finite and respects the Lloyd bound for standard charged black holes.
- Divergences cancel out in phantom field cases, yielding finite action growth.
- Lloyd bound is violated in black holes with phantom Maxwell fields.

## 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 $\beta^2 Q^2<1/4$. 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.