Reparameterization Dependence and Holographic Complexity of Black Holes

TL;DR

This paper refines holographic complexity calculations for various black holes by introducing a new criterion that ensures finiteness and fixes the reference state, leading to consistent bounds across different black hole solutions.

Contribution

It introduces a criterion based on causal diamond action vanishing, fixing the reference state and ensuring finite, consistent holographic complexity calculations for multiple black hole types.

Findings

The new criterion fixes the reference state as empty AdS.

Holographic complexity becomes explicitly finite under the new criterion.

The generalized Lloyd bound always holds for Reissner-Nordstr"om-AdS and BTZ black holes.

Abstract

We refine the calculation of holographic complexity of black holes in the complexity equals action approach by applying the recently introduced criterion that the action of any causal diamond in static vacuum regions must vanish identically. This criterion fixes empty anti-de Sitter (AdS) spacetime as the reference state with vanishing complexity and renders holographic complexity explicitly finite in all the cases we consider. The cases considered here include the Reissner-Nordstr\"om-AdS black hole, the rotating BTZ black hole, the Kerr-AdS black hole, and AdS-Vaidya spacetime. The criterion is equivalent to imposing that the corner contributions vanish. Contrary to earlier results, we find that the generalized Lloyd bound always holds in the Reissner-Nordstr\"om-AdS and BTZ cases.