Noether charge, black hole volume, and complexity

TL;DR

This paper explores the thermodynamic volume of AdS black holes using Noether charge formalism, linking it to holographic complexity and proposing a new complexity measure based on spacetime volume, with implications for the Lloyd bound.

Contribution

It introduces a geometric interpretation of black hole thermodynamic volume in the context of holographic complexity and proposes a novel complexity measure called 'complexity = volume 2.0'.

Findings

Late time complexity rate decomposes into thermodynamic quantities.

Thermodynamic volume has a holographic geometric interpretation.

Proposed 'complexity = volume 2.0' aligns with Lloyd bound in some cases.

Abstract

In this paper, we study the physical significance of the thermodynamic volumes of AdS black holes using the Noether charge formalism of Iyer and Wald. After applying this formalism to study the extended thermodynamics of a few examples, we discuss how the extended thermodynamics interacts with the recent complexity = action proposal of Brown et al. (CA-duality). We, in particular, discover that their proposal for the late time rate of change of complexity has a nice decomposition in terms of thermodynamic quantities reminiscent of the Smarr relation. This decomposition strongly suggests a geometric, and via CA-duality holographic, interpretation for the thermodynamic volume of an AdS black hole. We go on to discuss the role of thermodynamics in complexity = action for a number of black hole solutions, and then point out the possibility of an alternate proposal, which we dub "complexity = volume 2.0". In this alternate proposal, the complexity would be thought of as the spacetime volume of the Wheeler-DeWitt patch. Finally, we provide evidence that, in certain cases, our proposal for complexity is consistent with the Lloyd bound whereas CA-duality is not.