Noether charge, black hole volume, and complexity

TL;DR

The paper derives the thermodynamic volume of AdS black holes from the Noether-Wald formalism in extended thermodynamics and explores its holographic relevance to quantum complexity. It shows that the late-time growth in complexity under CA-duality admits a Smarr-like decomposition and introduces Complexity = Volume 2.0, linking complexity to the spacetime volume of the Wheeler-DeWitt patch with a PV scaling. The analysis extends to charged and rotating black holes, revealing that complexity growth can be governed by differences of horizon volumes and that Lloyd-bound considerations constrain both action- and volume-based proposals in nontrivial ways. The work thus provides a geometric, holographic interpretation of thermodynamic volume, connects black-hole interior structure to quantum information, and outlines avenues for further exploration, including tensor-network descriptions and bound refinements.

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.

Figures (3)

• Figure 1: The Wheeler-DeWitt patch of the AdS-Schwarzschild black hole (depicted in orange). When $t_{L}$ is shifted to $t_{L} + \delta t_{L}$, the patch loses a sliver and gains another one (depicted in darker orange). The contributions from the Gibbons-Hawking term are in blue.
• Figure 2: The Penrose diagram of a charged and/or rotating black hole and a Wheeler-DeWitt patch (depicted in orange). When $t_{L}$ is shifted to $t_{L} + \delta t_{L}$, the patch loses a sliver and gains another one (depicted in darker orange). The singularity is in red, and the horizons are dashed.
• Figure 3: Given $M$ and $L$, we vary the angular momentum to mass ratio $a$ and for each value solve numerically for $V=V_+-V_-$. Notice that $a=0$, which reduces to the Schwarzschild case, has the maximal $V$. As we approach extremality, which here occurs as the plots flatten out on the left (In flat space exremality occurs for $a = 1$, but this is modified by the AdS length dependence of the metric), $V$ tends towards zero. In the plot green is for $M=5,L=1$, blue is for $M=2,L=3$, and red is for $M=1,L=2$.