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Noether charge, black hole volume, and complexity

Josiah Couch, Willy Fischler, Phuc H. Nguyen

arXiv:1610.02038v4hep-thgr-qc

TL;DR

The paper asks what AdS black-hole thermodynamic volume means geometrically and holographically, particularly beyond simple static geometries. It generalizes the Iyer–Wald formalism, connects volume to complexity and CA-duality, and proposes Complexity = Volume 2.0; in certain cases, this proposal respects the Lloyd bound while CA-duality does not.

  • Problem

    The paper addresses the missing geometric interpretation of thermodynamic volume, especially for rotating or hairy AdS black holes.

    Source: §1 · pp. 2–3

  • Method

    The paper generalizes the Iyer–Wald Noether-charge formalism to derive thermodynamic volume and applies it to charged BTZ and R-charged black holes.

    Source: §1 · p. 3, §6 · p. 27

  • Results

    The paper relates thermodynamic volume and the Smarr relation to CA-duality complexity growth and proposes Complexity = Volume 2.0 as WDW-patch spacetime volume.

    Source: §1 · p. 3, §6 · p. 27

  • Takeaways & Limitations

    In certain cases, the proposed complexity is consistent with the Lloyd bound whereas CA-duality is not.

    Source: §5.1 · p. 24, §5.3 · p. 26, §5.4 · p. 27

    The discussion is bounded by quantum recurrences at doubly exponential times and by Lifshitz solutions whose matter profiles depend explicitly on the cosmological constant.

    Source: §3.3 · p. 17, §6 · p. 28

Abstract

from arXiv · show

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.

1 Introduction

The paper investigates the geometric meaning and holographic relevance of AdS black-hole thermodynamic volume, especially for complicated geometries and quantum complexity. It motivates an Iyer–Wald derivation, connects volume to CA-duality, and proposes Complexity = Volume 2.0.

Motivation

  • Extended black-hole thermodynamics identifies pressure with the cosmological constant and reinterprets ADM mass as enthalpy, but the pressure–volume pair is absent from traditional thermodynamics.A coordinate-invariant definition of black-hole volume is difficult because naive interior integration depends on spacetime foliation.Source: §1 · pp. 2–3
  • The central problem is explaining thermodynamic volume geometrically when rotating or hairy black holes make it less intuitive than simple static examples.The paper asks whether volume can arise from integrating a local quantity over a spacetime region.Source: §1 · p. 3
  • The paper uses the Iyer–Wald formalism to derive thermodynamic volume and relates that quantity to holographic quantum complexity.This extends a formalism already used to connect bulk geometry with boundary information-theoretic quantities.Source: §1 · pp. 3–4
  • Bulk volume is relevant to holography because Einstein–Rosen-bridge size, subregion complexity, and fidelity have been related to bulk geometric quantities.The paper therefore investigates whether thermodynamic volume also admits a quantum-information interpretation.Source: §1 · pp. 3–4

Related proposals

  • CA-duality identifies complexity with the Wheeler–DeWitt action, while the paper proposes Complexity = Volume 2.0, identifying complexity with the WDW patch’s spacetime volume.CV-duality requires a length scale chosen by hand, whereas CA-duality avoids that issue but motivates the alternative proposal developed here.Source: §1 · p. 4

Scope

  • The paper applies the formalism to charged BTZ and R-charged black holes, studies complexity for neutral and charged or rotating holes, and compares proposals with the Lloyd bound.Its sections cover derivation, complexity connections, bound issues, and future directions.Source: §1 · p. 4

2 Volume and Iyer-Wald Formalism

The generalized Iyer–Wald construction derives the extended first law and thermodynamic volume by allowing the cosmological constant to vary. Applications to charged BTZ and R-charged black holes clarify how volume emerges from horizon, infinity, and scalar-potential contributions.

2 Volume and Iyer-Wald Formalism

  • Allowing the cosmological constant to vary in a diffeomorphism-invariant Iyer–Wald formalism produces an identity whose black-hole evaluation yields the extended first law and V dP.Stokes’ theorem is applied on a constant-time region bounded by the bifurcate Killing horizon and conformal infinity.Source: §2 · pp. 5–6
  • Noether’s theorem supplies the conserved current and charge, while the generalized variation form χ organizes the horizon and asymptotic contributions.The horizon integral reproduces TδS through the bifurcation-surface area contribution.Source: §2 · p. 5, §2.2 · p. 10

2.1 Application to Einstein-Maxwell: charged BTZ black hole

  • For the charged BTZ black hole, evaluating the formalism at the outer and inner horizons yields corresponding first laws and thermodynamic volumes.The calculation treats the two bifurcate Killing horizons separately and includes the appropriate inner-horizon contribution.Source: §2.1 · pp. 6–8

2.2 Application to Einstein-Maxwell-dilaton: the R-charged Black Hole

  • For the R-charged black hole, varying the coupling leaves matter profiles, ADM mass, and charges unchanged, so the cosmological-constant variation isolates the volume contribution.The pressure is identified with the cosmological constant, which is also the bottom of the scalar potential.Source: §2.2 · pp. 8–10
  • The R-charged calculation makes the thermodynamic-volume integrand manifest as the scalar potential rather than requiring the indirect expression used previously.The extended Iyer–Wald formalism exposes this scalar-potential integral directly.Source: §2.2 · pp. 10–11
  • Divergent asymptotic terms cancel pairwise, leaving a finite result composed of the finite infinity contribution and the horizon term.The lower integration limit arises from the finite χ contribution at infinity.Source: §2.2 · pp. 10–11
  • The R-charged volume is best viewed as a regularized scalar-potential integral over the black-hole exterior, not simply as the volume of its interior.For coincident charges, the result reduces to the familiar AdS-RN form after a radial-coordinate redefinition, while the thermodynamic volume remains coordinate invariant.Source: §2.2 · pp. 9, 11

3 Thermodynamic volume and Complexity: Schwarzschild-AdS

This section reviews CA-duality for Schwarzschild-AdS, showing how black-hole thermodynamic quantities organize late-time complexity growth and motivating complexity = volume 2.0.

3.1 Review of Brown et al.

  • The late-time CA-duality complexity rate approaches the black-hole mass, providing evidence linked to the Lloyd bound.The ADM mass plays the role of energy, and the bound is saturated at late time.Source: §3.1 · pp. 13–14

3.2 CA-duality, through the lens of black hole chemistry

  • In AdS-Schwarzschild, boost symmetry cancels the exterior sliver contributions, leaving the WdW action variation determined by the black-hole interior.The surviving integral yields the thermodynamic volume contribution.Source: §3.2 · p. 15
  • The WdW action calculation decomposes complexity growth into M, TS, and PV contributions, with PV representing the black-hole interior.The singularity contributes M, horizon corners contribute TS, and the interior away from the singularity contributes PV.Source: §3.2 · pp. 14–16

3.3 Complexity = Volume 2.0

  • The proposed holographic complexity duals rely on late-time linear growth, but proving a matching lower bound for complexity remains non-trivial.Quantum recurrence eventually invalidates the stated late-time behavior at doubly exponential timescales.Source: §3.3 · pp. 16–17
  • The paper proposes complexity = volume 2.0, identifying complexity with the spacetime volume of the WdW patch multiplied by pressure.At late time, the spacetime-volume growth is related to the thermodynamic volume, giving a PV complexification rate.Source: §3.3 · pp. 16–18
  • For large black holes, M and PV become equal while TS differs from M only by an O(1) factor, making the late-time prescriptions parametrically similar.The comparison applies in the high-temperature regime r+ >> L.Source: §3.3 · pp. 18–19

4 Thermodynamic Volume and Complexity: Conserved Charges

For charged and rotating AdS black holes, the late-time complexity growth in the volume proposal is governed by differences between outer- and inner-horizon thermodynamic volumes. The WDW patch approaches the inner horizon at late time, and this structure extends across electrically charged and rotating solutions.

4 Thermodynamic Volume and Complexity: Conserved Charges

  • The volume proposal extends the Schwarzschild-AdS connection to charged and rotating black holes, with pressure and thermodynamic volume emerging naturally in late-time complexity growth.The CA calculation is not as clean for these solutions and does not respect the Lloyd bound, so the section focuses on complexity=volume 2.0.Source: §4 · p. 19
  • The charged and rotating WDW patches approach the inner horizon at late time rather than the singularity, distinguishing them qualitatively from the Schwarzschild-AdS patch.Shifting tL removes one sliver of the patch and adds another, as depicted in the Penrose diagram.Source: §4 · p. 19, Figure 2 · p. 20

4.1 Electrically charged black holes

  • For charged black holes, the late-time complexification rate is proportional to the difference V+ − V− between outer- and inner-horizon thermodynamic volumes.The same form persists for the charged BTZ case because the additional volume term cancels in V+ − V−.Source: §4.1 · p. 21

4.2 Rotating Black Hole

  • For rotating black holes, the late-time complexification rate is again proportional to the difference between the two thermodynamic volumes.The result is obtained by relating the bulk Einstein-Hilbert action to the WDW-patch spacetime volume and defining the inner-horizon volume by replacing r+ with r−.Source: §4.2 · pp. 21–22
  • In four-dimensional Kerr-AdS, the non-rotating-frame volume is called thermodynamic volume, while the rotating-frame quantity is called geometric volume and has a geometrical interpretation.The analysis distinguishes these two notions because rotating and non-rotating frames at infinity produce different thermodynamic descriptions.Source: §4.2 · pp. 21–22
  • In the rotating frame, the thermodynamic quantities satisfy the Smarr relation but not the first law, whereas non-rotating-frame quantities satisfy both.The non-rotating-frame quantities are also the ones derivable from the Iyer-Wald formalism.Source: §4.2 · p. 22
  • At fixed mass and AdS length, V+ − V− is maximal at zero angular momentum and tends toward zero as the rotating black hole approaches extremality.The plotted curves use (M,L)=(5,1), (2,3), and (1,2).Source: §4.2 · p. 22, Figure 3 · p. 23

5 Action or Volume?

The section compares action- and volume-based complexity proposals through the Lloyd bound for charged black holes. Near extremality both proposals violate the bound, while exact results show CV respects it for specified chemical-potential ranges and CA violates it in the four-dimensional μ≤1 regime.

5 Action or Volume?

  • The two proposals are compared because CA-duality and complexity=volume 2.0 have similar behavior, while the latter avoids boundary terms that can be problematic near singularities in higher-curvature theories.The section treats consistency with the Lloyd bound as a criterion for distinguishing the proposals.Source: §5 · p. 23

5.1 The Lloyd Bound with conserved charge

  • The charged Lloyd bound subtracts the ground-state value of M−μQ, choosing empty AdS or an extremal black hole according to the chemical potential.For μ≤1 the appropriate ground state is pure AdS, while for μ>1 it is an extremal black hole with the same chemical potential.Source: §5.1 · p. 24

5.2 Bound Violation: Near the Ground State

  • Near extremality, both CA-duality and CV-duality violate the Lloyd bound.For μ≤1 near empty AdS, CV satisfies the bound but is far from saturated, while CA saturates it at lowest order and requires subleading analysis.Source: §5.2 · p. 25

5.3 Bound Violation: Exact Results

  • For four-dimensional charged black holes with μ≤1, CA-duality always violates the Lloyd bound, whereas CV-duality respects it.The exact expressions establish the CA violation and the CV result over the region where μ≤1.Source: §5.3 · p. 26
  • In five dimensions, CV-duality respects the bound whenever μ^2 < 1, and the authors conjecture the condition μ^2 ≤ 1 in arbitrary dimensions.The arbitrary-dimensional statement is presented as a conjecture rather than a proof.Source: §5.3 · pp. 26–27

5.4 Altering the Bound by a Pre-Factor

  • Allowing an overall prefactor α, the choice α=1 remains consistent with the bound in the AdS-RN case, while other values are left for future study.The authors note that the Lloyd bound may only be trusted up to an overall factor.Source: §5.4 · p. 27

6 Conclusion

The paper develops an Iyer–Wald approach to thermodynamic volume and connects that volume to Wheeler–DeWitt spacetime volume and holographic complexity. It supports these connections across several AdS black holes while identifying limitations for varying the cosmological constant in Lifshitz solutions and proposing future directions.

Conclusion

  • The Iyer–Wald method systematically derives thermodynamic volume, illustrated for charged BTZ and R-charged black holes, with possible application to more complicated solutions.For Lifshitz black holes, varying the cosmological constant also changes matter-field profiles, removing the simplifying feature available for the R-charged example.Source: §6 · pp. 27–28
  • The late-time rate of Wheeler–DeWitt spacetime-volume growth equals the thermodynamic volume for one horizon and the difference of volumes for two horizons.The authors conjecture this correspondence generally and report checks in several examples, but do not provide a general proof.Source: §6 · p. 28
  • The thermodynamic volume is related to holographic complexity through the Wheeler–DeWitt patch, while the bulk action on that patch decomposes into pressure–volume and charge–potential work terms.These relationships connect the thermodynamic definition, the Iyer–Wald construction, and the complexity proposals, though the paper presents them as distinct routes to the same quantity.Source: §6 · p. 28
  • The paper suggests extending pressure–volume variables to tensor-network and quantum-circuit descriptions of black-hole interiors and to questions about emergent space.These are proposed future directions rather than results established by the paper.Source: §6 · p. 28
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