Does Complexity Equal Anything?

Alexandre Belin; Robert C. Myers; Shan-Ming Ruan; Gábor Sárosi; Antony J. Speranza

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Does Complexity Equal Anything?

Alexandre Belin, Robert C. Myers, Shan-Ming Ruan, Gábor Sárosi, Antony J. Speranza

arXiv:2111.02429v2Nov 3, 2021hep-thgr-qc

TL;DR

The paper asks whether the extremal-volume proposal is uniquely selected as a gravitational dual of complexity. It constructs an infinite class of codimension-one gravitational observables and shows that, for the thermofield-double state, they grow linearly at late times and reproduce the switchback effect, supporting all members as viable candidates.

Problem
The extremal-volume proposal is not uniquely fixed because holographic complexity permits conventional ambiguities analogous to choices in quantum complexity.
Source: §1 · p. 2, §6 · p. 6
Method
The paper defines an infinite family of codimension-one observables and studies their curvature-invariant realizations in thermofield-double and shock-wave geometries.
Source: §1 · p. 2, §6 · p. 5
Results
These observables grow linearly at late times and reproduce the switchback effect in shock-wave geometries.
Source: §1 · p. 2, §6 · p. 5
Takeaways & Limitations
Any member of the studied class is a viable gravitational candidate for complexity, while the universal time dependence matters more than the precise observable value.
Source: §1 · p. 2, §6 · p. 5
The universal behavior requires curvature-invariant couplings not to be too large, while more general observables are left for future work.
Source: §6 · p. 5

Abstract

from arXiv · show

We present a new infinite class of gravitational observables in asymptotically Anti-de Sitter space living on codimension-one slices of the geometry, the most famous of which is the volume of the maximal slice. We show that these observables display universal features for the thermofield-double state: they grow linearly in time at late times and reproduce the switch-back effect in shock wave geometries. We argue that any member of this class of observables is an equally viable candidate as the extremal volume for a gravitational dual of complexity.

1. Observables grow linearly with time at late times

The paper defines an infinite family of diffeomorphism-invariant observables on extremal codimension-one slices and shows that, in the thermofield-double state, they grow linearly at late times. This behavior follows when the effective potential has an interior local maximum, and extends to observables evaluated on slices determined by a different functional.

Interpretation

Because these observables share the universal complexity signatures, the paper concludes that any member of the class is a viable gravitational dual of complexity, with universal scaling more significant than its precise value.This parallels conventional ambiguities in quantum complexity, such as the choice of gate set.Source: §1 · p. 2

Setup

The analysis specializes to the eternal planar black hole in the thermofield-double state and to curvature-invariant functionals whose reduced factor depends only on the radial coordinate.This restriction removes derivative dependence from the reduced problem and simplifies the extremal-surface equations.Source: §1 · pp. 2–3

Late-time behavior

The generalized volume’s time derivative is controlled by the endpoint momentum, so its approach to a constant P∞ establishes linear growth at late times.The extremal-surface problem can be recast as classical particle motion in an effective potential, with the turning point fixed by the conserved momentum.Source: §1 · pp. 2–3
Linear late-time growth occurs when the effective potential has a local maximum inside the horizon, with the extremal surface approaching the corresponding constant-radius final slice.As the boundary time diverges, rmin approaches rf and the conserved momentum approaches a critical value P∞, making the growth rate asymptotically constant.Source: Figure 1 · p. 2, §1 · pp. 3–4

More general observables

The same late-time linear growth holds for observables with F1 ≠ F2, although the final-slice location is set by F2 while the growth rate is controlled by F1.The additional integral term vanishes when the two effective potentials coincide.Source: §1 · p. 4

6. Switchback Effect: Consider the perturbed TFD state

The generalized codimension-one observables reproduce the switchback effect in shockwave geometries: their contributions remain additive, exhibit late-time linear growth, and yield the expected scrambling-time subtraction after extremization.

Switchback effect

The switchback effect appears as a subtraction −2n t∗ from the total absolute-value time contour in the perturbed thermofield-double geometry.The absolute-value terms measure the contour length, while the subtraction reflects cancellation around uns scrambled switchbacks; the formula assumes each switchback lasts much longer than the scrambling time.Source: §6 · pp. 4–5

Shockwave geometry

The strong-shock state is represented by n+1 black-hole patches glued alternately across horizons, with null shifts controlled by the insertion times and scrambling time.The shifts are parameterized by α_i and depend on t∗ ± t_i, with t∗ proportional to β log G_N^-1.Source: §6 · p. 5

Relation to maximal volume

The switchback result relies on the same additive and late-time-linear-growth properties previously used for the maximal-volume slice.For the maximal volume, additivity in shockwave geometries and linear growth of each connecting extremal segment lead to the standard construction [9].Source: §6 · p. 5

Additivity

For generalized observables, diffeomorphism invariance prevents delta-function contributions from shockwaves, preserving the additivity needed for the switchback construction.Scalar contractions force derivatives to occur in balanced U- and V-derivative combinations, so potentially singular terms appear as Uδ(U), U^2δ′(U), and vanish on the horizon.Source: §6 · p. 5

Late-time growth

General observables remain dominated by linear growth in ingoing and outgoing time, extending the late-time behavior beyond the maximal-volume prescription.The generalized result is obtained using the effective-potential description of the observable.Source: §6 · p. 5

Extremization

Using the additive generalized volume and extremizing over intermediate horizon coordinates recovers the expected switchback effect.The extremization variables include V1, U2, …, U_n−1, and V_n, paralleling the maximal-volume construction.Source: §6 · p. 5

A. Explicit example with a Weyl-squared term for F1 = F2

The Weyl-squared example produces an effective potential whose extrema, and therefore late-time extremal surfaces, depend on the dimensionless coupling. Within a specified coupling range, the potential has the maximum required for the late-time analysis, while outside it the relevant surface is absent.

A. Explicit example with a Weyl-squared term for F1 = F2

The Weyl-squared functional uses the bulk Weyl-tensor square C2 as its nontrivial quadratic-curvature scalar.Source: A. Explicit example with a… · p. 7
The effective potential’s extrema are determined by solving ∂r eU(r) = 0 after introducing the dimensionless radial coordinate w and rescaled coupling ˜λ.Source: A. Explicit example with a… · p. 7
For −1 < ˜λ < ˜λcrt1, the potential has two extremal points inside the horizon, corresponding to a local minimum and a local maximum.Source: Figure 3 · p. 7, A. Explicit example with a… · p. 8
For ˜λ < −1 or ˜λ > ˜λcrt1, the effective potential has no extremal point inside the horizon and therefore lacks the corresponding late-time extremal surface.Source: Figure 3 · p. 7, A. Explicit example with a… · p. 8
When the maximum exists, the generalized-volume extremal surface is not unique because the potential diverges at the origin, and the maximal surface is the branch approaching constant radius.Source: A. Explicit example with a… · p. 8

B. Observables with F1̸ = F2

The paper also separates the functional being evaluated from the functional that determines the extremal hypersurface. This defines generalized observables whose evaluation formula reduces to the F1 = F2 case when the two effective potentials coincide.

B. Observables with F1̸ = F2

For F1 ≠ F2, the observable is evaluated using F1 on a hypersurface extremized by F2.Source: B. Observables with F1̸ = F2 · p. 8
The generalized-volume expression depends on the extremal-surface potential eU2 and reduces to the F1 = F2 result when eU2 = eU1.Source: B. Observables with F1̸ = F2 · p. 8

Time Derivative and Linear Growth

The late-time behavior is controlled by the effective potential near its interior maximum. Under the stated assumptions, the generalized observables have finite growth rates and exhibit linear late-time growth, with special cases determined by how the relevant potentials share their maxima.

Time Derivative and Linear Growth

The growth rate remains finite even though the observable is evaluated on a surface determined by F2 rather than F1.Source: Time Derivative and Linear Growth · p. 9
The generalized observable grows linearly at late times because the effective potential for the extremal hypersurface has a local maximum as rmin approaches rf.Source: Time Derivative and Linear Growth · p. 9
The analysis of the late-time growth rate uses the asymptotic expansion of Pv(τ) and the regulated integral representation of the observable.Source: Time Derivative and Linear Growth · p. 9
The late-time limit requires a final radius rf that is not an extremum of the evaluation potential eU1 in the general F1 ≠ F2 case.Source: Time Derivative and Linear Growth · p. 9
When eU1 and eU2 share the same maximal radius but have different maximum values, the late-time behavior follows a distinct case from the coincident-potential result.Source: Time Derivative and Linear Growth · p. 9