PaperlayerView exact revision on arXiv

Source-linked AI summary

Holographic Complexity in Vaidya Spacetimes I

Shira Chapman, Hugo Marrochio, Robert C. Myers

arXiv:1804.07410v1hep-thgr-qcquant-ph

TL;DR

Holographic entanglement entropy misses some late-time features of eternal black holes and their dual thermal states, while deriving bulk prescriptions for holographic complexity remains unclear. This paper studies CV and CA for a null shell forming a one-sided AdS black hole, finding that with a null-boundary counterterm both approaches recover the eternal-black-hole late-time growth rate.

  • Problem

    Holographic entanglement entropy misses certain late-time features of eternal black holes and their dual thermal states, and the bulk derivation of complexity prescriptions remains unclear.

    Source: §1 · p. 2

  • Method

    The paper evaluates CV and CA for Vaidya geometries sourced by a collapsing null shell, constructing a null-fluid action and examining its thin-shell contribution.

    Source: §1 · p. 3, §4 · p. 43

  • Results

    With the null-boundary counterterm, both CV and CA yield the eternal-black-hole late-time complexity growth rate for the studied one-sided black holes.

    Source: §4 · p. 44

  • Takeaways & Limitations

    The null-boundary counterterm is essential for CA to yield positive growth and match the expected late-time rate in the studied Vaidya formation.

    Source: §4 · p. 44

    The vanishing shell-action result leaves an ambiguity in the interior WDW action unless the null-boundary parameter κ is fixed; the authors identify κ = 0 as the natural choice.

    Source: §2.2 · p. 15

Abstract

from arXiv · show

We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity$=$volume (CV) and complexity$=$action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is essential if the gravitational action is to properly describe the complexity of the boundary state. For both the CV proposal and the CA proposal (with the extra boundary counterterm), the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole.

1 Introduction

The introduction motivates holographic complexity as a way to study late-time features that entanglement entropy misses, while noting that its bulk geometric derivation remains unclear. The paper investigates CV and CA in Vaidya black-hole formation and argues that a null-boundary counterterm is essential for the CA prescription in dynamical spacetimes.

1 Introduction

  • Entanglement of microscopic degrees of freedom is described as central to the emergence of semiclassical bulk spacetime geometry.The introduction places this within broader work connecting information theory and entanglement to quantum gravity.Source: §1 · p. 2
  • Because entanglement entropy misses certain late-time features of eternal black holes and thermal states, quantum circuit complexity was proposed as a complementary holographic observable.State complexity is framed as the resources needed to prepare a target state from a simple reference using elementary gates.Source: §1 · p. 2
  • The CV proposal identifies boundary-state complexity with an extremal bulk volume, while CA identifies it with the gravitational action on the Wheeler-DeWitt patch.The WDW patch is the causal development of the relevant bulk surface.Source: §1 · p. 3
  • Although CV and CA are viable candidates, how to derive either bulk prescription from a known boundary-theory complexity calculation remains unclear.Source: §1 · p. 3
  • The paper studies one-sided Vaidya black-hole formation with CV and CA, finding that the CA prescription for dynamical geometries requires a null-boundary counterterm.It constructs a null-fluid action that vanishes on shell and examines the thin-shell contribution; results cover general spacetime dimensions and planar or spherical horizons.Source: §1 · pp. 3–4

2 Null Fluid and the Vaidya Geometry

This section describes AdS-Vaidya collapse sourced by a null-fluid shell and constructs a null-fluid action for evaluating complexity=action. It examines the shell’s contribution to the Wheeler-DeWitt action, the treatment of its interior contribution, and the role of a null-boundary counterterm.

2 Null Fluid and the Vaidya Geometry

  • AdS-Vaidya metrics describe black-hole formation by null-fluid collapse; a step-profile shell creates a sharp transition between stationary geometries and can collapse into AdS vacuum to form a one-sided black hole.The profile is generally chosen positive and monotonically increasing so that mass and shell energy density are positive.Source: §2 · pp. 4–5

2.1 Action for a Null Fluid

  • The proposed null-fluid action vanishes on shell, leaving no explicit matter-field contribution to the CA calculation.The action’s stress tensor matches the null-fluid form, with λ proportional to the energy density (ε = 2λ).Source: §2.1 · p. 7

2.2 Null Fluids & Complexity=Action

  • In the thin-shell limit, the two nonvanishing null-boundary and joint contributions cancel, so the null shell contributes no net Wheeler-DeWitt action.The shell’s bulk and singularity terms also vanish as its width shrinks to zero; the cancellation uses the same normalization for the two shell-boundary normals.Source: §2.2 · pp. 9, 11, 14
  • The thin-shell WDW action reduces to separate evaluations of the regions outside and inside the shell.These are denoted I2 for v > vs and I1 for v < vs.Source: §2.2 · p. 14
  • Setting κ = 0 along the past null boundary fixes the normalization ambiguity in the interior WDW action contribution I1.This choice determines the inner normalization constant and removes the ambiguity in evaluating I1.Source: §2.2 · p. 15

2.3 Counterterm for Null Boundaries

  • In dynamical Vaidya geometries, the null-boundary counterterm is essential for reproducing key properties of holographic complexity.The counterterm removes dependence on the parametrization of null generators; prior stationary-spacetime studies found it did not change complexity of formation or the eternal black hole’s late-time growth rate, while it modifies transient behavior.Source: §2.3 · p. 16
  • The counterterm produces a surface contribution where the past boundary crosses the null shell, which is essential to the proper behavior of holographic complexity.Source: §2.3 · p. 17

3 Complexity in Black Hole Formation

The section sets up one-sided black-hole formation by a thin null-fluid shell collapsing in empty AdS and evaluates the resulting complexity with both CA and CV. In the boundary CFT, this collapse represents a quench from the vacuum to an excited state.

3 Complexity in Black Hole Formation

  • A thin null-fluid shell collapses in empty AdS to form a one-sided black hole, whose complexity is evaluated with the CA and CV proposals.The boundary interpretation is a quantum quench: a homogeneous operator injects energy into the vacuum state.Source: §3 · p. 17
  • The study considers planar and spherical shells and horizons, with k = 0 and k = +1 [17], and focuses on dimensions d ≥ 3; the special case d = 2 is treated separately.Source: §3 · pp. 17–18
  • For boundary times t > 0, the state energy is set by the black-hole mass, and the excited state is assigned an effective temperature and entropy.Boundary time is defined by t = v at the asymptotic boundary.Source: §3 · p. 18
  • The hyperbolic case k = −1 is not covered by the same boundary-slice setup, since a time slice covers only half of the global AdS boundary.The discussion could be extended if null-fluid shells were injected symmetrically from both halves.Source: §3 · p. 18
  • The outgoing coordinate u is discontinuous across the shell because the metric function changes, but the spacetime and outgoing null rays remain smooth there.The jump shown in the Penrose-like diagrams is a representational artifact; the diagrams also mark the WDW null boundary's shell-crossing radius r_s.Source: §3 · p. 19, Figure 2 · p. 20

3.1 Complexity=Action

These analyses evaluate CA complexity growth in thin-shell collapse and assess how the null-boundary counterterm affects the rate. They cover BTZ and higher-dimensional spherical-shell collapses, including late-time behavior and, in BTZ, the initial rate.

3.1.1 Action Calculation

  • In the thin-shell limit, the shell action vanishes, so the WDW action is the sum of the vacuum-region and black-hole-region actions; only bulk and future-singularity GHY terms contribute to its time derivative.With affine parametrization, the relevant joint contribution vanishes.Source: §3.1 · pp. 21–22, §3.1.1 · p. 23, §3.1.2 · p. 24

3.1.2 Time Dependence of Complexity, Version 1

  • Without the counterterm, the late-time rate is below the eternal-black-hole value 2M/π for planar and spherical black holes.For spherical black holes, the late-time rate is slightly larger than in the planar case, but remains below the eternal result.Source: §3.1.2 · p. 24

3.1.3 Time Dependence of Complexity, Version 2

  • With the null-boundary counterterm, the late-time CA growth rate approaches 2M/π, increases monotonically from below, and respects the proposed growth bound.The counterterm on the future boundary does not affect the rate; the relevant contribution comes from the past boundary.Source: §3.1.3 · pp. 27–28
  • For d ≥3, the counterterm-modified rate begins at M/π for both spherical and planar black holes, then approaches its late-time limit exponentially from below.The late-time decay is controlled by the thermal length scale 1/(2πT).Source: §3.1.3 · pp. 28–29
  • Without the counterterm in BTZ, the Ramond-vacuum collapse has zero growth, while the Neveu-Schwarz rate remains negative and approaches zero from below.Source: §3.1.3 · p. 30
  • With the counterterm in BTZ, the early rate matches M/π for the Ramond vacuum, while the Neveu-Schwarz rate is temperature-reduced and its correction vanishes at 2πLT = 1.The correction is positive above the Hawking-Page transition, where 2πLT > 1.Source: §3.1.3 · p. 30
  • In BTZ with the counterterm, growth increases monotonically toward 2M/π from below and essentially reaches that limit at t0 ∼ 1/T.Source: §3.1.3 · p. 31
  • With the boundary counterterm, spherical-shell CA growth starts at M/π after shell injection and approaches 2M/π from below at late times; without it, late-time growth is lower, temperature-dependent, and unlike eternal black holes.Source: §3.1.3 · p. 32

3.2 Complexity=Volume

The CV proposal evaluates complexity from maximal-volume surfaces in the Vaidya geometry, whose motion across the shell can be analyzed through an effective potential. Its growth rate approaches the eternal-black-hole late-time result, with the planar case constant at all times and the spherical late-time contribution dominated by surfaces near a fixed radius.

3.2 Complexity=Volume

  • The growth rate starts at half its late-time limit, rises on the thermal-length scale, and approaches the eternal-black-hole bound from below.Source: §3.2 · p. 33
  • The radial surface equation is a particle-like Hamiltonian constraint with energy P^2 moving in U(r); shell-crossing surfaces must satisfy the potential-barrier condition to reach the boundary.The surface’s momentum is positive, and the effective energy and potential can jump at the shell.Source: §3.2 · pp. 35–37
  • The extremal surface is fixed by avoiding a conical singularity at r = 0 and by anchoring it at boundary time t0, which determines its shell-crossing radius and black-hole-side momentum.Source: §3.2 · pp. 35–36
  • In planar geometry, complexity begins growing for t0 > 0 at a constant rate matching the eternal black hole’s late-time rate; in spherical geometry, late-time growth comes mainly from surfaces wrapping r = rm.Contributions from smaller radii and the portions reaching the boundary are approximately constant and do not affect the time derivative.Source: §3.2 · p. 38

Early Time Behaviour

The early-time CV growth rate is obtained in the limit where the extremal surface crosses the shell at large radius. It jumps to a positive, nonzero value for t0 > 0, and its early-time rate matches the planar result.

Early Time Behaviour

Late Time Behaviour

At late times, the growing volume comes from the extremal surface wrapping the constant-radius surface at r = r_m. The planar growth rate stays constant, while spherical curvature corrections reduce it.

Late Time Behaviour

  • At late times, the extremal surface wraps r = r_m, and this segment supplies the increasing volume while the rest remains approximately constant.The surface reaches its minimum r_s below r_m and the boundary above it; those contributions are approximately time-independent.Source: Late Time Behaviour · p. 39
  • For spherical geometry, curvature corrections reduce the late-time growth rate below its initial value of 8πM/(d −1).The rate begins at 8πM/(d −1) and decreases at late times.Source: Late Time Behaviour · p. 40
  • For planar geometry, the complexity growth rate remains constant.The large-temperature expansion provides a route to determining the late-time rate when the general equation cannot be solved analytically.Source: Late Time Behaviour · pp. 39–40

Two Boundary Dimensions (d = 2)

For d = 2, the collapse forms a BTZ black hole, and the late-time CV growth rate is independent of the vacuum parameter k. In the Neveu-Schwarz vacuum, the early growth rate exceeds the late-time rate.

Two Boundary Dimensions (d = 2)

  • The BTZ collapse is associated with the Ramond vacuum for k = 0 and the Neveu-Schwarz vacuum for k = 1.The text identifies these vacuum choices with planar and spherical geometries, respectively.Source: Two Boundary Dimensions (d = 2) · p. 40
  • The late-time growth rate is independent of k; for the Ramond vacuum it remains constant at all times, while the Neveu-Schwarz rate is higher early than late.The text reports numerical evaluation of the intermediate-time rate for the Neveu-Schwarz vacuum at several temperatures.Source: Two Boundary Dimensions (d = 2) · p. 40
  • The BTZ results agree with those reported in appendix A of.Source: Two Boundary Dimensions (d = 2) · p. 40
  • Figure 7 plots the CV rate for the BTZ black hole with the Neveu-Schwarz vacuum at TL = 0.16, 0.32, and 0.64.The curves show intermediate-time numerical evaluations.Source: Two Boundary Dimensions (d = 2) · p. 40, Figure 7 · p. 41

Numerical Results

Numerical results for spherical geometries in d = 3 and 4 show an early decrease in the complexity growth rate, with the late-time limit approached from above.

Numerical Results

  • The numerical results cover spherical geometries with d = 3 and 4.Source: Numerical Results · p. 41
  • In all cases shown, the rate decreases at early times and approaches its late-time limit from above.Source: Numerical Results · p. 41
  • The early and late rates match the previously discussed limits, and the early rate is highest for the Neveu-Schwarz vacuum in d = 2.The text also refers to Figure 7 for the d = 2 comparison.Source: Numerical Results · p. 41

4 Discussion

For a black hole formed by a collapsing shell, the null-boundary counterterm restores the CA late-time growth rate, while CV late-time behavior agrees with eternal black holes. The discussion also distinguishes their transients and outlines extensions and open comparisons.

4 Discussion

  • Planar CV growth stays constant at the eternal-black-hole late-time rate, while spherical curvature lowers the late-time rate, with a smaller reduction at high temperature.The spherical one-sided black-hole curvature corrections match those of the analogous eternal backgrounds.Source: §4 · p. 42, Figure 8 · p. 42
  • Unlike eternal black holes, the one-sided planar CV rate jumps to a constant at shell injection, while spherical CV growth decreases toward its limit from above.Eternal planar rates rise through a transient, and eternal spherical rates increase toward their late-time value.Source: §4 · p. 43
  • The null-fluid action vanishes on shell, and the infinitely thin shell’s WDW action contribution vanishes through cancellation of past-null-boundary surface and joint terms.The WDW action can therefore be evaluated as the sum of the regions inside and outside the shell.Source: §4 · p. 43
  • Adding the null-boundary counterterm makes the d≥3 CA late-time rate match the eternal value 2M/π and keeps d=2 growth positive, unlike standard CA.For d=2, the counterterm-modified rate reaches the expected late-time limit by t0∼1/T.Source: §4 · p. 44
  • Modified CA growth starts at half its late-time rate for d≥3, then rises monotonically with exponential relaxation on the thermal scale 1/(2πT).It has essentially reached the late-time limit by t∼1/T.Source: §4 · pp. 44–45
  • The counterterm restores the missing CA contribution through a term where the past WDW boundary crosses the shell, compensating for the absent past-boundary joint behind the horizon.That shell-crossing contribution supplies the extra time derivative needed to recover the late-time rate.Source: §4 · pp. 45–46

Future Directions

  • Future work includes charged, finite-width, and other-matter shells, higher-curvature corrections, localized shocks, and field-theory quenches; a companion study examines the switchback effect.The quench comparison remains incomplete: the cited field-theory quenches saturate quickly, unlike holographic late-time linear growth.Source: Future Directions · pp. 47–48
  • The modified CA growth rate for collapsing shells obeys dC′A/dt≤2M/π, but analogous bounds are not universal across other settings.The discussion cites transient violations for eternal black holes and stronger violations in noncommutative and hyperscaling-violating geometries.Source: Future Directions · p. 47
Loading 1804.07410v1…