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REVIEW 3 major objections 5 minor 90 references

Time dependence of complexity for Lovelock black holes

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that for Lovelock black holes, Complexity=Action predicts a late-time complexity growth rate that is a coupling-independent constant multiple of mass and does not reduce to Schwarzschild when couplings vanish.

desk verdict Useful general Noether-charge derivation of time-dependent CA complexity for Lovelock black holes, but the headline coupling-independent late-time rate is GHY-convention-dependent and the paper itself documents an alternative boundary term giving a different rate. read the letter →

arxiv 1908.09310 v3 pith:S7GDNGRJ submitted 2019-08-25 hep-th

Zhong-Ying Fan , Hua-Zhi Liang This is my paper · ORCID
classification hep-th
keywords holographiccomplexityComplexity=ActionLovelockblackholesGauss-BonnetgravityWheeler-DeWittpatchactiongrowthrateNoetherchargehigher-curvaturecorrections
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how holographic complexity grows with time under the Complexity=Action proposal when the bulk gravity is a Lovelock theory rather than Einstein gravity. By combining the Wald-Iyer Noether charge with a generalized Gibbons-Hawking-York boundary term, it derives the full action growth rate and shows that at late times the rate is a fixed multiple of the black hole mass, independent of the higher-order couplings: $2(D-3)M/(D-4)$ for Gauss-Bonnet and $6M$ for third-order Lovelock in seven dimensions. Because that multiple is not Schwarzschild's $2M$, the growth rate does not reduce to the Einstein result when the couplings vanish, even though the metric and action do; the two normalized rates differ by a constant through the whole evolution. The paper also shows that the late-time limit is approached from above, that the critical time for the onset of growth decreases with the couplings, and that charged Lovelock black holes show a related mismatch between their uncharged limit and the neutral result.

What carries the argument

The machinery has three pieces. The Noether-charge identity $\partial_r(\sqrt{-g}Q^{rt})=\sqrt{-g}L$ rewrites every bulk-action integral as boundary terms, so time derivatives of the action become values of a temperature function times a Wald entropy function at the moving joints. The generalized Gibbons-Hawking-York boundary term, Eq. (36), evaluated at the singularity contributes the constant $\Delta$. The joint term at the moving corner $r_m$, whose motion is fixed by $t=-2r_*(r_m)$, contributes the logarithmic term; the Wald entropy function $\hat{S}(r)$ runs through both bulk and joint pieces and cancels except for the derivative term that vanishes at late times.

What would settle it

Compute the late-time complexity growth for a Gauss-Bonnet black hole using the alternative boundary term cited in the paper's footnote 5 instead of Eq. (36); the paper reports this gives $4M/3\pi$ rather than $(D-3)M/((D-4)\pi)$, so choosing one boundary term decides whether the coupling-independent $\Delta$ is physical.

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Extended reading notes

Core claim

The central claim is that for neutral Lovelock black holes the full time evolution of the gravitational action on the Wheeler-DeWitt patch is captured by $$\frac{dI_{\rm grav}}{dt} = \$\Delta$ - \frac{\hat{S}'(r_m)}{4\pi}|f(r_m)|\log\left(\frac{|f(r_m)|}{\$alpha^{2}$}\right),$$ where $r_m(t)$ is the past joint of the patch and $\hat{S}$ is the Wald entropy function. At late times $r_m\to r_h$, the logarithm vanishes, and the growth rate becomes $dI_{\rm grav}/dt = \Delta$. Evaluating the generalized Gibbons-Hawking-York term at the future singularity gives $\Delta=2(D-3)M/(D-4)$ for Gauss-Bonnet black holes and $\Delta=6M$ for the third-order Lovelock case, so $\Delta/M$ is a pure number with no dependence on the higher-order couplings. Since the limits $\lambda\to 0$ and $r\to 0$ do not commute, the $\lambda\to0$ limit of $\Delta$ is not the Schwarzschild value $2M$; the paper shows numerically that the normalized growth rates differ by a fixed constant for the entire evolution after the critical time.

Load-bearing premise

Everything rests on Eq. (36) being the correct generalized Gibbons-Hawking-York boundary term for Lovelock gravity; the paper notes that another established boundary term yields a different late-time growth rate.

Editorial extensions

If this is right

  • At late times, the action and therefore complexity growth rate for neutral Lovelock black holes is $\Delta/\pi$, with $\Delta$ a coupling-independent constant multiple of the mass.
  • In the vanishing-coupling limit, the normalized growth rate differs from Schwarzschild by a fixed constant for every time after the critical time, so perturbative treatments around Einstein gravity miss a finite offset.
  • Because the next-to-leading-order late-time correction is positive, the growth rate reaches its limiting value from above, violating any conjectured upper bound set by that late-time rate.
  • The critical time $t_c$ measured in thermal time decreases with the Lovelock couplings, so higher-curvature corrections make complexity start growing earlier than in Schwarzschild.
  • For charged Lovelock black holes, the uncharged limit approaches the universal rate $2M/\pi$, yet this remains offset from the neutral black hole's rate by a constant over the whole evolution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If a fundamental principle later fixes the boundary-term convention, the coupling-independent $\Delta$ would become a sharp Complexity=Action prediction distinguishing it from other complexity proposals; until then, the absence of a smooth Einstein limit should be read as convention-dependent.
  • The derivation assumes a single central singularity, so Lovelock solutions with an alternative finite-radius singularity may behave differently; testing those backgrounds would show whether coupling independence survives.
  • The observed inequality $T_+\le T_-$ for charged Lovelock black holes, if universal, predicts that all such charged solutions approach late-time complexity growth from above, a property that can be checked in other charged geometries.
  • The early-time growth laws, constant for five-dimensional Gauss-Bonnet and logarithmic for higher dimensions, could be compared with boundary circuit-complexity computations in CFTs with finite $N$ corrections, where Lovelock couplings correspond to subleading $1/N$ effects.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the full time dependence of holographic complexity, under the CA proposal, for neutral and charged Lovelock black holes. The authors first derive, in Section 3, a general formula for the action growth rate in higher-order gravities by combining the bulk Noether-charge identity (33), a generalized GHY boundary term (36), and the joint/counter-term actions. For planar Lovelock black holes this leads to Eq. (45) and the late-time result Eq. (46), where the asymptotic rate is the constant Δ coming from the GHY term at the singularity. They evaluate Δ for Gauss-Bonnet and third-order Lovelock black holes and find that Δ/M is a pure number independent of the higher-order couplings (e.g., Δ = 2(D-3)M/(D-4) for Gauss-Bonnet). The paper emphasizes that, in the vanishing coupling limit, this rate does not reduce to the Schwarzschild value, that the two nevertheless differ by a constant during the whole evolution, and that the late-time limit is approached from above, violating the conjectured upper bound. For charged Lovelock black holes the authors recover the known late-time rate (98) from reference [34] and study numerically the approach to the uncharged limit, finding a mismatch with the neutral case. Numerical results for the full time dependence are presented for several families of Lovelock black holes.

Significance. If the late-time rate were a robust prediction of the CA proposal for Lovelock gravity, the paper would provide an interesting extension of previous results [39] and a concrete, falsifiable statement about how higher-curvature corrections affect complexity growth. The derivation is coherent, uses no fitted parameters, and is cross-checked against the independent late-time result [34] for charged black holes. However, the central claim is not stable under the choice of surface term: the paper itself notes in Section 4, footnote 5, that an alternative established Gauss-Bonnet GHY term [90] gives a different late-time rate (4M/3π instead of Δ = 2(D-3)M/(D-4)). The coupling-independent constant and the non-smooth Einstein limit are therefore properties of the convention (36), not robust CA predictions. This caveat does not destroy the value of the paper as a careful derivation under a specific prescription, but it requires the conclusions to be reframed and the ambiguity to be addressed in the main text.

major comments (3)
  1. [Section 4, footnote 5] The paper's central late-time claim, Eq. (46) with the coupling-independent Δ, rests entirely on the generalized GHY term (36). Footnote 5 concedes that the alternative, also-established Gauss-Bonnet boundary term of reference [90] gives dC/dt = 4M/3π for the same black holes, rather than Δ = 2(D-3)M/(D-4) from Eq. (61). This means that the constant Δ, the non-smooth λ→0 limit, and the statement that the rate 'does not reduce to Schwarzschild' are not invariants of the CA proposal but depend on the chosen surface-term convention. This issue should be moved from the footnote into the main text, and the corresponding claims in the abstract and Section 7 should be explicitly qualified as holding within the prescription (36).
  2. [Section 4, Eq. (62)] The argument that the non-commutativity of the limits ϵ→0 and λ→0 makes the result 'mathematically sound' explains the origin of the mismatch but does not resolve the physical ambiguity: the same order-of-limits reasoning applies to the alternative GHY term [90], which would give a different value of Δ. To make the central claim robust, the paper needs a physical criterion for selecting the surface term (36) over other admissible boundary terms, or it must explicitly present the result as one possible convention. Without such a criterion, Eq. (46) cannot be described as the complexity growth rate of Lovelock black holes.
  3. [Abstract and Section 7] The statement that the difference from the Schwarzschild result is 'a constant during the whole time evolution' inherits the same surface-term ambiguity. Because Δ enters Eqs. (45), (61), and (77), and because all numerical plots in Sections 4 and 5 normalize by Δ, the curves in Figs. 4, 7, 8, and 9 would be different, including the constant offset from the Schwarzschild curve, if the boundary term of [90] were used. The convention-dependence of these results should be acknowledged explicitly wherever they are summarized.
minor comments (5)
  1. [Section 3.5 and Section 6.3] The numerical procedure is described only verbally; no code or detailed discretization/error estimates are provided. Given that the central results are numerical, it would improve reproducibility to include pseudocode or state the integration method and tolerances.
  2. [Section 3.4, Eqs. (43)-(45)] The arbitrary normalization α of the null normals affects the logarithmic term in Eq. (45) and hence the early-time results such as Eqs. (70) and (81). The text states this only implicitly; it should be stated explicitly which observables (e.g., the late-time rate Δ) are α-independent and which early-time curves depend on the choice α.
  3. [Section 5, paragraph 1] The sentence 'In this section, we further examine the complexity growth rate for charged Lovelock black holes' appears in the introduction of Section 5, but the charged analysis is in Section 6; this organization error should be corrected.
  4. [Section 7, Conclusion] The concluding paragraph states that the paper studied 'third order Lovelock gravities' and then refers to 'section 6' for the organizational structure; the previous organization lists section 6 as the charged case and section 7 as the conclusion. The cross-reference should be updated for consistency.
  5. [Section 4, Fig. 2 caption] The caption reports 'the complexity difference δC = C(t)−C(tc)' but the surrounding text and the left panel indicate that the plotted quantity is the rate dC/dt; please verify that the captions correctly label each panel.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the late-time growth rate is computed from the stated generalized GHY term and cross-checked against independent results; the footnote 5 boundary-term ambiguity is a disclosed robustness caveat, not a circular step.

full rationale

The derivation is self-contained. Section 3 obtains Eq. (45) by adding three separately derived pieces: the bulk piece (34) from the Wald-Iyer Noether identity (33), the GHY piece (37) from the standard generalized boundary term (36) of ref. [86], and the joint piece (44) from the corner-term results [31,35,87]. Delta is not a fitted parameter and is not an input renamed as a prediction; it is defined in (37) as the constant remainder of dIGHY/dt at the singularity and then evaluated explicitly for the Gauss-Bonnet and third-order Lovelock metrics, yielding (61) and Delta = 6M as genuine computations from the metric functions. The charged-sector late-time result (98)-(99) is explicitly checked against the independent result of ref. [34], and the arbitrary normalization alpha drops out at late times because the logarithm term in (45) vanishes as f(rm) -> 0. The relevant in-scope caveat is Section 4, footnote 5, where the paper concedes that the alternative established GB boundary term of ref. [90] gives dC/dt = 4M/3pi instead. This is a disclosed convention-dependence: the CA prediction depends on which boundary term defines the gravitational action, and its robustness as a physical prediction is therefore limited. But that is a premise-ambiguity about the action, not a circular reduction. The paper never fits Delta to data, never imports a uniqueness theorem from its own prior work, and its self-citations [48,49,77] concern the Noether-charge identity (33), which is standard, externally checkable Iyer-Wald mathematics and is unpacked transparently in the Appendix. The order-of-limits explanation (62) is an honest mathematical observation. Hence the derivation is self-contained and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The calculation rests on standard Wald-Iyer Noether charge formalism and the CA conjecture; the most fragile inputs are the single-singularity assumption and the choice of the generalized GHY boundary term. No free parameters are fitted to data, but α is an arbitrary normalization affecting early-time formulas.

free parameters (1)
  • α (null-normal normalization) = arbitrary positive constant
    Introduced in Eq. (42)-(43) to normalize null normals; enters the logarithmic joint term in Eq. (45) and therefore affects the time-dependent part of the action growth rate, but not the late-time limit.
assumptions (5)
  • standard math Wald-Iyer Noether charge identity (33): for a Killing horizon, ∂_r(√-g Q^rt)=√-g L
    Invoked in Section 3.1 to convert bulk action integrals into boundary terms.
  • domain assumption The CA conjecture C=I_grav/π (Eq. 1)
    The whole paper uses Complexity=Action as the definition of holographic complexity.
  • domain assumption The WDW patch terminates at a single central singularity (no alternative singularity before r=0)
    Stated before Fig. 1; constrains couplings and is required for the neutral-black-hole calculation.
  • domain assumption The generalized GHY term (36) from [86] is the correct surface term for Lovelock gravity
    Used to compute Δ; footnote 5 documents an alternative surface term leading to a different rate, so this choice is load-bearing and unresolved.
  • domain assumption For charged black holes, an inner horizon exists and no alternative singularity lies before it
    Section 6; needed for the two-joint WDW patch and Eq. (98).

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Pith. "Pith review of Time dependence of complexity for Lovelock black holes." pith.science (2026). https://pith.science/paper/S7GDNGRJ

@misc{pith2026190809310,
  author       = {Pith},
  title        = {Pith review of: Time dependence of complexity for Lovelock black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7GDNGRJ}},
  note         = {Machine review of arXiv:1908.09310}
}
read the original abstract

We study the general time dependence of complexity for holographic states dual to Lovelock black holes using the "Complexity=Action" (CA) proposal. We observe that at early times, the critical time at which the complexity begins to increase is a decreasing function of the higher order coupling constants, which implies that the complexity evolves faster than that of Schwarzschild black holes. At late times, the rate of change of complexity is essentially determined by the generalised Gibbons-Hawking-York boundary term evaluated at the future singularity. In particular, its ratio to black hole mass is a characteristic constant, independent of the higher order couplings. Thus, in the vanishing coupling limit, the result in general does not reduce to that of Schwarzschild black holes, in spite of that the metric reduces to the latter as well as the gravitational action. In fact, the two differ by a constant during the whole time evolution. Including the next-to-leading order term around late times, we find that as the Einstein case, the late time limit is always approached from above, thus violating any conjectured upper bound given by the late time result. For charged Lovelock black holes, we find that with sufficient charge, the complexity roughly behaves the same as the Einstein case. However, for smaller charges, the two have some significant differences. In particular, unlike the Einstein case, in the uncharged limit the complexity growth rate does not match with the neutral case, differing by a constant in the whole time evolution.

Figures

Figures reproduced from arXiv: 1908.09310 by the authors.

Figure 1
Figure 1. The Wheeler-DeWitt (WDW) patch for a neutral two-sided AdS black hole. It [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Left panel: complexity growth rate for different sizes of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. The critical time measured by 1/T as a function of the Gauss-Bonnet coupling for D = 5 (blue solid), D = 6 (green dashed) and D = 7 (red dotdashed) dimensional black holes, respectively. The orange points denote the value of Schwarzschild black holes in the same dimensions. In Fig.4, we show the numerical results for the time derivative of complexity for D = 5 and D = 7 dimensional Gauss-Bonnet black holes with vari… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: The time derivative of complexity for D = 5 (left) and D = 7 (right) dimensional Gauss-Bonnet black holes with various couplings in the causal region. The dashed lines stand for the Schwarzschild results which have been moved along the vertical axis properly. We have s…
Figure 5
Figure 5. Figure 5: The higher order coupling constants (λ , µ) are strongly constrained by micro￾causality of the theories. The allowed parameters are described by the striped region between the blue curve and the green curve. The black curve is a special case µ = λ 2 and the interval be…
Figure 6
Figure 6. Figure 6: The critical time measured by the thermal time 1 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: The time derivative of complexity for the Lovelock black hole (78) with various [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Left panel: the critical time T tc is plotted as a function of λ. The orange point denotes the value of D = 7 dimensional Schwarzschild black holes. Right panel: the evolution of complexity for the Lovelock black hole (82) with various couplings λ in the causal region …
Figure 9
Figure 9. Figure 9: The time derivative of complexity for the Lovelock black hole (85) with various [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: Fig.10. Since the WDW patch does not terminate at a singularity, the generalised Gibbons [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 10
Figure 10. Figure 10: The WDW patch for a charged AdS black hole with an inner horizon. There are [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Left panel: The time derivative of complexity for [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]
Figure 12
Figure 12. Figure 12: The time derivative of complexity for charged Gauss-Bonnet black hole in [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]
Figure 13
Figure 13. Figure 13: The time derivative of complexity for the charged black hole (112). [PITH_FULL_IMAGE:figures/full_fig_p033_13.png]

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