Holographic Timelike c-function

Dimitrios Giataganas

REVIEW 4 major objections 4 minor 2 cited by

Holographic Timelike c-function

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Timelike entanglement entropy supplies a monotonic c-function for non-relativistic RG flows.

desk verdict New timelike-entanglement c-function with a real proof gap that narrows the claimed generality. read the letter →

arxiv 2505.20459 v2 pith:CUH4OYNO submitted 2025-05-26 hep-th cond-mat.stat-mech

Dimitrios Giataganas This is my paper

classification hep-thcond-mat.stat-mech

keywords timelikeentanglemententropyholographicc-theoremrenormalizationgroupflowLifshitztheorieshyperscalingviolationnullenergyconditionnon-relativisticholographypseudoentropy

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

The paper proposes a new c-function built from holographic timelike entanglement entropy, a pseudoentropy defined by extremal surfaces anchored on a boundary time interval. The claim is that this quantity is monotonic along renormalization-group flow in any homogeneous holographic theory, including Lifshitz and hyperscaling-violating theories where ordinary entanglement entropy fails. If correct, it gives a measure of the number of degrees of freedom that behaves sensibly in non-Lorentz-invariant systems, where no such universal measure was known. The monotonicity is shown to follow from the null energy conditions, thermodynamic stability, and an effective-dimension inequality, $d_x \le d_{xr}$.

What carries the argument

The central object is the timelike entanglement entropy defined by extremal surfaces anchored to a time interval, split into a spacelike part (real contribution) and a timelike part (imaginary contribution). The argument is carried by the identity $c = c_{\mathrm{Re}} + c_{\mathrm{Im}}$, whose derivative along the radial flow $r_m$ cancels the boundary terms because the spacelike and timelike surfaces merge smoothly in the deep IR ($t'^2_{\mathrm{Re},b} \simeq t'^2_{\mathrm{Im},b}$, $T_\infty \to \infty$), leaving the compact integral (20). The effective dimension $d_x$, fixed by scaling symmetry or by $d_{xr} = \Lambda'(r_f)^2/(\Lambda''(r_f) + B'(r_f)\Lambda'(r_f))$, controls the sign of the integrand.

What would settle it

For a homogeneous holographic background that satisfies the null energy conditions and thermodynamic stability but contains a horizon or infrared wall, compute $\partial c/\partial r_m$ and search for a sign change or non-monotonicity in $c$; a concrete counterexample would settle the generality of the claim.

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Extended reading notes

Core claim

The central discovery is that the combination $c = c_{\mathrm{Re}} + c_{\mathrm{Im}}$, with $c_{(\mathrm{Im},\mathrm{Re})} = t_{(\mathrm{Im},\mathrm{Re})}^{d_x}\,\partial S_{(\mathrm{Im},\mathrm{Re})}/\partial t_{(\mathrm{Im},\mathrm{Re})}$, built from the real and imaginary parts of holographic timelike entanglement entropy, is monotonic along the holographic RG flow. The derivative with respect to the radial flow parameter reduces to $\partial c/\partial r_m = 2 e^{\Lambda_m} T_\infty^{d_x-1} d_x \Lambda'_m \int_0^{T/2} dt\, \frac{1}{\Lambda'}\bigl(\frac{\Lambda'}{d_x} - \frac{\Lambda''}{\Lambda'} - B'\bigr)$, which is nonnegative when the null energy conditions, thermodynamic stability, and $d_x \le d_{xr}$ hold. The proof works in Poincaré-invariant theories as a consistency check, and extends to Lifshitz and hyperscaling-violating fixed points, where previous entanglement-entropy c-functions fail because entanglement monotonicity is violated.

Load-bearing premise

The proof of monotonicity assumes the spacelike and timelike extremal surfaces meet smoothly at a deep-infrared point, with no horizon or infrared wall, so the boundary terms in the derivative cancel exactly.

Editorial extensions

If this is right

If the claim holds, every homogeneous holographic theory satisfying the null energy conditions and thermodynamic stability has a monotonic c-function, extending the holographic c-theorem beyond Lorentz-invariant fixed points.
In Lifshitz theories the c-theorem reduces to $z \ge z_r$ for a slowly varying Lifshitz exponent, giving a concrete constraint on allowed non-relativistic RG flows.
In hyperscaling-violating theories the monotonicity is tied to the effective spatial dimension $d-\theta$, so the c-function sees the reduced dimensionality of modes.
Because the two definitions of the c-function are equivalent, the quantity can be read directly from the boundary time interval $T$ and the total timelike entropy $S$, which makes it a potentially observable probe.
The bounds on $\partial c/\partial T$ derived by the author show that the same conditions that guarantee monotonicity in $r_m$ also guarantee monotonicity when flowing to larger boundary time intervals.

Reading between the lines

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

A possible direct test is to compute the timelike pseudoentropy of a free non-relativistic field theory, such as a Lifshitz scalar at $z=2$, under a relevant perturbation and check whether $c$ is monotonic; this would probe the claim outside holography.
If the merging condition also holds in backgrounds with a horizon, the construction could yield an RG monotone for finite-temperature or finite-density systems, which the paper does not analyze.
The inequality $d_x \le d_{xr}$ may be connected to a quantum-information bound in the dual theory, though the paper does not explore that link.
In confining geometries with IR walls the boundary terms may not cancel; checking whether monotonicity survives there would test the robustness of the construction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a genuinely new construction of a holographic c-function from timelike entanglement entropy, and the derivative computation is explicit enough to check. The gap is that the monotonicity proof relies on a merging condition that the paper itself restricts to spacetimes without horizons or IR walls, while the abstract claims applicability to all homogeneous non-relativistic theories. That gap is central, not cosmetic.

What the paper does well: the definition (9)-(10) is natural, and the recasting in terms of an effective dimension d_x is useful. The paper works out the derivative ∂c/∂r_m in detail, and the end matter fills in most of the algebra that the main text skips. The consistency check for Poincaré-invariant theories is clean. The Lifshitz and hyperscaling-violating examples are worked out concretely, and the conditions NEC + thermodynamic stability + d_x ≤ d_xr are plausible physical inputs. The citation pattern is fine, building on the timelike entropy literature without overclaiming novelty.

Where the soft spots are. First and foremost, Eq. (18) — the smooth merging of the spacelike and timelike surfaces — is asserted, not proven, and the text says it holds only in theories without horizons or IR walls. For a black hole horizon or a domain wall ending at finite Λ_b, the boundary terms in Eq. (19) survive, and Eq. (20) does not follow. The abstract's claim of generic homogeneous theories is therefore too broad. The author should either prove the merging condition for those cases or explicitly restrict the theorem. Second, the condition d_x ≤ d_xr is assumed for general RG flows; the examples show it can be satisfied, but the paper does not derive it as a consequence of the physical conditions. Third, some sign arguments in the hyperscaling-violating section are compressed; they are plausible if Eq. (20) holds, but that brings you back to the first issue.

Verdict: this deserves peer review. The problem is real, the construction is novel, and the gaps are fixable. I'd send it out and ask a referee to focus on the merging condition and the generality claim. I wouldn't cite it in its current form, but a revised version that narrows the scope or proves the condition would be worth citing.

Referee Report

4 major / 4 minor

Summary. The paper proposes a holographic c-function built from timelike entanglement entropy in homogeneous holographic spacetimes. The construction defines c = c_Re + c_Im with c_(Im,Re) = t_(Im,Re)^{d_x} ∂S_(Im,Re)/∂t_(Im,Re), or equivalently c = T^{d_x} ∂S/∂T, and claims that ∂c/∂r_m ≥ 0 along RG flows. The central formula is Eq. (20), which expresses ∂c/∂r_m as an integral involving geometric data, and the monotonicity claim is then checked for Poincaré-invariant, Lifshitz, and hyperscaling-violating theories under NEC, thermodynamic stability, and the condition d_x ≤ d_xr. The proof of Eq. (20) relies on cancellation of boundary terms using the deep-IR merging of spacelike and timelike extremal surfaces.

Significance. If the monotonicity claim is fully established, the paper would supply a much-needed RG monotone for non-relativistic holographic theories, where standard entanglement entropy fails to be monotonic. The construction is explicit and natural, the final formula (20) is compact, and the end matter provides useful algebraic detail. The use of NEC and thermodynamic stability as physical constraints is sensible. However, the proof as written establishes only a conditional statement, and the claimed genericity for all homogeneous holographic theories is not supported by the derivation.

major comments (4)

[Sec. 2 and Sec. 3.C, Eq. (19)] The derivation of the main result (20) from (19) cancels the boundary term (1/Λ'_b)(t'_Re,b − t'_Im,b) using condition (18) and the assumption T_Re ≃ T_Im ≃ ∞. The paper explicitly restricts this assumption to 'theories without horizons or IR walls' (Sec. 2). For the generic homogeneous metric (1), and for Lifshitz or hyperscaling-violating RG flows that end at a finite IR point or contain a horizon, e^{2Λ_b} → 0 is not guaranteed; the NEC (23)-(24) and thermodynamic stability (26) do not imply it. Therefore the boundary term survives in general and Eq. (20), and hence inequality (22), is not established for the class of theories claimed in the abstract.
[Sec. 3.C, Eq. (18)] The smooth-merging condition t'^2_Re,b ≃ t'^2_Im,b is not a consequence of the normal-vector expression (17). When e^{2Λ_b} → 0, Eq. (17) gives |T|² → −s^{-1}, which fixes the signature of the normal vector for each branch but does not equate the two derivatives t'_Re,b and t'_Im,b. The argument from Ref. [21] is quoted but not reproduced in a way that establishes (18) for a general homogeneous flow. Since the cancellation in (19) depends entirely on (18), this is a load-bearing gap.
[Sec. 4.B.2, hyperscaling-violating subregime C] For the subregime 0 ≤ α ≤ 1, the paper concludes ∂c/∂r_m ≤ 0 and calls this the c-theorem, but Eqs. (20)-(22) were derived with the UV boundary located at infinity. When the UV boundary is at r = 0, the boundary term t'_Re,∂/Λ'_∂ in (19), the integration limits, and the sign convention for monotonicity must all be re-evaluated. The paper does not provide this separate derivation, so the claim for subregime C is not supported by the preceding calculation.
[Sec. 3.E and Sec. 4.B.1, Eq. (31)] The monotonicity condition is ultimately reduced to d_x ≤ d_xr, but this is an additional assumption rather than a consequence of NEC or thermodynamic stability. For Lifshitz flows with a slowly varying exponent z_r(r), taking d_x from the IR fixed point makes z ≥ z_r hold by construction; if d_x is instead fixed at the UV fixed point, the inequality can fail. The paper therefore proves a conditional statement, not the unconditional monotonicity for 'all such theories' claimed in the abstract. The value of d_x for a generic non-fixed-point flow is also never specified, since the scaling argument (30) applies only at scale-covariant points.

minor comments (4)

[Sec. 1, paragraph 4] The sentence 'it is not surprising that fails to support the existence of a proper c-function' is missing a subject and should read 'that entanglement entropy fails to support...'.
[Sec. 4.B.1] The text states A = r and B = zr with the boundary at r → ∞ and the deep IR at r_b → 0. With this coordinate choice e^{2Λ_b} → 1, which contradicts the assumption e^{2Λ_b} → 0 used in Sec. 3.C. The coordinate convention (for example ilde r = e^r with deep IR r → −∞) should be stated explicitly.
[Sec. 3, Eq. (10)] The phrase 'a unique and naturally motivated c-function' is stronger than supported, since the definition depends on the choice of d_x and on normalization conventions; d_x is specified only for scale-covariant examples.
[End Matter EM.2, Eq. (52)] The statement that 'the c-theorem gives ... ∂T c ≤ 0' is asserted without derivation; from (51)-(52) this requires conditions on the denominator and on the sign of d_x that should be spelled out.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the timelike c-function and its monotonicity are derived from the holographic entropy functional, with explicit geometric and energy conditions, not from fitted data or self-referential definitions.

full rationale

The paper's central object, c = t^dx ∂S/∂t in Eq. (10), is constructed directly from the holographic timelike entanglement entropy action (2), and its derivative along the RG flow is computed explicitly in Eqs. (15), (16), and (20) using the equations of motion, Leibniz rule, and integration by parts. The final monotonicity statement is conditional on the null energy conditions, thermodynamic stability, and the effective-dimension condition d_x ≤ d_xr, as made explicit in Eqs. (31) and the surrounding text. These are stated physical/geometric conditions, not parameters fitted to data, and the derivation does not assume the conclusion. The citation to the author's earlier work [21] supplies the geometric merging condition (18), but that condition is a property of the extremal surfaces used in the timelike entropy construction, not an assertion equivalent to the c-theorem itself. The paper also explicitly acknowledges the restriction to theories without horizons or IR walls, which is an honest limitation rather than a circularity. There are self-citations, but none is load-bearing in the sense that the claimed result reduces to those citations by construction. Overall, the derivation is self-contained once the geometric inputs and stated assumptions are accepted, and no circular step was identified.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The analysis introduces no new particles or forces. The free parameter dx is an effective dimension that is fixed by scaling arguments, not fitted to data. The main axioms are the holographic setup and the smooth merging assumption of the extremal surfaces, which is essential but not fully proven.

free parameters (2)

dx (effective dimension) = dx = (beta + alpha(d-2))/(beta-delta-1), e.g., dx = 1 + (d-2)/z for Lifshitz
The parameter dx is introduced in (10) and later identified as an effective dimension via scaling arguments (30). It is not fitted to data, but it is a parameter chosen so that the c-function is dimensionless and the monotonicity condition dx <= dxr holds. It is determined from the metric scalings, not from a best fit.
dxr (radial effective dimension) = dxr = Lambda'(r_f)^2 / (Lambda''(r_f) + B'(r_f) Lambda'(r_f))
Defined in (27) as the value of dx for which the integrand in (20) vanishes. It is a derived function of the metric, not a free fit parameter. It is used to state the monotonicity condition dx <= dxr.

assumptions (5)

domain assumption The bulk spacetime is homogeneous and can be written as ds^2 = -e^{2B} dt^2 + e^{2A} dx^2 + dr^2.
This restricts the analysis to homogeneous backgrounds, excluding anisotropic or inhomogeneous RG flows. It is stated in section 2.
domain assumption The null energy conditions (23) and (24) hold.
These are imposed as physical constraints on the bulk geometry. They are standard but for Lifshitz and hyperscaling-violating theories they are not always satisfied.
domain assumption Thermodynamic stability, as expressed by the specific heat condition (26), holds.
This is imposed to avoid instabilities in the black hole background. It is a standard but nontrivial condition.
ad hoc to paper The spacelike and timelike extremal surfaces merge smoothly at r_b, with t'^2_{Re,b} ≈ t'^2_{Im,b} and T_Re ≈ T_Im ≈ infinity.
This is the key geometric assumption in section 3.C. It is stated without a proof and is essential for the cancellation of boundary terms in (19).
domain assumption The UV boundary is at r = infinity, and the surface intersects the boundary orthogonally, so t'_{Re,∂} = 0.
This is standard in holographic setups, but for hyperscaling-violating subregime C the boundary is at r=0, so this assumption must be adjusted.

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Cite this review

Pith. "Pith review of Holographic Timelike c-function." pith.science (2026). https://pith.science/paper/CUH4OYNO

@misc{pith2026250520459,
  author       = {Pith},
  title        = {Pith review of: Holographic Timelike c-function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUH4OYNO}},
  note         = {Machine review of arXiv:2505.20459}
}

read the original abstract

The integration of high-energy degrees of freedom along the renormalization group (RG) flow in Poincar\'e-invariant theories can be captured by a monotonic c-function. For such theories, holographic monotonic c-functions have been constructed using entanglement entropy. However, in theories with broken Lorentz invariance, such constructions generally fail, reflecting both the violation of the entanglement RG monotonicity and its limitations in capturing certain properties of non-relativistic RG flows. Since many quantum many-body systems lack Lorentz invariance, it is of significant importance to identify a quantity that reflects the decrease in degrees of freedom along non-relativistic RG flows. We show that the recently introduced holographic timelike entanglement entropy naturally gives rise to a new c-function applicable to all such theories. We further demonstrate the existence of this c-function in theories with Lifshitz and hyperscaling-violating fixed points, showing that, provided the null energy conditions and thermodynamic stability are satisfied, the proposed c-function exhibits the expected monotonic behavior along the RG flow.

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Forward citations

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [21]

Timelike entanglement entropy,

Kazuki Doi, Jonathan Harper, Ali Mollabashi, Tadashi Takayanagi, and Yusuke Taki, “Timelike entanglement entropy,” JHEP05, 052 (2023), arXiv:2302.11695 [hep- th]

arXiv 2023

[2] [1]

Introduction.The renormalization group (RG) flow in quantum field theories reveals a rich structure and serves as a powerful tool for their analysis. One of its most well-known features is the existence of a so- called c-function, which decreases along RG flows and coincides with the central charge at the conformal fixed points of the theory. This propert...

work page

[3] [2]

Ho- mogeneous spacetimes can always be diagonalized and brought to the form of (1); therefore, this represents the most generic case

Holographic Setup.We consider the homogeneous holographic spacetime in the following coordinate system ds2 d+1 =−e 2B(r) dt2 +e 2A(r)dx2 +dr 2 ,(1) whered−1 is the number of dimensions of the spatial plane spanned by⃗ xand the boundary of the space-time is taken to be atr→ ∞, without loss of generality. Ho- mogeneous spacetimes can always be diagonalized ...

arXiv 2025

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Holographic Timelike c-function.The timelike entropy supports a holographic, monotonicc-function for any type of homogeneous theory. To construct a suitable function, we consider the derivatives oft s andS s, cor- responding to the timelike and spacelike surfaces, with respect to the turning pointr m of the timelike surface. We will demonstrate that a uni...

work page

[5] [4]

Holographic Timelikec-theorem. 4.A. Timelikec-theorem in Poincare Invariant Theories.Let us consider a conformal theory with A(r) =B(r), with a bulk geometry described by the metric (1). The evolution of the geometry along the RG flow is encoded in the conformal factorA(r). Such ge- ometries are typical solutions of Einstein gravity coupled to a scalar fi...

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We demonstrated that non- Lorentz-invariant renormalization group flows obey time- like entanglement RG monotonicity, thereby establish- ing a deep connection between the two

Discussion.We have constructed ac-function via holographic timelike entanglement entropy, applicable to non-relativistic theories. We demonstrated that non- Lorentz-invariant renormalization group flows obey time- like entanglement RG monotonicity, thereby establish- ing a deep connection between the two. This represents a significant improvement over pre...

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A. B. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett.43, 730–732 (1986)

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Is There a c Theorem in Four- Dimensions?

John L. Cardy, “Is There a c Theorem in Four- Dimensions?” Phys. Lett. B215, 749–752 (1988)

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