REVIEW 4 major objections 5 minor 12 references
Extreme Compactness, Extreme Gravity: Higher-Derivative Corrections to ECOs
T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read In Gauss-Bonnet gravity, the ECO compactness scale shifts by order alpha/r0^2, and by alpha^2/r0^4 in EdGB gravity.
desk verdict A readable, well-scoped essay that puts known higher-derivative black hole metrics into the Mathur-Mehta ECO compactness framework, but the EdGB correction it advertises does not follow from the displayed equations. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the compactness scale $s_c$, the proper distance from the ECO surface at $R_{\rm ECO}=r_0+\delta r$ to the horizon radius $r_0$ of a black hole with the same mass. In GR the scale follows from three ingredients: the collapse condition $g_{tt}(R_{\rm ECO},M_{\rm ECO})=0$ with $M_{\rm ECO}=M+E_{\rm vac}$; the near-horizon scaling of the vacuum energy $E_{\rm vac}\propto s^{-(d-1)}$ in $D=d+1$ spacetime dimensions; and the Rindler relation $\delta r\propto s^2$ between coordinate and proper distance. The paper applies these same three ingredients to the corrected EGB and EdGB metrics, then expands the resulting equation in powers of the coupling $\alpha$ to obtain the corrected scales (14)-(17).
What would settle it
Directly compute the vacuum-energy scaling and the proper-distance relation $\delta r(s)$ in the corrected EGB and EdGB metrics to next order; if either acquires an $\alpha$-dependent correction, the factors $1-\frac{4}{d+1}\alpha/r_0^2$ and $1-\alpha^2/r_0^4$ are incomplete. A numerical construction of a static horizonless ECO in these theories with the same surface conditions would settle whether the collapse condition actually yields the claimed scales.
Extended reading notes
Core claim
The central claim is that the compactness scale of an ECO is not fixed once and for all by the GR-based heuristic: it inherits a theory-dependent correction from higher-curvature terms. Inserting the small-coupling EGB metric (Eq. (3)) into the collapse condition $g_{tt}(R_{\rm ECO}, M_{\rm ECO})=0$ gives $s_c^{\rm EGB}\sim (r_0/\ell_p)^{2/(d+1)}\,\ell_p\,[1-\frac{4}{d+1}\alpha/r_0^2+\cdots]$. Inserting the leading EdGB correction (Eqs. (5)-(6)) into the same condition gives $s_c^{\rm EdGB}\sim (r_0/\ell_p)^{1/2}\,\ell_p\,[1-\alpha^2/r_0^4+\cdots]$, with the first correction appearing only at order $\alpha^2$ because the EdGB field equations are first corrected at that order. The paper argues that these shifts are potentially observable under current constraints on $\alpha$ and that the same derivation can be applied to other modified-gravity models.
Load-bearing premise
The derivation assumes that the GR-based collapse condition—setting $g_{tt}$ to zero at the ECO radius with the vacuum-energy-inflated mass—remains valid unchanged once Gauss-Bonnet corrections modify the metric; Section 3.2 asserts the relations $E_{\rm vac}\propto s^{-(d-1)}$ and $\delta r\propto s^2$ for the corrected geometries without re-deriving them.
Editorial extensions
If this is right
- In EGB gravity the ECO surface sits closer to or farther from $r_0$ by a relative amount $\frac{4}{d+1}\alpha/r_0^2$, so the shift is largest for low-mass ECOs with small $r_0$.
- In EdGB gravity the first shift is suppressed by $\alpha^2/r_0^4$, so deviations from the GR scale are much smaller at fixed coupling, and detecting them requires larger $\alpha$ or higher-precision near-horizon probes.
- The two theories predict different powers of $\alpha$ in the compactness correction, so a measurement sensitive to the scale could distinguish EGB from EdGB corrections rather than merely bounding $\alpha$.
- The same derivation is stated to extend to other modified-gravity models such as $f(R)$ gravity, yielding an ECO compactness scale for each theory.
Reading between the lines
- The pattern suggests a rule: the order in $\alpha$ of the first compactness correction matches the order at which the theory's metric is first corrected—linear for EGB, quadratic for EdGB—so other higher-curvature models can be classified by the same criterion.
- The reliability of the corrected scales hinges on the unstated assumption that the vacuum-energy scaling $E_{\rm vac}\propto s^{-(d-1)}$ and the Rindler relation $\delta r\propto s^2$ survive the Gauss-Bonnet deformation of the metric; if they do not, the claimed $\alpha$-scalings would be replaced by whatever the corrected near-horizon geometry dictates.
- Because the compactness scale controls the depth of the near-horizon well, a shifted $s_c$ would propagate into the thermodynamic and radiation properties of ECOs derived in the GR framework, providing an indirect observational handle beyond direct metric measurements.
- A direct numerical construction of a horizonless solution with the corrected EGB/EdGB metric, rather than the perturbative expansion around GR, would test whether the corrected scale is robust or an artifact of the leading-order expansion.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This essay-style paper claims to derive corrections to the compactness scale of horizonless extremely compact objects (ECOs) in Einstein-Gauss-Bonnet (EGB) gravity and Einstein-dilaton-Gauss-Bonnet (EdGB) gravity. Using the heuristic collapse condition g_tt(R_ECO,M_ECO)=0 together with vacuum-energy scaling relations from earlier work by Mathur and Mehta, the author writes down compactness scales s_c^{EGB} and s_c^{EdGB} in Eqs. (14) and (16), expands them in the coupling alpha, and concludes that the corrections are of order alpha/r0^2 in EGB and alpha^2/r0^4 in EdGB (D=4). The abstract emphasizes that observational constraints on alpha could make these corrections relevant for astrophysical ECO candidates.
Significance. If the claimed scaling were established, this would be a useful extension of the universal-thermodynamics program for ECOs to higher-derivative gravity, with a concrete prediction for how the near-horizon structure of horizonless objects deviates from the GR-ECO compactness scale. The qualitative ordering (alpha/r0^2 versus alpha^2/r0^4) is plausible on dimensional grounds, since the EGB metric is corrected at order alpha while the EdGB metric first changes at order zeta^2 = alpha^2/(GM)^4. However, the manuscript is an essay that relies almost entirely on a heuristic derivation from previous work; the central formulas (14)-(17) are asserted without derivation, and the displayed algebra contains concrete inconsistencies. The paper does not provide machine-checked derivations, numerical checks, or parameter-free predictions beyond the scaling itself. Its value as it stands is therefore prospective rather than established.
major comments (4)
- [Section 3.3, Eqs. (14) and (16)] The central formulas for s_c^{EGB} and s_c^{EdGB} are introduced with no derivation. The text says 'Using the heuristic derivation for higher-order gravity models, we derive...' but does not show how the collapse condition (9), the vacuum-energy scaling (12), and the modified metrics (3) or (5)-(6) are combined to produce Eqs. (14) and (16). This is a load-bearing gap because all quantitative claims rest on these expressions. Additionally, Eq. (16) fails a basic consistency check: for zeta=0 the EdGB metric reduces to Schwarzschild, but Eq. (16) with the stated r0≈(2-zeta^2)GM gives a denominator 1-4GM/(3r0)≈1/3, so s_c^{EdGB}(zeta=0) is 3^{1/4} times the standard GR compactness scale. A correct formula must reduce to s_c^{GR} when the coupling vanishes.
- [Section 3.3, Eqs. (16)-(17)] The expansion of Eq. (16) into Eq. (17) is algebraically inconsistent. Using the paper's own horizon r0=(2-zeta^2)GM and keeping terms through O(zeta^2), the fourth root of the ratio in Eq. (16) gives 3^{1/4}(1+7zeta^2/32), and sqrt(r0/l_p) gives sqrt(2GM/l_p)(1-zeta^2/4). Combining these yields s_c ≈ 3^{1/4} sqrt(2M l_p) (1 - zeta^2/32) (for G=l_p^2), i.e. a prefactor 3^{1/4} and a coefficient -1/32 in front of alpha^2/(GM)^4, not the claimed 1 - alpha^2/(GM)^4. The denominator 1-4GM/(3r0) is O(1) (about 1/3) and cannot be dropped or approximated by 1, and the numerator gives a zeta^2/8 contribution, not a coefficient of unity. Therefore the displayed alpha^2/r0^4 statement in Eq. (17) is not supported by Eq. (16).
- [Section 2.2 and Section 3.3] The approximation A2(r0)≈1, used to simplify Eq. (16), is not justified by the displayed A2(r). With r around 2GM, the terms in A2 evaluate to 1 + 13 + 66/20 + 96/40 - 5 ≈ 14.7, so A2(r0) is of order ten, not close to one. Since Eq. (16) explicitly depends on setting A2(r0)=1, the omitted O(1) terms in A2 can change the O(zeta^2) coefficient of the compactness-scale correction. This issue must be addressed before the EdGB result can be accepted.
- [Section 3.3, Eq. (14)] Eq. (14) has a dimensional inconsistency. The factor (r0^d/M)^{1/(d+1)} already has dimensions of length (because M has dimension 1/length in natural units), so multiplying by the explicit l_p makes the right-hand side dimensionally length^2. This is also inconsistent with the first expression in Eq. (15), whose prefactor (M/m_p)^{2/((d+1)(d-2))} l_p has dimension length. The explicit l_p in Eq. (14) should be removed (or the prefactors in Eq. (15) corrected consistently).
minor comments (5)
- [Eq. (3)] The term 'α4G^2M^2' should be typeset as '4αG^2M^2' to avoid ambiguity.
- [References] Reference [6] has a typo in the title: 'Einstein-dilation-Gauss-Bonnet' should be 'Einstein-dilaton-Gauss-Bonnet'.
- [Section 3.2] The scaling relations δr∝s^2 and Evac∝1/s^{d-1} are asserted without derivation and with only a pointer to prior work; since the modified-gravity analysis depends on these relations holding for the corrected metrics, the paper should at least state the assumptions explicitly.
- [Sections 2.2 and 3.3] The treatment of O(1) factors is loose: Eq. (15) uses r0∼(GM)^{1/(d-2)} while Eq. (16) relies on r0≈(2-zeta^2)GM, and the text also says 'used r0∼GM' for EdGB. This ambiguity matters because, for example, the denominator 1-4GM/(3r0) in Eq. (16) changes from about 1/3 to a negative value if r0 is approximated as GM rather than 2GM.
- [Figure 1] The caption for Fig. 1 is present in the text, but the figure itself does not appear in the manuscript; please ensure the figure is included in the final version.
Circularity Check
No significant circularity: the alpha-correction orders come from externally computed GB/EdGB metrics, not from the paper's assumptions.
full rationale
The central scaling results are obtained by inserting the known Boulware-Deser (EGB) and Mignemi-Stewart/Yunes-Stein (EdGB) metrics, which are external to the authors, into the collapse condition g_tt(R_ECO,M_ECO)=0. The compactness-scale framework itself is taken from the authors' prior work and Section 3.2 is explicitly labeled 'heuristic,' but the target alpha/r0^2 and alpha^2/r0^4 orders are not assumed in that framework; they follow from the structure of the modified metric functions. No parameter is fitted to data, and no equation is defined in terms of the claimed result, so the derivation does not reduce to its inputs by construction. The main issue is a non-circular correctness concern: the expansion of Eq. (16) into Eq. (17) appears algebraically inconsistent, for example the denominator 1-4GM/(3r0) is of order one at r0 approximately (2-zeta^2)GM and A2(r0) is not approximately 1, but an algebraic error is not a tautology. Therefore the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The ECO criteria ECO1-ECO3 and the collapse condition g_tt(R_ECO,M_ECO)=0 define the compactness scale s_c.
- domain assumption The near-horizon relations delta r proportional to s^2 and E_vac proportional to s^{-(d-1)} remain valid after Gauss-Bonnet corrections.
- domain assumption The perturbative metrics Eq. (3) and Eqs. (5)-(6) accurately describe the geometry at the ECO radius and its horizon.
Cite this review
Pith. "Pith review of Extreme Compactness, Extreme Gravity: Higher-Derivative Corrections to ECOs." pith.science (2026). https://pith.science/paper/UXXKXVIY
@misc{pith2026250509049,
author = {Pith},
title = {Pith review of: Extreme Compactness, Extreme Gravity: Higher-Derivative Corrections to ECOs},
year = {2026},
howpublished = {\url{https://pith.science/paper/UXXKXVIY}},
note = {Machine review of arXiv:2505.09049}
}
abstract
Higher-derivative gravity theories offer insights into the behavior of extremely compact objects (ECOs). Focusing on Gauss-Bonnet (GB) and Einstein-dilaton-Gauss-Bonnet (EdGB) gravity, we derive the compactness scale in these models and demonstrate how higher-curvature corrections lead to deviations from the standard ECO compactness scale. The corrections are of order $\alpha/r_0^2$ in EGB gravity and $\alpha^2/r_0^4$ in EdGB gravity, where $\alpha$ is the coupling constant and $r_0$ is the horizon radius corresponding to the mass of the ECO. Observational constraints suggest these effects could be significant in certain astrophysical systems, providing a new perspective on the nature of extremely compact objects in models of modified gravity.
Figures
Reference graph
Works this paper leans on
-
[1]
The Einstein tensor and its generalizations,
D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. 12 (1971) 498–501
1971
-
[2]
Towards a Non-singular Paradigm of Black Hole Physics,
R. Carballo-Rubio et al. , “Towards a Non-singular Paradigm of Black Hole Physics,” arXiv:2501.05505 [gr-qc]
-
[3]
S. D. Mathur and M. Mehta, “The Fuzzball Paradigm,” arXiv:2412.09495 [hep-th]
-
[4]
Renormalization of higher-derivative quantum gravity,
K. S. Stelle, “Renormalization of higher-derivative quantum gravity,” Phys. Rev. D 16 (Aug, 1977) 953–969. https://link.aps.org/doi/10.1103/PhysRevD.16.953
-
[5]
F. Moura and R. Schiappa, “Higher-derivative corrected black holes: Perturbative stability and absorption cross-section in heterotic string theory,” Class. Quant. Grav. 24 (2007) 361–386, arXiv:hep-th/0605001
arXiv 2007
-
[6]
B. Gao, S.-P. Tang, H.-T. Wang, J. Yan, and Y.-Z. Fan, “Constraints on Einstein-dilation-Gauss-Bonnet gravity and the electric charge of compact binary systems from GW230529,” Phys. Rev. D 110 no. 4, (2024) 044022, arXiv:2405.13279 [gr-qc]
arXiv 2024
-
[7]
The universality of black hole thermodynamics,
S. D. Mathur and M. Mehta, “The universality of black hole thermodynamics,” Int. J. Mod. Phys. D 32 no. 14, (2023) 2341003, arXiv:2305.12003 [hep-th]
arXiv 2023
-
[8]
String-generated gravity models,
D. G. Boulware and S. Deser, “String-generated gravity models,” Phys. Rev. Lett. 55 (Dec, 1985) 2656–2660. https://link.aps.org/doi/10.1103/PhysRevLett.55.2656
Show all 12 references
-
[9]
Schwarzschild field in n dimensions and the dimensionality of space problem,
F. Tangherlini, “Schwarzschild field in n dimensions and the dimensionality of space problem,” Nuovo Cimento 27 no. 1, (1963) 636–651
1963
-
[10]
Charged black holes in effective string theory,
S. Mignemi and N. R. Stewart, “Charged black holes in effective string theory,” Phys. Rev. D 47 (1993) 5259–5269, arXiv:hep-th/9212146. 10
1993 arXiv
-
[11]
Non-Spinning Black Holes in Alternative Theories of Gravity,
N. Yunes and L. C. Stein, “Non-Spinning Black Holes in Alternative Theories of Gravity,” Phys. Rev. D 83 (2011) 104002, arXiv:1101.2921 [gr-qc]
2011 arXiv
-
[12]
The universal thermodynamic properties of Extremely Compact Objects,
S. D. Mathur and M. Mehta, “The universal thermodynamic properties of Extremely Compact Objects,” arXiv:2402.13166 [hep-th] . 11
Reviewed August 15, 2026 · model on record in the stance chip above.
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