Energy conditions for regular black holes in EFT of gravity

Alexey S. Koshelev; Anna Tokareva; Yang Liu; Ziyue Zhu

REVIEW

Energy conditions for regular black holes in EFT of gravity

Ziyue Zhu, Alexey S. Koshelev, Yang Liu, Anna Tokareva

General Relativity and Quantum Cosmology · Submitted 21 Aug 2025 · 1 reference

Pith's verdict: Useful conditional result on EFT-supported regular black holes, but the paper's own 'given that' condition is load-bearing and unverified in the visible text.

T0 · AI review

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arXiv ID2508.15745 (version 1)

Pith IDZ7O3C3R6, Pith's permanent record of this paper and its review

CitationAlexey S. Koshelev, Anna Tokareva, Yang Liu, Ziyue Zhu. "Energy conditions for regular black holes in EFT of gravity." arXiv:2508.15745 (2025). https://pith.science/paper/Z7O3C3R6

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Claim paper

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Editor's note

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The short version: this is a sensible, honestly-framed paper that does something new—checking WEC/NEC for regular black hole metrics embedded in an EFT of gravity with cubic and quartic Riemann corrections, and deriving constraints on the EFT coefficients. The headline result, that matter can satisfy WEC and NEC outside the horizon for a tuned but finite region of coefficient space, is a legitimate existence argument. The negative result, NEC violated inside the horizon in all cases examined, is the sturdiest takeaway and likely to survive contact with referees.

Credit where due: the authors do not oversell. They explicitly say a vacuum solution without extra matter is beyond scope, and they flag the EFT-validity condition in the abstract itself. The strategy of defining matter as the residual of the EFT field equations is standard for this kind of embedding problem, and the parameter scan appears to be the appropriate tool.

Now the soft spots. The stress-test concern is on target: the claim that the EFT description is valid in the interior is conditioned on the maximal curvature being below the cutoff, but the visible text does not show that the WEC-satisfying parameter regions overlap with the curvature-bounded regions. That is not a minor omission—it is the difference between a conditional statement with no known instance and an actual construction. The phrase 'given that' in the abstract is doing a lot of work. A related but smaller issue is that 'wide range of parameters' is never quantified; the reader cannot tell whether that range is a sliver or a substantial region.

I also want to flag that the EFT expansion parameter needs to be small, not merely below the cutoff. If the maximal interior curvature is a sizable fraction of the cutoff, the cubic and quartic terms are not obviously a controlled truncation. The paper does not show this.

The matter sector being prescribed rather than derived is a limitation, but a standard one; I would not call it a flaw. The citation pattern looks reasonable for what I could see, and the paper engages honestly with the regular-BH literature.

My bottom line: this deserves a serious referee. The conditional claim is coherent, the negative result is interesting, and the missing overlap can in principle be checked. I would send it to review, with the expectation that the authors be asked to demonstrate the overlap explicitly and quantify the allowed parameter range. I would not cite it myself until that is done.

Referee report

Major objections (3)

[Abstract; §III–IV] The abstract conditions interior EFT validity on the maximal curvature being bounded below the cutoff, but the visible text never verifies that this condition is met at any parameter point where WEC/NEC hold. Boundedness below the cutoff is necessary but not sufficient: the dimensionless expansion parameters of the cubic and quartic Riemann terms must be small. The overlap between the WEC/NEC-allowed region of coefficient space and the EFT-validity region is never demonstrated. Without this, the paper proves only a conditional existence statement, not the claimed regular black hole in EFT.
[§IV] The matter sector is defined only as the residual of the EFT field equations, with no independent microphysical content. The construction is therefore a co-tuning: coefficients, metric ansatz, and residual matter are chosen together so that energy conditions hold. WEC/NEC are necessary but not sufficient for an acceptable matter source; the paper should discuss stability, hyperbolicity, and whether the residual stress-energy tensor can be derived from a matter Lagrangian, or at least state explicitly why the residual-matter interpretation is physically meaningful.
[Abstract; §V] The claim that WEC is satisfied 'in a wide range of parameters' is not quantified in the provided text. No explicit ranges for the EFT couplings or the regularization/deformation parameter are given, and no plots or tables are shown. To make the result reproducible and falsifiable, the authors should provide the parameter-space region in closed form or give a precise numerical scan with the metric ansatz and coefficient ranges stated.

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The paper

The paper claims

The paper argues that in an effective field theory of gravity with cubic and quartic Riemann corrections, regular black holes are supported by matter obeying the weak and null energy conditions outside the horizon, while the interior stays

The reading

The paper asks whether a black hole whose curvature stays finite everywhere—so it never leaves the regime where effective field theory applies—can be supported by matter that does not violate basic energy conditions. Working with gravity actions that include Riemann-tensor cube and fourth-power corrections, the authors treat the matter needed to sustain a regular black hole metric as the residual of the higher-curvature field equations and check the weak and null energy conditions. They find that both energy conditions hold everywhere outside the horizon; inside, the weak energy condition holds for a wide range of parameters while the null energy condition fails in every case examined. If correct, regular black holes are viable solutions within the effective field theory of gravity rather than requiring new physics at a singularity.

What carries the argument

The central object is the effective field theory action containing Einstein gravity plus parity-even Riemann-tensor cube and fourth-power terms. The mechanism is to use the EFT field equations, interpret the difference between the Einstein tensor and the higher-curvature corrections as the matter stress-energy tensor required to support a chosen regular black hole metric, and then impose the WEC and NEC inequalities on that tensor as constraints on the EFT coefficients.

Extended reading notes

Core claim

The central claim is that regular black hole metrics can be embedded as solutions of the effective field theory of gravity with cubic and quartic Riemann corrections, with the supporting matter satisfying the weak energy condition (WEC) and null energy condition (NEC) everywhere outside the horizon. Inside the horizon, the matter obeys WEC in a broad parameter range, but NEC is always violated; the interior remains within EFT validity because its maximal curvature is finite and bounded by the cutoff. This is established by deriving the effective matter stress tensor from the higher-curvature field equations and evaluating the energy conditions on that tensor.

Load-bearing premise

The central premise is that the regions of EFT coefficient space where the matter energy conditions are satisfied also keep the maximal interior curvature below the energy scale where the effective theory stops working; if those regions fail to overlap, the regular solution cannot be both supported by physical matter and fully described within the effective field theory.

Editorial extensions

If this is right

If regular black holes with bounded curvature can be embedded in EFT of gravity, the classical singularity theorems are bypassed not by abandoning general relativity but by allowing matter and higher-curvature corrections to smooth the interior.
The outer region of such a black hole is consistent with standard energy conditions, so the solution does not require exotic matter for a distant observer.
The universal NEC violation inside the horizon means the interior matter is nevertheless exotic, which may feed into instability or geodesic-incompleteness questions.
The requirement that maximal curvature stays below the cutoff ties the admissible EFT parameter space to geometric features of the regular metric.

Reading between the lines

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

A natural next step is to scan the full EFT coefficient space for an explicit overlap between the WEC-satisfying regions and the regions where the central curvature is below the cutoff; the paper's conditional wording suggests this overlap is assumed rather than proven.
The same residual-stress-tensor method could be applied to other regular black hole metrics to see whether the pattern—WEC outside, NEC violation inside—is a generic feature of higher-curvature EFT.
Because NEC is violated inside the horizon in all cases, the interior matter cannot be described by a conventional perfect fluid with positive pressure; any microphysical completion would need non-standard couplings, leaving the physicality of the interior an open question.

Original abstract

As Einstein's gravity is a non-renormalizable theory, it can be a good description of physics only at the scales of energy or spacetime curvature below the Planck mass. Moreover, it requires the presence of an infinite tower of higher-derivative corrections, as required in the framework of effective field theory (EFT). Black holes, known to be vacuum solutions in Einstein's gravity, necessarily have singularities in the center, where both Einstein's gravity and low-energy EFT expansions break down. In this work, we address the question of whether, in the presence of matter, regular solutions looking like black holes from outside do exist. We show that the matter distribution supporting the regular black hole solution in the presence of Riemann tensor cube and Riemann tensor to the fourth power EFT corrections satisfies positivity of energy (also called weak energy condition, WEC) and null energy condition (NEC) everywhere outside the horizon. Unlike the case of singular solutions, the EFT description is also valid in the interior of such an object, given that the maximal curvature is bounded and does not exceed the cut-off scale. We found that in a wide range of parameters, WEC is satisfied inside the horizon, but NEC is violated inside the horizon in all cases.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,712 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.56 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

As Einstein's gravity is a non-renormalizable theory, it can be a good description of physics only at the scales of energy or spacetime curvature below the Planck mass. Moreover, it requires the presence of an infinite tower of higher-derivative corrections, as required in the framework of effective field theory (EFT). Black holes, known to be vacuum solutions in Einstein's gravity, necessarily have singularities in the center, where both Einstein's gravity and low-energy EFT expansions break down. In this work, we address the question of whether, in the presence of matter, regular solutions looking like black holes from outside do exist. We show that the matter distribution supporting the regular black hole solution in the presence of Riemann tensor cube and Riemann tensor to the fourth power EFT corrections satisfies positivity of energy (also called weak energy condition, WEC) and null energy condition (NEC) everywhere outside the horizon. Unlike the case of singular solutions, the EFT description is also valid in the interior of such an object, given that the maximal curvature is bounded and does not exceed the cut-off scale. We found that in a wide range of parameters, WEC is satisfied inside the horizon, but NEC is violated inside the horizon in all cases.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	As Einstein's gravity is a non-renormalizable theory, it can be a good description of physics only at the scales of energy or spacetime curvature below the Planck mass.	background 1.00	0.04	0.02	asserted 0.60
s1	Moreover, it requires the presence of an infinite tower of higher-derivative corrections, as required in the framework of effective field theory (EFT).	background 1.00	0.05	0.02	asserted 1.00
s2	Black holes, known to be vacuum solutions in Einstein's gravity, necessarily have singularities in the center, where both Einstein's gravity and low-energy EFT expansions break down.	background 0.97	0.05	0.02	asserted 1.00
s3	In this work, we address the question of whether, in the presence of matter, regular solutions looking like black holes from outside do exist.	contribution 0.61	0.27	0.04	unclear 0.80
s4	We show that the matter distribution supporting the regular black hole solution in the presence of Riemann tensor cube and Riemann tensor to the fourth power EFT corrections satisfies positivity of energy (also called weak energy condition, WEC) and null energy condition (NEC) everywhere outside the horizon.	result 0.99	0.96	0.58	asserted 1.00
s5	Unlike the case of singular solutions, the EFT description is also valid in the interior of such an object, given that the maximal curvature is bounded and does not exceed the cut-off scale.	interpretation 0.53	0.57	0.11	asserted 0.99
s6	We found that in a wide range of parameters, WEC is satisfied inside the horizon, but NEC is violated inside the horizon in all cases.	result 1.00	0.97	0.46	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Geometric No-Go Theorem for Regular Black Holes in Scalar-Tensor Theories
gr-qc 2026-11 accept novelty 8.0 of 10

Static two-horizon regular black holes with a radial scalar field are ruled out in a broad class of scalar-tensor theories because the interior maximum of the scale ratio conflicts with a positive odd-parity tensor speed.
Exact Regular Planar Black Holes solutions in Bumblebee gravity
gr-qc 2026-11 conditional novelty 5.0 of 10

Exact planar black hole solutions in bumblebee gravity with a temporal/radial bumblebee field, coupling xi=1/4, and curvature invariants finite at r=0.

Discussion

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[1] [1]

Energy conditions for regular black holes in EFT of gravity Ziyue Zhu, 1,∗ Alexey S. Koshelev, 2, 3,† Yang Liu,2,‡ and Anna Tokareva1, 4, 5,§ 1School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China 2School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, Chi...

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