arxiv: 2604.18004 · v2 · submitted 2026-04-20 · ✦ hep-th · gr-qc

Recognition: unknown

Eikonal, nonlocality and regular black holes

Mariano Cadoni , Lorenzo Herres , Leonardo Modesto , Lorenzo Orlando , Mirko Pitzalis

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:48 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords nonlocal gravityeikonal approximationregular black holesde Sitter coreSchwarzschild deformationasymptotically flat spacetimesscalar scattering

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The pith

Nonlocal gravity yields singularity-free black holes with de Sitter cores through eikonal analysis.

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

The paper examines the leading gravitational eikonal in nonlocal D-dimensional gravity theories via tree-level 2 to 2 scalar scattering. It interprets the results as effective geometries: generalized Aichelburg-Sexl metrics for massless particles and smeared linearized Schwarzschild metrics for massive probes. These linearized solutions are then completed nonlinearly by imposing general conditions on the core behavior. The outcome is a class of asymptotically flat deformations of the Schwarzschild solution that remain regular everywhere and feature a de Sitter core. The authors also compute the main geometric and thermodynamic properties of the resulting spacetimes.

Core claim

In nonlocal D-dimensional gravity the eikonal limit of scalar scattering produces linearized geometries that, when completed nonlinearly using requirements on core behavior, become singularity-free asymptotically flat deformations of the Schwarzschild metric possessing a de Sitter core.

What carries the argument

Nonlinear completion of linearized eikonal geometries (generalized Aichelburg-Sexl for massless, smeared Schwarzschild for massive) guided by core-behavior requirements.

If this is right

The completed spacetimes contain no curvature singularities.
They remain asymptotically flat and approach the Schwarzschild metric at large distances.
Their interiors are described by a de Sitter geometry.
Geometric invariants and thermodynamic quantities such as temperature and entropy can be computed explicitly.
The construction applies to both massless and massive probe scattering in D dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The de Sitter core sets a finite upper bound on curvature near the center.
Similar completions might be possible for other choices of nonlocal form factors.
Thermodynamic stability could be altered relative to classical Schwarzschild black holes because of the regular core.

Load-bearing premise

General requirements about the behavior of solutions in the core suffice to determine a unique nonlinear completion without an explicit derivation from the nonlocal action.

What would settle it

Direct solution of the nonlinear field equations from the underlying nonlocal action to check whether the proposed geometries satisfy them exactly.

read the original abstract

We investigate the leading gravitational eikonal in nonlocal $D$ dimensional theories of gravity. We analyze the simplest cases of $2\rightarrow2$ massless and massive scalar scattering at tree level, studying the effects of nonlocal form factors in the gravitational sector. We give an interpretation of our results in terms of geodesic motion in effective generalized Aichelburg-Sexl geometries for the massless case, and in smeared linearized Schwarzschild metrics for the massive case in the probe limit. Combining our results for the geometries at linearized level with general requirements about the behaviour of the solutions in the core, we propose a nonlinear completion of the geometries. The resulting spacetimes describe singularity-free, asymptotically flat deformations of the Schwarzschild solution with a de Sitter core. We also analyze the main geometric and thermodynamic features of these solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Eikonal scattering gives explicit linearized geometries in nonlocal gravity, but the nonlinear regular black hole completions rest on heuristic core requirements instead of solving the integro-differential equations.

read the letter

The paper computes tree-level eikonal amplitudes for 2-to-2 scalar scattering in D-dimensional nonlocal gravity and maps them to effective linearized metrics: generalized Aichelburg-Sexl for massless particles and smeared Schwarzschild for massive probes. That part is concrete and follows standard methods with the nonlocal form factors inserted directly into the amplitudes. The interpretation in terms of geodesic motion is clear and useful for connecting scattering to geometry at linear order. The authors then combine these linearized results with general requirements on core behavior to propose nonlinear completions that are asymptotically flat, singularity-free, and carry de Sitter cores, plus some thermodynamic checks. This proposal is the main new element they highlight. The eikonal calculations themselves look solid and extend existing nonlocal gravity techniques without obvious errors in the linear regime. The geometric analysis of the proposed metrics is straightforward. The soft spot is the nonlinear step. The underlying action produces integro-differential equations, yet the completed metrics are not shown to satisfy them at higher orders; they are constructed by hand using external assumptions about the core rather than derived or verified from the theory. Similar de Sitter-core regular black holes already appear in other modified-gravity papers, so the link to nonlocal eikonal scattering is the distinctive part but remains loose. This work is mainly for people already working on nonlocal gravity or singularity resolution who want to see how eikonal methods feed into regular solutions. The concrete amplitude calculations give it enough substance to go to peer review, though the heuristic completion would need more justification or explicit checks in revisions.

Referee Report

2 major / 2 minor

Summary. The manuscript computes the leading gravitational eikonal for tree-level 2→2 scattering of massless and massive scalars in nonlocal D-dimensional gravity theories. It interprets the results as effective linearized geometries: generalized Aichelburg-Sexl metrics for massless cases and smeared Schwarzschild metrics for massive cases in the probe limit. Using these linearized geometries combined with general requirements on the core behavior, the authors propose nonlinear completions that yield singularity-free, asymptotically flat black hole spacetimes with de Sitter cores, and analyze their geometric and thermodynamic properties.

Significance. The explicit tree-level eikonal calculations provide concrete, reproducible results on how nonlocal form factors modify effective geometries at linear order, which is a clear strength of the work. If the proposed nonlinear completions can be shown to satisfy the equations of motion from the nonlocal action, the paper would offer a useful route to constructing regular black holes in such theories. However, the heuristic character of the completion step means the strongest claims about singularity resolution remain provisional until further justification is supplied.

major comments (2)

[§4] §4 (nonlinear completion): The proposal that the completed metrics are singularity-free deformations of Schwarzschild with de Sitter cores rests on 'general requirements about the behaviour of the solutions in the core' rather than an explicit derivation or verification that these metrics solve the integro-differential equations of motion implied by the nonlocal form factors beyond linear order. This step is load-bearing for the central claim in the abstract.
[§3.2] §3.2 (massive case, probe limit): While the smeared linearized Schwarzschild metric is obtained from the eikonal amplitude, the subsequent nonlinear completion invokes external core requirements without demonstrating consistency with the underlying nonlocal action at quadratic or higher order in the metric perturbation.

minor comments (2)

[§2] Notation for the nonlocal form factors (e.g., the precise definition of the exponential or entire function in the gravitational sector) should be stated explicitly in §2 to avoid ambiguity when comparing to prior literature on infinite-derivative gravity.
[§5] The thermodynamic analysis in §5 would benefit from a direct comparison table of the Hawking temperature and entropy for the proposed metrics versus the standard Schwarzschild case at fixed mass.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major points below, clarifying the scope and limitations of our analysis while defending the approach taken in the paper.

read point-by-point responses

Referee: [§4] §4 (nonlinear completion): The proposal that the completed metrics are singularity-free deformations of Schwarzschild with de Sitter cores rests on 'general requirements about the behaviour of the solutions in the core' rather than an explicit derivation or verification that these metrics solve the integro-differential equations of motion implied by the nonlocal form factors beyond linear order. This step is load-bearing for the central claim in the abstract.

Authors: We agree that the nonlinear completions proposed in §4 are constructed by combining the linearized geometries obtained from the tree-level eikonal with general requirements on core regularity and asymptotic flatness, rather than by solving the full integro-differential equations of the nonlocal theory at nonlinear order. The manuscript explicitly frames this as a proposal ('we propose a nonlinear completion'), not a derivation from the complete action. This is a standard strategy when the nonlinear nonlocal equations are intractable, and the eikonal result supplies the physically motivated linear-order input. The abstract claim is limited to the proposal of such spacetimes, which we believe is accurately stated. We will revise §4 to more explicitly note the heuristic character and the provisional nature of the singularity-resolution claim. revision: partial
Referee: [§3.2] §3.2 (massive case, probe limit): While the smeared linearized Schwarzschild metric is obtained from the eikonal amplitude, the subsequent nonlinear completion invokes external core requirements without demonstrating consistency with the underlying nonlocal action at quadratic or higher order in the metric perturbation.

Authors: The nonlinear completion in the massive probe-limit case follows the same logic as in §4: the eikonal supplies the smeared linearized Schwarzschild geometry, which is then completed using the same general core requirements. We do not claim or demonstrate consistency at quadratic or higher order in the metric perturbation, as this would require solving the nonlocal equations beyond the leading eikonal approximation, which lies outside the scope of the present work. The probe limit and tree-level calculation are used precisely because they yield concrete, reproducible results at linear order. We will add a clarifying remark in §3.2 emphasizing these limitations. revision: partial

standing simulated objections not resolved

Explicit verification that the proposed nonlinear metrics satisfy the full integro-differential equations of motion from the nonlocal action at quadratic and higher orders.

Circularity Check

0 steps flagged

No significant circularity: linearized geometries from explicit amplitudes; nonlinear completion is a separate proposal

full rationale

The paper derives linearized geometries explicitly from tree-level eikonal scattering amplitudes in the nonlocal theory, interpreting them as generalized Aichelburg-Sexl metrics (massless case) and smeared Schwarzschild metrics (massive probe limit). The nonlinear completion is introduced separately by combining these results with external 'general requirements about the behaviour of the solutions in the core' to yield singularity-free asymptotically flat spacetimes with de Sitter cores. This step does not reduce any claimed prediction or first-principles result to the inputs by construction, nor does it rely on self-citation load-bearing, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation. The derivation chain is self-contained, with the scattering calculations providing independent content and the completion presented as a proposal rather than a forced or fitted outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper assumes a standard nonlocal gravity setup with form factors and introduces a proposed nonlinear completion without independent evidence or derivation.

axioms (1)

domain assumption The leading gravitational eikonal can be extracted from tree-level 2-to-2 scalar scattering in nonlocal D-dimensional gravity
Invoked as the starting point for both massless and massive cases.

invented entities (1)

Nonlinear completion of the linearized geometries no independent evidence
purpose: To produce singularity-free spacetimes with de Sitter cores
Proposed from linearized results plus general core requirements; no independent falsifiable handle given in abstract.

pith-pipeline@v0.9.0 · 5441 in / 1295 out tokens · 67401 ms · 2026-05-10T04:48:34.834283+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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