Nonlocal gravity in a proper tetrad frame: traversable wormholes

Rocco D'Agostino; Vittorio De Falco

REVIEW 4 major objections 5 minor 1 cited by

Nonlocal gravity in a proper tetrad frame: traversable wormholes

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Nonlocal gravity can make traversable wormholes without exotic matter.

desk verdict First systematic wormhole solutions for the revised Deser-Woodard model, but only the constant-redshift case is fully reliable; the perturbative f(Y) is not actually first-order. read the letter →

arxiv 2501.18007 v3 pith:UKHLEKEH submitted 2025-01-29 gr-qc astro-ph.COhep-th

Rocco D'Agostino , Vittorio De Falco This is my paper · ORCID

classification gr-qcastro-ph.COhep-th

keywords nonlocalgravityDeser-Woodardmodeltraversablewormholestetradframedistortionfunctionauxiliaryscalarfieldsnullenergyconditionmodified

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 claims that in the revised Deser-Woodard nonlocal gravity theory, traversable wormholes can exist in vacuum, supported by the gravitational field itself rather than by exotic matter. The authors reformulate the nonlocal field equations in a proper orthonormal tetrad frame, which reduces the system to two independent equations for a combined scalar $W(r)=1+U(r)+f(Y(r))$. Postulating the redshift function $\Phi(r)$ lets them solve for the shape function and then reconstruct, step by step, the auxiliary scalars and the distortion function $f(Y)$. They exhibit three static, spherically symmetric wormhole solutions (analytic, perturbative, and numerical) and show that in each case the effective stress-energy tensor violates the null energy condition, mimicking exotic matter. If correct, this gives a concrete pathway from nonlocal gravity to compact astrophysical objects with distinctive observable signatures.

What carries the argument

The proper tetrad frame (11), with basis $e_{\hat\mu}^{\;\alpha}=\mathrm{diag}(1/\sqrt{-g_{tt}},1/\sqrt{g_{rr}},1/r,1/(r\sin\theta))$, recasts the nonlocal field equations into three diagonal components that combine into two independent equations, (19a) and (19b). These equations eliminate the $K_{\mu\nu}$ terms and relate Einstein-tensor components to derivatives of $W(r)=1+U(r)+f(Y(r))$, packaging the entire nonlocal sector into one radial function. The step-by-step reconstruction chain $\Phi \to (W,b) \to R \to X \to Y \to V \to U \to f(Y)$ carries the argument.

What would settle it

Substitute the explicit WH1 and WH2 fields (and the numerical WH3 data) into the original field equations (9) and evaluate the residual for $r>r_0$; the claim fails if the residual is nonzero beyond numerical error. A sharper test is a symbolic or high-precision numerical comparison showing that Eqs. (19a)-(19b) are equivalent to Eq. (9) for the Morris-Thorne metric; if the two systems disagree for even one solution, the reconstruction method is invalid.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a reconstruction method: instead of fixing a nonlocal action and solving for the metric, one postulates the $g_{tt}$ component of a static spherical metric and uses the tetrad-frame equations to solve for the shape function $b(r)$ and the combined field $W(r)$. From these, the Ricci scalar yields $X$, then $Y$, $V$, and $U$ through a chain of Klein-Gordon equations, and finally $f(Y)$ is obtained by inverting $Y(r)$. Applied to $\Phi=\text{const}$, $\Phi\simeq -B/2r$, and $\Phi=\frac12\ln(1-B/r)$, the method yields three traversable wormhole spacetimes (WH1, WH2, WH3) with throat radius $r_0=1$ and asymptotic flatness, and the authors fix integration constants by requiring that the reconstructed fields satisfy the original field equations. The result is that nonlocal gravity can sustain traversable wormholes without any matter source, with the nonlocal terms acting as an effective stress-energy tensor that violates the null energy condition.

Load-bearing premise

The load-bearing premise is that the two combined tetrad equations (19a) and (19b), taken from a companion paper, are exactly equivalent to the original Deser-Woodard field equations (9) for static, spherically symmetric spacetimes; if that equivalence fails, the reconstructed $W(r)$, $b(r)$, and $f(Y)$ do not solve the theory and none of the three wormhole solutions stands.

Editorial extensions

If this is right

If the three wormhole solutions are genuine, the revised Deser-Woodard theory predicts that vacuum nonlocal gravity can produce traversable wormholes without any exotic matter source.
The reconstruction procedure makes the approach inverse: any prescribed redshift function with consistent integration constants yields a nonlocal Lagrangian, so observations of compact-object shadows could in principle be inverted to constrain $f(Y)$.
For the allowed parameters, the WH2 and WH3 solutions have no photon sphere and no critical impact parameter, so their shadow would coincide with the throat—a distinctive, observable signature.
The same tetrad reduction should apply to other static, spherically symmetric spacetimes and to other nonlocal gravity models, extending the method beyond wormholes.

Reading between the lines

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

Because the theory is vacuum, the 'no exotic matter' statement is partly a restatement of the modified-geometry setup: the nonlocal terms act as a geometric fluid, and the physically nontrivial question is whether the reconstructed $f(Y)$ satisfies the ghost-free and boundary conditions assumed when deriving the field equations.
The method could be stress-tested by checking whether the reconstructed $f(Y)$ stays single-valued and smooth across the whole radial domain, since the inversion $r(Y)$ is analytic only for WH1; for WH2 and WH3 the inversion is approximate or numerical.
A natural next step the paper does not take is a linear stability analysis of these wormholes under axial perturbations; if unstable, the throat-shadow signature would not be observable.
One might extend the tetrad reduction to rotating wormholes, though the angular components of the tetrad equations would no longer collapse as cleanly.

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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

You should know this paper gives the first traversable wormhole solutions in the revised Deser-Woodard nonlocal gravity model, and the constant-redshift WH1 solution is exact and clean. But the perturbative WH2 case has a real internal inconsistency that the authors half-admit, and the numerical WH3 case lacks the checks needed to trust it.

What is actually new: the reconstruction method—postulate g_tt, solve the tetrad-reduced field equations for the shape function and W, then compute the auxiliary scalars and recover f(Y) from f(r)—produces explicit solutions. WH1, with constant redshift and b(r)=1/r, is exact, and the closed-form f(Y) in Eq. (43) is a genuine result. The observational quantities (photon sphere, ISCO, shadow) for each solution are a useful starting point, and the NEC analysis is a fair way to show that nonlocal effects mimic exotic matter.

The soft spots matter. First, the flaring-out condition as written in Eq. (27) is b(r) − r b'(r) < 1. That is not the standard Morris-Thorne condition, and it fails for WH1 at the throat (b − r b' = 2 there). The usual inequality is > 0, which WH1 satisfies. So either (27) is a typo or the authors are using a nonstandard condition; either way the Section V parameter constraints that follow from it are suspect. Second, the stress-test critique is correct: for WH2, the paper converts f(r) to f(Y) using the zeroth-order inverse Eq. (42) rather than the first-order-correct inverse. That drops an O(B) term from r(Y) and makes the reconstructed f(Y) only approximate, not a first-order solution of the theory. The authors acknowledge the numerical discrepancy (~40% near the throat), but that is not the same as showing the omitted term leaves the field equations satisfied to O(B). Third, WH3 is numerical, and the paper does not show residual errors of the full field equations for that solution. Finally, the tetrad reduction is imported from the companion paper [84]; the equivalence with Eq. (9) is plausible but not rederived here.

Who gets value from this: anyone working on nonlocal gravity or wormhole physics. The reconstruction strategy is worth knowing, and WH1 is a solid exact solution. But the paper as it stands overstates the status of WH2 and WH3, and the flaring-out confusion needs fixing.

My recommendation: send it to peer review. The core idea is sound, the exact solution deserves publication, and the issues are fixable—but a referee should insist that the O(B) consistency of WH2 be properly addressed and the numerical residuals for WH3 be reported.

Referee Report

4 major / 5 minor

Summary. The paper reformulates the Deser-Woodard nonlocal gravity field equations in a proper tetrad frame, reducing the vacuum equations to two independent equations (19). The authors propose an inverse reconstruction method: one postulates the g_tt component of a static, spherically symmetric metric, solves for W(r) and the shape function b(r), and then reconstructs the auxiliary scalar fields and the distortion function f(Y). Three wormhole solutions are presented: WH1 (exact, Phi constant, b=1/r), WH2 (first-order perturbative in a parameter B, Phi=(1/2)ln(1-B/r)), and WH3 (numerical for generic B). The paper also studies energy conditions, showing NEC violations, and discusses photon-sphere and ISCO radii. The central claim is that these wormholes are supported purely by nonlocal gravity without exotic matter.

Significance. If the technical issues are resolved, this paper offers a tractable method for solving the revised Deser-Woodard theory in static spherical symmetry and provides the first wormhole solutions in that framework. The WH1 solution is exact, the reconstruction pipeline is explicit, and the analytic expressions for the scalar fields and f(Y) are given in closed form for WH1. The paper also correctly identifies that the effective stress-energy tensor violates the NEC, which is the expected analog of exotic matter. However, the significance is presently limited by internal inconsistencies in the flaring-out condition and by the approximate nature of the WH2 distortion function, as detailed below.

major comments (4)

[Sec. V and Eq. (27)] The flaring-out condition as written in Eq. (27), b(r)-r b'(r)<1, is not satisfied by the WH1 solution: with b(r)=1/r and r0=1, one obtains b-rb'=2>1 at the throat. For WH2, using Eq. (45) one finds b'(1) approximately equal to -(1+B), so b-rb' is approximately 2+B>1 for all B>0. These facts contradict the statement in Sec. V that the flaring condition entails B>1 for WH2 and 3B/2<1 for WH3, and they also contradict the parameter values used in Figs. 1, 3, and 4 (B=0.01 for WH2 and B=0.8 for WH3). The standard Morris-Thorne flaring condition is b-rb'>0, not <1; with that criterion WH1 and WH2 would satisfy it, but then the constraints in Sec. V need to be recomputed. As written, the wormhole interpretation of the presented solutions is not supported by the paper's own condition, so Eq. (27) and Sec. V must be corrected and the allowed B-ranges must be reconciled with the plotted solutions.
[Sec. IVB, Eqs. (42), (49), (B3)] The conversion from f(r) to f(Y) for WH2 uses the inverse of the zeroth-order function Y0(r) given in Eq. (42), whereas the field equations were solved with the full first-order Y(r)=Y0(r)+B Y1(r). The omitted O(B) correction to r(Y) generates a term in f(Y) of exactly the same order as the retained B f1 term. The authors acknowledge a relative error of about 40% near the throat (Sec. IVB), so the resulting f(Y) is not the distortion function corresponding to the WH2 solution. Because the nonlocal action is defined by f(Y), substituting this approximate f(Y) into Eq. (9) together with the WH2 metric and scalar fields has not been shown to satisfy the field equations to first order; the O(B) residuals are not evaluated. The paper should either compute the corrected first-order inverse or present a direct check of Eq. (9) with the approximate f(Y).
[Sec. III, Eqs. (18)-(19)] The reduction from the tetrad components (18) to the two independent equations (19), and the claim that solving (19) together with the auxiliary scalar equations (2), (3), (7), and (8) is equivalent to solving the original field equations (9), is taken from Ref. [84] without derivation. Since this equivalence is load-bearing for all three wormhole solutions, the manuscript should include a derivation or a precise statement of the conditions under which (19) is equivalent to (9); alternatively, a direct verification of Eq. (9) for the final solutions would also address this concern.
[Sec. IVB, WH3 numerical solution] For the numerical solution WH3, the paper states that the field equations and all wormhole properties are satisfied, but no numerical residual, convergence test, or error estimate is reported. Given that the distortion function is obtained as a parametric plot [Y(r), f(r)], a residual check of Eq. (9) is necessary to support the claim that WH3 is a solution of the nonlocal theory. Please include such a check or a quantitative measure of how accurately the numerical fields satisfy the field equations.

minor comments (5)

[Sec. II] There is a typo in the sentence defining the action: 'detrminant' should be 'determinant'.
[Sec. IVB] In the paragraph introducing the numerical solution, the sentence 'In this case, the grr metric cannot be calculated analytically' is missing a verb and should read 'the grr metric cannot be calculated analytically'.
[Fig. 2 caption] The caption describes the quantity as an 'Absolute difference' while the vertical axis label is 'Delta f [%]'; please clarify whether the plot shows an absolute difference or a relative percentage difference.
[Sec. V] The phrase 'for B to 0, 1 > rps = 0' is confusing; it should be written as 'rps = B/2 tends to 0 as B tends to 0'.
[Table I] The entry for WH2's f1(Y) states that r(Y) is given by Eq. (42), which is exactly the zeroth-order inverse whose inadequacy is the subject of a major comment; a footnote explaining the approximation would help the reader.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the distortion function is reconstructed from the assumed g_tt via the field equations, with no fitted parameter presented as a prediction; the only self-citation (tetrad reduction from [84]) is parameter-free and does not predetermine the wormhole solutions.

full rationale

The derivation is an inverse construction, not a prediction from fitted inputs. The authors postulate the redshift function Phi(r), solve the tetrad-reduced field equations for W(r) and b(r), and then reconstruct f(Y) from Eq. (21) after obtaining the auxiliary scalar fields from Eqs. (2)-(8). No parameter is fitted to a target quantity and then renamed a prediction; the constants v1 and v2 are integration constants fixed by requiring the field equations to hold, and the resulting f(Y) is presented as a reconstruction rather than as an independent prediction. The only self-citation is the companion paper [84] for the non-vanishing tetrad components (18a-c), which feed the central equations (19) and (31). This is load-bearing but not circular in the prohibited sense: the cited reduction is a parameter-free algebraic manipulation of the same field equations, and the wormhole results are not assumed in it. The admitted WH2 approximation in Sec. IVB, where Eq. (42) inverts the zeroth-order function Y0(r) rather than the full first-order Y(r), and the relative error is stated to go from about 40% near the throat to about 1% for r >= 1.08, is an accuracy limitation rather than a circular step, because it does not make f(Y) equal by construction to the input Phi(r) or b(r). Overall, no significant circularity is present.

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

The model already contains four auxiliary scalar fields X, Y, U, V from the Deser-Woodard action; no new particles or fields are introduced. The only new free parameter is B, a solution parameter. The paper's main additions are the reconstruction procedure and the wormhole solutions, which rely on the assumed tetrad equivalence and the chosen metric ansatze.

free parameters (1)

B (wormhole redshift parameter) = 0<B<1 (figures use B=0.01, 0.05, 0.1 for WH2 and B=0.8 for WH3)
Introduced by hand in the redshift ansatz Phi(r)=1/2 ln(1-B/r) or its perturbative form. It is not fitted to data, but the ranges quoted in Sec. V are internally inconsistent with the plotted values.

assumptions (5)

domain assumption The revised Deser-Woodard action with four auxiliary scalar fields, Eq. (5), is the starting theory.
The action and auxiliary fields are taken from prior literature [41] and are not derived in this paper.
domain assumption Auxiliary fields obey retarded boundary conditions and vanish at spatial infinity.
Used to fix integration constants and to recover GR at infinity; justified by ghost-avoidance arguments from [86].
domain assumption The tetrad-reduced equations (18) and (19) are equivalent to the original field equations (9).
The reduction is taken from the same authors' companion paper [84] and is not independently derived or machine-checked here.
domain assumption The static spherically symmetric Morris-Thorne metric ansatz (25) with two asymptotically flat regions is assumed.
This is the standard wormhole geometry used throughout the paper.
ad hoc to paper Specific ansatze are chosen: Phi=constant with b=r^{-n} for WH1, Phi=1/2 ln(1-B/r) with first-order expansion for WH2, and full Phi for WH3.
These ansatze are not derived from the theory; they are chosen to make the reconstruction tractable.

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Cite this review

Pith. "Pith review of Nonlocal gravity in a proper tetrad frame: traversable wormholes." pith.science (2026). https://pith.science/paper/UKHLEKEH

@misc{pith2026250118007,
  author       = {Pith},
  title        = {Pith review of: Nonlocal gravity in a proper tetrad frame: traversable wormholes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UKHLEKEH}},
  note         = {Machine review of arXiv:2501.18007}
}

abstract

We investigate the revised Deser-Woodard model of nonlocal gravity involving four auxiliary scalar fields, introduced to explain the standard cosmological background expansion history without fine-tuning issues. In particular, we simplify the complex field equations within a proper tetrad frame, thereby recasting the original system into a more tractable equivalent differential problem. We show that, by initially postulating the form of the $g_{tt}$ metric component, it is possible to reconstruct the distortion function of the gravitational model. We then describe a step-by-step procedure for solving the vacuum field equations in the case of a static and spherically symmetric spacetime. We apply our technique to find new traversable wormholes supported purely by gravity by employing either analytical, perturbative, or numerical methods. Furthermore, we demonstrate that the role of the nonlocal effects is analogous to that of exotic matter in general relativity, owing to their quantum nature. Finally, we discuss the main geometric properties of the obtained solutions. Our results present a feasible avenue for identifying novel compact objects while enhancing the comprehension of nonlocal gravitational theories.

Figures

Figures reproduced from arXiv: 2501.18007 by the authors.

**Figure 3.** FIG. 3. Embedding of the WH2 and WH3 solutions in a three-dimensional Euclidean space for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** FIG. 4. Three-dimensional plots of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

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