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One finite-mass two-throat wormhole metric yields a complete light-ring phase map and a sum rule for cross-throat strong lensing.

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2026-07-10 16:57 UTC pith:7DXSSRYB

load-bearing objection Solid finite-mass two-throat benchmark: complete light-ring phases and a clean transmitted-lensing sum rule, all from one metric.

arxiv 2607.07839 v1 pith:7DXSSRYB submitted 2026-07-08 gr-qc

Finite mass two throat wormholes: global light rings, branch resolved strong lensing, and scalar transmission

Anirudh Pradhan , K. Ghaderi , M. Zeyauddin , K. Karimizadeh This is my paper
classification gr-qc
keywords traversable wormholeslight ringsstrong gravitational lensingenergy conditionswave scatteringtwo-throat geometryLyapunov exponents
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper builds a static, horizonless, asymptotically flat wormhole with two symmetric throats and one intermediate equator from a single global metric whose areal profile is a finite-mass deformation of a known quartic embedding. That same metric fixes the conserved anisotropic source, the exact complete radial averaged null energy functional (which is strictly negative), every circular null orbit, the reflected and transmitted strong-deflection asymptotics, and the scalar scattering potential. The central result is a complete classification of the global null potential over all positive masses and nonnegative redshift deformation: the unstable set can be throat rings, four off-throat rings that share one critical impact parameter but have unequal Lyapunov exponents, or exterior rings plus a weaker equatorial ring. Analytic logarithmic strong-deflection coefficients follow from the local instability rates, and the cross-throat coefficient is exactly the sum of the local coefficients of every global maximum the ray traverses; direct numerical integration confirms the slopes. A sympathetic reader cares because topology, energy-condition diagnostics, branch-resolved lensing, and wave transmission are no longer treated as separate constructions: they are derived from one spacetime whose asymptotic mass is finite and unambiguous at both ends.

Core claim

From one smooth finite-mass two-throat metric, the null potential admits a complete phase classification for all m > 0 and χ ≥ 0, and the logarithmic coefficient of cross-throat strong deflection equals the sum of the local coefficients of every global maximum the ray crosses. In the four-ring phase those rings share a single critical impact parameter yet split into two Lyapunov classes, so same-side and opposite-side lensing resolve unequal local instabilities.

What carries the argument

The global null potential U(z̄) = A[r̄(z̄)]/r̄²(z̄) together with the local logarithmic weights ā_i = Ω_i/λ_i. Circular null orbits sit at stationary surfaces of the areal radius or at C(r̄_ph) = 1; the phase map of its maxima, and the additive rule for reflected versus transmitted deflection coefficients (Eqs. 52–54), carry the optical content of the whole family.

Load-bearing premise

Everything rests on one hand-chosen areal profile and redshift function; the stress-energy is whatever the Einstein tensor of that ansatz requires, with no independent matter model or dynamical stability analysis.

What would settle it

Recompute the phase-II transmitted logarithmic slope by independent high-precision numerical integration of the azimuthal equation for the benchmark parameters; if the numerical slope fails to approach 2(ā_in + ā_out) as ε_− → 0, the claimed sum rule is false for this metric.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper constructs a static, horizonless, asymptotically flat two-throat wormhole from a finite-mass deformation of a quartic embedding profile together with a redshift function Φ = −m/r̄ − χ/r̄². From this single metric the authors derive the Misner–Sharp/Komar mass, the Einstein-tensor effective source, an exact complete radial averaged null functional that is strictly negative, a full light-ring phase classification for m > 0 and χ ≥ 0 (throat rings; four off-throat rings sharing one critical impact parameter but with unequal Lyapunov exponents; exterior rings plus a subcritical equatorial ring), analytic logarithmic strong-deflection coefficients for reflected and transmitted channels (with the cross-throat coefficient equal to the sum of local ā_i over every global maximum traversed), numerical verification of those slopes, and phase-dependent scalar barriers with resonant transmission.

Significance. If the derivations hold, the work supplies a clean finite-mass setting in which topology, averaged null energy diagnostics, multi-ring null dynamics, branch-resolved strong lensing, and two-ended scalar transmission are obtained from one spacetime. The most distinctive optical result is the phase-II structure: four unstable rings degenerate in critical impact parameter and angular frequency but split into two Lyapunov classes, so that the same-side logarithmic coefficient samples only the outer class while the transmitted coefficient sums inner and outer weights twice. The exact identity A = −2 ∫ e^{-Φ} (r̄,ℓ/r̄)² dℓ < 0 is a useful global complement to the local flare-out theorem. Analytic coefficients with direct numerical slope checks and a controlled scalar transfer-matrix calculation are concrete strengths of the manuscript.

minor comments (6)
  1. In Sec. 7 the authors correctly state that the stress tensor is the effective Einstein-tensor source of the ansatz. A single explicit sentence noting that no independent matter model or dynamical linear-stability analysis is supplied would help readers place the construction relative to more phenomenological wormhole literature without changing any claim.
  2. Near the phase boundaries χ = χ_th and χ = χ_* the maxima become quartically degenerate and the leading deflection is nonlogarithmic (as the authors note, citing [21,22]). A brief pointer in Sec. 5 that the tabulated coefficients apply only away from those codimension-one loci would prevent misapplication of Eqs. (52)–(54).
  3. Table 1 lists only inequivalent nonnegative z̄ branches; a short footnote that the full unstable set is completed by the Z_2 images would make the counting in phase II fully explicit for readers skimming the table.
  4. In Fig. 3(d) and Table 2 the transmitted coefficient in phase II is more than five times the reflected one. Adding the numerical values of ā_in and ā_out next to the sum in the caption or table would make the sum rule immediately checkable without recomputing Lyapunov exponents.
  5. Appendix C reports excellent flux residuals and grid convergence for the scalar problem. Stating the tortoise cutoff Z and maximum h/r_0 used for the published spectra already in the main-text caption of Fig. 5 would improve reproducibility for readers who do not consult the appendix.
  6. A few typographical items: “Completelightringphasestruc-ture” in the Sec. 4 heading lacks spaces; “prescribedareal” and similar run-ons appear in the introduction; and the arXiv identifier style in the reference list is inconsistent with journal DOI formatting for published items.

Circularity Check

0 steps flagged

No significant circularity: one explicit metric ansatz yields derived phase map, sum-rule lensing coefficients, and scattering spectra by direct calculation, not by redefining targets as inputs.

full rationale

The paper’s load-bearing chain is: prescribe the global areal profile (4) and redshift (14); read geometry, Misner–Sharp/Komar mass, Einstein-tensor source, null potential U, light-ring loci, Lyapunov exponents, deflection integrals, and scalar potential from that single metric. The complete radial averaged null identity (34) is an integration by parts of the Einstein combination, not a fitted constraint. The three-phase light-ring classification follows from the sign of U,zz at stationary surfaces and the off-throat root C=1. The transmitted logarithmic coefficient as the sum of local āi over every global maximum traversed (52)–(54) is the standard singular-part expansion of the deflection integral applied branch by branch; numerical slopes are checks, not calibration targets. There is no external-data fit renamed as prediction, no uniqueness theorem imported from the authors’ prior work, and no self-citation that carries the central optical claims. The embedding and Φ are openly ansatz-level (a scope limitation of the genre), which is not circularity under the stated criteria. Score 0 with empty steps is therefore the correct finding.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 1 invented entities

The paper is a geometric GR construction. Load-bearing inputs are the hand-chosen areal profile and redshift function, standard Einstein equations with an effective anisotropic fluid, and classical null/scalar test-field dynamics. No observational fit is performed; free parameters set the shape of the family and the benchmark plots. No new fundamental entity is postulated beyond the metric ansatz itself.

free parameters (4)
  • ā (dimensionless throat-separation parameter)
    Sets the locations of the two throats at z̄ = ±√ā; chosen by hand (benchmark ā = 8). Controls geometry, not fitted to data.
  • K̄ (embedding amplitude)
    Controls flare-out strength and equatorial radius; benchmark K̄ = 7.5×10^{-3}. Free shape parameter of the family.
  • m = M/r0 (dimensionless mass)
    Asymptotic Misner–Sharp/Komar mass in units of reference throat radius; free positive parameter scanned in the phase diagram.
  • χ (redshift deformation)
    Nonnegative coefficient in Φ = −m/r̄ − χ/r̄²; free parameter that drives phase transitions χ_th(m) and χ_*(m).
axioms (4)
  • domain assumption Einstein equations hold with the effective stress-energy equal to G_μν/8πG of the given metric; no additional matter Lagrangian is required.
    Section 3 defines T from the Einstein tensor and uses Bianchi identities for conservation; standard for wormhole constructions but leaves the source as exotic anisotropic fluid.
  • domain assumption Static spherical symmetry and asymptotic flatness with a global regular radial coordinate z̄ covering both ends.
    Metric ansatz Eq. (2) and profile Eq. (4); assumed throughout.
  • domain assumption Classical null geodesics and minimally coupled massless scalar field on a fixed background (no backreaction).
    Sections 4–6; standard test-field approximation.
  • standard math Nondegenerate unstable light rings produce logarithmic strong-deflection divergence with coefficient ā_i = Ω_i/λ_i; quartic mergers require separate marginal expansions.
    Used in Sec. 5 with citations to Bozza and marginal-photon-sphere literature; applied away from phase boundaries.
invented entities (1)
  • Finite-mass deformed quartic two-throat wormhole metric family (Eqs. 2–4, 14) no independent evidence
    purpose: Provide a single smooth, horizonless, finite-mass geometry with two throats and an equator on which source, light rings, lensing, and scalar transmission can all be computed.
    The specific profile is introduced by the authors as a deformation of Cataldo et al.’s quartic embedding; it is not independently measured or derived from a matter model.

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read the original abstract

We introduce a static, horizonless, asymptotically flat two throat wormhole family in which one global metric determines the geometry, effective source, null dynamics, lensing, and test field propagation. A finite mass deformation of a quartic embedding profile produces two symmetric throats and an intermediate equator, while the associated Misner--Sharp mass tends to the same finite value at both asymptotic ends. The temporal sector yields the same mass parameter, and the Einstein tensor defines a single conserved anisotropic source. In addition to the local flare out identity, the complete radial averaged null functional is obtained exactly and is strictly negative. The global null potential admits a complete phase classification for all positive masses and nonnegative redshift deformation. Depending on the parameters, the global unstable set consists of throat rings, four off throat rings, or exterior rings accompanied by a subcritical equatorial ring. In the four ring phase, all unstable rings have the same impact scale but the inner and outer branches have different Lyapunov exponents. We derive the logarithmic strong deflection coefficients analytically and show that the cross throat coefficient is the sum of the contributions from every global maximum traversed by the ray. Direct numerical integration verifies these coefficients. The scalar scattering problem exhibits phase dependent barriers and resonant transmission. The resulting model provides a finite mass setting in which topology, averaged energy conditions, branch resolved strong lensing, and wave transmission are derived from one spacetime.

Figures

Figures reproduced from arXiv: 2607.07839 by Anirudh Pradhan, K. Ghaderi, K. Karimizadeh, M. Zeyauddin.

Figure 1
Figure 1. Figure 1: Geometry and regularity diagnostics. (a) Global areal radius for the benchmark (18); dotted lines mark [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effective source and averaged null diagnostics for the benchmark geometry. (a) Density and principal [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Global light ring phase structure for a¯ = 8 and K¯ = 7.5 × 10−3 . (a) Complete phase map in the (m, χ) plane. (b) Normalized null potentials for the three benchmark phases; filled and open markers denote unstable and stable rings. (c) Coordinate time Lyapunov exponents of the inequivalent unstable branches at m = 0.3. (d) Reflected and transmitted logarithmic strong deflection coefficients. Vertical dotte… view at source ↗
Figure 4
Figure 4. Figure 4: Strong lensing and asymptotic validation. (a) Same side deflection as the reflected critical branch is [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scalar scattering for ℓ = 2. (a) Phase dependent effective potentials in the tortoise coordinate. (b) Transmission probabilities for the three benchmark models; the dotted line marks unit transmission. (c) Transmission phase derivative for phase II on a symmetric logarithmic vertical scale; markers identify local transmission maxima. The multi barrier geometry produces a sequence of resonant transmission f… view at source ↗

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Reference graph

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