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Wormhole throat leaves optical fingerprint that black holes cannot mimic

2026-07-09 02:38 UTC pith:ELGXPCQY

load-bearing objection Solid, useful paper with one real algebra error in the intermediate steps of the light deflection derivation. The final results are correct, but anyone reproducing the work step-by-step will hit a contradiction. the 2 major comments →

arxiv 2607.07679 v1 pith:ELGXPCQY submitted 2026-07-08 gr-qc

On relativistic observables in black bounce spacetimes

V\'ictor Ovejero-Berm\'udez , \'Alvaro de la Cruz-Dombriz , Riccardo Della Monica This is my paper
classification gr-qc PACS 04.50.Kd98.80.-k98.80.Cq
keywords blackbounceparameterrelativisticschwarzschildspacetimesthroatalpha
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.

This paper asks whether a specific family of singularity-free spacetimes — the Simpson-Visser black bounce geometries, which use a single parameter α to replace the central singularity with a finite-radius throat and continuously interpolate between a Schwarzschild black hole (α = 0) and a two-way traversable wormhole (α > 2M) — produces observable signatures distinct from standard black holes. The authors derive weak-field analytical expressions for the periastron advance of massive particles and the deflection angle of light, then verify these against numerical geodesic integrations. They find that both the periastron precession and the light deflection receive positive corrections proportional to α² on top of the Schwarzschild values. The most distinctive result concerns the critical impact parameter b_c that separates scattered photons from captured (or throat-transmitted) ones: it stays locked at the Schwarzschild value 3√3 M throughout the entire regular black hole branch (α ≤ 2M), because the throat is hidden behind the horizon and the exterior photon sphere at r = 3M is unaffected, but it increases monotonically with α once the horizon disappears and the throat becomes a timelike hypersurface in the wormhole branch (α > 2M). This means the shadow radius is identical to Schwarzschild for any regular black hole in this family, but grows for traversable wormholes, providing what the authors argue is a clean observational discriminator between the two topological branches.

Core claim

The critical impact parameter for photon capture/transmission is invariant under changes in the bounce parameter α as long as an event horizon is present (α ≤ 2M), remaining exactly at the Schwarzschild value 3√3 M, but increases monotonically once the geometry transitions to a horizonless traversable wormhole (α > 2M). This occurs because the photon sphere governing optical capture sits at r = 3M in the exterior region, unaffected by the throat hidden behind the horizon; only when the throat is exposed does it begin to govern photon dynamics directly. Simultaneously, the periastron advance and light deflection angle both receive additive positive corrections proportional to α², meaning the喉

What carries the argument

The Simpson-Visser black bounce metric, which modifies the Schwarzschild radial coordinate by replacing r with √(ξ² + α²), introducing a throat at ξ = 0 with areal radius r = α. The parameter α controls the causal structure: α ≤ 2M yields a regular black hole with horizons at ±√(4M² - α²); α > 2M yields a horizonless traversable wormhole. The Hamilton-Jacobi formalism is used for timelike geodesics and a Lagrangian approach for null geodesics, with weak-field perturbative expansions in σ = M²/h² for massive particles and α²/b² for light.

Load-bearing premise

The analytical formulas for periastron advance and light deflection are derived under weak-field conditions (M²/h² ≪ 1 and α² ≪ b²), which break down for compact orbits or large throat sizes; the paper's own comparisons show percent-level errors already at α = 5M for light deflection at small impact parameters, and the claim that these deviations are 'measurable' depends on whether the valid parameter ranges overlap with realistic astrophysical scales.

What would settle it

Measure the shadow radius of a compact object of known mass M. If it equals 3√3 M, the object could be a Schwarzschild black hole or any regular black hole in this family (α ≤ 2M). If it exceeds 3√3 M without a spin or charge explanation consistent with observations, the wormhole branch (α > 2M) is a candidate. If periastron precession of orbiting stars shows excess beyond what GR predicts for Schwarzschild (or Kerr), the α² correction is a possible contributor.

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

If this is right

  • If the critical impact parameter (shadow radius) of a compact object is measured to exceed 3√3 M for a given mass, and no other explanation (spin, charge, exotic matter) suffices, the black bounce wormhole branch becomes a candidate explanation.
  • The positive α² corrections to periastron precession could be constrained by S-star monitoring data around Sgr A*, since the cumulative phase shift grows over multiple orbital periods and is degenerate with other effects only if considered in isolation.
  • The constancy of b_c in the black hole branch means that shadow-size measurements alone cannot distinguish a regular black hole from a Schwarzschild black hole in this family; complementary observables (precession, redshift phase shifts) are required.
  • Future ray-tracing and accretion disk models in the wormhole branch must account for photons transmitted to a second asymptotic region, which would produce a fundamentally different shadow morphology than absorption at a horizon.

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

2 major / 6 minor

Summary. This manuscript studies relativistic observables in the Simpson–Visser black bounce spacetime, which interpolates between a Schwarzschild black hole and a traversable wormhole via a regularization parameter α. The authors derive weak-field analytical expressions for the periastron advance of massive particles (Hamilton–Jacobi formalism, Eqs. 9–22) and the deflection angle of light (Lagrangian formalism, Eqs. 28–39). These are compared against numerical geodesic integrations using the PyGRO code. The central result concerns the critical impact parameter b_c: it remains equal to the Schwarzschild value 3√3 M throughout the regular black hole branch (α ≤ 2M) but increases monotonically with α in the traversable wormhole branch (α > 2M), providing an optical signature of the throat. The paper also analyzes relativistic redshift of bound orbits, finding cumulative phase shifts with increasing α.

Significance. The paper provides parameter-free analytical derivations (the bounce parameter α is a property of the metric, not a fitted constant) and falsifiable predictions (b_c behavior, α² corrections to precession and deflection) that are systematically checked against reproducible numerical integrations with PyGRO. The identification of b_c as an observational discriminator between the black hole and wormhole branches is a clean, testable result. The constancy of b_c in the black hole branch is supported by an analytical argument (Eqs. 40–41) and is consistent with prior ray-tracing studies. The combined study of spatial (precession, deflection) and temporal (redshift) observables is well-motivated by the degeneracy problem discussed in the introduction.

major comments (2)
  1. Eqs. (37)–(38): There is a sign error in the intermediate derivation of the light deflection angle. Starting from the perturbative solution Eq. (36) and setting u = 0 at the incoming asymptote, the correct algebra yields φ∞ = −2M/b · 1/(1 − α²/(16b²)), i.e., the factor (1 − α²/(16b²)) should appear in the denominator, not as a multiplicative factor. Consequently, Eq. (38) as written, 4M/b(1 − α²/(16b²)), is incorrect; the correct expression is 4M/b · 1/(1 − α²/(16b²)). The text following Eq. (38) states 'we can expand the denominator in Eq. (38),' but Eq. (38) as written contains no denominator. Expanding the correct expression gives ∆φ ≈ 4M/b + Mα²/(4b³), which matches the final result Eq. (39) and is confirmed by the numerics in Fig. 5. The final formula is thus correct, but the intermediate steps in Eqs. (37)–(38) are wrong: a reader reproducing the derivation step-by-step would find
  2. §IV.C and Fig. 5: The claim that the weak-field formulas produce 'measurable deviations' from Schwarzschild depends on whether the parameter ranges where the formulas are valid overlap with realistic astrophysical scales. The paper itself notes (Fig. 3 inset) that for a = 500M the relative error exceeds the percent level, and for α = 5M the deflection formula diverges from numerics at small impact parameters. The manuscript would benefit from a brief, quantitative discussion of whether the valid parameter regime (σ = M²/h² ≪ 1 and α² ≪ b²) is accessible for objects like Sgr A*, given current observational precision from GRAVITY and EHT. Without this, the abstract's claim of 'measurable deviations' is not fully substantiated.
minor comments (6)
  1. Eq. (22): The periastron advance formula contains a term proportional to (1 − E²) in the denominator. For bound orbits, E < 1, so this is positive, but the physical meaning of this factor and its origin from the substitution h² = aM(1 − e²) could be clarified.
  2. Fig. 3: The inset showing relative error uses a different horizontal axis range (0–20 M) than the main plot (0–20 M but with different tick labels). This is slightly confusing; consider using consistent axis ranges or adding a clear label.
  3. Fig. 5: The caption states that dashed lines show the theoretical approximation 'derived from Eq. (28),' but the deflection formula is Eq. (39). This appears to be a typo.
  4. §II.A: The term 'aeral radius' (appearing after Eq. (4) and elsewhere) should be 'areal radius.'
  5. References [41] and [46] appear to be duplicate citations of the same Visser (1989) paper. Consider consolidating.
  6. The abstract states 'measurable deviations from standard black hole predictions' — given that the black hole branch (α ≤ 2M) is shown to be indistinguishable from Schwarzschild via b_c, the abstract could more precisely state that measurable deviations arise primarily in the wormhole branch.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for identifying a genuine algebraic error in the intermediate derivation of the light deflection angle. We address both major comments below.

read point-by-point responses

  1. Referee: Eqs. (37)–(38): There is a sign error in the intermediate derivation of the light deflection angle. Starting from the perturbative solution Eq. (36) and setting u = 0 at the incoming asymptote, the correct algebra yields φ∞ = −2M/b · 1/(1 − α²/(16b²)), i.e., the factor (1 − α²/(16b²)) should appear in the denominator, not as a multiplicative factor. Consequently, Eq. (38) as written, 4M/b(1 − α²/(16b²)), is incorrect; the correct expression is 4M/b · 1/(1 − α²/(16b²)). The text following Eq. (38) states 'we can expand the denominator in Eq. (38),' but Eq. (38) as written contains no denominator. Expanding the correct expression gives ∆φ ≈ 4M/b + Mα²/(4b³), which matches the final result Eq. (39) and is confirmed by the numerics in Fig. 5. The final formula is thus correct, but the intermediate steps in Eqs. (37)–(38) are wrong.

    Authors: We thank the referee for identifying this error. The referee is entirely correct. Setting u = 0 in Eq. (36) and applying the weak-field approximations sin φ ≈ φ, cos φ ≈ 1, cos(2φ) ≈ 1, sin(3φ) ≈ 3φ, one obtains φ(1/b − α²/(16b³)) = −2M/b², which gives φ∞ = −2M/b · 1/(1 − α²/(16b²)). The factor (1 − α²/(16b²)) should indeed appear in the denominator, not as a multiplicative factor as we wrote in Eqs. (37)–(38). We also confirm the internal inconsistency the referee notes: the text following Eq. (38) refers to expanding 'the denominator in Eq. (38),' yet Eq. (38) as written contains no denominator — this is a direct consequence of the error. The final result, Eq. (39), is correct because expanding 1/(1 − α²/(16b²)) ≈ 1 + α²/(16b²) to first order yields ∆φ = 4M/b + Mα²/(4b³), which is what we report and what the numerics in Fig. 5 confirm. We will correct Eqs. (37)–(38) and the surrounding text in the revised manuscript so that the intermediate steps are consistent with the final formula. revision: yes

  2. Referee: §IV.C and Fig. 5: The claim that the weak-field formulas produce 'measurable deviations' from Schwarzschild depends on whether the parameter ranges where the formulas are valid overlap with realistic astrophysical scales. The paper itself notes (Fig. 3 inset) that for a = 500M the relative error exceeds the percent level, and for α = 5M the deflection formula diverges from numerics at small impact parameters. The manuscript would benefit from a brief, quantitative discussion of whether the valid parameter regime (σ = M²/h² ≪ 1 and α² ≪ b²) is accessible for objects like Sgr A*, given current observational precision from GRAVITY and EHT. Without this, the abstract's claim of 'measurable deviations' is not fully substantiated.

    Authors: This is a fair point. The abstract's claim of 'measurable deviations' is stronger than what the manuscript currently substantiates, and we agree that a quantitative discussion of the accessible parameter regime would strengthen the paper. We will add a brief discussion in §IV.C (and adjust the abstract wording accordingly). Concretely: for Sgr A* with M ≈ 4×10⁶ M☉, the S2 star has a pericenter distance of roughly 120 AU ≈ 3000 r_g, placing it well within the weak-field regime σ ≪ 1. For the deflection observable, EHT probes impact parameters of order b ~ 5–10 r_g, where the weak-field expansion is marginal; the α²/b² correction is reliable only for b ≫ α, which for the wormhole branch (α > 2M) requires b ≳ 10M at minimum. Thus, the weak-field formulas are most directly testable through stellar-orbit precession (GRAVITY) rather than photon deflection (EHT) at current precision. The shadow-radius result (b_c increasing with α in the wormhole branch) is, by contrast, a strong-field result obtained numerically and is directly relevant to EHT observations. We will clarify these distinctions and soften the abstract language to reflect that the 'measurable deviations' are most accessible through the combination of precession measurements and the shadow-radius signature, rather than through weak-field deflection alone. revision: yes

Circularity Check

0 steps flagged

No circularity found: derivations are self-contained perturbative calculations from the metric, with numerical validation independent of analytical results

full rationale

The paper's derivation chain is self-contained and free of circularity. The periastron advance (Eq. 22) is derived from the Hamilton-Jacobi equation applied to the black bounce metric (Eq. 6), using standard perturbation theory in σ = M²/h². The light deflection angle (Eq. 39) is derived from the Lagrangian formalism applied to the same metric, again via perturbation theory. In both cases, the sole input is the metric (Eq. 3, from Simpson-Visser [17], an external citation) and standard GR techniques; no parameter is fitted to data and then presented as a prediction. The critical impact parameter b_c for α ≤ 2M is shown analytically to equal 3√3 M using standard shadow-radius formulas (Eqs. 40-41) cited from Psaltis [49, 50] (external), and for α > 2M it is determined numerically via bisection in PyGRO. The numerical results serve as independent validation of the analytical formulas, not as inputs to them. Self-citations are non-load-bearing: Ref [32] (Della Monica & de Martino) provides observational context, Ref [33] (Della Monica) is the PyGRO code tool, and Ref [23] is mentioned only as agreeing with the b_c result. None of these are invoked as theoretical premises that would make the derivation circular. The skeptic's identified sign error in Eqs. (37)-(38) is a correctness/presentation issue, not a circularity issue, and the final result (Eq. 39) is independently confirmed by numerics.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The paper introduces no new particles, forces, fields, or dimensions. The black bounce metric and its parameter α are inherited from Simpson & Visser (2019). The 'two universes' connected by the wormhole are a geometric feature of the inherited metric, not a new postulate. The PyGRO code is a tool, not an entity.

free parameters (2)
  • α (bounce parameter)
    The regularization parameter α is a free parameter of the Simpson-Visser metric (Eq. 3), not fitted to data in this paper. It is treated as an input varied across [0, 10M]. Its physical origin (what matter source produces it) is not addressed.
  • M (mass)
    The mass of the central object, set to M=1 in dimensionless units throughout. Standard input from prior literature, not fitted.
axioms (4)
  • domain assumption The Simpson-Visser metric (Eq. 3) is a valid solution to the GR vacuum field equations with an unspecified matter source.
    The paper treats this metric as given (from Ref. [17]) and does not derive it from an action or matter Lagrangian. The matter source supporting the geometry is not specified.
  • domain assumption Weak-field perturbation: σ = M²/h² ≪ 1 and α² ≪ b².
    Invoked in Sec. III.B (after Eq. 10) and Sec. IV.B (before Eq. 30) to justify truncating the perturbative expansion at first order.
  • standard math The PyGRO code correctly integrates the geodesic equations.
    Used throughout Secs. III.C and IV.C for numerical trajectories. The code is open-source (Ref. [33]) but no independent verification of integration accuracy is provided.
  • standard math The shadow radius formula r_sh = r_ph √(-g_00(r_ph)) (Eq. 40) applies to the black bounce geometry.
    Invoked in Sec. IV.C to explain the constancy of b_c in the black-hole branch. This is a standard result from Psaltis (2008, Ref. [50]) for spherically symmetric spacetimes.

pith-pipeline@v1.1.0-glm · 20476 in / 2980 out tokens · 556578 ms · 2026-07-09T02:38:51.410291+00:00 · methodology

0 comments
read the original abstract

We investigate the phenomenology of black bounce spacetimes through a combined analytical and numerical study of relativistic observables associated with both time-like and null geodesics. Black bounce geometries provide a continuous interpolation between Schwarzschild black holes and traversable wormholes by introducing a regularization bounce parameter $\alpha$, which removes the central singularity by replacing it with a finite-radius throat. Using the Hamilton--Jacobi and Lagrangian formalisms, we derive weak-field analytical expressions for the periastron advance of massive particles and the deflection angle of light, highlighting the leading corrections induced by the bounce parameter. These results are systematically compared with numerical integrations performed with the PyGRO code. We show that increasing $\alpha$ enhances both the periastron precession and the light deflection with respect to the Schwarzschild case. Additionally, we analyze the relativistic redshift of bound orbits and identify cumulative temporal phase shifts associated with the modified geometry. For null geodesics, we also determine numerically the critical impact parameter separating scattered and captured (or transmitted through the throat) photon trajectories. Although this quantity remains identical to the Schwarzschild value in the regular black hole branch, it increases significantly in the traversable wormhole regime, providing a clear observational signature of the throat structure. Our results demonstrate that black bounce spacetimes can produce measurable deviations from standard black hole predictions, opening the possibility of constraining these geometries with current and future observations of strong gravitational fields.

Figures

Figures reproduced from arXiv: 2607.07679 by \'Alvaro de la Cruz-Dombriz, Riccardo della Monica, V\'ictor Ovejero-Berm\'udez.

Figure 1
Figure 1. Figure 1: Time-like geodesic in a two-way traversable black bounce wormhole with [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bound timelike geodesics in the black bounce space [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the redshift as a function of time for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Deflection angle versus impact parameter for photons [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Null geodesics in the black bounce spacetime for two different regimes of the regularization bounce parameter [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

discussion (0)

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