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arxiv: 2606.06339 · v2 · pith:QFSCUGUA · submitted 2026-06-04 · gr-qc

tt GrayHawk tt v2: wormholes and numeric extension

Marco Calz\'a This is my paper

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classification gr-qc
keywords GrayHawktortoise coordinateswormholesscatteringnumerical methodsblack holesgeneral relativityfield propagation
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The pith

GrayHawk v2 enables numerical computation of tortoise coordinates and wormhole scattering.

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

The paper presents an updated version of the GrayHawk tool that supports fully numerical evaluation of the tortoise coordinate integral. This change allows analysis of metrics where analytic expressions for the integral are unavailable or impractical. The update also adds support for wormhole solutions in the scattering problem. A reader would care because these features expand the set of spacetimes that can be studied for phenomena like field propagation and Hawking radiation using the same software framework. The modular design is preserved to allow further customizations.

Core claim

GrayHawk v2 enlarges the capabilities of the tool by enabling a fully numerical computation of the tortoise coordinates integral and by extending the scattering problem to wormhole solutions, while enriching the pool of pre-loaded metrics and maintaining the modular structure of the code.

What carries the argument

The GrayHawk software tool for studying field propagation on curved manifolds, extended with numerical tortoise integral and wormhole support.

If this is right

  • The tool can now handle metrics lacking analytic tortoise coordinate expressions.
  • Scattering problems can be solved for wormhole spacetimes.
  • New pre-loaded metrics allow immediate testing of the features.
  • The modular code structure facilitates easy modifications for additional capabilities.

Where Pith is reading between the lines

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

  • This numerical approach may allow detailed studies of Hawking radiation in wormhole backgrounds.
  • Comparisons of scattering properties between black holes and wormholes become feasible with the same code.
  • Further extensions to other types of curved spacetimes could be implemented due to the maintained modularity.

Load-bearing premise

The numerical integration routines and wormhole implementations are assumed to produce correct and accurate results without any reported validation against known analytic cases.

What would settle it

Running the numerical tortoise coordinate computation for the Schwarzschild metric and comparing the output to the known analytic expression would test the accuracy of the new feature.

Figures

Figures reproduced from arXiv: 2606.06339 by Marco Calz\'a.

Figure 1
Figure 1. Figure 1: Transmission coefficient of a field of spin-2 and mode [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Transmission coefficient of a field of spin-2 and mode [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Transmission coefficient of a field of spin-2 and mode [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Transmission coefficient of a field of spin-2 and mode [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Residuals between the gray-body factors of a wave of spin-2 and mode [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

We enlarged the capabilities of the publicly available Mathematica-based tool $\tt GrayHawk$. This second version expands the spectrum of metrics that can be considered in two distinct and disjoined directions. First, it enables a fully numerical computation of the tortoise coordinates integral, allowing the user to account for many metrics for which an analytic computation was impractical. Second, it extends the scattering problem to wormhole solutions. The pool of pre-loaded metrics is enriched, enabling immediate testing of the new features, and the code's modular structure is maintained to facilitate easy modification. This implementation proves $\tt GrayHawk$ adaptability and makes it an even more powerful tool for studying black holes, wormholes, Hawking radiation, and other features involving field propagation on curved manifolds. The codes described in this work are publicly available at \href{https://github.com/marcocalza89/GrayHawk-v2}{\faGithub}.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript describes GrayHawk v2, an updated publicly available Mathematica package that adds two main capabilities to the original tool: a fully numerical routine for evaluating the tortoise coordinate integral (to handle metrics lacking closed-form expressions) and an extension of the scattering problem to wormhole backgrounds. The pool of pre-loaded metrics is enlarged, the modular structure is preserved, and the code is released on GitHub.

Significance. A verified, modular, open-source tool for numerical studies of wave propagation on black-hole and wormhole spacetimes would be a useful community resource. The significance is currently limited by the absence of any reported validation of the new numerical components against known analytic results.

major comments (2)
  1. [Description of the numerical tortoise integral routine] The central claim that the new numerical tortoise-coordinate routine 'enables' reliable computation for metrics without analytic forms rests on an untested implementation. No comparison is provided to the exact Schwarzschild result r* = r + 2M ln|r/2M − 1| or to any other standard analytic case, leaving open the possibility of integration, boundary-condition, or discretization errors.
  2. [Wormhole scattering extension] The extension of the scattering problem to wormhole solutions is presented without any quantitative benchmark against known wormhole scattering cross-sections or transmission/reflection coefficients. This omission is load-bearing for the assertion that the wormhole implementation works correctly.
minor comments (2)
  1. [Abstract] The abstract states that the implementation 'proves' adaptability; the wording should be softened to 'demonstrates' since the paper is a code description rather than a formal proof.
  2. [Abstract] The GitHub link is given with a font-awesome icon that may not render in all PDF viewers; a plain URL would be clearer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive comments. We address the two major comments point by point below. Both concerns are valid given the current manuscript content, and we will revise the paper accordingly.

read point-by-point responses

  1. Referee: The central claim that the new numerical tortoise-coordinate routine 'enables' reliable computation for metrics without analytic forms rests on an untested implementation. No comparison is provided to the exact Schwarzschild result r* = r + 2M ln|r/2M − 1| or to any other standard analytic case, leaving open the possibility of integration, boundary-condition, or discretization errors.

    Authors: We agree that the manuscript lacks explicit validation of the numerical tortoise-coordinate routine. In the revised version we will add a dedicated subsection comparing the numerical output for the Schwarzschild metric against the known analytic expression r* = r + 2M ln|r/2M − 1|, including error estimates and convergence tests with respect to integration parameters. This will directly address concerns about integration, boundary-condition, and discretization accuracy. revision: yes

  2. Referee: The extension of the scattering problem to wormhole solutions is presented without any quantitative benchmark against known wormhole scattering cross-sections or transmission/reflection coefficients. This omission is load-bearing for the assertion that the wormhole implementation works correctly.

    Authors: We concur that quantitative benchmarks are required to substantiate the wormhole scattering extension. The revised manuscript will include a new figure and accompanying text showing transmission and reflection coefficients for at least one standard wormhole metric, compared against existing analytic or high-precision numerical results from the literature. This will provide the necessary verification of the implementation. revision: yes

Circularity Check

0 steps flagged

No circularity: code-description paper with no derivations or fitted predictions

full rationale

The manuscript is a description of software extensions to GrayHawk v2 for numerical tortoise-coordinate integration and wormhole scattering. No derivation chain, no fitted parameters, no predictions, and no self-citation load-bearing steps exist. The central claims concern code capabilities and modularity; these are not reduced to inputs by construction and require only standard validation (absent here but irrelevant to circularity).

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, physical axioms, or invented entities are introduced; the work is a software engineering update.

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discussion (0)

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Forward citations

Cited by 2 Pith papers

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

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  2. $\tt BlackHawk$ $\tt v3.0$: Hawking Radiation from Regular Black Holes

    gr-qc 2026-06 unverdicted novelty 5.0

    BlackHawk v3.0 adds Hawking temperatures and greybody factors for multiple regular black hole metrics to an existing public code via numerical routines.

Reference graph

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