Cela fait plusieurs années que mon collègue Jean-Pierre Petit et ses collaborateurs clament l’inconsistance mathématique et physique  des modèles relativistes de trou noir, dans le but de promouvoir leur alternative dite des « plugstars », qui selon eux serait soutenue par leur modèle cosmologique Janus.

Après s’être heurté durant des années à un rejet de publication de la part des « referees » des grandes revues scientifiques, en peaufinant leur argumentaire ils ont fini par réussir à publier récemment deux articles dans Journal of Modern Physics (2025), 16(10), 1479-1490 et (2026), 17(2), 199 -239. Bravo pour leur persévérance ! Mais cela valide-t-il pour autant leur propos ? Certainement pas. Dans ce premier billet je me contenterai de montrer explicitement les failles principales de leur travail sur l’incohérence mathématique de la théorie des trous noirs. dans un second billet j’étendrai ma critique aux fondements du “modèle Janus” qui prétend rebâtir toute la cosmologie.

Je précise d’emblée que ce n’est pas parce que j’ai consacré quelques dizaines d’années de recherches aux trous noirs, objets aux propriétés étranges et fascinantes dont j’ai réalisé dès 1978 les premières visualisations exactes, validées 40 ans plus tard par les observations télescopiques de l’Event Horizon Telescope (contestées par JP Petit et al.) que je « crois ferme » à leur existence dans l’univers réel.   Mais ce ne sont certainement pas les arguments erronés avancés par l’équipe de JPP qui me feront douter de leur pertinence, au minimum théorique, sinon observationnelle.

Avant de critiquer en détail les deux articles susmentionnés, je voudrais rappeler que l’idée de contester la validité mathématique de la solution de Schwarzschild n’est pas nouvelle. Je m’étais déjà donné le mal de l’expliquer il y a quelques années, mais sans résultat. J’y disais que la première traduction en anglais de l’article de Schwarzschild (originellement en allemand), publiée en 1999 par S. Antoci et Loinger [arXiv:physics/9905030], avait réactivé des doutes sur l’interprétation du calcul de Schwarzschild chez certains chercheurs instinctivement allergiques au concept de trou noir (défini correctement par un horizon des événements, et non, comme beaucoup le croient à tort, par une singularité gravitationnelle). En effet, Antoci et Loinger avaient cru bon d’agrémenter leur traduction d’un bref « foreword » affirmant péremptoirement et sans démonstration technique:

“This fundamental memoir contains the ORIGINAL form of the solution of Schwarzschild’s problem. It is regular in the whole space-time, with the only exception of the origin of the spatial co-ordinates; consequently, it leaves no room for the science fiction of the black holes. (In the centuries of the decline of the Roman Empire people said: “Graecum est, non legitur”…).”

Or, contrairement à ce qu’avancent JPP et ses collègues, ce n’est pas parce que les articles de Schwarzschild n’étaient pas traduits en anglais qu’ils étaient ignorés de la communauté des théoriciens, lesquels se seraient contentés de répéter depuis un siècle une « erreur » de Hilbert sans vérifier à la source. Avant cette traduction de 1999, plusieurs chercheurs avaient déjà glosé fallacieusement sur cette prétendue « erreur » de Hilbert qui, selon eux, aurait invalidé l’extension analytique maximale de Kruskal et Fronsdal et le concept de trou noir. Voir par exemple L. Abrams (Can. J. Phys. 67 (1989) 919 – arXiv :gr-qc/0102055 ). Jean-Pierre Petit et al. reprennent la même argumentation dans le but essentiel d’écarter la possibilité de formation d’un horizon des événements et de promouvoir leur modèle alternatif de “plugstar”.  Or, concernant du moins les interprétations de la solution de Schwarszchild, Hilbert, etc., le débat faussement ouvert par Antoci et Loinger a été clos rapidement:

(a) En 2011, C. Corda publie “A clarification on the debate on ‘the original Schwarzschild solution’” [Arxiv 1010.6031 [gr-qc] Electron.J.Theor.Phys. 8 (2011) 25, 65-82], dont voici l’abstract :

“Now that English translations of Schwarzschild’s original paper exist, that paper has become accessible to more people. Historically, the so-called ‘standard Schwarzschild solution’ was not the original Schwarzschild’s work, but it is actually due to J. Droste and, independently, H. Weyl, while it has been ultimately enabled like correct solution by D. Hilbert. Based on this, there are authors who claim that the work of Hilbert was wrong and that Hilbert’s mistake spawned black-holes and the community of theoretical physicists continues to elaborate on this falsehood, with a hostile shouting down of any and all voices challenging them. In this paper we re-analyse ‘the original Schwarzschild solution’ and we show that it is totally equivalent to the solution enabled by Hilbert. Thus, the authors who claim that ‘the original Schwarzschild solution’ implies the non existence of black holes give the wrong answer. We realize that the misunderstanding is due to an erroneous interpretation of the different coordinates. In fact, arches of circumference appear to follow the law dl = rd{\phi}, if the origin of the coordinate system is a non-dimensional material point in the core of the black-hole, while they do not appear to follow such a law, but to be deformed by the presence of the mass of the central body M if the origin of the coordinate system is the surface of the Schwarzschild sphere.”

(b) En 2013, P. Fromholz, E. Poisson et C. Will publient « The Schwarzschild metric: It’s the coordinates, stupid!” [arXiv:1308.0394 – American Journal of Physics 82, 295 (2014)]

 dont j’extrais ces quelques lignes:

“But Schwarzschild went on to address the integration constant b. He demanded that the metric be regular everywhere except at the location of the mass-point, which he assigned to be at ρ = 0, where the metric should be singular. This fixed b = (2M)³. This choice resulted in considerable confusion about the nature of the “Schwarzschild singularity”, which was not cleared up fully until the 1960s. Because we now are attuned to the complete arbitrariness of coordinates, we understand that ρ = 0, or r_S = 2M is not the origin, but is the location of the event horizon, while ρ = −2M, or r_S = 0 is the location of the true physical singularity inside the black hole. The unusual radial coordinate x was forced on Schwarzschild by Einstein’s constraint g = −1, nevertheless it led to a quite simple derivation of the exact solution.”

Dans une autre section, Fromholz et al. rappellent à juste titre que les équations d’Einstein peuvent être reformulées de différentes manières, par exemple dans le formalisme de Landau et Lifshitz ou encore celui d’Arnowitt-Deser-Misner (ADM). Dans tous les cas, la solution pour la métrique sphérique statique est strictement équivalente à la solution « standard » de Schwarzschild, telle qu’elle est reformulée dans les monographies classiques.

Comme cela ne semble pas avoir convaincu JPP et al., je reviens sur la question et j’étends ma critique à leur interprétation alternative des images de M87* et SgrA* obtenues par l’Event Horizon Telescope.

Pour les lecteurs motivés, je passe maintenant exclusivement à la langue anglaise.

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Critical analysis of the article “Questionable Black Holes” by JP Petit and G. d’Agostini

General outline

The paper’s central pattern is this: it takes legitimate historical or mathematical facts — Schwarzschild used different coordinates, Hilbert influenced later notation, Eddington–Finkelstein coordinates contain cross terms, the interior Schwarzschild solution has a pressure singularity for an ideal incompressible fluid — and then converts them into much stronger physical claims that are not justified. The biggest flaws are category errors: confusing coordinate choices with physical topology, coordinate singularities with true boundaries, metric sign conventions with imaginary time, mathematical extensions with physical interiors, and image brightness/temperature maps with gravitational redshift measurements.

The paper itself states its thesis clearly: standard black holes allegedly arise from Hilbert’s “wrong” interpretation of Schwarzschild, from an imposed contractible topology, from Birkhoff’s theorem excluding cross terms, and from a choice of Lagrangian; it then proposes “plugstars” with mass inversion and redshift ratio 3 as alternatives to M87* and Sgr A* . That chain has several weak links that I develop below.

1. Coordinate choices do not change the physics!

The paper argues that the “so-called Schwarzschild solution” is not Schwarzschild’s original solution and that Hilbert’s coordinate choice led people to wrongly treat r as a radial variable. There is a real historical distinction here: Schwarzschild’s original radial coordinate differs from the later areal-radius coordinate. But the JPP’s paper overstates the consequence. In standard general relativity, changing coordinates does not change the spacetime geometry. The modern Schwarzschild coordinate r is not “ordinary Euclidean radius”; it is the areal radius, defined so that spheres have area 4πr². That is a perfectly geometric definition. Calling it “not the original Schwarzschild coordinate” does not invalidate the modern Schwarzschild solution!

So the flaw is that the paper turns a coordinate-history point into a physical objection to black holes, without showing an invariant geometric contradiction. A serious critique would need curvature invariants, causal structure, geodesic completeness, trapped surfaces, or horizon properties — not merely the fact that another coordinate system exists.

2. The Schwarzschild horizon is NOT a physical boundary!

The paper repeatedly suggests that the Schwarzschild sphere should be treated as a boundary or throat rather than as a horizon in an extended spacetime. It claims the original Schwarzschild geometry is “non-contractile” and has a boundary at the Schwarzschild sphere.

But the event horizon at r=2M is not a curvature singularity. Curvature scalars remain perfectly finite there. The “singularity” at r=2M in Schwarzschild coordinates is a coordinate singularity, removable by Eddington–Finkelstein, Kruskal–Szekeres, Painlevé–Gullstrand, or other coordinates.

So the flaw is that the paper treats a coordinate-limited chart as evidence that spacetime itself must end or change topology at r=2M. But that does not follow : a coordinate patch can fail without the manifold ending.

3. The criticism of Birkhoff’s theorem is misleading

The paper says Birkhoff’s theorem “immediately prohibits the presence of a cross term in drdt” and ties this to an allegedly unjustified uniqueness assumption.

This is not right. Cross terms can appear in perfectly valid coordinate systems for the same Schwarzschild spacetime. Eddington–Finkelstein coordinates, Painlevé–Gullstrand coordinates, and other forms include dtdr -type terms. That does not mean Birkhoff’s theorem is false, nor does it create a new physical solution.

Birkhoff’s theorem says that any spherically symmetric vacuum solution is locally isometric to Schwarzschild. It does not say every coordinate representation must be diagonal! A cross term may be a coordinate artifact, not new physics.

So the flaw is that the paper confuses “a metric has a cross term in this coordinate system” with “Birkhoff uniqueness fails” or “there is a physically distinct spacetime.”

4. The Lagrangian argument is confused

The paper argues that the modern use of a quadratic geodesic Lagrangian changes the topology and permits geodesics “inside” the Schwarzschild sphere, whereas Schwarzschild’s original length functional supposedly forbids such curves.

This is a serious problem. In Lorentzian geometry, geodesics are not generally “shortest paths” in the Riemannian sense. Timelike geodesics extremize proper time; null geodesics have zero interval; spacelike, timelike, and null curves behave differently. Using the quadratic Lagrangian L=g_μν x’_μ x’_ν is standard and gives the same affinely parametrized geodesic equations as the square-root action where applicable. It does not by itself determine global topology.

So the flaw is that the paper wrongly claims that the choice between ds and ds² as a variational object determines whether the black-hole interior is physically allowed.

Topology is not chosen by the geodesic Lagrangian. It is part of the manifold structure and global extension, constrained by field equations, causal structure, and physical boundary conditions.

5. It conflates metric signature, imaginary time, and physical time reversal

The paper suggests that Hilbert’s approach preserved a “purely imaginary time” legacy and that modern black-hole theory inherits this problem . This is historically and technically weak.

Modern general relativity does not require physical time to be imaginary. The use of different metric signatures, such as (+– – –) or (– +++), is convention. Wick rotation or imaginary time appears in some mathematical and quantum-field contexts, but standard Lorentzian black-hole geometry uses real time coordinates.

So the flaw is that the paper treats sign convention and historical notation as if they implied a physical error in black-hole theory. That is not a valid inference.

6. The “time factor becomes negative, therefore mass becomes negative” claim is unsupported

This is probably the most speculative part. The paper argues that beyond “physical criticality,” a time factor f becomes negative; then dt becomes negative; then time reverses; then energy and mass reverse; then matter transfers to a PT-symmetric sheet.

However this chain is not at all established. A negative metric coefficient does not automatically mean that physical time reverses. In Schwarzschild coordinates inside the horizon, the roles of t and r change character, but that is not equivalent to matter acquiring negative mass. Similarly, PT symmetry in quantum or field-theoretic contexts does not imply gravitational mass inversion merely because a coordinate changes sign.

So the flaw is that the paper jumps from a sign change in a metric component to negative mass and another universe without deriving this from Einstein’s equations plus a physically defined stress-energy tensor.

To make this credible, it would need a full dynamical solution with energy conditions, matching conditions, conservation laws, stability analysis, and observable predictions. The paper does not provide that.

7. The use of Schwarzschild’s interior solution is physically inappropriate for neutron stars

The paper leans heavily on the interior Schwarzschild solution for an incompressible fluid and its pressure divergence. It then interprets that divergence as evidence for variable light speed, topological surgery, or mass inversion.

But the interior Schwarzschild solution assumes constant density and an idealized perfect fluid. Real neutron stars do not have constant density. They require an equation of state, relativistic hydrostatic equilibrium through the Tolman–Oppenheimer–Volkoff equation, nuclear physics, causality constraints, rotation, magnetic fields, and possibly phase transitions.

So the flaw is that the paper treats a known idealized-model pathology as a message about new topology or variable speed of light c, rather than as a sign that the incompressible-fluid model has exceeded its physical domain. The pressure singularity in the constant-density model is not evidence that mass inversion occurs!

8. The proposed neutron-star mass limit is asserted, not derived

The paper claims the plugstar mechanism would limit neutron-star masses to “just over 2.5 solar masses”. But it does not provide a proper stellar-structure calculation.

A credible mass-limit claim would require solving the TOV equations with a specified equation of state, stability conditions, perturbation analysis, and comparison with observed neutron-star masses and merger constraints.

So the flaw is that the 2.5-solar-mass limit appears here as a qualitative consequence of the proposed mechanism, not as a rigorously derived astrophysical bound. This is especially problematic because observed compact-object mass limits are strongly equation-of-state dependent.

9. The EHT redshift argument is not valid

The paper claims that the chromaticity bar in EHT images of M87* and Sgr A* gives a wavelength ratio close to 3, and that this supports plugstars because black holes would produce infinite gravitational redshift .

This is a major observational flaw. EHT images are reconstructed radio-intensity maps, not direct surface-temperature photographs. The color scale is a visualization of brightness/intensity or model-dependent emission, not a direct gravitational-redshift ruler. Also, black-hole observations do not require radiation to be emitted from the event horizon itself. The observed radiation comes mainly from hot plasma outside the horizon, in an accretion flow, jet base, or photon-ring region.

So the flaw is that the paper compares a plugstar surface-redshift prediction to an EHT image color scale as though the latter directly measured gravitational redshift. That is not how EHT data work.

Also, saying black holes predict “infinite redshift” is misleading in this context. Infinite redshift applies to light emitted exactly from the horizon by a static emitter, but realistic observed emission comes from outside the horizon.

10. The paper ignores stronger black-hole evidence

The paper focuses on M87* and Sgr A* images, but black-hole evidence is broader: stellar orbits around Sgr A*, compactness bounds, X-ray binaries, gravitational-wave ringdowns, accretion-disk spectra, relativistic jets, tidal disruption events, and binary black-hole mergers.

The paper’s alternative model would need to explain all of these at least as well as the standard model. Instead, it mostly targets selected features and gives qualitative reinterpretations.

So the flaw is that the paper attacks a simplified version of black-hole evidence rather than the full observational case.

11. The Hoag’s Object / quasar section is highly speculative and disconnected

The later section jumps from plugstars to Seyfert galaxies, Hoag’s Object, Hubble constant anomalies, density waves, quasars, cosmic rays, M87 jet structure, and metric fluctuations. These claims are not developed from the earlier equations in a controlled way. The paper expands into large cosmological and astrophysical speculation without deriving testable quantitative predictions. This weakens the paper because it mixes a (wrong) mathematical critique of Schwarzschild geometry with broad conjectures about quasars and galaxy evolution.

Conclusion

The JPP et al paper’s central weakness is that it repeatedly promotes coordinate, notation, and model-dependent features into physical claims. It treats Schwarzschild coordinate choices as physical topology, cross terms as violations of Birkhoff uniqueness theorem, a quadratic geodesic Lagrangian as a topology-changing assumption, an ideal-fluid pressure divergence as evidence for variable light speed or topological surgery, and EHT image color scales as direct redshift measurements. The proposed plugstar mechanism therefore rests on several unsupported transitions rather than on a complete solution of Einstein’s equations with a physically defined stress-energy tensor and observationally testable predictions.