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REVIEW 3 major objections 4 minor 86 references

The hyperbolically symmetric black hole

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Pith's one-line read This paper argues that the region inside a black hole's horizon can be described by a static, hyperbolically symmetric spacetime in which gravity is repulsive, test particles never reach the center, and information can escape outward along

desk verdict Clear, honest summary of an earlier model, but the central junction between the exterior Schwarzschild and the hyperbolically symmetric interior cannot be realized: the induced transverse geometry is a sphere on one side and a hyperboloid on the other, so no Israel shell can join them. read the letter →

arxiv 2509.02621 v1 pith:RCJBDFCE submitted 2025-09-01 gr-qc astro-ph.GA

L. Herrera , L. Witten This is my paper · ORCID
classification gr-qcastro-ph.GA MSC 83C5783C10 PACS 04.70.-s04.20.Jb
keywords hyperbolicallysymmetricblackholerepulsivegravityinformation-lossparadoxLandauerprinciplegeodesicshorizonthinshellshadowrelativisticjets
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 authors are trying to establish that the standard picture of a black hole interior — a non-static region in which everything inevitably falls to the center — can be replaced by a different vacuum solution of Einstein's equations. In their model, the exterior is ordinary Schwarzschild, but inside the horizon the geometry is static and hyperbolically symmetric rather than spherically symmetric. The consequence is repulsive gravity inside the horizon: test particles are pushed away from the center, and they can cross the horizon outward along a single symmetry axis. If this picture is right, a collapsed object is not truly 'black': information encoded in matter can leave along that axis, so the thermal evaporation of the hole need not end in an information-loss paradox. The paper also suggests the escaping axial flux could be observable, as a collimated high-energy outflow or as an imprint on the shadow image of the collapsed object.

What carries the argument

The load-bearing object is the hyperbolically symmetric line element (Eq. 4), the vacuum solution that replaces the Schwarzschild interior: it is static, admits a timelike Killing vector plus three Killing vectors of hyperbolic symmetry (Eq. 5), and has the same R=2M horizon radius as the exterior. Its role is to supply an interior that stays static while abandoning spherical symmetry; the repulsive sign of the radial four-acceleration (Eq. 7) is what generates the geodesic structure — no capture at the center, outward escape only along the axis — on which all the information-flow and observational consequences rest. The junction between the two manifolds is not smooth in the Darmois sense,

What would settle it

A concrete settling check is the junction problem the authors flag as open: write down the Israel conditions for the thin shell at R=2M connecting metric (1) to metric (4); if the required surface stress-energy violates every energy condition or forbids the outward axial geodesic crossing, the HSBH interior cannot be realized. Observationally, the model predicts a collimated, high-energy outflow along the symmetry axis of a Schwarzschild-like collapsed object; a clean sample of black-hole shadow images and jet morphologies in which no such axially collimated counterpart appears would count aga

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the interior of a Schwarzschild black hole is not the usual non-static extension of the Schwarzschild metric; instead it is the static, hyperbolically symmetric line element obtained by the complex transformation θ → iθ of the exterior metric. In that interior, the four-acceleration of a static observer is a^r = −M/r^2, meaning gravity repels, so test particles never reach r = 0 and are pushed back from the deep interior. Geodesic analysis of this interior shows that particles can cross the horizon outward only along the θ = 0 axis. Because matter — and therefore information — can leave the hole through that axis, the paper argues the HSBH

Load-bearing premise

The load-bearing premise is that the Schwarzschild exterior and the hyperbolically symmetric interior can actually be joined at the horizon into a single black-hole spacetime; the authors concede the two manifolds do not match smoothly in the Darmois sense and that a thin shell governed by the Israel conditions must mediate the junction, and if no physically admissible shell exists that permits the axial escape, the model's distinctive consequences fail.

Editorial extensions

If this is right

  • If the interior is hyperbolically symmetric, test particles inside the horizon are repelled and never reach the center, so the classical 'singularity as final state' picture is replaced by a bounce in the deep interior.
  • Matter crossing the horizon outward along the θ=0 axis means collapsed objects are not information-tight; information can escape even if the radiation itself is thermal, removing the need for a unitary-evaporation resolution of the information-loss paradox.
  • The Landauer argument gives a concrete, in-principle observable: an energy flux along the symmetry axis associated with the erasure of information inside the horizon.
  • A Schwarzschild exterior with an axis-selective interior escape predicts a collimated, high-energy outflow, which the authors connect to the puzzle of extragalactic relativistic jets.
  • Shadow images of the collapsed object at galactic centers could carry imprints of the axial material flow, giving an observational route to confirm or dismiss the model.

Reading between the lines

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

  • A natural extension the paper leaves implicit: the same interior geometry with angular momentum, where the escape axis would likely align with the spin axis, making the predicted outflow directional in an observable way.
  • The model suggests a testable demographic correlation: collapsed objects whose shadows match a Schwarzschild exterior should also show axially collimated outflows if the HSBH picture holds; compiling shadow and jet samples would test that correlation directly.
  • If the repulsive interior is right, accretion inside the horizon would differ dramatically from classical expectations (no central pile-up, matter accumulating near a bounce radius), which in principle changes the emitted gravitational-wave and neutrino signatures of collapse — a consequence the paper does not develop.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper advocates a picture of a black hole in which the exterior is the usual Schwarzschild solution (1) but the interior (R < 2M) is described by the static, hyperbolically symmetric vacuum metric (4). The authors claim that this interior is static, that test particles there experience repulsive gravity (four-acceleration a^r = -M/r^2, Eq. 7), never reach the center, and may cross the horizon outward only along the symmetry axis θ=0. They further use the Landauer principle in the form of Eqs. (13)-(14) to argue that information can leave the interior in this scenario, so that no information-loss paradox arises. The paper also proposes that such an outflow could produce collimated relativistic jets and leave imprints in black-hole shadow observations. The manuscript is largely a summary of the authors' previous work: the metric is from Ref. [1], the geodesic results from Ref. [19], and the Landauer-in-GR formulas from Refs. [41,42].

Significance. If the construction were a genuine solution of Einstein's equations representing a single black-hole spacetime, the paper would challenge the standard interior picture and propose a concrete mechanism for information escape and jet collimation. The paper honestly presents some of its own limitations, including the lack of a smooth Darmois match at the horizon. However, the central construction is not established: the junction between the two metrics is not merely non-smooth, but impossible as stated because the induced geometries on the two sides are intrinsically different. The paper also contains no new derivations; its main physical conclusions are inherited from earlier works by the same authors. The four-acceleration calculations in Eqs. (6)-(9) are straightforward and correct for the isolated metric (4), but their physical meaning depends entirely on the unproven global spacetime. The observational section is explicitly speculative and does not provide a quantitative test.

major comments (3)
  1. [Section III, first bullet; Eqs. (1) and (4)] The junction between the exterior and interior is the load-bearing assumption of the paper, and it is not viable as stated. On a surface of constant (t,R) at R=2M, metric (1) induces 4M^2(dθ^2 + sin^2θ dφ^2), a round sphere with curvature +1/(4M^2), while metric (4) induces 4M^2(dθ^2 + sinh^2θ dφ^2), a hyperbolic plane with curvature -1/(4M^2). These are not locally isometric and are not even diffeomorphic (S^2 versus the noncompact hyperbolic plane). The Israel formalism requires the equality of the first fundamental forms on the junction surface; no choice of shell stress-energy can change the induced metric. The authors' statement that the two manifolds 'do not match smoothly in the Darmois sense' is therefore an understatement: the Israel shell they appeal to cannot exist because the first junction condition fails identically. Without a valid junction, the combined spacetime is not a
  2. [Section I.B, Eqs. (13)-(14) and following text] The claim that no information-loss paradox appears in the HSBH is asserted rather than derived. Even granting the mass-of-a-bit formula (14), the existence of a classical test-particle trajectory crossing the horizon outward along θ=0 does not by itself imply that quantum information initially in the collapsing matter is returned to infinity in a unitary way. One would need a quantum field theory on the proposed background and a computation of the late-time radiation state; the paper provides none. Thus the central advertised conclusion about information loss is a non sequitur unless one assumes, without argument, that classical particle escape is equivalent to information recovery. This gap is central to the paper's significance and cannot be filled by the geodesic results quoted from Ref. [19].
  3. [Section I.A and Section III, discussion of negative mass] The repulsive-gravity interpretation is tied to the claim that the interior contains negative mass-energy, but metric (4) is a vacuum solution with zero Ricci tensor, so any 'negative mass' is an interpretation rather than a property derived from a matter source. More importantly, the geodesic claims quoted from Ref. [19] are statements about the metric (4) in isolation. Since the global manifold is not constructed (see major comment 1), the phrase 'test particles may cross the horizon outward' is ambiguous: the metric (4) alone has a coordinate singularity at R=2M and does not specify the continuation. The paper should either provide a well-defined global atlas with a valid junction or clearly state that the model is only a local interior metric; in the latter case, the black-hole and information-loss claims do not follow.
minor comments (4)
  1. [Eq. (4) and Section I.A] The coordinate ranges for θ and φ in metric (4) are not specified. For the hyperbolic angular part to be regular, θ∈[0,∞) with φ identified modulo 2π is required; the 'axis' θ=0 then has a different global structure from the exterior's θ axis. The paper should clarify the topology and what 'along the axis' means precisely.
  2. [Section I.B, Eqs. (12)-(13)] The generalization from the weak-field Landauer expression (12) to the strong-field formula (13) is not derived. The Tolman temperature is normally T/√(-g_tt) (or with sign conventions), not T√|g_tt|; the sign and coordinate invariance should be checked and stated.
  3. [References] There are several typographical issues in the reference list, e.g., 'Caroll' for Carroll and 'Lema ˆ ıtre' for Lemaître. Also, the reference to 'Hawking radiation is completely thermal' [52] is cited in a way that does not distinguish between the semiclassical approximation and the full information paradox.
  4. [Section II] The observational discussion for EHT shadows and relativistic jets is explicitly speculative. It would be more useful if the paper identified a specific observable (e.g., a predicted shadow feature or a jet power/opening angle) that could distinguish the HSBH from the standard model, rather than merely suggesting that such imprints 'could' exist.

Circularity Check

2 steps flagged · score 7.0 of 10

The HSBH information-escape conclusion is inherited from the authors' own prior geodesic paper; the model's central premise is also set by a same-author citation, so the argument is a self-referential chain.

  1. self citation load bearing [Section I.A (Geodesics in HSBH) and Section I.B (Flow of Information and Landauer Principle)]
    "In [19], a general study of geodesics in the spacetime described by (4) is presented ... leading to some interesting conclusions ... • Unlike the CBH , test particles can cross the horizon outward, but only along the θ = 0 axis. ... As mentioned before, in such a case, the crossing of massive particles through the horizon outwardly is allowed along the θ = 0 axis, thereby implying the existence of a flux of information from the inside of the horizon to the outside. Thus, in this scenario, no information loss paradox appears."

    The paper's central resolution of the information-loss paradox is not derived in this paper. The premise that particles can cross the horizon outward along θ=0 is quoted from Ref. [19], a prior paper by the same group (Herrera, Di Prisco, Ospino, Witten), analyzing geodesics in the metric proposed by the same authors in Ref. [1]. The conclusion 'no information loss paradox' is a direct restatement of this self-cited escape property. No independent derivation, external benchmark, or quantitative prediction is supplied; the argument reduces to: our previous model has the escape property we put into it.

  2. self citation load bearing [Section I.B (Flow of Information and Landauer Principle)]
    "The link between the Landauer principle and general relativity is discussed in detail in [41]. Here, we resort to some results found in that reference ... a mass given by Mbit = kT c2 ln 2 ... was assigned to any bit of information ... [42]."

    The generalized Landauer formula ΔE = kT√|gtt| ln2 and the 'mass of a bit' Mbit are imported from Refs. [41] and [42], both authored by L. Herrera. These self-cited results are used to argue that information escaping along the axis could carry observable energy. This is secondary to the no-paradox claim, but it is another load-bearing self-citation: the paper does not independently re-derive or test these formulas, and the observable-consequences discussion relies on them.

full rationale

This paper is essentially an application/review of the authors' own HSBH model. The interior metric (4) is adopted from Ref. [1], where it was obtained from Schwarzschild by θ→iθ; it is an ansatz from a same-author citation, not a first-principles derivation in this paper. The three distinctive physical properties—repulsive gravity, no reaching the center, and outward horizon crossing only along θ=0—are quoted from Ref. [19], also by the same group, rather than re-derived here. The central claim that the HSBH avoids the information-loss paradox follows directly from that self-cited geodesic property. The Landauer-in-GR and bit-mass formulas used in the information discussion are likewise from same-author references [41,42]. There is some independent content: the four-acceleration computation (7) is performed in this paper and is a straightforward consequence of (4), and the observational discussion is explicitly speculative. However, the load-bearing premise for the information conclusion is a self-citation chain. The authors themselves flag the Darmois matching failure and the need for an Israel thin shell (Section III, first bullet); that is a serious physical correctness risk, but it is not circularity and does not by itself raise the circularity score. Overall, the central claim reduces to results taken from the same authors' prior model, giving a circularity score of 7.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central claims rest on the asserted validity of metric (4), the imported geodesic results, the Landauer/Tolman generalization, the existence of negative mass as the source of repulsion, and the unverified junction across the horizon. No free parameters are fitted to data; all parameters are the Schwarzschild mass M and physical constants.

assumptions (5)
  • domain assumption Metric (4) is a static vacuum solution of Einstein's equations inside the horizon.
    The paper asserts (4) follows from (1) via theta to i theta and is a solution, but does not verify Ricci-flatness here; all interior claims depend on it.
  • domain assumption The gravitational version of the Landauer principle, Eq. (13): Delta E = kT sqrt(|g_tt|) ln 2, and the mass of a bit, Eq. (14).
    Imported from refs [41,46] (one authored by the current first author) and used to convert axis escape into an information flow that removes the paradox.
  • domain assumption Negative mass/energy is present inside the horizon and is the origin of repulsive gravity.
    Stated in the Discussion with citations to the authors' own fluid studies [18,21,22,30]; no independent evidence is given in this paper.
  • ad hoc to paper A horizon shell (Israel layer) can physically join the spherical exterior to the hyperbolic interior.
    Section III admits the Darmois matching fails and a shell is required, but no shell stress-energy tensor or stability analysis is provided.
  • domain assumption The geodesic results from ref [19] apply to physical test particles and are correct.
    The three bullet properties in Section I.A are cited from ref [19] rather than derived in this text.
invented entities (3)
  • Hyperbolic interior spacetime (HSBH)
    purpose: Provides a static region inside the horizon in place of the dynamic Schwarzschild interior.
    Defined by the authors in ref [1]; the paper offers qualitative observational ideas (jets, shadows) but no quantitative signature that could falsify this specific interior.
  • Negative mass/energy inside the horizon
    purpose: Explains why gravity is repulsive in the HSBH interior.
    Invoked from the authors' earlier fluid papers; no independent detection or mechanism is proposed here.
  • Outward flux of matter/information along the symmetry axis
    purpose: Resolves the information-loss paradox and provides a collimation mechanism for relativistic jets.
    A consequence of the model's geodesics, not an observed signature; the paper calls the jet connection speculation.

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Cite this review

Pith. "Pith review of The hyperbolically symmetric black hole." pith.science (2026). https://pith.science/paper/RCJBDFCE

@misc{pith2026250902621,
  author       = {Pith},
  title        = {Pith review of: The hyperbolically symmetric black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCJBDFCE}},
  note         = {Machine review of arXiv:2509.02621}
}
abstract

We describe some properties of the hyperbolically symmetric black hole (hereafter referred to as the $HSBH$) proposed a few years ago. We start by explaining the main motivation behind such an idea, and we determine the main differences between this scenario and the classical black hole (hereafter referred to as the $CBH$) scenario. Particularly important are the facts that, in the $HSBH$ scenario, (i) test particles in the region inside the horizon experience a repulsive force that prevents them from reaching the center, (ii) test particles may cross the horizon outward only along the symmetry axis, and (iii) the spacetime within the horizon is static but not spherically symmetric. Next, we examine the differences between the two models of black holes in light of the Landauer principle and the Hawking results on the eventual evaporation of the black hole and the paradox resulting thereof. Finally, we explore what observational signature could be invoked to confirm or dismiss the model.

Discussion (0). Continue with ORCID to comment.

Reference graph

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