Atemporality from Conservation Laws of Physics in Lorentzian-Euclidean Black Hole

Emmanuele Battista; Salvatore Capozziello; Silvia De Bianchi

REVIEW 4 major objections 5 minor 1 cited by

Atemporality from Conservation Laws of Physics in Lorentzian-Euclidean Black Hole

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read In Lorentzian-Euclidean black holes, conservation laws force infalling bodies to stop at the horizon, so no singularity is ever reached.

desk verdict The paper's central theorem is a restatement of the authors' Lorentzian-Euclidean model with an unjustified energy-scaling step; the philosophical discussion is competent, but the derivation does not hold. read the letter →

arxiv 2504.17570 v1 pith:2S2VNGUF submitted 2025-04-24 gr-qc astro-ph.HEhep-th

Silvia De Bianchi , Salvatore Capozziello , Emmanuele Battista This is my paper · ORCID

classification gr-qcastro-ph.HEhep-th

keywords Lorentzian-EuclideanblackholeAtemporalityImaginarytimeConservationlawsNoethersymmetriesSignaturechangeSingularityavoidanceCausality

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

This paper argues that atemporality—the mechanism by which infalling bodies in a Lorentzian-Euclidean black hole stop at the event horizon and never reach the singularity—is not an ad hoc feature but a direct consequence of conservation laws. In this class of solutions, the metric changes signature at the horizon, time becomes imaginary inside, and the paper claims this preserves the energy-conservation law associated with time-translation symmetry. The authors state an Atemporality Theorem: preserving a conservation law keeps events causally connected and avoids singularities, while violating it would make time imaginary and measurements impossible. If correct, this would mean that singularity avoidance follows from Noether symmetries in general relativity, and that a singularity-free physics is classically consistent.

What carries the argument

The central object is the signature-changing Lorentzian-Euclidean metric $ds^2 = -\varepsilon(1-2M/r)\,dt^2 + dr^2/(1-2M/r) + r^2 d\Omega^2$, with $\varepsilon = \mathrm{sign}(1-2M/r)$, together with the Noether-conserved energy $E = -\varepsilon g_{\mu\nu}\xi^{\mu}u^{\nu}$ built from the static Killing vector $\xi^{\mu}$. The paper writes $E^2 = \alpha^2 \varepsilon^4$ with $\alpha$ a bounded positive function, which makes the radial velocity $\dot r$ vanish exactly at the horizon and become imaginary inside; this identity is what ties atemporality to conservation of energy.

What would settle it

Integrate the radial geodesic equation with the Killing energy held constant across the horizon, without assuming $E^2=\alpha^2\varepsilon^4$: if $\dot r$ does not vanish at $r=2M$ for any allowed $0<E^2<1$, then infalling bodies can enter the imaginary-time region and the Atemporality Theorem's conclusion fails.

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Extended reading notes

Core claim

The central claim is the Atemporality Theorem: atemporality is the dynamical mechanism which, by preserving a conservation law, allows events in a Lorentzian-Euclidean spacetime to remain causally connected; consequently any singularity is avoided and time can only be defined through real values; if the conservation law is violated, a singularity emerges, time becomes imaginary, and relativistic measurements are impossible. In the model, the metric is Schwarzschild-like outside $r=2M$, but the sign function $\varepsilon = \mathrm{sign}(1-2M/r)$ changes from $+1$ to $-1$ at the horizon, making the interior Euclidean. Energy defined through the static Killing vector is conserved only if infalling particles halt at the horizon rather than entering the imaginary-time region, and the paper presents this as a derivation of atemporality from Noether symmetries.

Load-bearing premise

The argument hinges on redefining the particle's conserved energy as $E^2=\alpha^2\varepsilon^4$ so that the radial velocity vanishes exactly at the horizon; if that redefinition is not legitimate for a genuine constant of motion, infalling particles would not necessarily stop there.

Editorial extensions

If this is right

No massive infalling body reaches $r=0$ in this model, so the geodesic structure is complete and the singularity is avoided within general relativity.
Atemporality becomes a consequence of Noether symmetries rather than a manually inserted Wick rotation, giving a principled basis for signature change.
Wherever time-translation symmetry is conserved, time must remain real and causality is preserved; the Euclidean interior marks a breakdown of measurability.
The Kretschmann scalar at the horizon, $K(r=2M)=3/(4M^4)$, provides a finite, mass-dependent quantity that could quantify the degree of atemporality.
The theorem supports a classical 'Singularity-Free Physics' in which conservation laws enforce the inaccessibility of singularities.

Reading between the lines

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

If atemporality follows from conservation laws, the same reasoning might extend to rotating or charged signature-changing black holes, where additional Noether charges could enforce similar horizon-halting behavior; the paper does not establish this extension.
The theorem suggests a broader criterion: physical time remains real exactly where a time-translation symmetry is conserved, so imaginary time could be read as the signature of a violated conservation law in any signature-changing spacetime.
A testable extension would be to ask whether the predicted absence of infalling matter crossing the horizon leaves observable signatures, for instance in accretion or gravitational-wave ringdown of such hypothetical objects; the paper does not address observational consequences.
The generality of the Atemporality Theorem would be sharpened by proving that the relation $E^2=\alpha^2\varepsilon^4$ follows directly from Noether's theorem rather than being imposed to obtain the desired horizon behavior.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

What you should know: the main result of this paper, the 'Atemporality Theorem' in Section 4, is not actually derived. The proof recaps features of the authors' earlier Lorentzian-Euclidean black hole model [11] and then asserts the converse. The load-bearing step is the relation E^2 = alpha^2 epsilon^4 in Section 2. That step is wrong as stated. Along a radial geodesic starting at rest at r_i > 2M, Eq. (7) fixes E^2 = 1 - 2M/r_i, a positive constant. Since epsilon -> 0 at the horizon, the relation forces alpha^2 = E^2/epsilon^4 to diverge, contradicting the claim that alpha^2 is bounded. If instead alpha^2 is bounded, then E is a function of position and not the Noether-conserved Killing energy. Either way, the claimed vanishing of \dot r at r = 2M is an artifact, not a consequence of conservation laws. The stress-test note holds up.

The paper does have real virtues. It is clearly written and honest about its scope: the abstract calls it a philosophical perspective, and footnote 6 concedes that a complete treatment requires quantum effects. The discussion of Noether's theorems and the philosophy of physics literature (Earman, Read, Brading, Brown, Gomes/Roberts/Butterfield) is competent and genuinely useful. The Kretschmann bound K(r=2M)=3/(4M^4) is standard, and the authors do not overclaim about new calculations. The notion that atemporality is 'naturally related to conservation laws' is conceptually suggestive, even if it is immediate once energy is defined through the Killing vector.

The soft spots, in proportion: the 'Theorem' is a restatement of [11] with a philosophical gloss, and the counterfactual ('if the conservation law is violated, a singularity emerges, time becomes imaginary...') is asserted without argument. The proof of the theorem assumes the conclusion when it says 'any particle remains in the region r > 2M'. Self-citation is not the problem; the problem is that the new content is essentially a translation, not a derivation. The paper itself says it 'expounded in which sense we can talk of a derivation' — a candid admission.

Who is this for? Philosophers of physics working on signature change, singularity theorems, and Noether symmetries. They will get a clear statement of the model's conceptual implications, but they should not take the theorem as proven. It deserves a serious referee because the underlying PRD model and the philosophical stakes are real; but the appropriate outcome is major revision, not acceptance as-is. A referee should require the authors to either justify or abandon the energy-scaling step and to recast the 'theorem' as a definition or a conjecture.

Referee Report

4 major / 5 minor

Summary. The paper argues that the avoidance of singularities in Lorentzian-Euclidean Schwarzschild black holes, which the authors call "atemporality", is a direct consequence of Noether conservation laws. It reviews the radial geodesic equations of the model from [11], claims that infalling massive particles halt at the event horizon and take infinite proper time to do so, defines atemporality as a dynamical mechanism, states an "Atemporality Theorem", and draws philosophical conclusions about causality, measurement, and imaginary time. The central claim is that conservation laws force the transition from a Lorentzian to an Euclidean region, thereby preventing particles from reaching the singularity and preserving real time.

Significance. If the central derivation were sound, the paper would offer a novel philosophical foundation for singularity avoidance: conservation laws would not merely be compatible with signature change but would require it. The manuscript is well read in the relevant literature on Noether theorems and philosophy of spacetime, and the conceptual discussion of imaginary time and measurement is stimulating. However, the key physical step supporting the theorem is invalid as presented, so the significance is currently conditional; the paper would need a corrected derivation or an explicitly weakened interpretive claim to contribute as claimed.

major comments (4)

[Section 2, Eq. (5)] The radial geodesic equation does not follow from the stated metric and energy definition. With E = -ε g_{μν} ξ^μ u^ν and Eq. (6), one has E = ε^2 (1 - 2M/r) \dot t, and the timelike normalization gives \dot r^2 = E^2/ε^3 - (1 - 2M/r). Eq. (5) instead gives -ε (1 - 2M/r) + E^2/ε^3. These expressions agree only for ε = 1; for ε = -1 they differ by 2(1 - 2M/r). Thus the equations used to conclude that \dot r becomes imaginary inside the horizon are not the geodesic equations of metric (1).
[Section 2, after Eq. (7)] The relation E^2 = α^2 ε^4 is incompatible with E being the Noether-conserved energy. For a particle starting from rest at r_i > 2M, Eq. (7) fixes E^2 = 1 - 2M/r_i, which is constant along the geodesic. Since ε tends to zero at the horizon, a bounded α^2 would force E to tend to zero, whereas keeping E constant forces α^2 = E^2/ε^4 to diverge, contradicting the boundedness assumption. With the actual conserved E, Eq. (5) gives \dot r^2 → E^2/ε^3 → ∞ as ε → 0, so the claimed vanishing of \dot r at r = 2M is an artifact of redefining E rather than a consequence of conservation laws. The Atemporality Theorem in Section 4 explicitly relies on this halt ('any particle remains in the region r > 2M'), so the central claim is not established.
[Section 4, Atemporality Theorem] The Proof is not a proof but a summary of the model's properties. It asserts that geodesic completeness implies conservation laws and that particles remain in r > 2M, which is precisely the conclusion to be derived. The counterfactual 'If the conservation law is violated, a singularity emerges' is asserted without derivation and is not a logical consequence of Noether's theorems. The paper should either supply a genuine derivation from conservation laws or state the result as a conjecture or interpretive claim, with the model assumptions made explicit.
[Section 2, Eq. (2) and Eqs. (5)-(14)] ε is defined as a step function, yet the paper repeatedly takes limits such as ε → 0± and treats ε as a continuous parameter, for example in E^2 = α^2 ε^4 and in the proper-time divergence argument. Since ε^4 = 1 for all r ≠ 2M and is zero only at r = 2M, these limiting arguments require a regularization of the step function that is never specified. Without such a regularization, the horizon-stopping mechanism and the geodesic-completeness claim are not mathematically well-defined.

minor comments (5)

[Abstract and Introduction] There is a typo in the abstract: 'a temporality' should be 'atemporality'.
[Section 2, Eqs. (12)-(13)] The notation σ_η is introduced without definition; it would be clearer to state explicitly that the subscript denotes differentiation with respect to η.
[Section 3] The phrase 'the fundamental ontology of both GR and SR is conserved in our model' is vague; please clarify which ontology is meant and in what sense it is conserved.
[Section 4, proof of the Atemporality Theorem] The statement 'Noether symmetries are preserved for time t ∈ R and violated for t ∈ I' uses the unexplained symbol I; define it (presumably imaginary time) and make the claim precise.
[Section 5] The sentence 'causation not only can, but must be defended in general relativity' is more rhetorical than analytic; consider softening or supporting it with a concrete argument.

Circularity Check

2 steps flagged · score 8.0 of 10

Atemporality Theorem is not derived from Noether conservation; it is imposed by defining E^2 ∝ ε^4 so that ṙ vanishes at the horizon, and by restating the authors' own [11] model as a proof.

self definitional [Section 2, after Eq. (13), where E^2=α^2ε^4 is introduced]
"This issue can be solved if we recall that, in the model, the energy is defined as E = −εgµν ξµ uν , ξµ being the static Killing vector field. This means that for a given motion having 2M < r_i < ∞, one can write E2 = α2ε4, where α2 is some positive-definite bounded function depending on Eq. (8). Thus, either from Eq. (5) or Eq. (10), one sees that ˙r becomes zero on the change surface"

E is introduced as the conserved Noether energy along the geodesic, and Eq. (7) fixes E^2 = 1 − 2M/r_i > 0 for any finite r_i. Setting E^2 = α^2 ε^4 with bounded α^2 forces E→0 as ε→0 at the horizon; the only alternative, α^2 = E^2/ε^4, diverges and violates boundedness. Therefore E is not the conserved Killing energy but a position-dependent function chosen so that Eq. (5) gives ṙ=0 at ε=0. The claimed 'outcome' that infalling particles halt at the horizon is inserted by definition, not derived from Noether conservation.
self citation load bearing [Section 4, Atemporality Theorem, proof]
"Proof: In the Lorentzian-Euclidean Schwarzschild black hole, any proper observer takes an infinite time to reach the event horizon. The geodesic structure results as complete and this implies that energy and momenta are conserved according to Noether’s theorems. This also means that any particle remains in the region r > 2M. As a consequence, the Kretschmann curvature invariant is finite and cannot become infinite because observers in free fall and observers with any acceleration approaching the event horizon cannot reach the singularity at r = 0."

The theorem is supposed to show that atemporality is the mechanism by which a conservation law is preserved. Its proof simply lists properties—infinite proper time to the horizon, geodesic completeness, confinement to r > 2M—that are the atemporality mechanism itself, imported from the authors' own Ref. [11]. The inference 'geodesic completeness implies energy-momentum conservation' is not a Noether argument; conservation follows from the static Killing field, and the boundary facts doing the work are the model's already-constructed signature-change dynamics. The conclusion restates the model, so the derivation is circular: the conservation law does not explain the horizon stopping, it is assigned to it after the fact.

full rationale

The paper's central claim—that atemporality follows from Noether conservation laws—fails to be a derivation in two connected ways. First, Section 2's halt-at-horizon conclusion, which is the physical content of atemporality, is obtained by writing E^2 = α^2 ε^4 with α^2 bounded. Since the Killing energy E is fixed by Eq. (7) to a positive constant for an observer starting at rest at finite r_i, and ε→0 at r=2M, the only way to keep α^2 bounded is to make E position-dependent; the 'conserved' energy is redefined so that it vanishes exactly where the horizon is. The vanishing of ṙ at ε=0 is thus an artifact of this definition, not a consequence of the geodesic equation or Noether's theorem. Second, the Atemporality Theorem's proof does not derive conservation laws from anything; it assumes the Lorentzian-Euclidean model's already-established properties—infinite proper time to the horizon, geodesic completeness, particles remaining in r>2M—all taken from the authors' own Ref. [11], and restates them as the conclusion. No independent argument shows that conservation of the Killing energy produces the horizon-crossing obstruction. The cited Ref. [11] is by the same three authors, so the load-bearing premise is a self-citation whose content is exactly the atemporality mechanism being 'explained'. These two moves make the central result true by construction rather than by derivation. The paper does contain independent philosophical analysis (e.g., discussion of measurement, imaginary time, and causal structure), and the score is therefore 8 rather than 10.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The paper introduces no new free parameters beyond the unspecified function alpha; all other quantities come from the prior Lorentzian-Euclidean model. The axioms include standard Noether theory plus two ad hoc philosophical premises: the physical requirement that time be real, and the asserted converse connection between conservation-law violation and singularity generation.

free parameters (1)

alpha = unspecified bounded function of the orbit
Introduced to rewrite E^2 = alpha^2 epsilon^4 so that the radial velocity vanishes at the horizon; no explicit form is given and the rewrite appears to conflict with E being a constant of motion.

assumptions (4)

standard math Noether's first theorem associates conserved charges with continuous symmetries; for the Schwarzschild exterior, the time translation Killing field yields energy conservation.
Invoked in Section 4 to identify energy conservation with the Noether symmetry of the model.
domain assumption The Lorentzian-Euclidean Schwarzschild metric (Eq. 1) is a valid vacuum solution of an extended version of GR allowing degenerate metrics.
The paper adopts the solution from [11] without re-derivation; all later conclusions depend on this metric.
ad hoc to paper Time must be real-valued for physical meaning; imaginary time implies loss of physical meaning and of causality.
This premise, stated in Section 3, is needed to equate imaginary radial velocity with the mechanism of atemporality and to claim that the singularity is thereby avoided.
ad hoc to paper If the relevant conservation law were violated, a singularity would emerge.
This converse direction of the Atemporality Theorem is asserted without proof and is not established by the cited singularity theorems.

invented entities (1)

Atemporality
purpose: Dynamical mechanism preventing infalling bodies from acquiring imaginary radial velocity and thus avoiding singularities while preserving conservation laws.
Defined conceptually; no falsifiable handle beyond the Lorentzian-Euclidean model. The paper states that future observational verification is needed but provides no specific observable.

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Pith. "Pith review of Atemporality from Conservation Laws of Physics in Lorentzian-Euclidean Black Hole." pith.science (2026). https://pith.science/paper/2S2VNGUF

@misc{pith2026250417570,
  author       = {Pith},
  title        = {Pith review of: Atemporality from Conservation Laws of Physics in Lorentzian-Euclidean Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2S2VNGUF}},
  note         = {Machine review of arXiv:2504.17570}
}

read the original abstract

Recent results have shown that singularities can be avoided from the general relativistic standpoint in Lorentzian-Euclidean black holes by means of the transition from a Lorentzian to an Euclidean region where time loses its physical meaning and becomes imaginary. This dynamical mechanism, dubbed ``atemporality'', prevents the emergence of black hole singularities and the violation of conservation laws. In this paper, the notion of atemporality together with a detailed discussion of its implications is presented from a philosophical perspective. The main result consists in showing that atemporality is naturally related to conservation laws.

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Forward citations

Cited by 1 Pith paper

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

Null geodesics, causal structure, and matter accretion in Lorentzian-Euclidean black holes
gr-qc 2025-07 reject novelty 5.0 of 10

In the Lorentzian-Euclidean black hole, photons and massive particles are claimed to be unable to cross the event horizon, making the spacetime geodesically complete and avoiding the central singularity.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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Capozziello, S., De Bianchi, S., Battista, E.: Avoiding singularities in L orentzian- Euclidean black holes: The role of atemporality. Phys. Rev. D 109(10), 104060 (2024) https://doi.org/10.1103/PhysRevD.109.104060 arXiv:2404.17267 [gr-qc]

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