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Null geodesics, causal structure, and matter accretion in Lorentzian-Euclidean black holes

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Pith's one-line read This paper argues that in the Lorentzian-Euclidean Schwarzschild spacetime, null geodesics—like timelike ones—cannot cross the event horizon, so causal geodesics are complete and the central singularity is never reached.

desk verdict The geodesic completeness claim rests on treating the step function ε as a continuous parameter, invalidating the central result. read the letter →

arxiv 2507.08431 v1 pith:KYSV5IZH submitted 2025-07-11 gr-qc astro-ph.HEhep-th

Salvatore Capozziello , Emmanuele Battista , Silvia De Bianchi This is my paper
classification gr-qcastro-ph.HEhep-th MSC 83C5783C7583C10 PACS 04.70.-s04.20.-q
keywords Lorentzian-Euclideanblackholesignature-changingmetricnullgeodesicsgeodesiccompletenesseventhorizoncausalbarrieratemporalitymatteraccretionPenrosediagram
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 extends the Lorentzian-Euclidean black hole—a spherically symmetric vacuum solution whose metric changes from Lorentzian to Euclidean signature at the event horizon—to photons and to the flow of accreting matter. Its central claim is that null geodesics, like the timelike geodesics studied earlier, cannot cross the horizon: the radial velocity of a photon vanishes at $r = 2M$ and becomes imaginary inside, so no causal curve reaches the central singularity. From this the authors conclude that the spacetime is geodesically complete, that the event horizon acts as a causal barrier rather than a one-way membrane, and that infalling matter and light accumulate at the horizon instead of being swallowed. The exterior geometry is the ordinary Schwarzschild one, so standard observed features such as accretion disks and gravitational blueshift are preserved, while the interior atemporality—time becoming imaginary—keeps the Kretschmann invariant bounded along radial paths.

What carries the argument

The central object is the signature-changing metric $ds^2 = -\varepsilon(1-2M/r)dt^2 + dr^2/(1-2M/r) + r^2 d\Omega^2$, where $\varepsilon = \mathrm{sign}(1-2M/r)$ equals $1$ outside, $0$ at, and $-1$ inside the horizon. The argument runs through the choice of photon energy $\Omega = -F g_{\mu\nu}\xi^{\mu}k^{\nu}$ with the constraint $F = \sqrt{F^2}$, solved by $F = \sqrt{\varepsilon}$, and through signature-adaptive Lorentzian-Euclidean boosts with Lorentz factor $\gamma = 1/\sqrt{1 - V^2/\varepsilon}$ that connect static and free-fall frames; these transformations become ill-defined exactly where the metric degenerates. The load-bearing step for the no-crossing result is the radial null equation $k^{r} = -\alpha_p \sqrt{\varepsilon}$, from which the affine parameter to the horizon is computed and found to diverge in the $\varepsilon \to 0$ limit.

What would settle it

Integrate the radial null geodesic equation (63) with $\varepsilon(r) = 1$ for all $r > 2M$, as fixed by Eq. (2), without extracting $\sqrt{\varepsilon}$ from the integral: the affine parameter to reach $r = 2M$ is $\lambda_* = (r_0 - 2M)/\alpha_p$, a finite number, which contradicts the claim that the horizon is reached only as $\lambda \to \infty$.

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Extended reading notes

Core claim

The paper's discovery, stated on its own terms, is that the Lorentzian-Euclidean Schwarzschild geometry blocks all causal geodesics at the horizon. For radially infalling photons, the null condition $g_{\mu\nu} k^{\mu} k^{\nu} = 0$ combined with a generalized conserved energy $\Omega = -F g_{\mu\nu}\xi^{\mu}k^{\nu}$ yields a radial four-velocity $k^{r} = -\alpha_p \sqrt{\varepsilon}$; since $\varepsilon = 1$ outside, $0$ at $r = 2M$, and $-1$ inside, the radial velocity tends to zero at the horizon and is imaginary in the Euclidean interior. The same behavior was previously found for massive bodies. Integrating the radial null equation gives $\lambda_* = (r_0 - 2M)/(\alpha_p \sqrt{\varepsilon})$, which the authors take to diverge as $\varepsilon \to 0^+$, meaning the horizon is reached only at infinite affine parameter; hence causal geodesics are complete and $r = 0$ is dynamically inaccessible. The event horizon is therefore not a one-way membrane but an asymptotic causal boundary, and the Penrose diagram consists only of the two exterior Lorentzian regions.

Load-bearing premise

The conclusion that light and matter reach the horizon only after infinite time rests on pulling the signature factor $\varepsilon$ out of the radial integral and then sending $\varepsilon$ to zero, even though $\varepsilon$ is defined to be exactly $1$ everywhere outside the horizon.

Editorial extensions

If this is right

  • No causal geodesic—massive or massless—enters the interior $r < 2M$, so the $r = 0$ curvature singularity is dynamically unreachable.
  • The Lorentzian and Euclidean regions are causally disconnected; the horizon is a barrier, not a one-way membrane, and no white-hole emission can occur.
  • The Penrose diagram of the model contains only the two asymptotically flat exterior regions, with $H^{\pm}$ drawn as asymptotic boundaries.
  • Accreted matter halts at $r = 2M$ with finite physical density and pressure, yet the black hole mass still grows according to the standard accretion formula, so disk formation in the exterior is unchanged.
  • Observed near-horizon effects of ordinary Schwarzschild black holes—divergent energy at the horizon, gravitational blueshift, finite free-fall frequency—are reproduced by different mechanisms.

Reading between the lines

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

  • If the step function $\varepsilon$ is kept equal to $1$ throughout the exterior, as its definition requires, the integration in Eq. (64) gives a finite affine parameter to reach $r = 2M$; the geodesic-completeness conclusion would then depend on treating $\varepsilon$ as a limit variable rather than as the fixed signature factor.
  • A direct observational test of the model would be the near-horizon emission profile: a flow that stalls at $r = 2M$ should produce a distinctive accumulation region, possibly distinguishable from standard thin-disk models at horizon scales.
  • The same signature-adaptive transformation machinery could be carried over to the promised Kerr-like extension; if the horizon barrier persists there, the model would predict that rotating accretion flows also stall, with consequences for jet formation that the paper does not address.
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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

4 major / 5 minor

Summary. The paper investigates the Lorentzian-Euclidean Schwarzschild black hole, a signature-changing vacuum solution introduced in the authors' earlier work, in which the metric becomes degenerate at r = 2M and Euclidean (ultrahyperbolic) inside. The central new claims are that radial null geodesics, like timelike ones, cannot cross the event horizon; that causal geodesics reach the horizon only at infinite affine parameter; that the spacetime is therefore geodesically complete and the r = 0 singularity is avoided; and that the causal structure is captured by a Penrose diagram consisting only of the two exterior diamonds. The paper also develops signature-adaptive Lorentz transformations and presents an accretion model for an isothermal perfect fluid, concluding that accretion proceeds as in General Relativity while the fluid halts at the horizon.

Significance. If the geodesic-completeness claim were correct, the model would offer a conceptually striking mechanism for singularity avoidance through metric degeneracy and signature change, and the paper's detailed derivations of signature-adaptive transformations, null geodesics, causal diagrams, and accretion dynamics would constitute a substantial extension of the authors' prior program. The paper is also commendable for carrying through a large number of explicit computations and for connecting the model to observational probes such as accretion disks. However, the central claim is undermined by an internal inconsistency in the treatment of the step function ε in the geodesic integration. The conclusion that photons require infinite affine parameter to reach r = 2M is obtained by a limiting procedure that contradicts the paper's own definition of ε as a step function, and the same error affects the timelike-particle argument. Because the completeness and singularity-avoidance claims are the core of the paper, the significance of the work as presented cannot be sustained.

major comments (4)
  1. [§IV.A, Eqs. (63)–(65)] The derivation of the infinite affine parameter for radial null geodesics is invalid. Equation (63) gives dr/dλ = −α_p√ε, and ε is defined in Eq. (2) as the step function ε = sign(1 − 2M/r), which equals 1 for every r > 2M. Along any exterior radial null geodesic, ε = 1 on the entire interval, so dr/dλ = −α_p and the affine parameter to reach r = 2M is λ* = (r0 − 2M)/α_p, which is finite. The limit ε → 0+ in Eq. (65) treats ε as a parameter that can be sent to zero along a geodesic, but ε is a fixed function of r and does not tend to zero anywhere in the exterior region, including in the limit r → (2M)+. Consequently, the claim that photons reach the horizon only at infinite affine parameter is not supported, and the geodesic-completeness conclusion based on Eq. (65) collapses.
  2. [§II.B, Eq. (11) and surrounding text] The analogous claim for massive particles that infinite proper time is required to reach the horizon is also contradicted by the paper's own equations. For radial infall from rest, Eq. (11) gives u^r = −√ε√(2M/r), and since ε = 1 for r > 2M, the radial geodesic equation is dr/dσ = −√(2M/r). The proper time to reach r = 2M from r0 is finite, e.g. (2/(3√(2M)))(r0^{3/2} − (2M)^{3/2}), not infinite. The paper's assertion that 'an infinite amount of proper time σ is required to reach the event horizon' (Sec. II.B) is therefore unsupported by the equations presented, and the interpretation of atemporality as enforcing the cutoff r ≥ 2M rests on this erroneous conclusion.
  3. [§V.B.1 and Fig. 5] The causal-structure analysis inherits the error in the affine-parameter calculation. The statement that 'any causal geodesic evolving in the Lorentzian domain reaches the event horizon at arbitrarily large values of its affine parameter' (final paragraph of Sec. V.B.1) is directly contradicted by the finite affine parameter computed from Eqs. (63) and (11) with ε = 1. Moreover, the geodesic-completeness claim is conceptually problematic: if the manifold is taken to end at r = 2M, the exterior region is geodesically incomplete because causal geodesics reach the boundary at finite affine parameter; if the manifold is extended to r < 2M, the metric changes signature and real causal geodesics do not exist in the interior, but that does not make the exterior geodesics complete. The Penrose diagram in Fig. 5, which depicts H± as asymptotic boundaries, is therefore not a valid representation of a geodesically complete spacetime.
  4. [§VI.A, Eqs. (148)–(153)] The accretion analysis also relies on the same ε-step-function treatment: Eq. (148) gives U^r = −(√ε/(η+1))√((εA4)^2 − (1 − 2M/r)(η+1)^2), and the statement that the fluid velocity vanishes at the horizon and becomes imaginary inside is used to conclude that matter accumulates at r = 2M. Since ε = 1 throughout the exterior, the radial velocity does not smoothly tend to zero as a limiting process along the flow; it is simply undefined at the degenerate surface. The claim that 'the motion of the perfect fluid halts at r = 2M' and the associated mass-accretion picture inherit the same internal inconsistency as the geodesic analysis.
minor comments (5)
  1. [§II.A, Eq. (3)] The smooth regularization (3) for ε is introduced but not used in the geodesic integration of Sec. IV.A; the paper switches back to the step function (2) without explaining why the regularized version would not alter the limiting procedure in Eq. (65).
  2. [§II.B, Eq. (5)] The parameter σ is defined by u^μ = dx^μ/dσ and is said to be an affine parameter that reduces to proper time in the Lorentzian region, but the distinction between σ and τ is not consistently maintained in the text; clarify whether σ is affine along all causal geodesics or only proper time in D+.
  3. [§III.B.2] The statement that the local transformations (44) are 'formally valid also for r < 2M' is not explained, since γ in Eq. (43) becomes imaginary when ε = −1; the domain of validity of the boosted tetrad should be stated explicitly.
  4. [Fig. 5 caption] The caption states that the resulting spacetime is geodesically complete, but this is the central claim under dispute; the caption should be revised to reflect the actual status of the completeness argument rather than asserting it as a conclusion.
  5. [Global] There are numerous typographical errors and awkward phrasings (e.g., 'thethe', 'in the the', inconsistent spacing in equations). A careful editing pass would improve readability.

Circularity Check

3 steps flagged · score 7.0 of 10

The central claim that causal geodesics reach r=2M only at infinite affine parameter is manufactured by treating the step-function epsilon as a vanishing parameter; in the exterior epsilon=1, so the affine and proper times to the horizon are finite and geodesic completeness is not established.

  1. self definitional [Sec. IVA, Eqs. (63)-(65); cf. Eq. (2)]
    "where we have assumed that ∫ dλ αp√ε = αp√ε ∫ dλ. Therefore, photons get to the event horizon at the affine parameter value λ∗ ... λ∗ = (r0 − 2M)/(αp√ε) −−−−−→ ε→0+ ∞. This result shows that null geodesics exhibit the same behavior as timelike ones, as the event horizon is reached only in the limit of infinite affine parameter."

    By Eq. (2), epsilon(r) is the step function equal to 1 for every r > 2M, 0 only at r = 2M, and −1 inside. Along the whole exterior radial null geodesic used to compute lambda*, epsilon = 1; it is not a free parameter that can be pulled out of the integral and then sent to 0. With epsilon = 1, Eq. (63) is dr/dlambda = −alpha_p and the affine parameter from r0 to 2M is finite, lambda* = (r0 − 2M)/alpha_p. The smooth approximation (3) does not repair this, since epsilon(r) is r-dependent and cannot be factored out; its integral is finite. The divergence is therefore created by the limiting move epsilon → 0+, not by the geodesic equation; the geodesic-completeness conclusion is assumed through that move rather than derived.

  2. self definitional [Sec. IIA, Eq. (2); Sec. IVA, Eq. (63)]
    "Bearing in mind Eq. (2), we thus find that the behavior of photons mirrors that of massive particles, as kr vanishes on the event horizon and attains imaginary values inside it."

    The signature of the metric was defined by the same epsilon in Eq. (2): epsilon = 0 at the horizon and epsilon = −1 inside. Since Eq. (63) gives kr = −alpha_p sqrt(epsilon), the statements 'kr vanishes at r = 2M' and 'kr is imaginary inside' are direct consequences of the definition of epsilon in the metric ansatz (1). The paper presents this as a dynamical prediction that causal geodesics cannot cross the horizon, but the no-crossing behavior is installed in the signature-changing step function from the outset.

1 more flagged steps
  1. self citation load bearing [Sec. IIB; Refs. [73,74]]
    "As detailed in Ref. [73], ... like in the standard Schwarzschild solution, the coordinate time t attains unboundedly large values when r = 2M; however, in contrast to the standard scenario, a similar behavior occurs also in the proper reference frame of the body, as an infinite amount of proper time sigma is required to reach the event horizon."

    The massive-particle analogue of the geodesic-completeness claim is imported from the authors' own Ref. [73] and is not re-derived here. The present paper's Eq. (11) gives u^r = −sqrt(epsilon) sqrt(2M/r), i.e. dr/dsigma = −sqrt(2M/r) in the exterior, whose integral from r0 to 2M is finite. Thus the load-bearing premise that timelike observers need infinite proper time to reach the horizon is asserted by self-citation, and the downstream conclusions (Penrose diagram in Fig. 5, accretion halting at r = 2M) inherit that unjustified premise.

full rationale

The paper is not circular in its accretion calculation (Sec. VI), which reduces to the standard Michel/Babichev result in the epsilon = 1 exterior, nor in the F-function algebra of Sec. IVB, which only fixes a sign convention. The circularity is concentrated in the central causal claim. The exterior region r > 2M is, by the authors' own statement, the ordinary Schwarzschild geometry, for which radial null geodesics reach r = 2M at finite affine parameter and radial timelike geodesics at finite proper time. The paper's opposite conclusion is obtained in Eq. (65) by factoring sqrt(epsilon) out of the integral and then sending epsilon → 0+, even though Eq. (2) fixes epsilon = 1 at every exterior point of the integration interval. The same manufactured limit is inherited from the authors' prior work for massive particles and is then used to declare the spacetime geodesically complete and to draw the Penrose diagram of Fig. 5. Because the result that causal observers never cross the horizon is built into the signature ansatz and the infinite affine-parameter claim is generated by an invalid limiting move rather than by the geodesic equations, the central prediction reduces by construction and receives a score of 7.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The model rests on the authors' own prior construction of the signature-changing solution and its Hadamard regularization (Ref. [73]). The new results add several ad hoc choices: the arbitrary function F for photon energy, the equation-of-state parameter \tilde{η}, and the interpretation of the horizon as a causal barrier. No free parameters are fitted to data; the central claims follow analytically from the metric ansatz, but the geodesic-completeness claim relies on an invalid limiting procedure.

free parameters (3)
  • ε-regularization parameter ϱ = small positive quantity with dimensions of length squared (unspecified); κ a large positive integer
    Introduced in Eq. (3) to smooth the step function ε near r=2M. The paper uses this approximation in checks of the energy (Eq. 51) and in distributional identities (Eq. 92), but the main geodesic results use the step function directly; their values are arbitrary.
  • Equation-of-state parameter \tilde{η} = unspecified constant (η=1/3 for radiation gives \tilde{η}=1/(3ε))
    Introduced in Eq. (145) to make the pressure proportional to the physical density via p = \tilde{η} ε² ρ. The accretion velocity, density, and pressure formulas (Eqs. 148-153) depend on this ad hoc linear dependence of η on ε.
  • Photon energy function F(ε) = F = √ε chosen in Eq. (72), though F=√F² admits any positive F
    The general definition of photon energy in Eq. (53) introduces an arbitrary function F; the constraint Eq. (61) is F=|F|, and the specific solution Eq. (72) is one of infinitely many. The paper claims the main null results only use the constraint.
assumptions (5)
  • domain assumption The metric (1) with step-function ε is a well-defined solution of vacuum Einstein equations after Hadamard partie finie regularization of the distributional curvature at r=2M.
    This is the foundational assumption from Ref. [73], used throughout (Sec. IIA, Eqs. 1-3 and Sec. VB1). The present paper does not re-derive the regularization.
  • domain assumption The affine parameter σ can be made continuous across the event horizon and equals proper time in the Lorentzian region D+.
    Used to define u^μ and the geodesic equations (Sec. IIB, Eqs. 5-6); relies on Refs. [84,85] for geodesics in signature-changing spacetimes.
  • standard math The Hadamard prescription δ(x)|x|^{-n}=0, which makes the distributional contribution of d√ε vanish (Eq. 92).
    Invoked in Sec. VB1 to justify d(t√ε)=√ε dt across the change surface; the validity of this identity is assumed.
  • ad hoc to paper The accreting fluid is a perfect isothermal fluid with equation of state p = \tilde{η} ε² ρ (Eqs. 144-145).
    This EOS is chosen to keep the pressure real and proportional to the physical density across the signature change; it is not derived from microphysics and the accretion results depend on it.
  • ad hoc to paper Photon energy can be defined with an arbitrary function F as Ω = -F g_μν ξ^μ k^ν (Eq. 53), and the constraint F=√F² is treated as physically significant.
    The 'constraint' Eq. (61) is a tautology; the paper uses it to argue photons have imaginary radial velocity inside, though that behavior follows directly from the metric signature.
invented entities (1)
  • Atemporality (time becoming imaginary across the horizon)
    purpose: Proposed as the dynamical mechanism that prevents causal geodesics from reaching r=0, regularizing the black hole interior.
    Introduced in Ref. [73] and used throughout as the interpretation of the signature change. The paper itself notes imaginary time is not measurable (Sec. VII), so there is no falsifiable handle outside the model.

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Cite this review

Pith. "Pith review of Null geodesics, causal structure, and matter accretion in Lorentzian-Euclidean black holes." pith.science (2026). https://pith.science/paper/KYSV5IZH

@misc{pith2026250708431,
  author       = {Pith},
  title        = {Pith review of: Null geodesics, causal structure, and matter accretion in Lorentzian-Euclidean black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYSV5IZH}},
  note         = {Machine review of arXiv:2507.08431}
}
abstract

Recently, we introduced the Lorentzian-Euclidean black hole, a static and spherically symmetric solution of vacuum Einstein equations that exhibits a change in metric signature across the event horizon. In this framework, the analysis of radial trajectories of freely falling bodies proves that the central singularity can be avoided via a mechanism we interpret as atemporality, which is responsible for the shift of the time variable from real to imaginary values. In this paper, we further explore this model by first examining the behavior of null geodesics. Our investigation requires a set of signature-adaptive coordinate changes that generalize the local Lorentz transformations underlying General Relativity. We find that photon orbits, like their massive counterparts, cannot traverse the event horizon, thereby strengthening the previous result on the impossibility to reach the $r=0$ singularity. Additionally, we discuss the causal structure of the spacetime, provide the corresponding Penrose diagram, and analyze the process of matter accretion in the outer region of the black hole.

Figures

Figures reproduced from arXiv: 2507.08431 by the authors.

Figure 1
Figure 1. FIG. 1. The Penrose diagram for the Lorentzian-Euclidean [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The Kruskal diagram of the Lorentzian-Euclidean Schwarzschild spacetime. In regions I and III, the coordinates [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Finkelstein diagram in outgoing coordinates, with [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Finkelstein diagram in ingoing coordinates, with in [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Penrose diagram of the Lorentzian-Euclidean Schwarzschild geometry. The Lorentzian regions [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

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