Lorentzian-Euclidean Black Hole

• Lorentzian–Euclidean black hole is a signature-changing modification of Schwarzschild geometry where the metric shifts from Lorentzian outside the horizon to Euclidean inside, precluding access to the central singularity.
• The model employs smooth regularization techniques to handle metric degeneracy at r = 2M, resulting in a weak vacuum solution with vanishing Ricci and Einstein tensors.
• It predicts observable effects such as a slightly fuzzy inner shadow and horizon expansion from energy accumulation, offering potential tests for alternative black hole geometries.

Searching arXiv for the most relevant papers on Lorentzian–Euclidean black holes and closely related treatments. The Lorentzian–Euclidean black hole is a signature-changing modification of the Schwarzschild geometry in which the metric is Lorentzian outside the event horizon, degenerate on the horizon, and Euclidean or “Euclidean-like” inside. In the formulation introduced by Capozziello, De Bianchi, and Battista, the horizon $r=2M$ is not merely a null surface of a fully Lorentzian spacetime but a change surface beyond which the time coordinate loses its real-valued physical role. This transition is interpreted through “atemporality,” a dynamical mechanism intended to preclude causal access to the central $r=0$ singularity while preserving a vacuum Einstein solution in a weak, distributional sense. Subsequent work has developed the geometric meaning of the transition hypersurface, the status of geodesic motion, and potential observational signatures in black-hole shadows and horizon-scale energy accumulation (Capozziello et al., 2024, Bianchi et al., 24 Apr 2025, Battista et al., 15 Jan 2026).

1. Definition and basic metric structure

The standard form used for the Lorentzian–Euclidean black hole is a Schwarzschild line element multiplied by a sign controller $\varepsilon(r)$:

$$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$

with

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$

For $r>2M$, $\varepsilon=+1$ and the metric is Lorentzian. At $r=2M$, $\varepsilon=0$ and the metric determinant vanishes. For $r<2M$, $r=0$0; interpreted via $r=0$1, the interior acquires the structure of a Euclidean Schwarzschild region rather than a Lorentzian black-hole interior (Capozziello et al., 2024).

In this construction the spacetime manifold is split into an exterior Lorentzian region $r=0$2, an interior Euclidean region $r=0$3, and a common boundary

$r=0$4

The metric determinant is

$r=0$5

so $r=0$6. This degeneracy is not treated as a coordinate artifact but as the signature-change locus itself (Bianchi et al., 24 Apr 2025).

A horizon-regular presentation is obtained with Gullstrand–Painlevé-type coordinates. In the formulation summarized for the Lorentzian–Riemannian transition, the resulting metric is continuous at $r=0$7, smooth on each side, and still degenerate on the transition hypersurface. This coordinate rewriting separates the coordinate divergence of Schwarzschild coordinates from the geometric degeneracy associated with signature change (Bartolo et al., 19 Feb 2025).

2. Atemporality and the fate of causal geodesics

The central physical interpretation of the model is “atemporality.” In the authors’ formulation, atemporality is the dynamical mechanism by which real-valued motion cannot be prolonged into the region where time becomes imaginary. This is tied to the preservation of conservation laws, especially the conserved energy associated with the static Killing vector, and is presented as the reason singularities are avoided in the physically admissible sector (Bianchi et al., 24 Apr 2025).

For radial timelike geodesics, the conserved energy per unit mass is written as

$r=0$8

with $r=0$9 the static Killing vector. Earlier analyses gave radial and temporal equations of the form

$\varepsilon(r)$0

and emphasized that once $\varepsilon(r)$1 switches sign, the radial velocity becomes imaginary, so real timelike motion cannot be continued into $\varepsilon(r)$2. On this interpretation, massive particles do not reach $\varepsilon(r)$3, and the interior Euclidean domain is inaccessible to physical observers (Bianchi et al., 24 Apr 2025).

The status of the proper time needed to reach the horizon was later clarified. A subsequent analysis of the Lorentzian–Euclidean Schwarzschild metric proved that freely falling observers reach the transition hypersurface $\varepsilon(r)$4 in finite proper time, consistently with classical Schwarzschild. For radial timelike geodesics in the Lorentzian region,

$\varepsilon(r)$5

and the proper time to fall from $\varepsilon(r)$6 to $\varepsilon(r)$7 is

$\varepsilon(r)$8

The same work also gave a concavity argument in Gullstrand–Painlevé coordinates showing that infalling timelike geodesics cannot asymptote to $\varepsilon(r)$9 from above while remaining physically admissible; they arrive there in finite proper time (Bartolo et al., 19 Feb 2025).

This yields a distinctive internal tension in the literature. The original atemporality program emphasizes the impossibility of continuing real-valued causal motion into the Euclidean region; the later clarification preserves that conclusion but rejects the claim that proper time to the horizon is infinite. What remains stable across the series is the central assertion that the singular locus $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$0 is not reached by physical timelike motion.

3. Weak vacuum solution and transition geometry

A direct use of the sign function $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$1 generates distributional curvature contributions at $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$2. To regularize these terms, the model replaces $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$3 by a smooth family

$$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$4

with $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$5 and integer $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$6. The regularization is combined with the Hadamard partie finie prescription, in particular

$$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$7

which removes the ill-defined $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$8, $$ds^2 = -\varepsilon(r)\,f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2,\qquad f(r)=1-\frac{2M}{r},$$9, and $$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$0-type contributions generated by derivatives of $$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$1 (Capozziello et al., 2024).

After regularization, the Ricci tensor and Ricci scalar vanish:

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$2

so the Einstein tensor also vanishes in the regularized sense,

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$3

The model is therefore presented as a vacuum solution of Einstein’s equations in a weak or distributional framework that admits degenerate metrics, with no thin shell, no impulsive gravitational wave, and no surface stress localized at the horizon (Capozziello et al., 2024).

The Kretschmann invariant remains the Schwarzschild one,

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$4

Hence

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$5

while $$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$6 formally as $$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$7. The claimed singularity resolution is therefore not based on regularizing curvature invariants at the center; it is based on making the formally singular locus inaccessible to real-valued causal motion (Capozziello et al., 2024).

In the broader Lorentzian–Riemannian transition framework developed around the model, the signature-changing hypersurface is described as naturally spacelike and identifiable with the future or past causal boundary of the Lorentzian sector. Geometrically, degeneracy of the metric corresponds to collapse of causal cones into a line, while degeneracy of the dual metric corresponds to collapse into a hyperplane. The same framework points to induced Galilean and dual Galilean structures on the transition hypersurface, reflecting the non-Lorentzian geometry intrinsic to signature change (Bartolo et al., 19 Feb 2025).

4. Null geodesics, photon sphere, and shadow phenomenology

The shadow analysis of the Lorentzian–Euclidean black hole is based on equatorial null geodesics obeying

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$8

with conserved angular momentum

$$\varepsilon(r)=\mathrm{sign}\!\left(1-\frac{2M}{r}\right)=2\,H\!\left(1-\frac{2M}{r}\right)-1,\qquad H(0)=\tfrac12.$$9

and conserved energy

$r>2M$0

The resulting radial equation is

$r>2M$1

Introducing the impact parameter $r>2M$2, one obtains the angular integral

$r>2M$3

To integrate trajectories that approach $r>2M$4, the analysis smooths $r>2M$5 and uses the identity

$r>2M$6

described in the paper as a “magic trick” (Battista et al., 15 Jan 2026).

The photon-sphere condition is modified to

$r>2M$7

Because $r>2M$8 away from the horizon, the first-order correction to the photon-sphere radius is

$r>2M$9

so $\varepsilon=+1$0 is slightly smaller than the Schwarzschild value $\varepsilon=+1$1. The corresponding critical impact parameter satisfies

$\varepsilon=+1$2

and is likewise slightly smaller than in Schwarzschild. The shadow is therefore marginally smaller, though the paper stresses that this effect is likely degenerate with emission-model uncertainties (Battista et al., 15 Jan 2026).

The ray-tracing setup uses a distant observer at $\varepsilon=+1$3 in units $\varepsilon=+1$4, a geometrically and optically thin face-on accretion disk along $\varepsilon=+1$5, and no absorption. Radiative transfer is implemented through

$\varepsilon=+1$6

with bolometric observed intensity

$\varepsilon=+1$7

The emissivity is taken monochromatic with a GLM radial profile

$\varepsilon=+1$8

and parameters $\varepsilon=+1$9, chosen to provide nonvanishing emission down to the horizon (Battista et al., 15 Jan 2026).

The principal shadow signature is an excess inner-shadow intensity. In Schwarzschild, rays forming the inner shadow connect directly to the horizon and are dark because they do not intersect the disk. In the Lorentzian–Euclidean geometry, rays with $r=2M$0 never cross the horizon; instead they asymptote to $r=2M$1 and can intersect the disk repeatedly very close to the horizon, contributing a highly redshifted but nonzero intensity floor. For $r=2M$2, the dominant scaling is

$r=2M$3

so the excess grows with increasing $r=2M$4 and decreasing $r=2M$5. In image space this appears as a slightly fuzzy, not completely dark inner shadow rather than the sharp dark cutoff of Schwarzschild (Battista et al., 15 Jan 2026).

5. Horizon-scale energy accumulation and backreaction

Because the Lorentzian–Euclidean horizon functions as a barrier to causal penetration, the model predicts continuous accumulation of photons, particles, and energy near $r=2M$6. To analyze the backreaction of this pile-up, the geometry is modeled by a continuum of thin spherical shells outside the horizon, with metric

$r=2M$7

where the smooth mass function $r=2M$8 encodes the shells and $r=2M$9 depends on $\varepsilon=0$0, $\varepsilon=0$1, and $\varepsilon=0$2. The accumulation surface is defined by

$\varepsilon=0$3

Perturbing $\varepsilon=0$4 and the root $\varepsilon=0$5, the leading response is

$\varepsilon=0$6

Assuming $\varepsilon=0$7 is the outermost horizon so that $\varepsilon=0$8, a positive $\varepsilon=0$9 implies

$r<2M$0

The accumulation surface therefore expands under positive energy input (Battista et al., 15 Jan 2026).

This response is explicitly contrasted with the behavior of stable light rings in many horizonless exotic compact objects. In those systems, additional energy generally shrinks the ring radius and can enhance long-lived modes and geometric instabilities. In the Lorentzian–Euclidean black hole, by contrast, horizon growth under backreaction is the perturbative prediction. The paper presents this as a qualitative distinction between the LEBH horizon and the stable light-ring phenomenology often associated with ECOs (Battista et al., 15 Jan 2026).

A compact comparison is as follows:

Feature	Lorentzian–Euclidean black hole	Comparator
Interior structure	Ultrahyperbolic or Euclidean-like interior beyond $r<2M$1	Schwarzschild has Lorentzian interior
Access to $r<2M$2	Causal geodesics are precluded from reaching the central singularity	Schwarzschild has an inextendible singularity
Shadow core	Small residual inner-shadow brightness floor	Schwarzschild inner shadow is perfectly dark
Backreaction to added energy	Accumulation surface expands	Stable light rings in many ECOs generally shrink

This comparison suggests that the model is not merely a regularized Schwarzschild analogue but a distinct horizon-scale geometry with its own causal and photometric signatures.

6. Thermodynamics, conceptual status, and open questions

The thermodynamic status of the Lorentzian–Euclidean black hole is subtle. In the original regularized Schwarzschild-based treatment, the surface gravity is computed as

$r<2M$3

leading to the standard Hawking temperature and entropy,

$r<2M$4

On that account, the Euclidean interior does not modify the usual exterior thermodynamic quantities (Capozziello et al., 2024).

A later geometric clarification distinguishes this from the usual Euclidean Schwarzschild instanton logic. In that account, the transition hypersurface is treated as a degenerate boundary rather than as a regular bolt in a smooth Euclidean manifold, so the usual requirement of $r<2M$5-periodicity at the horizon is not built into the Lorentzian–Euclidean Schwarzschild construction itself. This does not negate the standard values quoted above, but it does shift their conceptual interpretation: they are inherited from the exterior Schwarzschild structure rather than derived from regularity of a globally smooth Euclidean section (Bartolo et al., 19 Feb 2025).

The conceptual core of the program remains the claim that singularity avoidance follows from signature change and atemporality rather than from modifying the curvature invariant $r<2M$6. Within that framework, the interior Euclidean region is not an ordinary dynamical extension of the Lorentzian exterior. It is a domain in which “time loses its physical meaning,” and the horizon becomes a causal or quasi-causal boundary of the real-time sector (Bianchi et al., 24 Apr 2025).

Several limitations are explicit in the current literature. The detailed construction is restricted to the static, spherically symmetric, neutral case. The shadow study assumes a stationary, optically thin, face-on disk with monochromatic GLM emissivity and no absorption. The backreaction result is perturbative, and the question of nonperturbative instability from prolonged energy build-up is left open. The interaction between Hawking radiation and an Euclidean interior is also deferred, as are rotating generalizations, realistic plasma effects, polarization, and parameter inference for the regularization parameters $r<2M$7 from synthetic or observational imaging (Battista et al., 15 Jan 2026).

In that sense, the Lorentzian–Euclidean black hole is best understood as a developing geometric program rather than a closed alternative to Schwarzschild. Its most distinctive features are the horizon-local signature change, the atemporal exclusion of the central singularity from the physically measurable sector, and the possibility of testing that modification through a nonzero inner-shadow floor, a fuzzy inner boundary, and a horizon that responds to accumulated energy by expansion rather than by light-ring contraction.