Strong Gravitational Lensing by Lorentzian-Euclidean Black Hole

We investigate the strong gravitational lensing properties of the Lorentzian Euclidean black hole, a spacetime in which the horizon at [Formula: see text] is not a coordinate singularity but a genuine surface of signature change, with the associated curvature singularities removed by two regularization parameters, [Formula: see text] and [Formula: see text]. Starting from the null geodesic equations, we derive the photon sphere, the critical impact parameter, and the strong deflection limit coefficients, and use them to obtain the deflection angle and the full set of strong lensing observables, namely the angular position of the relativistic images, their angular separation, the relative magnification, and the differential time delay between successive images. We show that the photon sphere and critical impact parameter increase with [Formula: see text] and decrease with [Formula: see text], indicating that the two parameters have opposite effects on the optical geometry, and evaluate the resulting observables numerically for the supermassive black holes Sgr A* and M87*. Comparison with the Event Horizon Telescope shadow measurements shows that the Schwarzschild limit is mildly disfavored for Sgr A*, whereas M87* places [Formula: see text]-dependent upper bounds on [Formula: see text], with the adopted fiducial values lying well below these limits. We further show that the shadow constrains only a combination of [Formula: see text] and [Formula: see text], and identify observables such as shadow circularity, higher order image time delays, and quasinormal mode spectra that are capable of breaking this degeneracy. These results identify gravitational lensing observables as effective probes of the Lorentzian–Euclidean scenario.