Shadow signatures and energy accumulation in Lorentzian-Euclidean black holes

TL;DR

This work investigates shadow signatures of a Lorentzian-Euclidean black hole, a geodesically complete spacetime with a signature change at the horizon $r=2M$, where light cannot reach the central singularity. Through equatorial photon dynamics and ray-tracing of a thin accretion disk, it finds that the shadow closely resembles Schwarzschild near the photon sphere, but exhibits a distinct inner-shadow excess caused by near-horizon photon piercings, with the excess controlled by regularization parameters $\kappa$ and $\rho$. The study also analyzes backreaction from accumulated energy near the horizon, showing the horizon effectively expands under perturbative energy build-up, in contrast to the contraction seen for stable light rings in horizonless models. These results indicate that inner-shadow features and horizon-scale backreaction offer potential observational probes of horizon modifications and possible quantum-gravity effects, motivating further nonperturbative analyses.

Abstract

The Lorentzian-Euclidean black hole has been recently introduced as a geodesically complete spacetime featuring a signature shift at the event horizon where causal geodesics are precluded from reaching the central $r=0$ singularity. In this paper, we investigate the shadows produced by this geometry to identify deviations from the standard Schwarzschild solution. Our analysis reveals an excess intensity in the inner shadow region that points to a potential observational signature of the novel behavior of light rays propagating near the event horizon. This excess could be a probe for horizon-scale modifications of black hole geometries. Furthermore, although the horizon surface of the Lorentzian-Euclidean black hole continuously accumulates photons and energy, we show that its backreaction response differs from that of stable light rings found in various exotic compact objects.

Figures (4)

• Figure 1: Photon trajectories in the equatorial plane of the Lorentzian-Euclidean black hole. The black dashed circle marks the photon sphere (see Eq. \ref{['photon-sphere-radius']}), while the green vertical line denotes the thin accretion disk plane (see Sec. \ref{['Sec:shadow']}). All orbits originate from the point $(X,Y)=(100,0)$; light rays with $b>b_c$ (resp. $b<b_c$), where $b_c$ is given in Eq. \ref{['impact-b-c']}, are shown in red (resp. blue). We set $\kappa=1$, $\rho=1/400$, and adopt units in which $2M=1$.
• Figure 2: Observed intensity $I_o$ as a function of the impact parameter $b$ for the GLM emission profile \ref{['GLM']} and with units $2M=1$. The black (resp. cyan dashed) curve represents the Schwarzschild (resp. Lorentzian-Euclidean) black hole. The two curves nearly coincide, with a slight difference appearing only near the inner-shadow boundary around $b\sim1.434$; a zoomed-in view of this region is provided in Fig. \ref{['fig:innershadowzoom']}.
• Figure 3: Shadow images of the Schwarzschild solution (left) and the Lorentzian-Euclidean black hole with $\kappa=1$ and $\rho/M^2=1/100$ (right). The Lorentzian-Euclidean model produces an inner shadow with a somewhat more diffuse boundary compared to the sharply defined one in the Schwarzschild geometry.
• Figure 4: Observed intensity $I_o$ in logarithmic scale near the inner shadow, assuming the GLM emission profile \ref{['GLM']} and units with $2M=1$. The intensity for the Schwarzschild black hole (black solid curve) falls to zero at the inner-shadow boundary (vertical line at $b=1.434$). Colored curves refer to the Lorentzian-Euclidean black hole, and show an excess intensity. The blueish curves represent $\kappa=1$, while the reddish ones stand for $\kappa=2$. Within each set, the curves from top to bottom correspond to $\rho=1/400$, $1/800$, $1/1200$, $1/1600$, $1/2000$, and $1/2400$, respectively. As $\kappa$ increases or $\rho$ decreases, the excess intensity inside the inner shadow diminishes. The shaded and dotted curves denote the analytic formula \ref{['analyticIo']} for $\kappa=1$ and $\kappa=2$, respectively; the upper (resp. lower) curve indicates $\rho=1/400$ (resp. $\rho=1/2400$).