Ray tracing for the Terrell-Penrose effect in black hole spacetime

TL;DR

The paper investigates the Terrell-Penrose aberration in black hole spacetimes using relativistic ray tracing, focusing on two setups: static emitters seen by moving observers and moving emitters seen by static observers. It derives celestial-coordinate transformations that are conformal for moving observers and demonstrates how gravity preserves or modifies these distortions via a conformal factor Ω^2, with explicit forms for corotating and other motions. A slow-light ray tracing extension to curved spacetime shows that, in BH spacetimes, gravity introduces non-conformal, lengthening effects for moving emitters and alters the apparent motion of extended sources. These results have potential implications for interpreting horizon-scale images from EHT/ngEHT, as they suggest that observed distortions encode both kinematic and gravitational information about spacetime geometry.

Abstract

Motivated by recent images of black holes in M87 and our galaxy, efficient relativistic ray tracing was developed to simulate the snapshots of variable emissions around the black holes. Half a century ago, the appearance of a moving emission source was addressed by Terrell and Penrose, who independently found that the aberration effect induces a conformal transformation on the observer's celestial sphere. Consequently, a snapshot of a moving sphere should remain circular. In this study, we examine the Terrell-Penrose effect with our ray-tracing simulations for two contrasting cases: i) static emission sources in the view of a moving observer, ii) and moving emission sources in the view of a static observer. In flat spacetime, it was believed that the images of the emission sources in these two cases are equivalent due to the relativity of motion. Our simulation demonstrates that although both cases remain apparent shape of the sphere, the apparent distortions of the images are different, and case ii) violates conformality on the observer's celestial sphere. Furthermore, we extended similar situations to a black hole spacetime. For case i), it is found that the conformal transformation induced by the aberration effect also holds in black hole spacetime, and is not restricted to observers in geodesic motion. For case ii), we study the slow-light effect on the moving sources, and show that the gravity introduces additional influence on the snapshots of a moving source.

Figures (13)

• Figure 1: Schematic diagram of imaging geometric objects onto projection plane. The spatial direction vectors of $k^{(\phi)}$, $k^{(\theta)}$ and $k^{(r)}$ are $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OC}$, respectively. We use the observer's celestial sphere centralized at O, and $x$, $y$ and $z$ axes given by the OA, OB and OC, respectively. The celestial coordinates are defined as $\Phi_0\equiv \angle\text{AOD}'$ and $\Psi_0\equiv\angle\text{COD}$. Light ray $\widetilde{SO}$ propagates from source S to observers O. At point O, the direction vector of the light ray $k$ is $\overrightarrow{OD}$, and the $\overrightarrow{OD'}$ represents the projection on plane AOB. We use Lambert azimuthal projection to map celestial coordinates $(\Phi_0,\Psi_0)$ onto image plane coordinated as $(\text{X},\text{Y})$.
• Figure 2: Optical appearance of sphere for low-mass and massive black holes with selected speed $\varsigma$.
• Figure 3: Optical appearance of jets for low-mass and massive black holes. We set $\varsigma=0$ in left panels and $\varsigma=0.999$ in right panels.
• Figure 4: Optical appearance of disks for low-mass and massive black holes with selected inclination angles. We set $\varsigma=0$ in top panels and $\varsigma=0.999$ in bottom panels.
• Figure 5: The schematic diagram illustrating the Lorentz contraction of sphere (left panel) and face-on disk (right panel).