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

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.