An analytic study of lensing by black holes in Kerr-de Sitter spacetimes - Omwoyo, Eunice Monyenye (2023)
The recent release of the images of M87 and Sagittarius A∗ (SgrA∗ ) black holes by the Event Horizon Telescope (EHT) collaboration has provided unprecedented insights into the emission structure on horizon scales. As technology advances, the aim is to capture even sharper and more detailed images which is among the main aims of the next generation Event Horizon Telescope (EHT). This raises the question of what can be expected and learned from highly resolved black hole images. In-depth studies using general relativistic magnetohydrodynamics simulations reveals that a highly resolved black hole image exhibits a distinct feature called the photon ring. This feature persists in the simulations, regardless of the nature of the astrophysical source profile surrounding the black hole. The photon ring is generated by photons on trajectories that have undergone extreme bending due to the strong gravity of the black hole, causing them to execute multiple orbits. As such, it is intricately connected to the specific properties and spacetime geometry in the vicinity of the black hole and is less sensitive to the astrophysical source profile around the black hole. Besides, the photon ring exhibits a nested sequence of self similar subrings that exponentially converge to the critical curve. The critical curve is purely a theoretical entity whose shape directly follows from General Relativity (GR) but is not in itself observable. However, the photon ring is in principle detectable in the near future observations. Given that this feature is contingent on spacetime geometry and black hole properties, its detection presents the potential for more robust tests of General Relativity (GR) and the Kerr hypothesis. Given the significance thereof, it is vital to conduct an extensive study and make predictions about the explicit nature of the photon ring in various black hole spacetimes. In this thesis, we present the photon ring structure in asymptotically de Sitter spacetimes, with emphasis on the Kerr-de Sitter (KdS) and Kerr-de Sitter Revisited (RKdS) spacetimes. Our analytical approach begins by obtaining solutions to the null geodesic equations in these spacetimes in terms of the Jacobi elliptic functions. These solutions shed light on the overall structure of bound and nearly bound photon orbits, which are the orbits central to this thesis. Subsequently, we delve into the analysis of the critical curve, for which we focus on observers located in the vicinity of the static radius. Moreover, utilizing the solutions we conduct an analytical ray-tracing to explore the properties of direct images, lensed rings, and photon rings. We also consider the special case of zero spin and zero cosmological constant. Our analysis takes into account locally static observers and assumes equatorial disks around the black holes. We compare the various images to the corresponding critical curves. Images arising from photons that have made 2 or more half orbits around the black hole exhibit a remarkable resemblance to the critical curve and are located in close proximity to this curve. Furthermore, these images demonstrate the same universal behavior as the critical curve, such as a more circular shape for small black hole spin and observer inclination angles, as well as a flattened appearance on one side for larger spin and inclination angles. From our study, these images demonstrate a more promising arena for tests of General Relativity (GR) than images arising from photons that have executed one half orbit around the black hole. Besides, we investigate the parameters that govern the subsequent rings’ exponential demagnification, rotation, and detection delay. These parameters are the Lyapunov exponent, the azimuthal angle change, and the time delay.