Following the trails of light in curved spacetime
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Light has played an instrumental role in the initial development of the theory of relativity. In this thesis, we intend to explore other physical phenomena that can be explained by tracing the path that light takes in curved spacetime. We consider the null geodesic and eikonal equations that are equivalent descriptions of the propagation of light rays within the framework of numerical relativity. We find that they are suited for different physical situations. The null geodesic equation is more suited for tracing the path of individual light rays. We solve this equation in order to visualize images of the sky that have been severely distorted by one or more black holes, the so-called gravitational lensing effect. We demonstrate that our procedure of solving the null geodesic equation is sufficiently robust to produce some of the world's first images of the lensing effect from fully dynamical binary black hole coalescence. The second formulation of propagation of light that we explore is the eikonal surface equation. Because this equation describes the propagation of whole surfaces of light, instead of individual light rays, we find it more apt in locating the event horizon of a black hole. We will show that our solution method of the eikonal surface equation to locate event horizons is also robust enough to find the event horizon of a black hole that has accreted some negative energy density. While both of these numerical simulations were able to achieve their basic goals for dynamical spacetimes where solution by analytical methods is impractical, both simulations are limited by computational requirements that we discuss.