Investigation of the gravitational lens effect with differential topology

Abstract

In this thesis, the methods of differential topology are used to study the gravitational lensing effect. First, the reader is taken on a brief tour through the basics of black holes, wormholes, gravitational lensing and the mathematical tools of Fermat's principle, Morse theory and the Gauss-Bonnet theorem. Then, the understanding is deepened by applying Morse theory and the Gauss-Bonnet theorem to gravitational lensing.

In the first part, we have fixed an observation event p and the worldline of a light source gamma and identified the set of all past-oriented lightlike geodesics from p to gamma. Since each such geodesic corresponds to an image of the light source on the observer’s sky, this allows us to examine the lensing properties of wormholes. As key results, we have proven with the help of Morse theory that under very mild conditions on gamma, the observer is able to see infinitely many images of gamma. Moreover, we have studied some qualitative features of the lightlike geodesics with the help of two potentials that determine the sum of the centrifugal and Coriolis forces of observers in circular motion for the case that the observers’ velocity approaches the velocity of light. We have exemplified the general results with two specific wormhole spacetimes.

In the second part, we have shown with the help of Fermat’s principle that every lightlike geodesic in the Brill metric projects onto a geodesic of a two-dimensional Riemannian metric, the so-called optical metric. The optical metric is defined on a (coordinate) cone, whose opening angle is determined by the impact parameter of the lightlike geodesic. We have shown that the optical metrics on cones with different opening angles are locally isometric. With the help of the Gauss-Bonnet theorem, we have demonstrated that the deflection angle of a lightlike geodesic is determined by an area integral over the Gaussian curvature of the optical metric.

Last but not least, we have investigated gravitational lensing of Brill wormholes by examining the existence of photon circles and propagation possibilities of light rays.