Nonspherical black holes and spacetime reconstructions

This thesis consists of three papers in mathematical general relativity. The first paper concerns inverse problems and reconstruction of spacetimes from boundary data. We consider boundary data in the form of broken geodesics with different causal types, which have the physical interpretation of relativistic fireworks. Using this data, we show that it is possible to completely reconstruct the topology, smooth structure, and metric of the spacetime. The second paper is about the topology of black holes. When black holes are shown in illustrations, they are typically shown as spherical, and a natural question is whether this spherical topology is necessary. Would it not be possible for black holes to have other topologies? It is previously known that an apparent horizon of a black hole must admit a metric of positive scalar curvature, and this imposes restrictions on the possible topologies. However, there is a large class of topologies which cannot be excluded by this result, but where there are no examples or constructions showing that they are actually possible. This means that it is possible that there are further restrictions which have not yet been found. In the second paper of the thesis, we describe a construction which gives rise to a large class of new examples of topologies for apparent horizons. This decreases the gap between the topologies which are known to be possible and those which are known to be impossible. The third paper in the thesis also concerns apparent horizons of black holes, and the main result is more technical in character than those of the other papers. It is a special case of a generalization of a regularity theorem for apparent horizons, which is previously known in low dimensions. It is not immediately obvious from the definition of an apparent horizon that it should have any particularly good properties, but it is previously known that it is a smooth hypersurface if the dimension is sufficiently small. We show in the third paper for time-symmetric initial data in any dimension that an outermost apparent horizon is a smooth hypersurface, apart from a singular set of large codimension.QC 20180905