| Summary: | We provide different mathematical frameworks to describe singularities in General Relativity. The main idea is to regard singularities as obstructions to the dynamics of different matter models. The first part of the thesis initiates our examination of spacetime by probing spacetime with matter that can be modelled as a point-particle. In particular, we discuss the case of two-dimensional Lorentzian metrics. We give concrete applications of the framework in the case of the Minkowski spacetime (which is regular) and the Friedmann-Robertson-Walker spacetime (which is geodesically incomplete). In the second part of the thesis, we probe the geometry of spacetime with classical scalar fields. The general motivation is to redefine a singularity in spacetime not as an obstruction to geodesics or curves but as an obstruction to the dynamics of test fields. We discuss curve-integrable spacetimes, spacetimes with surface layers, impulsive gravitational waves and brane-world scenarios, and spacetimes that contain string-like singularities. In the third part of the thesis, we present the outline of a framework to analyse the geometry of the spacetime by probing it with quantum scalar fields. The main focus of this chapter is to describe what is meant by a quantisation in a spacetime with finite differentiability. The fourth part of the thesis presents future outlooks and some open problems.
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