# Singularities in relativity

### Tatsuhiko Koike

Keio University, Yokohama, Japan

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## Abstract

Many phenomena of importance in general relativity theory are related to singularities in mathematics. For a simple example, the spacetime regions of extreme gravitational field such as the beginning of the Universe and the fate of a massive star are described by singularities in the differential-geometric sense, i.e., curvature singularities of pseudo-Riemannian manifolds. This type of singularity is one of the main objects of interest in general relativity. A less trivial example is that the formation of a black hole horizon can be described as a blow-up solution of some partial differential equations in a certain coordinate system, which is a singularity in the analytic sense. Another is that the “shape” of the black hole horizon is fully characterised by the set of its nondifferential points which are singularities in the sense of singularity theory. I will explain these connections between singularity and relativity with some comments on my related works.