Spacetime geometry is a foundational framework in general relativity that combines the three dimensions of space with the dimension of time into a single four-dimensional manifold. It describes how matter and energy influence the curvature of spacetime, which in turn dictates the motion of objects and the propagation of light.
A black hole singularity is a point in space where gravitational forces compress matter to an infinitely small point, resulting in infinite density and zero volume. This enigmatic region challenges the current understanding of physics, indicating a breakdown of spacetime as described by general relativity.
A Cartan connection generalizes the notion of linear connections by allowing for curved or deformed geometry, thus providing a framework for describing the geometry of a manifold with a wider scope. It enables a more flexible treatment of geometrical and physical contexts, such as in the theory of general relativity and gauge theories, by incorporating the group-theoretical approach to differential geometry.