Beyond the space-time boundary

Abstract

In General Relativity, a space-time M is regarded as singular if there is an obstacle that prevents an incomplete curve in M from being continued. Such a space-time is completed to form M¯=M∪∂M where ∂M is a singular boundary of M. The standard geometric tools on M do not allow one “to cross the boundary”. However, the so-called Synthetic Differential Geometry (SDG), a categorical version of standard differential geometry based on intuitionistic logic, has at its disposal tools permitting this to be done. Owing to the existence of infinitesimals, one is able to penetrate “germs of manifolds” that are not visible from the standard perspective. We present a simple model showing what happens “beyond the boundary” and when the singularity is finally attained. The model is purely mathematical and is mathematically rigorous but it does not pretend, at its present stage, to refer to the physical universe.