Beyond the Space-Time Boundary

TL;DR

This paper explores how Synthetic Differential Geometry enables crossing space-time boundaries and singularities, providing a new mathematical perspective beyond traditional geometric limitations in General Relativity.

Contribution

It introduces a simple, rigorous model using SDG to analyze what occurs beyond space-time singularities, expanding the mathematical understanding of boundaries in General Relativity.

Findings

SDG allows crossing of space-time boundaries

Infinitesimals enable penetration into manifold germs

Model demonstrates behavior beyond singularities

Abstract

In General Relativity a space-time $M$ is regarded singular if there is an obstacle that prevents an incomplete curve in $M$ to be continued. Usually, such a space-time is completed to form $\bar{M} = M \cup \partial M$ where $\partial M$ is a singular boundary of $M$. The standard geometric tools on $M$ do not allow "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 doing so. 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 to refer to the physical universe.