Through the Singularity
Michael Heller, Tomasz Miller, Wiesław Sasin
TL;DR
The paper investigates the problem of crossing strong, hidden singularities in general relativity by leveraging Sikorski differential spaces and structured spaces to redefine smoothness beyond classical manifolds. It conceptualizes malicious singularities as $b$-boundary phenomena where the extended differential structure may collapse to constants, highlighting pathologies in prior boundary constructions. The authors then develop a generalized framework for curves that pass through the singularity, showing how two smooth curves can be joined through a singular point via a unified parameter space and a concatenated curve, while acknowledging potential energy- and reparametrization-related limitations. This work provides a mathematical foundation for discussing passage through singularities and informs future approaches to quantum gravity where a smooth passage through such regions might be conceptually meaningful.
Abstract
In this work, we propose a dangerous journey -- a journey through the strong singularity from one universe to another or from inside of a black hole to its 'inverse' as a white hole. Such singularities are hidden in the Friedman and Schwarzschild solutions; we call them malicious singularities. The journey is made possible owing to two generalizations. The first generalization consists in considering spaces with differential structures on them (the so-called ringed spaces) rather than the usual manifolds. This entails a generalization of the concept of smoothness, which allows us to think about a smooth passage through the singularity. The second generalization is related to the concept of curve. We show that if a kind of singularity is implanted in the set of curve's parameters, along with an appropriate topology, in such a way that the structure of the set of parameters corresponds to the structure of the singular space-time, the curve can smoothly -- in a generalized sense -- pass through the singularity.