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Universe 2017, 3(1), 16; doi:10.3390/universe3010016

Infinitesimal Structure of Singularities

Copernicus Center for Interdisciplinary Studies, ul. Szczepańska 1/5, PL-31011 Cracow, Poland
Institute of Physics, University of Silesia, PL-40007 Katowice, Poland
Author to whom correspondence should be addressed.
Academic Editors: Mariusz P. Dąbrowski, Manuel Krämer and Vincenzo Salzano
Received: 9 January 2017 / Revised: 19 February 2017 / Accepted: 20 February 2017 / Published: 27 February 2017
(This article belongs to the Special Issue Varying Constants and Fundamental Cosmology)
View Full-Text   |   Download PDF [232 KB, uploaded 27 February 2017]


Some important problems of general relativity, such as the quantisation of gravity or classical singularity problems, crucially depend on geometry on very small scales. The so-called synthetic differential geometry—a categorical counterpart of the standard differential geometry—provides a tool to penetrate infinitesimally small portions of space-time. We use this tool to show that on any “infinitesimal neighbourhood” the components of the curvature tensor are themselves infinitesimal, and construct a simplified model in which the curvature singularity disappears, owing to this effect. However, one pays a price for this result. Using topoi as a generalisation of spaces requires a weakening of arithmetic (the existence of infinitesimals) and of logic (to the intuitionistic logic). Is this too high a price to pay for acquiring a new method of solving unsolved problems in physics? Without trying, we shall never know the answer. View Full-Text
Keywords: general relativity; category theory; synthetic differential geometry; infinitesimal formal manifold; curvature; space-time singularity general relativity; category theory; synthetic differential geometry; infinitesimal formal manifold; curvature; space-time singularity
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Heller, M.; Król, J. Infinitesimal Structure of Singularities. Universe 2017, 3, 16.

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