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Title: Infinitesimal structure of singularities
Authors: Heller, Michael
Król, Jerzy
Keywords: general relativity; category theory; synthetic differential geometry; infinitesimal formal manifold; curvature; space-time singularity
Issue Date: 2017
Citation: Universe (Basel), Vol. 13, iss. 1, art. no 16 (2017)
Abstract: 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.
DOI: 10.3390/universe3010016
ISSN: 2218-1997
Appears in Collections:Artykuły (WNŚiT)

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