http://hdl.handle.net/20.500.12128/8681
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. |
URI: | http://hdl.handle.net/20.500.12128/8681 |
DOI: | 10.3390/universe3010016 |
ISSN: | 2218-1997 |
Appears in Collections: | Artykuły (WNŚiT) |
File | Description | Size | Format | |
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Heller_Infinitesimal_Structure_of_Singularities.pdf | 407,61 kB | Adobe PDF | View/Open |
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