Infinitesimal Structure of Singularities
AbstractSome 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
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Heller, M.; Król, J. Infinitesimal Structure of Singularities. Universe 2017, 3, 16.
Heller M, Król J. Infinitesimal Structure of Singularities. Universe. 2017; 3(1):16.Chicago/Turabian Style
Heller, Michael; Król, Jerzy. 2017. "Infinitesimal Structure of Singularities." Universe 3, no. 1: 16.