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.
Highlights
A tacit assumption is that all mathematical tools we use are founded on some theory of sets
We tried to pave the way towards a broader use of categorical methods—in particular, those related to synthetic differential geometry—in general relativity
The great advantage of SDG is the existence of infinitesimals; they offer a unique opportunity to deal with all these questions in which very small scales of space-time are involved
Summary
A tacit assumption is that all mathematical tools we use are founded on some theory of sets. One of the essential differences between them is that in SDG, infinitesimals appear which substantially enrich the usual real line Owing to this fact, geometry acquires a tool to penetrate infinitesimally small portions of a given manifold, which are invisible in the usual approach (in SET, they do not exist). One should study the transition (a suitable functor) from the SET environment of standard general relativity to a suitable topos providing the logic of the “infinitesimal behaviour”. Some work in this respect is underway.
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