Gravity in the smallest

Abstract

Synthetic Differential Geometry (SDG) is a categorical version of differential geometry based on enriching the real line with infinitesimals and weakening of classical logic to intuitionistic logic. We show that SDG provides an effective mathematical tool to formulate general relativity in infinitesimally small domains. Such a domain is modelled by a monad around a point $x$ of a manifold $M$, defined as a collection of points in $M$ that differ from $x$ by an infinitesimal value. Monads have rich enough matematical structure to allow for the existence of all geomeric quantities necesary to construct general relativity "in the smallest". We focus on connection and curvature. We also comment on the covariance principle and the equivalence principle in this context. Identification of monads with what happens "beneath the Planck threshold" could open new possibilities in our search for quantum gravity theory.