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.

arXiv comment19 pages

References in corpus (7)

• A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
• A Topos Foundation for Theories of Physics: IV. Categories of Systems
• Toposes in General Theory of Relativity
• Synthetic Differential Geometry: A Way to Intuitionistic Models of General Relativity in Toposes
• Bohrification: From classical concepts to commutative algebras
• Solution of the vacuum Einstein equations in Synthetic Differential Geometry of Kock-Lawvere
• Remarks on the Fact that the Uncertainty Principle Does Not Determine the Quantum State