# Gravity in the smallest

## TL;DR

This paper explores how Synthetic Differential Geometry enables modeling of general relativity at infinitesimally small scales using monads, potentially informing quantum gravity research.

## Contribution

It demonstrates the application of SDG to formulate general relativity within infinitesimal domains, emphasizing monads' structure for geometric quantities.

## Key findings

- Monads model infinitesimal neighborhoods in manifolds.
- SDG provides tools to define connection and curvature at small scales.
- Implications for quantum gravity and foundational principles discussed.

## 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.