# Minimum Spacetime Length and the Thermodynamics of Spacetime

## TL;DR

This paper explores how spacetime might have a smallest possible size and connects this idea to thermodynamics and quantum gravity theories.

## Contribution

The paper introduces a bi-tensorial quantum metric to describe spacetime's minimum length and derive gravity's field equations from entropy.

## Key findings

- Quantum gravity theories suggest a minimum length in spacetime.
- A bi-tensorial quantum metric provides a finite geodesic distance in the coincidence limit.
- Entropy from microscopic spacetime degrees of freedom leads to gravity's field equations via a thermodynamic principle.

## Abstract

Theories of emergent gravity have established a deep connection between entropy and the geometry of spacetime by looking at the latter through a thermodynamic lens. In this framework, the macroscopic properties of gravity arise in a statistical way from an effective small-scale discrete structure of spacetime and its information content. In this review, we begin by outlining how theories of quantum gravity imply the existence of a minimum length of spacetime as a general feature. We then describe how such a structure can be implemented in a way that is independent from the details of the quantum fluctuations of spacetime via a bi-tensorial quantum metric qαβ(x,x′) that yields a finite geodesic distance in the coincidence limit x→x′. Finally, we discuss how the entropy encoded by these microscopic degrees of freedom can give rise to the field equations for gravity through a thermodynamic variational principle.