Minimum spacetime length and the thermodynamics of spacetime

TL;DR

The paper develops a phenomenological framework, the q-metric, to implement a minimum spacetime length in a covariant, semiclassical setting using a two-point bitensor $q_{\\alpha\\beta}(x, x')$. In the coincidence limit, the q-Ricci scalar does not recover the classical curvature but maps to an entropy-density functional tied to horizons, aligning with thermodynamic emergent-gravity programs. By computing space-like, time-like, and null cases, the work shows that a minimal length yields a finite minimal area but not a finite volume and predicts dimensional reduction to $D_{eff} \\ o 2$ at small scales, which resonates with multiple quantum-gravity approaches. The framework connects horizon thermodynamics, entropy bounds, and Padmanabhan’s emergent gravity, offering a viable phenomenological route to encode quantum-gravity effects without a full microscopic theory, while leaving open foundational tasks such as a complete stress-energy description in the q-metric and concrete observational implications.

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 $q_{αβ}(x, x')$ that yields a finite geodesic distance in the coincidence limit $x\rightarrow x'$. 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.