Minimum Spacetime Length and the Thermodynamics of Spacetime

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 $q_{\alpha \beta }(x,x^{\prime })$ that yields a finite geodesic distance in the coincidence limit $x\to x^{\prime }$. 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.