Gravity from entropy

TL;DR

This paper proposes a novel entropic approach to gravity, deriving Einstein's equations from a quantum relative entropy framework that links spacetime geometry with matter fields, potentially offering new insights into quantum gravity and dark matter.

Contribution

It introduces an entropic action based on quantum relative entropy, relating spacetime metric to matter-induced metric, leading to modified Einstein equations with a small positive cosmological constant.

Findings

Modified Einstein equations reduce to standard form at low coupling

Emergent small positive cosmological constant from G-field

Second order equations in metric and G-field

Abstract

Gravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a quantum operator, playing the role of an renormalizable effective density matrix and to describe the matter fields topologically, according to a Dirac-K\"ahler formalism, as the direct sum of a zero-form, a one-form and a two-form. While the geometry of spacetime is defined by its metric, the matter fields can be used to define an alternative metric, the metric induced by the matter fields, which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the G-field, which acts as a set of Lagrangian multipliers, the proposed entropic action reduces to a dressed Einstein-Hilbert action with an emergent small and positive cosmological constant only dependent on the G-field. The obtained equations of modified gravity remain second order in the metric and in the G-field. A canonical quantization of this field theory could bring new insights into quantum gravity while further research might clarify the role that the G-field could have for dark matter.