Covariant Information Theory and Emergent Gravity

Abstract

Informational dependence between statistical or quantum subsystems can be described with Fisher matrix or Fubini-Study metric obtained from variations of the sample/configuration space coordinates. Using these non-covariant objects as macroscopic constraints we consider statistical ensembles over the space of classical probability distributions or quantum wave-functions. The ensembles are covariantized using dual field theories with either complex or real scalar fields identified with complex wave-functions or square root of probabilities. We argue that a full space-time covariance on a field theory side is dual to local computations on the information theory side. We define a fully covariant information-computation tensor and show that it must satisfy conservation equations. Then we switch to a thermodynamic description and argue that the (inverse of) space-time metric tensor is a conjugate thermodynamic variable to the ensemble-averaged information-computation tensor. In the equilibrium the entropy production vanishes and the metric is not dynamical, but away from equilibrium the entropy production gives rise to an emergent dynamics of the metric. This dynamics can be described by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry the Onsager tensor is expected to be symmetric. We show that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein-Hilbert term. This proves that general relativity is equivalent to a theory of non-equilibrium (thermo)dynamics of the metric which is expected to break down far away from equilibrium where the symmetries of the Onsager tensor are to be broken.