Gravity induced from quantum spacetime

TL;DR

This paper explores how quantum spacetime geometry constraints limit possible metrics, revealing a unique moduli space, and extends classical Einstein geometry concepts into the quantum realm with new connections and tensors.

Contribution

It introduces a fully solved noncommutative Riemannian geometry model in 2D, identifying quantum corrections to classical Einstein tensors and metrics, and proposes a new quantum Einstein tensor concept.

Findings

Tensoriality constraints reduce metric moduli to a single parameter.

Classical geometry in 4D can match perfect fluid Einstein tensors.

Quantum corrections to Einstein tensor are of order λ^2, with metric corrections of order λ.

Abstract

We show that tensoriality constraints in noncommutative Riemannian geometry in the 2-dimensional bicrossproduct model quantum spacetime algebra [x,t]=\lambda x drastically reduce the moduli of possible metrics g up to normalisation to a single real parameter which we interpret as a time in the past from which all timelike geodesics emerge and a corresponding time in the future at which they all converge. Our analysis also implies a reduction of moduli in n-dimensions and we study the suggested spherically symmetric classical geometry in n=4 in detail, identifying two 1-parameter subcases where the Einstein tensor matches that of a perfect fluid for (a) positive pressure, zero density and (b) negative pressure and positive density with ratio w_Q=-{1\over 2}. The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the quantum spacetime model as a quadratic algebra. The noncommutative Riemannian geometry is fully solved for $n=2$ and includes the quantum Levi-Civita connection and a second, nonperturbative, Levi-Civita connection which blows up as \lambda\to 0. We also propose a `quantum Einstein tensor' which is identically zero for the main part of the moduli space of connections (as classically in 2D). However, when the quantum Ricci tensor and metric are viewed as deformations of their classical counterparts there would be an O(\lambda^2) correction to the classical Einstein tensor and an O(\lambda) correction to the classical metric.