Emergent Gravity from Noncommutative Spacetime

TL;DR

This paper explores how gravity can emerge from noncommutative electromagnetism, showing that gauge fields in fuzzy spacetime can give rise to gravitational phenomena, linking noncommutative geometry with gravity.

Contribution

It demonstrates that self-dual electromagnetism in noncommutative spacetime is equivalent to self-dual Einstein gravity, revealing gravity as an emergent collective phenomenon from gauge fields.

Findings

Gravity emerges from NC electromagnetism as a nonlinear realization of diffeomorphisms.

NC electromagnetism is realized through a spontaneously broken diffeomorphism symmetry.

Evidence suggests emergent gravity contains structures of generalized complex geometry and NC gravity.

Abstract

We showed before that self-dual electromagnetism in noncommutative (NC) spacetime is equivalent to self-dual Einstein gravity. This result implies a striking picture about gravity: Gravity can emerge from electromagnetism in NC spacetime. Gravity is then a collective phenomenon emerging from gauge fields living in fuzzy spacetime. We elucidate in some detail why electromagnetism in NC spacetime should be a theory of gravity. In particular, we show that NC electromagnetism is realized through the Darboux theorem as a diffeomorphism symmetry G which is spontaneously broken to symplectomorphism H due to a background symplectic two-form $B_{\mu\nu}=(1/\theta)_{\mu\nu}$, giving rise to NC spacetime. This leads to a natural speculation that the emergent gravity from NC electromagnetism corresponds to a nonlinear realization G/H of the diffeomorphism group, more generally its NC deformation. We also find some evidences that the emergent gravity contains the structure of generalized complex geometry and NC gravity. To illuminate the emergent gravity, we illustrate how self-dual NC electromagnetism nicely fits with the twistor space describing curved self-dual spacetime. We also discuss derivative corrections of Seiberg-Witten map which give rise to higher order gravity.