Emergent Gravity from Noncommutative Spacetime

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 photons 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 diffeomorphism symmetry G which is spontaneously broken to symplectomorphism H due to a background symplectic two-form Bμν = (1/θ)μν , 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. As a corollary, we find evidences that the emergent gravity contains the structure of generalized complex geometry and NC gravity. To illuminate the emergent gravity from NC spacetime, 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. PACS numbers: 11.10.Nx, 02.40.Gh, 04.50.+h