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Emergent Gravity, Spacetime Topology Change and Resolution of Singularity
Emergent gravity is aimed at constructing a Riemannian geometry from U(1) gauge fields on a noncommutative (NC) spacetime. It is based on the Darboux theorem or the Moser lemma in symplectic geometry stating that electromagnetic force can always be eliminated by a local coordinate transformation as long as the U(1) gauge theory is defined on a spacetime with symplectic structure. In this set up, the spacetime geometry is defined by U(1) gauge fields on NC spacetime and accordingly the topology of spacetime is determined by the topology of NC U(1) gauge fields. I will try to demonstrate using ADHM construction that the topology change of spacetime is ample in emergent gravity and the subsequent resolution of spacetime singularity is possible in NC spacetime. Therefore emergent gravity provides us with a well-defined mechanism for topology change of spacetime without suffering from any singularity in sharp contrast to general relativity.