Emergent Gravity from Quantized Spacetime

We examine the picture of emergent gravity arising from a mass deformed matrix model. Due to the mass deformation, a vacuum geometry turns out to be a constant curvature spacetime such as d-dimensional sphere and (anti-)de Sitter spaces. We show that the mass deformed matrix model giving rise to the constant curvature spacetime can be derived from the d-dimensional Snyder algebra. The emergent gravity beautifully confirms all the rationale inferred from the algebraic point of view that the d-dimensional Snyder algebra is equivalent to the Lorentz algebra in (d + 1)-dimensional flat spacetime. For example, a vacuum geometry of the mass deformed matrix model is completely described by a G-invariant metric of coset manifolds G/H defined by the Snyder algebra. We also discuss a nonlinear deformation of the Snyder algebra. PACS numbers: 11.10.Nx, 02.40.Gh, 04.50.+h