Noncommutative Spaces and Gravity

This is an introduction to an algebraic construction of a gravity theory on noncommutative spaces which is based on a deformed algebra of (infinitesimal) diffeomorphisms. We start with some fundamental ideas and concepts of noncommutative spaces. Then the $\theta$-deformation of diffeomorphisms is studied and a tensor calculus is defined. A deformed Einstein-Hilbert action invariant with respect to deformed diffeomorphisms is given. Finally, all noncommutative fields are expressed in terms of their commutative counterparts up to second order of the deformation parameter using the $\star$-product. This allows to study explicitly deviations to Einstein's gravity theory in orders of $\theta$.