Fuzzy Gravity: Four-Dimensional Gravity on a Covariant Noncommutative Space and Unification with Internal Interactions

In the present work, an extended description of the covariant noncommutative space is presented, which accommodates the Fuzzy Gravity model constructed previously. It is based on the historical lesson that the use of larger algebras containing all generators of the isometry of the continuous one helped in formulating a fuzzy covariant noncommutative space. Specifically a further enlargement of the isometry group leads the authors, in addition to the construction of the covariant noncommutative space, also to the suggestion of the group that should be gauged on such a space in order to construct a Fuzzy Gravity theory. As a result, two Fuzzy Gravity models are obtained, one in de Sitter and one in anti-de Sitter space, depending on the extension of the isometry group, and their spontaneous symmetry breaking leading to fuzzy versions of the noncommutative $SO(1,3)$ gravity are discussed. In addition, how to introduce fermions in the fuzzy gravity is discussed for the first time, and even more importantly, how to unify the constructed noncommutative-fuzzy gravity with internal interactions based on $SO\left(10\right)$ or $SU\left(5\right)$ as grand unified theories.