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

In the present work we present an extended description of the covariant noncommutative space, 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 us, 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, we obtain two Fuzzy Gravity models, one in de Sitter and one in anti-de Sitter space, depending on the extension of the isometry group, and we discuss their spontaneous symmetry breaking leading to fuzzy versions of the noncommutative $SO(1,3)$ gravity. In addition we discuss for the first time how to introduce fermions in the fuzzy gravity and even more importantly how to unify the constructed noncommutative-fuzzy gravity with internal interactions based on $SO(10)$ or $SU(5)$ as grand unified theories.