External symmetry in general relativity

We propose a generalization of the isometry transformations to the geometric context of the spin field theories where the local frames are explicitly involved. We define the external symmetry transformations as isometries combined with suitable tetrad gauge transformations and we show that they form a group which is locally isomorphic with the isometry one. We point out that these symmetry transformations leave invariant the field equations in local frames and have generators with specific spin terms that represent new physical observables. The examples we give are the generators of the central symmetry and those of the maximal symmetries of the de Sitter and anti-de Sitter spacetimes for which we derive the spin terms in different tetrad gauge fixings.