  Connes showed that spectral triples encode (noncommutative) metric
information. Further, Connes and Moscovici in their metric bundle construction
showed that, as with the Takesaki duality theorem, forming a crossed product
spectral triple can substantially simplify the structure. In a recent paper,
Bellissard, Marcolli and Reihani (among other things) studied in depth metric
notions for spectral triples and crossed product spectral triples for
$Z$-actions, with applications in number theory and coding theory. In the work
of Connes and Moscovici, crossed products involving groups of diffeomorphisms
and even of \'{e}tale groupoids are required. With this motivation, the present
paper develops part of the Bellissard-Marcolli-Reihani theory for a general
discrete group action, and in particular, introduces coaction spectral triples
and their associated metric notions. The isometric condition is replaced by the
contractive condition.
