A generalized Riemannian geometry is studied where the metric tensor is replaced by a matrix g of metrics. In this context new geometric quantities arise, which are trivial in ordinary Riemannian geometry. An application of this formalism to many-body alignments in general relativity is proposed, where the sub-constituents of the overall gravitational field are described by the components of g. The mutual gravitational interactions between the individual particles are encoded in specific tensors. In particular, very specific approximation schemes for Einstein’s field equations may be considered, which exclusively approximate those terms in the field equations which are due to interactions. The Newtonian limit as well as the first post-Newtonian approximation of the presented formalism is studied in order to display the interpretability of the presented formalism in terms of many-body alignments and in order to deduce a physical interpretation of the new geometric quantities.
