We present an improvement to the Classical Effective Theory approach to the
non-relativistic or Post-Newtonian approximation of General Relativity. The
"potential metric field" is decomposed through a temporal Kaluza-Klein ansatz
into three NRG-fields: a scalar identified with the Newtonian potential, a
3-vector corresponding to the gravito-magnetic vector potential and a 3-tensor.
The derivation of the Einstein-Infeld-Hoffmann Lagrangian simplifies such that
each term corresponds to a single Feynman diagram providing a clear physical
interpretation. Spin interactions are dominated by the exchange of the
gravito-magnetic field. Leading correction diagrams corresponding to the 3PN
correction to the spin-spin interaction and the 2.5PN correction to the
spin-orbit interaction are presented.Comment: 10 pages, 3 figures. v2: published version. v3: Added a computation
of Einstein-Infeld-Hoffmann in higher dimensions within our improved ClEFT
which partially confirms and partially corrects a previous computation. See
notes added at end of introductio