  The Geroch-Hansen and Thorne (ACMC) formalisms give rigorous and equivalent
definitions for gravitational multipoles in stationary vacuum spacetimes.
However, despite their ubiquitous use in gravitational physics, it has not been
shown that these formalisms can be generalized to non-vacuum stationary
solutions, except in a few special cases. This paper shows how the
Geroch-Hansen formalism can be generalized to arbitrary non-vacuum stationary
spacetimes for metrics that are sufficiently smooth at infinity. The key is the
construction of an improved twist vector, which is well-defined under a mild
topological condition on the spacetime (which is automatically satisfied for
black holes). Ambiguities in the construction of this improved twist vector are
discussed and fixed by imposing natural gauge fixing conditions, which also
immediately lead to the equivalence between the Geroch-Hansen and Thorne
formalisms for such arbitrary stationary spacetimes.
