Logarithmic Newman-Penrose constants for arbitrary polyhomogeneous spacetimes

A discussion in how to calculate asymptotic expansions for polyhomogeneous spacetimes using the Newman-Penrose formalism is done. The existence of logarithmic Newman-Penrose constants for a general polyhomogeneous spacetime (i.e. a polyhomogeneous spacetime such that $\Psi_0=O(r^{-3}\ln ^{N_3})$) is addressed. It is found that these constants exist for the generic case. As an application stationary polyhomogneous spacetimes are studied. It is found that for these spacetimes $\Psi_0=O(r^{-5}\ln ^{N_5} r)$. The logarithmic NP constants are calculated for these stationary spacetimes. A connection between these logarithmic constants and the Hansen multipole moments is done.