  The Hartle-Thorne metric defines a reliable spacetime for most astrophysical
purposes, for instance for the simulation of slowly rotating stars. Solving the
Einstein field equations, we added terms of second order in the quadrupole
moment to its post-linear version in order to compare it with solutions found
by Blanchet in the frame of the multi-polar post-Minkowskian framework. We
first derived the extended Hartle-Thorne metric in harmonic coordinates and
then showed agreement with the corresponding post-linear metric from Blanchet.
We also found a coordinate transformation from the post-linear Erez-Rosen
metric to our extended Hartle-Thorne spacetime. It is well known that the
Hartle-Thorne solution can be smoothly matched with an interior perfect fluid
solution with physically appropriate properties. A comparison among these
solutions provides a validation of them. It is clear that in order to represent
realistic solutions of self-gravitating (axially symmetric) matter
distributions of perfect fluid, the quadrupole moment has to be included as a
physical parameter.
