Rotating neutron stars: an invariant comparison of approximate and numerical spacetime models

We compare three different models of rotating neutron star spacetimes: the Hartle-Thorne (HT) slow-rotation approximation at second order in rotation, the exact analytic vacuum solution of Manko et al. and a numerical solution of the full Einstein equations. We integrate the HT structure equations for five representative equations of state. Then we match the HT models to numerical solutions of the Einstein equations, imposing that the mass and angular momentum of the models be the same. We estimate the limits of validity of the HT expansion computing relative errors in the spacetime's quadrupole moment Q and in the ISCO radii. We find that ISCO radii computed in the HT approximation are accurate to better than 1%, even for the fastest observed ms pulsar. At the same rotational rates the accuracy on Q is of order 20%. In the second part of the paper we focus on the exterior vacuum spacetimes. We introduce a physically motivated `quasi-Kinnersley' Newman-Penrose frame. In this frame we evaluate the speciality index S, a coordinate-independent quantity measuring the deviation of each model from Petrov Type D. On the equatorial plane this deviation is smaller than 5%, even for the fastest rotating models. Our main conclusion is that the HT approximation is very reliable for most astrophysical applications.