Revising the Multipole Moments of Numerical Spacetimes and its Consequences

Identifying the relativistic multipole moments of a spacetime of an astrophysical object that has been constructed numerically is of major interest, both because the multipole moments are intimately related to the internal structure of the object, and because the construction of a suitable analytic metric that mimics a numerical metric should be based on the multipole moments of the latter one in order to yield a reliable representation. In this Letter, we show that there has been a widespread delusion in the way the multipole moments of a numerical metric are read from the asymptotic expansion of the metric functions. We show how one should read correctly the first few multipole moments (starting from the quadrupole mass moment) and how these corrected moments improve the efficiency of describing the metric functions with analytic metrics that have already been used in the literature, as well as other consequences of using the correct moments.

Figure 1

A typical log-log plot of the relative difference between the numerical and the analytic metric [$(g_{ij}^n-g_{ij}^a)/g_{ij}^n$] for a specific numerical model (model 16 of EOS FPS of the SM [20]), before (dashed curve) and after (dashed-dotted curve) the correction of $M_2$. The left plot is for $g_{tt}$ and the right one for $g_{t\varphi }$. We note that the corresponding overall improvement for this particular neutron star model was 6.6 in $g_{tt}$ and 15.1 in $g_{t\varphi }$. This was a model with a medium improvement in $g_{tt}$, compared to the whole set of models which were examined. The dips in the curves are simply due to coincidental crossings between the numerical and the analytic metrics.