Revising the multipole moments of numerical spacetimes, and its consequences

TL;DR

This paper identifies a systematic misread of relativistic multipole moments from asymptotic metric expansions and provides coordinate-invariant corrections to the quadrupole $M_2$ and octupole $S_3$ for numerically constructed neutron-star spacetimes using Ryan's framework. It derives explicit corrected relations $M_2^{GH}=M_2-rac{4}{3}ig(rac{1}{4}+big)M^3$ and $S_3^{GH}=S_3-rac{12}{5}ig(rac{1}{4}+big)jM^4$, highlighting the role of the parameter $b$ and showing that Kerr in isotropic coordinates does not obey simple Schwarzschild-like asymptotics. The authors demonstrate that applying the corrected moments substantially improves the fidelity of analytic metrics (e.g., Manko et al.) in matching numerical neutron-star exteriors, affects the inferred relationships between higher multipoles and spin, and refines ISCO predictions. Supplemental analyses provide fitting formulas for higher-order moments as functions of spin and quantify improvements in metric mismatches and ISCO calculations across multiple equations of state. Overall, the work enhances the reliability of analytic representations of numerical spacetimes and informs interpretations linking multipole structure to neutron-star interior physics.

Abstract

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 note 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.

Figures (5)

• 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 complement), before (dashed curve) and after the correction of $M_2$ (dashed-dotted curve). The top plot is for $g_{tt}$ and the bottom one for $g_{t \phi}$. We note that the corresponding overall improvement for this particular neutron star model was 6.6 in $g_{tt}$ and 15.1 in $g_{t \phi}$. This was a model with a medium improvement in $g_{tt}$, compared to the whole set of models which were examined.
• Figure 2: Plot of the reduced moments and the corresponding fits for the three EOSs (AU on the left, FPS in the middle and L on the right). In the case of $q$ and $s_3$ (two upper rows), red is the fit for sequence (i), green is the fit for sequence (ii), blue is the fit with one parameter for sequence (iii), and cyan is the fit with two parameters for the same sequence. The bottom row shows the plots and corresponding fits for the parameter $b$ with the same color coding.
• Figure 3: Plot of the comparison of the analytic against the numerical metric functions $g_{tt}$ (upper plots) and $g_{t\phi}$ (lower plots) for EOS AU. We show three typical models, one from each sequence. From left to right, the models are: $\#6$, $\#18$ and $\#29$. The dashed curves are the ones without the correction in the quadrupole and the dashed-dotted curves are the ones with the corrected quadrupole.
• Figure 4: Same as Fig. \ref{['figmanko1']} for EOS FPS. We show typical models from the three sequences of neutron stars. From left to right, the models are: $\#6$, $\#16$ and $\#27$.
• Figure 5: Same as Fig. \ref{['figmanko1']} for EOS L. We show typical models from the three sequences of neutron stars. From left to right, the models are: $\#6$, $\#16$ and $\#26$.