On the accuracy of the IWM-CFC approximation in differentially rotating relativistic stars

TL;DR

This paper assesses the accuracy of the IWM-CFC approximation in modeling strongly differentially rotating relativistic stars, finding deviations below 10% for local quantities and providing practical error estimation tools.

Contribution

It introduces a simple empirical relation to estimate the approximation error based on star flattening and lapse function, enhancing numerical relativity simulations.

Findings

Deviation from full GR is below 5% for integrated quantities.

Deviation is below 10% for local quantities like angular velocity.

A local metric-based error indicator is more reliable for most models.

Abstract

We determine the accuracy of the conformal flatness (IWM-CFC) approximation for the case of single, but strongly differentially rotating relativistic stars. We find that for the fastest rotating and most relativistic polytropic models, the deviation from full general relativity is below 5% for integrated quantities and below 10% for local quantities, such as the angular velocity. Furthermore, we study the deviation of the IWM-CFC approximation from full general relativity by evaluating and comparing different error indicators. We find that for the models that are not near the maximum mass, a simple error indicator constructed from local values of the metric potentials is more indicative of the accuracy of the IWM-CFC approximation than an error indicator that is based on the Cotton-York tensor. Furthermore, we construct a simple, linear empirical relation that allows for the estimation of the error made by the IWM-CFC approximation and which only involves the flattening of the star due to rotation and the minimum value of the lapse function. Thus, in any numerical simulation involving rotating relativistic stars, one can readily know the deviations from full general relativity due to the IWM-CFC approximation.