On the Post-linear Quadrupole-Quadrupole Metric

TL;DR

This paper extends the Hartle-Thorne metric by including second-order quadrupole terms, compares it with Blanchet's post-Minkowskian solutions, and discusses its relevance for modeling realistic astrophysical objects.

Contribution

It derives an extended Hartle-Thorne metric with second-order quadrupole terms and demonstrates its agreement with Blanchet's post-linear solutions, enhancing modeling accuracy for rotating stars.

Findings

Derived the extended Hartle-Thorne metric in harmonic coordinates.

Established agreement with Blanchet's post-linear metric.

Provided a coordinate transformation from Erez-Rosen to Hartle-Thorne.

Abstract

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.