Wahlquist's metric versus an approximate solution with the same equation of state

TL;DR

This paper compares Wahlquist's exact solution for a rotating perfect fluid with an approximate global metric, analyzing conditions for vanishing twist and Petrov type D classification.

Contribution

It demonstrates how to relate Wahlquist's solution to an approximate metric under specific conditions, clarifying the role of parameters and coordinate transformations.

Findings

Vanishing twist in Wahlquist's metric requires parameter r_0 to approach zero.

The approximate metric matches Wahlquist's solution when parameters are chosen for Petrov type D.

Coordinate changes align all metric components, revealing the connection between the solutions.

Abstract

We compare an approximation of the singularity-free Wahlquist exact solution with a stationary and axisymmetric metric for a rigidly rotating perfect fluid with the equation of state $\mu + 3p= \mu_0$, a sub-case of a global approximate metric obtained recently by some of us. We see that to have a fluid with vanishing twist vector everywhere in Wahlquist's metric the only option is to let its parameter $r_0\rightarrow0$ and using this in the comparison allows us in particular to determine the approximate relation between the angular velocity of the fluid in a set of harmonic coordinates and $r_0$. Through some coordinate changes we manage to make every component of both approximate metrics equal. In this situation, the free constants of our metric take values that happen to be those needed for it to be of Petrov type D, the last condition that this fluid must verify to give rise to the Wahlquist solution.