Wahlquist fluid

In general relativity, the Wahlquist fluid is an exact rotating perfect fluid solution to Einstein's equation with equation of state corresponding to constant gravitational mass density.

Introduction

The Wahlquist fluid was first discovered by Hugo D. Wahlquist in 1968.^[1] It is one of few known exact rotating perfect fluid solutions in general relativity. The solution reduces to the static Whittaker metric in the limit of zero rotation.

Metric

The metric of a Wahlquist fluid is given by

${\displaystyle ds^{2}=f(dt-{\tilde {A}}d\varphi )^{2}-r_{0}^{2}(\zeta ^{2}+\xi ^{2})[{\frac {d\zeta ^{2}}{(1-{\tilde {k}}^{2}\zeta ^{2}){\tilde {h}}_{1}}}+{\frac {d\xi ^{2}}{(1+{\tilde {k}}^{2}\xi ^{2}){\tilde {h}}_{2}}}+{\frac {{\tilde {h}}_{1}{\tilde {h}}_{2}}{{\tilde {h}}_{1}-{\tilde {h}}_{2}}}d\varphi ^{2}]}$

where

${\displaystyle f={\frac {{\tilde {h}}_{1}-{\tilde {h}}_{2}}{\zeta ^{2}+\xi ^{2}}}}$

${\displaystyle {\tilde {A}}=r_{0}({\frac {\xi ^{2}{\tilde {h}}_{1}+\zeta ^{2}{\tilde {h}}_{2}}{{\tilde {h}}_{1}-{\tilde {h}}_{2}}}-\xi _{A}^{2})}$

${\displaystyle {\tilde {h}}_{1}(\zeta )=1+\zeta ^{2}+{\frac {\zeta }{\kappa ^{2}}}[\zeta _{+}{\frac {1}{\tilde {k}}}{\sqrt {1-{\tilde {k}}^{2}\zeta ^{2}}}\arcsin({\tilde {k}}\zeta )]}$

${\displaystyle {\tilde {h}}_{2}(\xi )=1-\xi ^{2}-{\frac {\xi }{\kappa ^{2}}}[\xi _{-}{\frac {1}{\tilde {k}}}{\sqrt {1+{\tilde {k}}^{2}\xi ^{2}}}\sinh ^{-1}({\tilde {k}}\xi )]}$

and ${\displaystyle \xi _{A}}$ is defined by ${\displaystyle {\tilde {h}}_{2}(\xi _{A})=0}$. It is a solution with equation of state ${\displaystyle \mu +3p=\mu _{0}}$ where ${\displaystyle \mu _{0}}$ is a constant.

Properties

The pressure and density of the Wahlquist fluid are given by

${\displaystyle p={\frac {1}{2}}\mu _{0}(1-\kappa ^{2}f)}$

${\displaystyle \mu ={\frac {1}{2}}\mu _{0}(3\kappa ^{2}f-1)}$

The vanishing pressure surface of the fluid is prolate, in contrast to physical rotating stars, which are oblate. It has been shown that the Wahlquist fluid can not be matched to an asymptotically flat region of spacetime.^[2]