Wahlquist fluid

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.

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]