Wahlquist fluid

Metric

The metric of a Wahlquist fluid is given by

${\displaystyle d{s}^2=f(dt-\tilde{A}d\phi {)}^2-r_0^2({\zeta }^2+{\xi }^2)[\frac{d{\zeta }^2}{(1-{\tilde{k}}^2{\zeta }^2){\tilde{h}}_1}+\frac{d{\xi }^2}{(1+{\tilde{k}}^2{\xi }^2){\tilde{h}}_2}+\frac{{\tilde{h}}_1{\tilde{h}}_2}{{\tilde{h}}_1-{\tilde{h}}_2}d{\phi }^2]}$

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.