Non-existence of expansion-free dynamical stars with rotation and spatial twist

Abstract

Extending a previous work by the same authors, we investigate the existence of expansion-free dynamical stars with non-zero spatial twist and rotation and show that such stars cannot exist. Firstly, it is shown that a rotating expansion-free dynamical star with zero twist cannot exist. This is due to the fact that such stars cannot radiate and they are shear-free, in which case the energy density $\rho$ is time independent. Secondly, we prove that a non-rotating expansion-free dynamical star with non-zero spatial twist also cannot exist, as either the strong energy condition must be violated, i.e. $\rho+3p<0$, or the star must be shear-free in which case the star is static ($\Theta=\Omega=\Sigma=0$). Finally, if we insist that the rotation and spatial twist are simultaneously non-zero, then the star cannot be shear-free in which case we obtain a quadratic polynomial equation in $\phi$ and $\Sigma$ with no real solutions. Therefore such stars cannot exist.