black holes - Why can't a particle rotate opposite to the central mass within the ergosphere?

Wiki says about the Kerr metric:

A moving particle experiences a positive proper time along its worldline, its path through spacetime. However, this is impossible within the ergosphere, where $g_{tt}$ is negative, unless the particle is co-rotating with the interior mass $M$ with an angular speed at least of Ω. Thus, no particle can rotate opposite to the central mass within the ergosphere.

I don't get it. Can it be somehow seen that $g_{tt}$ is negative from

$$ \begin{align} g^{\mu\nu}\frac{\partial}{\partial{x^{\mu}}}\frac{\partial}{\partial{x^{\nu}}} = & \frac{1}{c^{2}\Delta}\left(r^{2} + \alpha^{2} + \frac{r_{s}r\alpha^{2}}{\rho^{2}}\sin^{2}\theta\right)\left(\frac{\partial}{\partial{t}}\right)^{2} + \frac{2r_{s}r\alpha}{c\rho^{2}\Delta}\frac{\partial}{\partial{\phi}}\frac{\partial}{\partial{t}} \\ & - \frac{1}{\Delta\sin^{2}\theta}\left(1 - \frac{r_{s}r}{\rho^{2}}\right)\left(\frac{\partial}{\partial{\phi}}\right)^{2} - \frac{\Delta}{\rho^{2}}\left(\frac{\partial}{\partial{r}}\right)^{2} - \frac{1}{\rho^{2}}\left(\frac{\partial}{\partial{\theta}}\right)^{2} \color{red}{?} \end{align} $$

And why can't no particle can rotate opposite to the central mass within the ergosphere?