On the instability of some k-essence space-times

We study the stability properties of static, spherically symmetric configurations in k-essence theories with the Lagrangians of the form $F(X)$, $X\equiv {\varphi }_{,\alpha }{\varphi }^{,\alpha }$. The instability under spherically symmetric perturbations is proved for two recently obtained exact solutions for $F(X)=F_0{X}^{1\mathrm{/}3}$ and for $F(X)=F_0{X}^{1\mathrm{/}2}-2\mathrm{\Lambda }$, where $F_0$ and $\mathrm{\Lambda }$ are constants. The first solution describes a black hole in an asymptotically singular space-time, the second one contains two horizons of infinite area connected by a wormhole. It is argued that spherically symmetric k-essence configurations with $n<1\mathrm{/}2$ are generically unstable because the perturbation equation is not of hyperbolic type.