Instability of some k -essence spacetimes

We study the stability properties of static, spherically symmetric configurations in k-essence theories with the Lagrangians of the form F(X), X φ,αφ,α. The instability under spherically symmetric perturbations is proved for the two recently obtained exact solutions for F(X) = F0X1/3 and for F(X) = F0X1/2 - 2Λ, where F0 and Λare constants. The first solution describes a black hole in an asymptotically singular spacetime, the second one contains two horizons of infinite area connected by a wormhole. It is argued that spherically symmetric k-essence configurations with n < 1/2 are generically unstable because the perturbation equation is not of hyperbolic type. © 2020 World Scientific Publishing Company.