We study the properties of possible static, spherically symmetric configurations in k-essence theories with the Lagrangian functions of the form F (X) , X ≡ φ ,α φ ,α . A no-go theorem has been proved, claiming that a possible black-hole-like Killing horizon of finite radius cannot exist if the function F (X) is required to have a finite derivative dF/dX . Two exact solutions are obtained for special cases of k-essence: one for F (X) = F 0 X 1/3 , another for F (X) = F 0 |X| 1/2 − 2Λ , where F 0 and Λ are constants. Both solutions contain horizons, are not asymptotically flat, and provide illustrations for the obtained no-go theorem. The first solution may be interpreted as describing a black hole in an asymptotically singular space-time, while in the second solution two horizons of infinite area are connected by a wormhole.

The k-essence theory with a power-law function of (∂φ) 2 and Rastall's non-conservative theory of gravity with a scalar field are shown to have the same solutions for the metric under the assumption that both the metric and the scalar fields depend on a single coordinate. This equivalence (called k-R duality) holds for static configurations with various symmetries (spherical, plane, cylindrical, etc.) and all homogeneous cosmologies. In the presence of matter, Rastall's theory requires additional assumptions on how the stressenergy tensor non-conservation is distributed between different contributions. Two versions of such nonconservation are considered in the case of isotropic spatially flat cosmological models with a perfect fluid: one (R1) in which there is no coupling between the scalar field and the fluid, and another (R2) in which the fluid separately obeys the usual conservation law. In version R1 it is shown that k-R duality holds not only for the cosmological models themselves but also for their adiabatic perturbations. In version R2, among other results, a particular model is singled out that reproduces the same cosmological expansion history as the standard Λ CDM model but predicts different behaviors of small fluctuations in the k-essence and Rastall frameworks. 1

We study the stability properties of static, spherically symmetric configurations in k-essence theories with the Lagrangians of the form F (X), X ≡ φ ,α φ ,α . The instability under spherically symmetric perturbations is proved for two recently obtained exact solutions for F (X) = F 0 X 1/3 and for F (X) = F 0 X 1/2 −2Λ , where F 0 and Λ 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 < 1/2 are generically unstable because the perturbation equation is not of hyperbolic type.

In this paper we investigate three theories characterised by non-vanishing divergence of the stress-energy tensor, namely f (R, LM ), f (R, T ), and Rastall theory. We show that it is not possible to obtain the third from the first two, unless in some very specific case. Nonetheless, we show that in the framework of cosmology in the f (R, T ) theory, a result similar to that found in the Rastall one is reproduced, namely that the dynamics of the ΛCDM model of standard cosmology can be exactly mimicked, even though the dark energy component is able to cluster.

We review some properties of black hole structures appearing in gravity with a massless scalar field, with both minimal and nonminimal coupling. The main properties of the resulting cold black holes are described. The study of black holes in scalar-gravity systems is extended to k-essence theories, and some examples are explicitly worked out. In these cases, even while the existence of horizons is possible, the metric regularity requirement on the horizon implies either a cold black type structure or a singular behavior of the scalar field. * kb20@yandex.ru †