Eur. Phys. J. C (2011) 71: 1791
https://doi.org/10.1140/epjc/s10052-011-1791-2

Regular Article - Theoretical Physics

On the stability of scalar-vacuum space-times

¹ Center for Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya 46, Moscow, 119361, Russia
² Institute of Gravitation and Cosmology, PFUR, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia
³ Departamento de Física, Universidade Federal do Espírito Santo, Avenida Fernando Ferrari 514, 29075-910, Vitória, ES, Brazil
⁴ Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Rua Santa Adélia, 166, 09210-170, Santo André, SP, Brazil

^* e-mail: kb20@yandex.ru

Received: 3 October 2011
Revised: 19 October 2011
Published online: 8 November 2011

Abstract

We study the stability of static, spherically symmetric solutions to the Einstein equations with a scalar field as the source. We describe a general methodology of studying small radial perturbations of scalar-vacuum configurations with arbitrary potentials V(ϕ), and in particular space-times with throats (including wormholes), which are possible if the scalar is phantom. At such a throat, the effective potential for perturbations V _eff has a positive pole (a potential wall) that prevents a complete perturbation analysis. We show that, generically, (i) V _eff has precisely the form required for regularization by the known S-deformation method, and (ii) a solution with the regularized potential leads to regular scalar field and metric perturbations of the initial configuration. The well-known conformal mappings make these results also applicable to scalar-tensor and f(R) theories of gravity. As a particular example, we prove the instability of all static solutions with both normal and phantom scalars and V(ϕ)≡0 under spherical perturbations. We thus confirm the previous results on the unstable nature of anti-Fisher wormholes and Fisher’s singular solution and prove the instability of other branches of these solutions including the anti-Fisher “cold black holes.”