Abstract
In this paper we construct new exact solutions in Einstein–Gauss–Bonnet and Lovelock gravity, describing asymptotically flat black strings. The solutions exist also under the inclusion of a cosmological term in the action, and are supported by scalar fields with finite energy density, which are linear along the extended direction and have kinetic terms constructed out from Lovelock tensors. The divergenceless nature of the Lovelock tensors in the kinetic terms ensures that the whole theory is second order. For spherically, hyperbolic and planar symmetric spacetimes on the string, we obtain an effective Wheeler’s polynomial which determines the lapse function up to an algebraic equation. For the sake of concreteness, we explicitly show the existence of a family of asymptotically flat black strings in six dimensions, as well as asymptotically hbox {AdS}_{5}times mathbb {R} black string solutions and compute the temperature, mass density and entropy density. We compute the latter by Wald’s formula and show that it receives a contribution from the non-minimal kinetic coupling of the matter part, shifting the one-quarter factor coming from the Einstein term, on top of the usual non areal contribution arising from the quadratic Gauss–Bonnet term. Finally, for a special value of the couplings of the theory in six dimensions, we construct strings that contain asymptotically AdS wormholes as well as rotating solutions on the transverse section. By including more scalars the strings can be extended to p-branes, in arbitrary dimensions.
Highlights
Black strings in general relativity (GR) in vacuum are easy to construct
The solutions exist under the inclusion of a cosmological term in the action, and are supported by scalar fields with finite energy density, which are linear along the extended direction and have kinetic terms constructed out from Lovelock tensors
The divergenceless nature of the Lovelock tensors in the kinetic terms ensures that the whole theory is second order
Summary
Black strings in GR in vacuum are easy to construct. they are obtained by a cylindrical oxidation of the Schwarzschild black hole in d dimensions by the inclusion of p extra flat coordinates. To successfully apply the procedure before described, we shall observe that the extended directions must be dressed with scalar fields of the type (2) that have non-minimal kinetic couplings, such that the extra higher curvature terms involved in Lovelock theories do not break the compatibility of the field equations. Such couplings must fulfil the following requirements: They must enjoy shift symmetry, which allows the inclusion of non-minimal kinetic couplings. The trace of the field equations on the d-dimensional manifold and the equation along the z direction respectively read
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