Static and rotating black strings in dynamical Chern–Simons modified gravity

Four-dimensional homogeneous static and rotating black strings in dynamical Chern–Simons modified gravity, with and without torsion, are presented. Each solution is supported by a scalar field that depends linearly on the coordinate that span the string. The solutions are locally AdS ₃ × R and they represent the continuation of the Bañados–Teitelboim–Zanelli black hole. Moreover, they belong to the so-called Chern–Simons sector of the space of solutions of the theory, since the Cotton tensor contributes nontrivially to the field equations. The case with nonvanishing torsion is studied within the first-order formalism of gravity, and it considers nonminimal couplings of the scalar fields to three topological invariants: Nieh–Yan, Pontryagin and Gauss–Bonnet terms, which are studied separately. These nonminimal couplings generate torsion in vacuum, in contrast to Einstein–Cartan theory. In all cases, torsion contributes to an effective cosmological constant that, in particular cases, can be set to zero by a proper choice of the parameters.