Slow-rotating black holes with potential in dynamical Chern-Simons modified gravitational theory

Abstract

The Chern-Simons amended gravity theory appears as a low-energy effective theory of string theory. The effective theory includes an anomaly-cancelation correction to the Einstein-Hilbert action. The Chern-Simons expression consists of the product φRR̃ of the Pontryagin density RR̃ with a scalar field φ, where the latter is considered a background field (dynamical construction or non-dynamical construction). Many different solutions to Einstein's general relativity continue to be valid in the amended theories. The Kerr metric is, however, considered an exceptional case that raised a search for rotating black hole solutions. We generalize the solution presented in Phys. Rev. D 77 (2008) 064007 by allowing the potential V to have a non-vanishing value, and we discuss three different cases of the potential, that is, V = const., V ∝ φ, and V ∝ φ 2 cases. This study presents, for the first time, novel solutions prescribing rotating black holes in the frame of the dynamical formulation of the Chern-Simons gravity, where we include a potential and generalize the previously derived solutions. We derive solutions in the slow-rotation limit, where we write the parameter of the slow-rotation expansion by ε. These solutions are axisymmetric and stationary, and they make a distortion of the Kerr solution by a dipole scalar field. Furthermore, we investigate that the correction to the metric behaves in the inverse of the fourth order of radial distance from the center of the black hole as V ∝ φ. This suggests that any meaningful limits from the weak-field experiments could be passed. We show that the energy conditions associated with the scalar field of the case V ∝ φ are non-trivial and have non-trivial values to the leading order. These non-trivial values come mainly from the contribution of the potential. Finally, we derived the stability condition using the geodesic deviations. We conclude this study by showing that other choices of the potential, i.e., V ∝ φn , where n > 2 are not allowed because all the solutions to these cases will be of order 𝒪(ε 2), which is not covered in this study.