Electrically charged dilatonic black rings

In this note we present (electrically) charged dilatonic black ring solutions of the EinsteinMaxwell-dilaton theory in five dimensions and we consider their physical properties. These solutions are static and as in the neutral case possess a conical singularity. We show how one may remove the conical singularity by application of a Harrison transformation, which physically corresponds to supporting the charged ring with an electric field. Finally, we discuss the slowly rotating case for arbitrary dilaton coupling. One of the most exciting recent results in higher dimensional gravity was Emparan and Reall’s discovery of the black ring in [1, 2]. Their solution describes the gravitational field of an isolated source equipped with an event horizon of S × S topology. Although the static black ring is plagued by a conical singularity, the authors were subsequently able to show the existence of a vacuum rotating black ring [2]. Since then, more solutions have been found in various five dimensional gravity theories. Elvang was able to apply a Hassan-Sen transformation to the [email protected] [email protected] solution [2] to find a charged black ring in the bosonic sector of the truncated heterotic string theory [3]. Furthermore, a supersymmetric black ring in five dimensional minimal supergravity was derived [4] and then generalized to the case of concentric rings in [5]. Finally, these solutions were placed into the context of string and M theory, corresponding to D1-D5-P supertube configurations [6]. Despite the number of interesting solutions, it is rather surprising that an electrically charged rotating black ring in the well-known Einstein-Maxwell (EM) theory remains to be found. In [7] a static black ring carrying electric charge was given, however their expression for the field strength is incorrect. In [8] Emparan was able to derive “dipole black rings” in the EinsteinMaxwell-dilaton (EMD) theory in five dimensions. Such solutions are supported by magnetic potential (though they are electrically charged by the Kalb-Ramond two form potential in the dual theory). In this paper we consider electrically charged black rings in the EMD theory. Our motivation is twofold: firstly, such solutions are interesting in their own right. Secondly, it has been shown that in a particular degenerate limit (rotating) black rings turn into (rotating) black holes. In particular the rotating black ring of pure 5D gravity reduces in this limit to the Myers-Perry black hole [9] with one non-vanishing angular momentum. As yet to the best of our knowledge there is no general Maxwell-charged Myers-Perry solution found in the literature; such a solution would be the higher dimensional counterpart of the Kerr-Newman solution in four dimensions‡. In this note we begin this line of attack by presenting a static charged dilatonic black ring valid for all values of the dilaton coupling. This paper is structured as follows. Firstly, we present our solutions and show that they may be interpreted as static, charged, dilatonic black rings. We discuss their physical properties and show they do not possess a sensible extremal limit. As in the neutral case, these static black rings suffer from the existence of a conical singularity. Next, we consider the possibility of removing the conical singularity and apply certain solution generating techniques in order to find a new static electrically charged black ring immersed in a background electric field. Finally, we conclude with a discussion on the possibility of finding rotating (charged electrically under the Maxwell field) dilatonic black rings. We consider the EMD system, in 4 + 1 dimensions defined by the following action: