Geometrothermodynamics of neutral black ring: large and small

Abstract

In this paper we study geometry of black ring in two cases of small or large. By using a Legendre invariant transformation we show that how we are able to recover the phase transitions between black hole to black ring in a self consistent geometrical way. Beyond the stability region of black ring we show that many singular pouts exist as the poles of Ricci scalar of the correspondent metric. It indicates that the phase transition from black hole to black ring is not unique and can be copied several times if we investigate the geometry of the mix system beyond the stability interval.