On the absence of scalar hair for charged rotating black holes in non-minimally coupled theories S SEN and N BANERJEE

Recently there has been a considerable resurgence in the no scalar hair theorem for black holes. Investigations regarding no hair theorem, however, had started about thirty years ago [1]. Inspired by Israel’s uniqueness theorem for Schwarzschild and Reissner–Nordstrom black holes [2] and Carter [3] and Wald’s [4] uniqueness theorem for Kerr black holes, Wheeler anticipated that gravitational collapse leads to black holes endowed with mass, charge and angular momentum and no other free parameters, which he summarized as ‘black holes have no hair’. The ‘no scalar hair theorem’ excludes the availability of any knowledge of a scalar field from the exterior geometry of a black hole even when a scalar field is present in the space-time along with gravity. The search for such scalar hair were initiated long back. Investigations, involving physical fields like massless scalar [5], massive vector [6], spinor [7] fields go in favour of Wheeler’s dictum as any information about these fields from a stationary [6] black hole exterior is excluded. These investigations were mainly limited to the cases where the scalar field is only minimally coupled to gravity. But in the early 90’s, solutions for stationary black holes with exterior non-abelian gauge field or Skyrmion field [8–10] have put strong challenge in front of the conjecture. Black hole solutions with new hair like Yang–Mills hair [8], Skyrme hair [9], dilaton hair [11] or others [12] act as counter examples to the conjecture. With a few exceptions [9] many of these black holes are unstable [13]. It is interesting to note that all the hair are not of similar stature [14]. The hair which act as new quantum numbers, i.e, independent of other quantum numbers are primary hair. Skyrme