Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories

Abstract

We consider a general Einstein-scalar-Gauss-Bonnet theory with a coupling function fðϕÞ. We demonstrate that black-hole solutions appear as a generic feature of this theory since a regular horizon and an asymptotically flat solution may be easily constructed under mild assumptions for fðϕÞ. We show that the existing no-hair theorems are easily evaded, and a large number of regular black-hole solutions with scalar hair are then presented for a plethora of coupling functions fðϕÞ. DOI: 10.1103/PhysRevLett.120.131102 Introduction.-The existence or not of black holes associated with a nontrivial scalar field in the exterior region has attracted the attention of researchers over a period of many decades. Early on, a no-hair theorem [1] appeared that excluded static black holes with a scalar field, but this was soon outdated by the discovery of black holes with Yang-Mills [2] or Skyrme fields [3]. The emergence of additional solutions where the scalar field had a conformal coupling to gravity [4] led to the formulation of a novel nohair theorem [5] (for a review, see [6]). Recently, this argument was extended to the case of standard scalar-tensor theories [7], and a new form was proposed that covers the case of Galileon fields [8].However, both novel forms of the no-hair theorem [5,8] were shown to be evaded: the former in the context of the Einstein-dilaton-Gauss-Bonnet theory [9] and the latter in a special case of shift-symmetric Galileon theories [10][11][12]. A common feature of the above theories was the presence of the quadratic Gauss-Bonnet (GB) term defined as R 2 GB ¼ R μνρσ R μνρσ − 4R μν R μν þ R 2 , in terms of the Riemann tensor R μνρσ , the Ricci tensor R μν , and the Ricci scalar R. In both cases, basic requirements of the no-hair theorems were invalidated, and this paved the way for the construction of the counterexamples.Here, we consider a general class of scalar-GB theories, of which the cases [9,11] constitute particular examples. We demonstrate that black-hole solutions, with a regular horizon and an asymptotically flat limit, may in fact be constructed for a large class of such theories under mild only constraints on the coupling function fðϕÞ between the scalar field and the GB term. We address the requirements of both the old and novel no-hair theorems, and we show

Large and ultracompact Gauss-Bonnet black holes with a self-interacting scalar field

“…for all forms of the coupling function f (φ), with D being the scalar charge, as found in [39]. For Λ < 0 and V (φ) = 1, black-hole solutions with an asymptotically Anti-de Sitter behaviour are expected to emerge as in [60,160].…”

“…To this end, we assume that, as r → r h , the metric function e A(r) should vanish (and e B(r) should diverge) whereas the scalar field must remain finite. As was explicitly shown in previous constructions [39,160], this amounts to working in the limit A (r) → ∞ while keeping φ (r) and φ (r) finite as the black-hole horizon is approached. Working in these limits, Eq.…”

“…Our numerical code has successfully reproduced the families of asymptotically-flat scalarised black-hole solutions derived in [39] as well as the asymptotically AdS solutions found in [160]. In the first part of the present analysis, we have chosen a specific form for the coupling function, namely the exponential (dilatonic) form f (φ) = αe φ , and supplemented it with different forms of the scalar-field potential V (φ).…”

“…Here, the indicative case of the quartic potential, V (φ) = φ 4 , is presented, however, the observed profile remains unaffected as the form of V (φ) varies. We may clearly see the characteristic behaviour caused by the presence of the GB in the theory: the radial pressure p = T r r is positive near the horizon while the energy density ρ = −T t t is negative -both these features cause the evasion of the scalar no-hair theorems [13,39] and the emergence of regular black-hole solutions with scalar hair. The depicted profiles of T µ ν also reveal the asymptotic flatness of spacetime as all components assume a zero value away from the black-hole horizon.…”

“…These solutions have the characteristic feature of scalar hair: a regular, non-trivial scalar field which is associated with the black hole, a feature forbidden by GR. The group of theories which may lead to such solutions was significantly expanded a few years ago, when it was demonstrated [39], both analytically and numerically, that the EsGB theories support black-hole solutions with a regular, non-trivial scalar hair for every form of the coupling function. In addition to this natural scalarisation of the solutions, spontaneous scalarisation of the corresponding GR solution was also shown to take place [40,41].…”