Hairy black-holes in shift-symmetric theories

Abstract

Scalar hair of black holes in theories with a shift symmetry are constrained by the no-hair theorem of Hui and Nicolis, assuming spherical symmetry, time-independence of the scalar field and asymptotic flatness. The most studied counterexample is a linear coupling of the scalar with the Gauss-Bonnet invariant. However, in this case the norm of the shift-symmetry current J 2 diverges at the horizon casting doubts on whether the solution is physically sound. We show that this is not an issue since J 2 is not a scalar quantity, since J µ is not a diff-invariant current in the presence of Gauss-Bonnet. The same theory can be written in Horndeski form with a non-analytic function G 5 ∼ log X. In this case the shift-symmetry current is diff-invariant, but contains powers of X in the denominator, so that its divergence at the horizon is again immaterial. We confirm that other hairy solutions in the presence of non-analytic Horndeski functions are pathological, featuring divergences of physical quantities as soon as one departs from time-independence and spherical symmetry. We generalise the no-hair theorem to Beyond Horndeski and DHOST theories, showing that the coupling with Gauss-Bonnet is necessary to have hair.

Effective Field Theory for the perturbations of a slowly rotating black hole

“…The scalar-Gauss-Bonnet model (and its variations) is a popular example. It circumvents earlier no-scalar-hair theorems [4][5][6][7] and gives interesting predictions for the background geometry and its perturbations, see e.g. [8][9][10][11][12][13].…”

“…9 Although possible in general, this procedure does not guarantee that the resulting effective 6 We stress that we are first taking the large-limit in (2.34) and then expanding the result at linear order in . 7 The fact that the equations become identical is a result of the eikonal limit (the same happens for instance with the Regge-Wheeler and Zerilli equations in general relativity). Note that this is a sufficient condition for isospectrality, but it is not necessary.…”

“…In particular, the coefficients of the ω 2 terms and the potentials equal (2.32) and (2.33), respectively. 7 In this sense, one recovers 'isospectrality' between the odd spectrum and one of the two even set of frequencies. 8 Given the action (2.28) with the coefficients (2.35), and given some educated guess on the form of the scalar profile Φ(r), one can in principle, by reverse engineering, find explicit examples of covariant Lagrangians that reduce to (2.28) in unitary gauge.…”

“…In fact, ref. [7] showed that in the context of scalar-tensor theories with no ghost degrees of freedom the no-hair theorem of Hui and Nicolis [4] can be evaded only in the presence of the operator ΦG, that is a linear coupling of the scalar to the…”

“…When considering the EFT in the narrower context of hairy black holes in shift-symmetric scalar-tensor theories, a scalar profile with Φ0 = 0 and Φ1 = 0 requires the presence of a linear coupling of the scalar field to both the Gauss-Bonnet invariant and the Pontryagin density[7].…”