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 J2 diverges at the horizon casting doubts on whether the solution is physically sound. We show that this is not an issue since J2 is not a scalar quantity, since Jμ is not a diffinvariant current in the presence of Gauss-Bonnet. The same theory can be written in Horndeski form with a non-analytic function G5∼ 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.
Highlights
Case of scalar hair in shift-symmetric theories
If a scalar field is, with the exception of a cosmological constant, the most plausible ingredient to be added to General Relativity to explain e.g. the accelerated expansion of the Universe, the presence of a shift symmetry represents the minimal choice to guarantee that such a field will be almost massless and relevant on cosmological scales
While the presence of a linear scalar Gauss-Bonnet coupling is a sufficient condition to guarantee that black holes have hair, the actual solution found in [2] seems puzzling
Summary
We want to understand whether the divergence of J2 at the horizon is a pathology of the sGB hairy solutions or not. Since loop corrections will induce λ = 0 even if we start with λ = 0 Another related way to see the pathology is that in general a particle will be coupled to the scalar. This means that one gets an effect on the dynamics of the particle (and on its stressenergy tensor) that diverges at the horizon This suggests that the solution is unstable when matter is included in the picture. The current JGμB satisfies ∇μJGμB = R2GB, but the form of the current is ambiguous and there is no privileged expression, even when a coordinate system is chosen [9] This statement is analogous to what happens in a (non-abelian) gauge theory for the term TrFμνFμν. In the appendix A we compute this current in the Schwarzschild spacetime and show that it is not covariant by writing it in different coordinate systems, in particular in Kruskal-Szekeres coordinates where the metric is regular at the horizon
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