Abstract
The general relativity vacuum black holes (BHs) can be scalarised in models where a scalar field non-minimally couples to the Gauss-Bonnet (GB) invariant. Such GB scalarisation comes in two flavours, depending on the GB sign that triggers the phenomenon. Hereafter these two cases are termed GB± scalarisation. For vacuum BHs, only GB+ scalarisation is possible in the static case, while GB− scalarisation is spin induced. But for electrovacuum BHs, GB− is also charged induced. We discuss the GB− scalarisation of Reissner-Nordström and Kerr-Newman BHs, discussing zero modes and constructing fully non-linear solutions. Some comparisons with GB+ scalarisation are given. To assess the generality of the observed features, we also briefly consider the GB± scalarisation of stringy dilatonic BHs and coloured BHs which provide qualitative differences with respect to the electrovacuum case, namely on the distribution and existence of regions triggering GB− scalarisation.
Highlights
GB scalarisation circumvents well-known no-hair theorems due to a certain class of non-minimal couplings between a real scalar field φ and the GB invariant
Let us remark, that such GB charge-induced scalarisation is different from the scalarisation of charged black holes (BHs) introduced in [27], where the non-minimal coupling occurs between the scalar field and the Maxwell field, with the GB term being absent
We have verified that spinning scalarised solutions exist for both signs of, we shall focus on the results for the more novel case of spin/charge scalarisation, = −1
Summary
We wish to consider the Einstein-Maxwell-scalar-GB (EMsGB) model, described by the following action. Where R is the Ricci scalar with respect to the spacetime metric gμν, R2GB is the GB invariant. Rαβμν is the Riemann tensor, Rαβ is the Ricci tensor, Fμν = ∂μ Aν − ∂ν Aμ is the Maxwell field strength tensor where A = Aμdxμ is the U(1) gauge potential, f (φ) is a coupling function of the real scalar field φ, λ is a constant of the theory with dimension of length and = ±1 is chosen for GB scalarisation. Varying the action (1) with respect to the metric tensor gμν, we obtain the Einstein equations, Rμν. The effective energy-momentum tensor Tμ(eνff) has three distinct components: Tμ(eνff) = Tμ(sν) + Tμ(Mν ) + Tμ(Gν B) ,
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