Aspects of Gauss-Bonnet Scalarisation of Charged Black Holes

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-Nordstrom 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.