Recently, no-go theorems for the existence of solitonic solutions in Einstein-Maxwellscalar (EMS) models have been established [1]. Here we discuss how these theorems can be circumvented by a specific class of non-minimal coupling functions between a real, canonical scalar field and the electromagnetic field. When the non-minimal coupling function diverges in a specific way near the location of a point charge, it regularises all physical quantities yielding an everywhere regular, localised lump of energy. Such solutions are possible even in flat spacetime Maxwell-scalar models, wherein the model is fully integrable in the spherical sector, and exact solutions can be obtained, yielding an explicit mechanism to de-singularise the Coulomb field. Considering their gravitational backreaction, the corresponding (numerical) EMS solitons provide a simple example of self-gravitating, localised energy lumps. ‡

Three non-existence results are established for self-gravitating solitons in Einstein-Maxwell-scalar models, wherein the scalar field is, generically, non-minimally coupled to the Maxwell field via a scalar function f (Φ). Firstly, a trivial Maxwell field is considered, which yields a consistent truncation of the full model. In this case, using a scaling (Derrick-type) argument, it is established that no stationary and axisymmetric self-gravitating scalar solitons exist, unless the scalar potential energy is somewhere negative in spacetime. This generalises previous results for the static and strictly stationary cases. Thus, rotation alone cannot support self-gravitating scalar solitons in this class of models. Secondly, constant sign couplings are considered. Generalising a previous argument by Heusler for electro-vacuum, it is established that no static selfgravitating electromagnetic-scalar solitons exist. Thus, a varying (but constant sign) electric permittivity alone cannot support static Einstein-Maxwell-scalar solitons. Finally, the second result is generalised for strictly stationary, but not necessarily static, spacetimes, using a Lichnerowicz-type argument, generalising previous results in models where the scalar and Maxwell fields are not directly coupled. The scope of validity of each of these results points out the possible paths to circumvent them, in order to obtain self-gravitating solitons in Einstein-Maxwell-scalar models.

In the simplest scalar-tensor theories, wherein the scalar field is non-minimally coupled to the Ricci scalar, spontaneous scalarization of electrovacuum black holes (BHs) does not occur. This ceases to be true in higher dimensional spacetimes, d > 4. We consider the scalarization of the higher dimensional Reissner-Nordström BHs in scalar-tensor models and provide results on the zero modes for different d, together with an explicit construction of the scalarized BHs in d = 5, discussing some of their properties. We also observe that a conformal transformation into the Einstein frame maps this model into an Einstein-Maxwel- scalar model, wherein the non-minimal coupling occurs between the scalar field and the Maxwell invariant (rather than the Ricci scalar), thus relating the occurence of scalarization in the two models. Next, we consider the spontaneous scalarization of the Schwarzschild- Tangherlini BH in extended-scalar-tensor-Lovelock gravity in even dimensions. In these models, the scalar field is non-minimally coupled to the (d/2)th Euler density, in d spacetime dimensions. We construct explicitly examples in d = 6, 8, showing the properties of the four dimensional case are qualitatively generic, but with quantitative differences. We compare these higher d scalarized BHs with the hairy BHs in shift-symmetric Horndeski theory, for the same d, which we also construct.