Gravitating solitons and black holes with synchronised hair in the four dimensional O(3) sigma-model

We consider the $\mathrm{O}(3)$ non-linear sigma-model, composed of three real scalar fields with a standard kinetic term and with a symmetry breaking potential in four spacetime dimensions. We show that this simple, geometrically motivated model, admits both self-gravitating, asymptotically flat, non-topological solitons and hairy black holes, when minimally coupled to Einstein's gravity, $without$ the need to introduce higher order kinetic terms in the scalar fields action. Both spherically symmetric and spinning, axially symmetric solutions are studied. The solutions are obtained under a ansatz with oscillation (in the static case) or rotation (in the spinning case) in the internal space. Thus, there is symmetry non-inheritance: the matter sector is not invariant under the individual spacetime isometries. For the hairy black holes, which are necessarily spinning, the internal rotation (isorotation) must be synchronous with the rotational angular velocity of the event horizon. We explore the domain of existence of the solutions and some of their physical properties, that resemble closely those of (mini) boson stars and Kerr black holes with synchronised scalar hair in Einstein-(massive, complex)-Klein-Gordon theory.