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We consider the $\mathrm{O}(3)$ non-linear sigma-model, composed of threereal scalar fields with a standard kinetic term and with a symmetry breakingpotential in four spacetime dimensions. We show that this simple, geometricallymotivated model, admits both self-gravitating, asymptotically flat,non-topological solitons and hairy black holes, when minimally coupled toEinstein's gravity, $without$ the need to introduce higher order kinetic termsin the scalar fields action. Both spherically symmetric and spinning, axiallysymmetric solutions are studied. The solutions are obtained under a ansatz withoscillation (in the static case) or rotation (in the spinning case) in theinternal space. Thus, there is symmetry non-inheritance: the matter sector isnot invariant under the individual spacetime isometries. For the hairy blackholes, which are necessarily spinning, the internal rotation (isorotation) mustbe synchronous with the rotational angular velocity of the event horizon. Weexplore the domain of existence of the solutions and some of their physicalproperties, that resemble closely those of (mini) boson stars and Kerr blackholes with synchronised scalar hair in Einstein-(massive, complex)-Klein-Gordontheory.

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