Abstract
We consider the 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 space-time 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.
Highlights
Of the scalar fields, as first realized in [4, 5]
We consider the 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
We explore the domain of existence of the solutions and some of their physical properties, that resemble closely those of boson stars and Kerr black holes with synchronised scalar hair in Einstein-(massive, complex)-Klein-Gordon theory
Summary
We consider the non-linear O(3) sigma model minimally coupled to Einstein’s gravity in 3+1 dimensions. Observe that by appropriately rescaling the coordinates xμ and G one can effectively set λ0 = λ1 = 1, leaving only one non-trivial parameter, α, where we define α2 ≡ 4πGλ1. Variation of (2.1) with respect to scalar field itself leads to the following field equations:. Which are solved together with the constraint equation (2.3). Observe that the potential term in (2.2) breaks the O(3) symmetry of the model to the SO(2) subgroup. Associated to the latter, a Noether current jμ exists, yielding a conserved Noether charge Q, given by jμ = −φ1∂μφ2 + φ2∂μφ , Q=
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