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.
Thus, there is symmetry non-inheritance : the matter sector is not invariant under the individual spacetime isometries. The solutions are obtained under a ansatz with oscillation (in the static case) or rotation (in the spinning case) in the internal space. Both spherically symmetric and spinning, axially symmetric solutions are studied. 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 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. In interpreting these results one should bear in mind, however, that the swampland criteria are not quantitatively strict. We also consider hairy black holes interpolating between these boson stars and the Kerr solution and establish the part of the domain of existence where the swampland criteria are violated. By contrast, in the region where they may be faithful black hole mimickers, in the sense they possess a light ring, the criteria fail (are obeyed) for static (rotating) ultracompact boson stars, which should thus be part of the swampland (landscape). Furthermore, we find the universal behaviour that in the region wherein the boson stars are expected to be perturbatively stable, the compact objects may be part of the landscape. Analysing several models (static, rotating, with and without self-interactions), we find that, in this context, the criteria are not independent.
Here we consider them in the context of solitonic compact objects made up of scalar fields: boson stars. So far, these criteria have been applied to inflationary and dark energy models. The former are the vacua that emerge from string compactifications the latter are not obtained by any such compactification and hence may not survive in a UV completed theory of gravity. Recently, two simple criteria were proposed to assess if vacua emerging from an effective scalar field theory are part of the string “landscape” or “swampland”.
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