We examined analytically and proposed a numerical model of an Abelian Higgsdark matter vortex in the spacetime of a stationary axisymmetric Kerr blackhole. In analytical calculations the dark matter sector was modeled by anaddition of a U (1)-gauge field coupled to the visible sector. The backreactionanalysis revealed that the impact of the dark vortex presence is far morecomplicated than causing only a deficit angle. The vortex causes an ergosphereshift and the event horizon velocity is also influenced by its presence. Thesephenomena are more significant than in the case of a visible vortex sector. Thearea of the event horizon of a black hole is diminished and this decline islarger in comparison to the Kerr black hole with an Abelian Higgs vortex case.After analyzing the gravitational properties for the general setup, we focusedon the subset of models that are motivated by particle physics. We retained theAbelian Higgs model as a description of the dark matter sector (this sectorcontained a heavy dark photon and an additional complex scalar) and added areal scalar representing the real component of the Higgs doublet in the unitarygauge, as well as an additional U (1)-gauge field representing an ordinaryelectromagnetic field. Moreover, we considered two coupling channels betweenthe visible and dark sectors, which were the kinetic mixing between the gaugefields and a quartic coupling between the scalar fields. After solving theequations of motion for the matter fields numerically we analyzed properties ofthe cosmic string in the dark matter sector and its influence on the visiblesector fields that are directly coupled to it. We found out that the presenceof the cosmic string induced spatial variation in the vacuum expectation valueof the Higgs field and a nonzero electromagnetic field around the black hole.
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