Abstract
We construct rotating boson stars and Myers–Perry black holes with scalar hair (MPBHsSH) as fully non-linear solutions of five dimensional Einstein gravity minimally coupled to a complex, massive scalar field. The MPBHsSH are, in general, regular on and outside the horizon, asymptotically flat, and possess angular momentum in a single rotation plane. They are supported by rotation and have no static limit. Such hairy BHs may be thought of as bound states of boson stars and singly spinning, vacuum MPBHs and inherit properties of both these building blocks. When the horizon area shrinks to zero, the solutions reduce to (in a single plane) rotating boson stars; but the extremal limit also yields a zero area horizon, as for singly spinning MPBHs. Similarly to the case of equal angular momenta, and in contrast to Kerr black holes with scalar hair, singly spinning MPBHsSH are disconnected from the vacuum black holes, due to a mass gap. We observe that for the general case, with two unequal angular momenta, the equilibrium condition for the existence of MPBHsSH is w=m1Ω1+m2Ω2, where Ωi are the horizon angular velocities in the two independent rotation planes and w,mi, i=1,2, are the scalar field's frequency and azimuthal harmonic indices.
Highlights
Introduction and motivationApart from vacuum and electro-vacuum, scalar-vacuum is the simplest model that may be considered in Einstein gravity
In its simplest form, this theory corresponds to couple to gravity one or more real massless scalar fields with standard kinetic terms and without self-interactions
If some amount of scalar field falls into a black hole (BH), at least classically, no memory of it is expected to be found in the exterior spacetime
Summary
Apart from vacuum and electro-vacuum, scalar-vacuum is the simplest model that may be considered in Einstein gravity. Adding a mass term in a theory with two massive real scalar fields, or equivalently, with a single massive complex scalar field, new regular, asymptotically flat BH solutions exist, both in four spacetime dimensions (D = 4) – Kerr BHs with scalar hair [3,4,5] – and in D = 5 – Myers–Perry BHs with scalar hair (MPBHsSH) [6] (see the recent work [7] for a D = 4 generalization). The corresponding D = 5 Myers– Perry BHs are akin to the D = 4 Kerr solution; in particular they are both a two parameter family of solutions – characterized, say, by the ADM mass, M, and horizon angular velocity, H – and have a regular, finite area, extremal limit In both cases the hairy BHs, just as the boson stars, have (a) monochromatic scalar field(s) whose frequency w is fixed by H and (in D = 4) an azimuthal winding number.
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