3.5 Rotating boson stars

Boson stars with rotation were not explored until the mid-1990s because of the lack of a strong astrophysical motivation and the technical problems with the regularization along the axis of symmetry. The first equilibrium solutions of rotating boson stars were obtained within the Newtonian gravity [203]. In order to generate axisymmetric time-independent solutions with angular momentum, one is naturally lead to the ansatz
Ο• (r,t) = Ο• (r,πœƒ )ei(ωt+kφ) , (67 ) 0
where Ο•0(r,πœƒ) is a real scalar representing the profile of the star, ω is a real constant denoting the angular frequency of the field and k must be an integer so that the field Ο• is not multivalued in the azimuthal coordinate φ. This integer is commonly known as the rotational quantum number.

General relativistic rotating boson stars were later found [193, 222] with the same ansatz of Eq. (67View Equation). To obtain stationary axially symmetric solutions, two symmetries were imposed on the spacetime described by two commuting Killing vector fields ξ = ∂t and η = ∂φ in a system of adapted (cylindrical) coordinates {t,r,πœƒ,φ }. In these coordinates, the metric is independent of t and φ and can be expressed in isotropic coordinates in the Lewis–Papapetrou form

[ ] l ( ) ( Ω )2 ds2 = − f dt2 +-- g dr2 + r2 dπœƒ2 + r2 sin2πœƒ dφ − --dt , (68 ) f r
where f,l,g and Ω are metric functions depending only on r and πœƒ. This means that we have to solve five coupled PDEs, four for the metric and one for the Klein–Gordon equation; these equations determine an elliptic quadratic eigenvalue problem in two spatial dimensions. Near the axis, the scalar field behaves as
lim Ο• (r,πœƒ ) = rk h (πœƒ) + O(rr+2) , (69 ) r→0 0 k
so that for k > 0 the field vanishes near the axis. Note that h k is some arbitrary function different for different values of k but no sum over k is implied in Eq. (69View Equation). This implies that the rotating star solutions have toroidal level surfaces instead of spheroidal ones as in the spherically symmetric case k = 0. In this case the metric coefficients are simplified, namely g = 1, Ω = 0 and f = f (r), l = l(r).

The entire family of solutions for k = 1 and part of k = 2 was computed using the self-consistent field method [222], obtaining a maximum mass 2 Mmax = 1.31 M Planckβˆ•m. Both families were completely computed in [141] using faster multigrid methods, although there were significant discrepancies in the maximum mass, which indicates a problem with the regularity condition on the z-axis. The mass M and angular momentum J for stationary asymptotically flat spacetimes can be obtained from their respective Komar expressions. They can be read off from the asymptotic expansion of the metric functions f and Ω

( ) ( ) 2GM 1 2J G 1 f = 1 − --r---+ O r2 , Ω = -r2--+ O r3 . (70 )
Alternatively, using the Tolman expressions for the angular momentum and the Noether charge relation in Eq. 15View Equation, one obtains an important quantization relation for the angular momentum [222]
J = kN , (71 )
for integer values of k. This remarkable quantization condition for this classical solution also plays a role in the work of [69Jump To The Next Citation Point] discussed in Section 6.3. Figure 7View Image shows the scalar field for two different rotating BSs.
View Image

Figure 7: The scalar field in cylindrical coordinates Ο•(ρ,z ) for two rotating boson-star solutions: (left) k = 1 and (right) k = 2. The two solutions have roughly comparable amplitudes in scalar field. Note the toroidal shape. Reprinted with permission from [141].

Recently, their stability properties were found to be similar to nonrotating stars [135Jump To The Next Citation Point]. Rotating boson stars have been shown to develop a strong ergoregion instability when rapidly spinning on short characteristic timescales (i.e., 0.1 seconds – 1 week for objects with mass M = 1– 106M βŠ™ and angular momentum J > 0.4 GM 2), indicating that very compact objects with large rotation are probably black holes [44].

More discussion concerning the numerical methods and limitations of some of these approaches can be found in Lai’s PhD thesis [141].

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