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

where and are metric functions depending only on and . 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 so that for the field vanishes near the axis. Note that is some arbitrary function different for different values of but no sum over is implied in Eq. (69). This implies that the rotating star solutions have toroidal level surfaces instead of spheroidal ones as in the spherically symmetric case . In this case the metric coefficients are simplified, namely , and , .The entire family of solutions for and part of was computed using the self-consistent field method [222], obtaining a maximum mass . 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 and angular momentum 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 and

Alternatively, using the Tolman expressions for the angular momentum and the Noether charge relation in Eq. 15, one obtains an important quantization relation for the angular momentum [222] for integer values of . This remarkable quantization condition for this classical solution also plays a role in the work of [69] discussed in Section 6.3. Figure 7 shows the scalar field for two different rotating BSs.Recently, their stability properties were found to be similar to nonrotating stars [135]. 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 and angular momentum ), 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].

Living Rev. Relativity 15, (2012), 6
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