Asymptotically flat spinning scalar, Dirac and Proca stars

Einstein's gravity minimally coupled to free, massive, classical fundamental fields admits particle-like solutions. These are asymptotically flat, everywhere non-singular configurations that realise Wheeler's concept of a geon: a localised lump of self-gravitating energy whose existence is anchored on the non-linearities of general relativity, trivialising in the flat spacetime limit. In arXiv:1708.05674 the key properties for the existence of these solutions (also referred to as stars or self-gravitating solitons) were discussed - which include a harmonic time dependence in the matter field -, and a comparative analysis of the stars arising in the Einstein-Klein-Gordon, Einstein-Dirac and Einstein-Proca models was performed, for the particular case of static, spherically symmetric spacetimes. In the present work we generalise this analysis for spinning solutions. In particular, the spinning Einstein-Dirac stars are reported here for the first time. Our analysis shows that the high degree of universality observed in the spherical case remains when angular momentum is allowed. Thus, as classical field theory solutions, these self-gravitating solitons are rather insensitive to the fundamental fermionic or bosonic nature of the corresponding field, displaying similar features. We describe some physical properties and, in particular, we observe that the angular momentum of the spinning stars satisfies the quantisation condition $J=m N,$ for all models, where $N$ is the particle number and $m$ is an integer for the bosonic fields and a half-integer for the Dirac field. The way in which this quantisation condition arises, however, is more subtle for the non-zero spin fields.