Asymptotically flat scalar, Dirac and Proca stars: Discrete vs. continuous families of solutions

The existence of localized, approximately stationary, lumps of the classical gravitational and electromagnetic field - geons - was conjectured more than half a century ago. If one insists on exactstationarity, topologically trivial configurations in electro-vacuum are ruled out by no-go theorems for solitons. But stationary, asymptotically flat geons found a realization in scalar-vacuum, where everywhere non-singular, localized field lumps exist, known as (scalar) boson stars. Similar geons have subsequently been found in Einstein-Dirac theory and, more recently, in Einstein-Proca theory. We identify the common conditions that allow these solutions, which may also exist for other spin fields. Moreover, we present a comparison of spherically symmetric geons for the spin 0, 1/2 and 1, emphasizingthe mathematical similarities and clarifying the physical differences, particularly between the bosonic and fermioniccases. We clarify that for the fermionic case, Pauli's exclusion principle prevents a continuous family of solutions for a fixed field mass; rather only a discrete set exists, in contrast with the bosonic case. (C) 2017 The Authors. Published by Elsevier B.V.