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We establish that massive complex Abelian vector fields (mass $\mu$) can formgravitating solitons, when minimally coupled to Einstein's gravity. Such Procastars (PSs) have a stationary, everywhere regular and asymptotically flatgeometry. The Proca field, however, possesses a harmonic time dependence(frequency $w$), realizing Wheeler's concept of geons for an Abelian spin 1field. We obtain PSs with both a spherically symmetric (static) and an axiallysymmetric (stationary) line element. The latter form a countable number offamilies labelled by an integer $m\in \mathbb{Z}^+$. PSs, like (scalar) bosonstars, carry a conserved Noether charge, and are akin to the latter in manyways. In particular, both types of stars exist for a limited range offrequencies and there is a maximal ADM mass, $M_{max}$, attained for anintermediate frequency. For spherically symmetric PSs (rotating PSs with$m=1,2,3$), $M_{max}\simeq 1.058 M_{Pl}^2/\mu$ ($M_{max}\simeq 1.568,\, 2.337,\, 3.247 \, M_{Pl}^2/\mu$), slightly larger values than those for (mini-)bosonstars. We establish perturbative stability for a subset of solutions in thespherical case and anticipate a similar conclusion for fundamental modes in therotating case. The discovery of PSs opens many avenues of research,reconsidering five decades of work on (scalar) boson stars, in particular aspossible dark matter candidates.

Proca Stars: gravitating Bose-Einstein condensates of massive spin 1 particles