Massive vector fields can form spatially localized, non-relativistic, stationary field configurations supported by gravitational interactions. The ground state configurations (p-solitons/vector solitons/dark photon stars/polarized Proca stars) have a time-dependent vector field pointing in the same spatial direction throughout the configuration at any instant of time, can carry macroscopic amounts of spin angular momentum, and are spherically symmetric and monotonic in the energy density. In this paper, we include general relativistic effects, and numerically investigate the stability of compact polarized Proca stars (linear and circularly polarized) and compare them to hedgehog-like field configurations (with radially pointing field directions). Starting with approximate field profiles of such stars, we evolve the system numerically using 3+1 dimensional numerical simulations in general relativity. We find that these initial conditions lead to stable configurations. However, at sufficiently large initial compactness, they can collapse to black holes. We find that the initial compactness that leads to black hole formation is higher for circularly polarized stars (which carry macroscopic spin angular momentum), compared to linearly polarized ones, which in turn is higher than that for hedgehog configurations.

10 pages, 6 figures