Spin Polarization of Proca Stars Formed by Gravitational Bose--Einstein Condensation

Abstract

We study the internal spin polarization of Proca stars formed by gravitational Bose--Einstein condensation of a three-component nonrelativistic vector field. In idealized periodic-box simulations, we decompose the aperture-averaged spin into a coherent net fraction, a local polarization fraction, and their ratio, thereby distinguishing genuine coherent core polarization from local spin density whose direction cancels inside the aperture. For independent vector components, condensation produces Proca stars that are sizably but not maximally polarized. Across an independent-component simulation ensemble, the coherent core-spin fraction has mean $\langleχ_{\rm net}\rangle\simeq0.62$, with substantial realization-to-realization scatter. We interpret this scatter as the outcome of random elliptical polarization of the dominant component-space mode, rather than as evidence for a universal Proca-star spin fraction. This interpretation is supported by the core polarization matrix: its leading eigenvector provides an estimate of the ideal single-mode spin fraction, while the difference between this estimate and the directly integrated coherent spin tracks the departure of the core from a rank-one component-space state. The measured leading-eigenvector spin fractions are broadly compatible with an isotropic random-complex-vector reference and less compatible with an equal-amplitude random-phase reference. Correlated and circular initial data drive the dominant component-space mode toward the circular-polarization bound, giving the ordering independent $\rightarrow$ correlated $\rightarrow$ circular. These results show that internal polarization is a genuine vector degree of freedom of gravitationally condensed nonrelativistic Proca stars, and that the resulting core spin is controlled by the polarization of the dominant condensed mode rather than by a fixed universal value.