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

TL;DR

This paper compares asymptotically flat geon solutions across scalar, Dirac, and Proca fields, highlighting the mathematical similarities and physical differences, especially the discrete versus continuous solution families due to quantum principles.

Contribution

It identifies common conditions for geon solutions across different spins and clarifies the impact of quantum principles on solution families.

Findings

Scalar and bosonic geons form continuous families of solutions.

Fermionic geons are limited to discrete solutions due to Pauli exclusion.

Mathematical similarities exist across different spin fields.

Abstract

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 exact stationarity, 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 $1$, emphasising the mathematical similarities and clarifying the physical differences, particularly between the bosonic and fermonic cases. 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.