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

Carlos A.R Herdeiro,Alexandre M Pombo,Eugen Radu

doi:10.1016/j.physletb.2017.09.036

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, emphasizing the mathematical similarities and clarifying the physical differences, particularly between the bosonic and fermionic 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.

Highlights

Introduction and overviewIn 1955 [1], John Wheeler investigated the existence, within general relativity (GR) coupled to classical electromagnetism, of “classical, singularity free, exemplar of the “bodies” of classical physics”
The main purpose of this work was to provide a comparative analysis of three different types of solitonic solutions of GR-matter systems, which can be interpreted as explicit realizations of Wheeler’s geon concept for matter fields of spin 0, 1 and 1/2, respectively
As classical field theory solutions, our results show that the existence of these self-gravitating, stable, energy lumps, composed of standing waves, does not distinguish between the fermionic/bosonic nature of the field, possessing a variety of similar features

Summary

Introduction and overview

In 1955 [1], John Wheeler investigated the existence, within general relativity (GR) coupled to classical electromagnetism, of “classical, singularity free, exemplar of the “bodies” of classical physics”. As classical GR solutions, all these three cases are qualitatively similar: in their domains of existence – Fig. 2; in their maximal mass, always of the form (1.1), with α1/2 0.709 and α1 1.058; and in the existence of a conserved Noether charge, always associated to a global U (1) symmetry and providing a measure of the particle number N in the single frequency state. This last point, unveils a sharp distinction between the bosonic and fermionic cases.

The three models

Units and scaling symmetries

The boundary conditions and numerical method

Numerical results: domain of existence and some properties

Conclusions and remarks