Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars
Abstract
:1. Introduction and Motivation
1.1. General Remarks
1.2. Conventions
- spin 0:is a complex scalar field, which is equivalent to a model with two real scalar fields, , via .
- spin 1/2:are massive spinors, with four complex components, the index corresponding to the number of copies of the Lagrangian. For a spherically symmetric configuration, one should consider (at least) two spinors (), with equal mass . A model with a single spinor necessarily possesses a nonzero angular momentum density and it cannot be spherically symmetric.
- spin 1:is a complex 4-potential, with the field strength . Again, the model can be described in terms of two real vector fields, .
2. The General Framework
2.1. The Action and Field Equations
2.2. The Ansätze and Explicit Equations
2.2.1. The Metric and Matter Fields
2.2.2. The Explicit Equations
2.3. Units and Scaling Symmetries
3. The Probe Limit: Flat Spacetime Solutions
3.1. Deser’s Argument and Virial-Type Identities
- scalar field:
- Dirac field:
- Proca field:
3.2. Numerical Results
3.2.1. General Remarks
3.2.2. Solutions with a Sextic Self-Interaction Term,
3.2.3. Solutions without a Sextic Self-Interaction Term,
4. Including the Gravity Effects
4.1. The Boundary Conditions
4.2. Virial Identities
- scalar field:
- Dirac field:
- Proca field:
4.3. General Features
5. Other Aspects
5.1. No Hair Results
- Scalar case
- Dirac case
- Proca caseA no hair theorem has been proven in [30] for a massive, non-selfinterating Proca field. In Appendix B, we generalize it for an arbitrary Proca potential .
5.2. The Issue of Particle Numbers: Bosons vs. Fermions
6. Further Remarks. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. The Dirac Field: Conventions
Appendix B. Self-Interacting Proca Field: A No Hair Result
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Herdeiro, C.A.R.; Radu, E. Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars. Symmetry 2020, 12, 2032. https://doi.org/10.3390/sym12122032
Herdeiro CAR, Radu E. Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars. Symmetry. 2020; 12(12):2032. https://doi.org/10.3390/sym12122032
Chicago/Turabian StyleHerdeiro, Carlos A. R., and Eugen Radu. 2020. "Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars" Symmetry 12, no. 12: 2032. https://doi.org/10.3390/sym12122032