Frozen Bardeen-Dirac stars and light ball

TL;DR

This paper introduces frozen Bardeen-Dirac stars, solutions combining magnetic charge and Dirac fields, featuring a light ball inside a critical horizon and distinct from traditional black holes.

Contribution

It presents the novel concept of frozen Bardeen-Dirac stars, solutions with no black hole horizon but with unique light ring and light ball structures, extending black hole physics.

Findings

Existence of frozen Bardeen-Dirac stars with no horizon

Presence of a light ball inside the critical horizon

Critical magnetic charge values determine solution types

Abstract

In this paper, we study solutions of a static spherically symmetric system, which is composed of the coupling with the Bardeen action and two Dirac fields. For the case where only the Bardeen action is present, the magnetic charge $q$ can be infinite, then when the magnetic charge is greater than a certain value $q_b$, there exists a black hole solution, which is called the Bardeen black hole (BBH). However, if the Dirac field is introduced, we find that the magnetic charge can only be smaller than the critical value $q_b$, in which there is no black hole solution. Moreover, in the region $q<q_b$, we find that if the magnetic charge exceeds another critical value $q_c$ (i.e., $q_c<q<q_b$), the frequency of the Dirac field can approach zero, and the solution where a critical horizon appears is similar to an extremal black hole outside the critical horizon but has a nonsingular interior. The Dirac fields are also almost concentrated within it. In fact, this is a frozen star solution, we call such solutions frozen Bardeen-Dirac stars (FBDSs). We analyze the light rings of FBDSs and find that there exists a ``true" light ring outside the critical horizon, but inside it, the velocity of photons is very close to zero, which leads to the formation of a ``light ball" inside the critical horizon.