Bardeen Spacetime with Charged Dirac Field

With the Event Horizon Telescope capturing the first photograph of a “black hole” EventHorizonTelescope:2019dse ; EventHorizonTelescope:2019uob ; EventHorizonTelescope:2019jan ; EventHorizonTelescope:2019ths ; EventHorizonTelescope:2019pgp ; EventHorizonTelescope:2019ggy , people belief in their existence has solidified. Black holes, a profound prediction of Einstein’s theory of general relativity, have long been theoretically associated with a singularity. At the singularity, all physical laws break down, leading scientists to hope for a unified theory of gravity and quantum mechanics to address the black hole singularity problem. Unfortunately, such a flawless theory of quantum gravity has yet to be realized.

As demonstrated by Hawking and Penrose, singularities are inevitable if matter adheres to the strong energy condition and certain other prerequisites Penrose:1964wq ; Hawking:1970zqf . However, this also implies that the introduction of a special type of matter that does not satisfy the strong energy condition might eliminate singularities, potentially leading to the formation of Regular Black Holes (RBHs). The first RBH model was proposed by Bardeen in 1968 Bardeen:1968 . It was not until 30 years later that the specific type of matter needed to form Bardeen black holes, which do not meet the strong energy condition, was discovered. E. Ayon-Beato and A. Garcia found that magnetic monopoles in the context of nonlinear electrodynamics could serve as the material source for Bardeen’s RBH Ayon-Beato:1998hmi ; Ayon-Beato:1999kuh ; Ayon-Beato:2000mjt . Recently, it has been discovered that nonlinear electromagnetic fields can provide various matter sources for RBHs Bronnikov:2000vy ; Dymnikova:2004zc ; Ayon-Beato:2004ywd ; Berej:2006cc ; Lemos:2011dq ; Balart:2014jia ; Balart:2014cga ; Fan:2016rih ; Bronnikov:2017sgg ; Junior:2023ixh .

Since we know which types of matter can form RBHs, we can broaden our scope beyond just solutions with the horizon. In fact, magnetic monopoles in nonlinear electromagnetic fields can also yield horizonless solutions. reference Wang:2023tdz has identified that introducing a massive complex scalar field into the Bardeen spacetime results in magnetic charge having an upper limit, thus precluding the formation of solutions with the horizon. Moreover, as the frequency of the scalar field approaches zero, a distinct type of solution emerges. In these solutions, there is a special location $r_{cH}$ within which the scalar field is distributed. For the metric field, $g_{tt}$ approaches zero for $r<r_{cH}$, and $1/g_{rr}$ is very close to zero at $r_{cH}$, so we refer to $r_{cH}$ as the critical horizon. It is noteworthy that because $1/g_{rr}$ is only very close to zero, and not exactly zero, this solution does not possess an event horizon and is not classified as a black hole but rather as a “frozen star”. These solutions can also be generalized, such as by replacing the Bardeen background with a Hayward background Yue:2023sep ; Chen:2024bfj , using different matter fields Huang:2023fnt ; Huang:2024rbg ; Zhang:2024ljd , or considering modified gravity Ma:2024olw .

Electromagnetic interactions, being another fundamental force, are often considered in gravitational systems Reissner:1916cle ; Newman:1965my ; Hartle:1972ya ; Maeda:2008ha ; Herdeiro:2018wub . In classical scenarios, the balance between Newtonian gravity and Coulomb repulsion leads to a critical charge $q_{c}$; if a particle’s charge $q$ exceeds $q_{c}$, it cannot condense due to excessive Coulomb repulsion. However, within the framework of general relativity, the gravitational binding energy allows for the existence of soliton solutions with $q>1$. These results have been corroborated by studies on charged boson stars and charged Dirac stars Jetzer:1989av ; Jetzer:1989us ; Pugliese:2013gsa ; Kumar:2017zms ; Collodel:2019ohy ; Lopez:2023phk ; Jaramillo:2023lgk ; Finster:1998ux ; Herdeiro:2021jgc . Research Huang:2024rbg has shown that in Bardeen spacetime, the critical charge $q_{c}$ for a charged scalar field changes. This paper will explore the interaction between Bardeen spacetime and charged Dirac fields. Similar to previous studies on spherically symmetric Dirac stars Finster:1998ws ; Herdeiro:2017fhv ; Dzhunushaliev:2018jhj ; Herdeiro:2020jzx ; Liang:2023ywv , this paper considers a pair of Dirac field with opposite spins to maintain spherical symmetry in the spacetime. We find that in Bardeen spacetime, the critical charge for Dirac fields also varies. Moreover, we define an effective frequency and discover that when this effective frequency approaches zero, a frozen star forms.

The structure of this paper is as follows: In section II, we present our model. Section III outlines the boundary conditions that the field functions must satisfy and details our numerical methods. The numerical results are displayed and discussed in section IV, followed by a summary in the concluding section.

We consider a system composed of charged Dirac fields minimally coupled to (3 + 1)-dimensional Einstein’s gravity and a nonlinear electromagnetic field (Bardeen field). The action is given by

where $R$ is the Ricci scalar, $G$ is the gravitational constant. $\mathcal{L}_{B}$, $\mathcal{L}_{D}$ and $\mathcal{L}_{M}$ represent the Lagrangian densities for the Bardeen field, Dirac field, and Maxwell field, respectively, with the following forms:

Here, we need to consider two Dirac fields ${\Psi}^{(j)}$ to construct a spherically symmetric spacetime. Two Dirac fields have the same mass $\mu$. $\bar{\Psi}^{(j)}=\Psi^{(j)\dagger}\xi$ are the Dirac conjugate of ${\Psi}^{(j)}$, with $\Psi^{(j)\dagger}$ denoting the usual Hermitian conjugate. For the Hermitian matrix $\xi$, we choose $\xi=-\hat{\gamma}^{0}$, where $\hat{\gamma}^{0}$ is one of the gamma matrices in flat spacetime. $\hat{D}_{\mu}=\partial_{\mu}-\Gamma_{\mu}+iqA_{\mu}$, where $\Gamma_{\mu}$ are the spinor connection matrices. In equation (2), $\mathcal{H}\equiv-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}$, the electromagnetic tensors $H_{\mu\nu}$ and $F_{\mu\nu}$ are defined by the electromagnetic four-potentials $B_{\mu}$ and $A_{\mu}$, respectively:

By varying the action, we can obtain the following equations of motion:

where, $T_{\mu\nu}=T_{\mu\nu}^{B}+T_{\mu\nu}^{D}+T_{\mu\nu}^{M}$, is the total energy-momentum tensor. The energy-momentum tensors of the Bardeen field, Dirac fields, and the Maxwell field take the following form:

In this article, we focus on spherically symmetric spacetime, thus we adopt the following metric ansatz:

The metric functions $F_{0}(r)$ and $F_{1}(r)$ depend only on the radial distance $r$. Additionally, we choose the ansatz for the Dirac fields as follows:

Here, the constant $\omega$ is the frequency of the Dirac fields, which means that all the spinor fields possess a harmonic time dependence. The part concerning the radial coordinate $r$ is

The real function $a(r)$ and $b(r)$ depend only on the radial distance $r$. For the electromagnetic four-potential $A_{\mu}$ and $B_{\mu}$, we employ the following ansatz:

By using the above ansatz, we find that the equations of motion for the Bardeen field naturally hold true. Two Dirac fields satisfy the same equations:

Maxwell’s equations read:

Equation (7) can yield three distinct, independent equations. In our solution process, we utilize two one-order ordinary differential equations concerning the metric functions, as follows:

while the remaining second-order equation serves as a constraint equation to verify the accuracy of the results.

The quantities $j^{\mu}_{(j)}$ defined in equation (10) are Noether density currents which arise from the invariance of the action (1) under a global $U(1)$ transformation $\Psi^{(j)}\rightarrow\Psi^{(j)}e^{i\alpha}$, where $\alpha$ is constant. This invariance implies the conservation of the total particle number. According to Noether’s theorem, we can obtain the particle number:

here, $S$ is a spacelike hypersurface. Additionally, the total electric charge in spacetime is $Q=qN$. Since the ansatz for the Dirac fields are given by equation (15) and (16), it follows that $J^{t}_{(1)}=J^{t}_{(2)}$, indicating that the particle numbers for the two Dirac fields are equal.

ADM mass is also an important global quantity, which can be obtained by integrating the Komar energy density on the spacelike hypersurface $S$:

Before numerically solving these ordinary differential equations, appropriate boundary conditions need to be proposed, which can be determined from the asymptotic behavior of the field functions. Firstly, we require them to be asymptotically flat at spatial infinity ($r\rightarrow\infty$). Thus, we need:

The asymptotic behavior of $c$ cannot be derived from the equations; for convenience, we set the electric potential at spatial infinity to zero, $c(\infty)=0$. At the origin, we need:

There are six input parameters, corresponding to Newton’s constant $G$, the mass of Dirac field $\mu$, the frequency $\omega$, the electric coupling constant $q$, the magnetic charge of the nonlinear electromagnetic field $p$ and the parameter $s$. We take the following field redefinition and scaling of both $r$ and some parameters:

During the solution process, this scaling is equivalent to selecting $\mu=1$ and $G=\frac{1}{4\pi}$. In all the following results, we set $s=0.3$. As such, we have three input parameters $p$, $q$, and $\omega$. To facilitate this, we introduce a new coordinate $x$, transforming the radial coordinate range from $[0,\infty)$ to $[0,1]$,

Our numerical calculations are all based on the finite element method, with 1000 grid points in the integration domain of $0\leq x\leq 1$. We employ the Newton-Raphson method as the iterative scheme. To ensure the accuracy of the computed results, we require the relative error to be less than $10^{-5}$.

In this paper, we investigated the model of two charged Dirac fields coupled to Einstein-Bardeen theory in a spherically symmetric spacetime. Through numerical methods, we obtained families of solutions with various magnetic charges and electric charges, analyzing how changes in these two parameters influence the properties of the solutions.

Firstly, we present the variation of the Dirac field function and the electric potential function with respect to $q$. We observe that, for any fixed frequency $\omega$ and magnetic charge $p$, the Dirac field function $a$ and $b$ exhibit the same variation, while the variation of the electric potential function $c$ is influenced by the magnetic charge $p$. Subsequently, we illustrate how the ADM mass $M$ varies with frequency for different values of $p$ and $q$, noting that all curves retain a spiraling trend. Additionally, we summarize three key factors that significantly affect the shape of the ADM frequency curves: critical charge, maximum frequency, and magnetic charge. If $q$ exceeds the critical charge, the solution cannot revert to pure Bardeen spacetime. If the maximum frequency $\omega_{max}=1$, the mass of the first branch increases monotonically with frequency; moreover, if $p$ is sufficiently large, the curve lacks a second branch.

We define an effective frequency and find that for all solutions, the effective frequency is positive. In reference Wang:2023tdz , a frozen star solution is obtained when the frequency approaches zero; in our charged model, this translates to the appearance of a frozen star solution when the effective frequency approaches zero. We present the field configurations and metric functions for frozen star solutions with varying magnetic charges, discovering that they exhibit some similarities to uncharged frozen stars. The $g_{rr}$ component approaches zero at a certain location (which we designate as the critical horizon) but does not fall below zero, while the Dirac field is concentrated within the critical horizon. Furthermore, when the magnetic charge is fixed, an increase in $q$ results in a larger critical horizon radius for the frozen star.

This model allows for several intriguing extensions. In this article, we have only considered the ground state of the Dirac field; exploring excited states with more nodes could unveil distinct characteristics. Furthermore, as we have focused on two Dirac fields, we conducted the two-particle picture, yet we could increase the particle count in the system by considering multiple pairs of Dirac fields, akin to $\kappa$ Dirac stars Sun:2024nuf . Lastly, we believe that the spacetime of frozen star solutions merits further investigation, particularly studying the motion of test particles within this spacetime, as well as the search for non-spherically symmetric spacetimes analogous to frozen star solutions.