Bardeen spacetime with charged scalar field

The singularity theorem proved by Penrose and Hawking Penrose:1964wq ; Hawking:1970zqf ; Hawking:1973uf suggests that when a star meets certain conditions, its collapse process will inevitably lead to the formation of a singularity. Therefore, the existence of singularities appears to be a property inherent to most circumstances in General Relativity. However, the singularity theorem does not deny the existence of singularity-free black holes. In fact, as Hawking and Ellis point out Hawking:1973uf , if there is no strong energy condition or global hyperbolicity, then the singularity theorem will become invalid.

Furthermore, the existence of a singularity will cause all laws of physics to fail there and predictability is also lost. Therefore, spacetime singularities are often regarded as a symptom of the limitations of general relativity, similar to the appearance of divergences in some other classical physical theories. It is expected that a suitable theory of quantum gravity will be able to resolve the singularity problem. Unfortunately, we have not yet found a mature and reliable quantum theory. Nevertheless, more phenomenological approaches have attempted to address these singularities in some way. In this context, an important research direction is the so-called regular black holes (RBHs) without singularity. These proposals about RBHs typically postulate the form of a regular black hole metric and then determine what stress tensor would be required to sustain it 1981NCimL161G ; Dymnikova:1992ux ; Borde:1994ai ; Hayward:2005gi ; Lemos:2011dq ; Bambi:2013ufa ; Simpson:2018tsi ; Rodrigues:2018bdc . Many of these models of the regular black hole can be embedded in models of GR coupled to nonlinear electrodynamics Bronnikov:2000vy ; Dymnikova:2004zc ; Ayon-Beato:2004ywd ; Berej:2006cc ; Balart:2014jia ; Fan:2016rih ; Bronnikov:2017sgg ; Junior:2023ixh .

The earliest ideas about RBHs were proposed in 1966 by Sakharov Sakharov:1966aja and Gliner Gliner:1966 , who pointed out that the fundamental singularity could be avoided if the vacuum is replaced by a vacuum-like medium endowed with a de Sitter metric. Later, the first model of the regular black hole was proposed by Bardeen Bardeen:1968 . On this basis, several RBH models with similar global spacetime properties to Bardeen spacetime have been studied in 1990s Barrabes:1995nk ; Mars:1996khm ; Frolov:1988vj , and these models are also often uniformly called “Bardeen black holes” Borde:1996df . It is worth noting that the Bardeen black hole, when initially proposed, is not the exact solution of Einstein’s equations and has no known physically reasonable source. The underlying physical motivation for this choice was proposed two decades ago by Ayon-Beato and Garcia, who reinterpreted the Bardeen model in terms of the gravitational field possessed by magnetic monopoles Ayon-Beato:1998hmi ; Ayon-Beato:2000mjt , namely, the Bardeen solution is a magnetic solution to Einstein’s field equations coupled to a nonlinear electromagnetic field. Apart from the Bardeen black hole, at present, there is a wealth of literature on the study of RBHs, garnering growing attention and undergoing continuous development (see review articles Ansoldi:2008jw ; Lan:2023cvz ; Torres:2022twv ).

However, the physical sources of RBHs have only considered the nonlinear electromagnetic field in most of the literature, with no consideration of other types of matter fields. Recently, Ref. Wang:2023tdz studied the model of the Einstein-Bardeen theory minimally coupled to a free complex scalar field and then found that after introducing a scalar field into the Bardeen spacetime, the magnetic charge of the nonlinear electromagnetic field will have an upper limit. Within this upper limit, the black hole solution is not found. In particular, when the frequency approaches zero, the scalar field of this model is almost concentrated inside a certain critical horizon $r_{cH}$ and the metric function inside the region $r\leq r_{cH}$ is nearly zero, while the other metric function $g^{rr}$ is nearly zero only at $r_{cH}$. Notice that since $g^{rr}$ is only nearly zero but not zero at $r_{cH}$, the surface represented by $r=r_{cH}$ is actually not an event horizon, and this solution is not a black hole but a frozen star. The term “frozen” is used because only $-g_{tt}$ is nearly zero within $r_{cH}$, leading objects within $r_{cH}$ taking a very long time to move a small distance from the perspective of an observer at infinity, as if they are frozen in place (see Ref. Oppenheimer:1939ue ; zeldovichbookorpaper ; Ruffini:1971bza for the early study on the frozen star). In fact, these solutions are essentially also a type of boson stars with a magnetic charge, in which boson stars are a theoretical concept in astrophysics characterized by their ability to remain stable and avoid gravitational collapse Wheeler:1955zz ; Power:1957zz ; Kaup:1968zz ; Ruffini:1969qy (see Refs. Schunck:2003kk ; Liebling:2012fv for a review). After Ref. Wang:2023tdz , the model of Einstein-Klein-Gordon theory coupled to another nonlinear electromagnetic field proposed by Hayward Yue:2023sep and the model of Bardeen spacetime with the free Dirac field Huang:2023fnt have also been studied. Same as  Wang:2023tdz , in these two works, the event horizon and black hole solution are not found after the matter field is introduced, and when the frequency of the matter field approaches zero, the frozen star also can be obtained.

It is worth noting that the matter fields considered in Ref. Wang:2023tdz ; Huang:2023fnt ; Yue:2023sep are all uncharged, so one may wonder, in addition to these free matter fields, whether the solution of the frozen star can still be found after introducing charged matter fields. Considering that, in gravitational physics, one of the simplest types of “charged matter” that is often considered is the charged scalar field. Thus, to answer this question preliminary, in this paper, we further study the influence of the charged scalar field on the Bardeen spacetime, i.e., the model of Einstein-Bardeen theory coupled with a charged scalar field. By solving the Einstein equations, we find that after introducing the linear electromagnetic field (i.e., $U(1)$ gauge field), two types of solutions can be obtained - the small $q$ solution and the large $q$ solution. For the small $q$ solution, we find that there is a maximum value for the electric charge $q$, and after introducing $U(1)$ gauge field, the frozen star can be obtained without the frequency very close to zero. For the large $q$ solution, the electric charge can tend to infinity, and as the $q$ increases, the large $q$ solution gradually transforms into the pure Bardeen solution. In addition, similar to Ref. Wang:2023tdz ; Huang:2023fnt ; Yue:2023sep , there is no event horizon and no black hole solution in our results.

The paper is organized as follows. In Sec. II, we provide an introduction to the model of Einstein-Bardeen theory coupled to a charged complex scalar field. Numerical results of this model and analysis of its physical properties are present in Sec. III. The conclusion and discussion are given in Sec. IV.

In this section, we wish to provide a concise introduction to the theoretical framework encompassing the Einstein-nonlinear electrodynamics model, coupled with a charged complex scalar field $\psi$, described by the following action (we use units with $c=1=\hbar$)

$$S=\int d^{4}x\sqrt{-g}\left[\frac{R}{16\pi G}+\mathcal{L}^{(1)}+\mathcal{L}^{(2)}+\mathcal{L}^{(3)}\right],$$

(1)

where $\mathcal{L}^{(1)}$, $\mathcal{L}^{(2)}$ and $\mathcal{L}^{(3)}$ are the Lagrangians of the nonlinear electromagnetic field Ayon-Beato:2000mjt , the linear electromagnetic field ($U(1)$ gauge field) and the scalar field, respectively,

$$\mathcal{L}^{(1)}=\frac{3}{2s}\left(\frac{\sqrt{p^{2}E^{\mu\nu}E_{\mu\nu}}}{\sqrt{2}+\sqrt{p^{2}E^{\mu\nu}E_{\mu\nu}}}\right)^{(5/2)},$$

(2)

$$\mathcal{L}^{(2)}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu},$$

(3)

$$\mathcal{L}^{(3)}=-\left(D^{\mu}\psi\right)^{*}\left(D_{\mu}\psi\right)-\mu^{2}\psi\psi^{*},$$

(4)

with $R$ the Ricci curvature, $G$ Newton’s constant. $E^{\mu\nu}$ and $F^{\mu\nu}$ are the electromagnetic field strength of the nonlinear electromagnetic field $A_{\mu}$ and the linear electromagnetic field $B_{\mu}$, respectively, given in terms of the potential 1-form Ayon-Beato:2000mjt ; Jetzer:1990wr ; Fulling ; BD as:

$$E_{\mu\nu}:=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$$

(5)

$$F^{\mu\nu}:=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$

(6)

and $D_{\mu}$ denotes the gauge invariant covariant derivative:

$$D_{\mu}:=\nabla_{\mu}+iqB_{\mu},$$

(7)

where $\nabla$ stands for the spacetime covariant derivative and the constant $q$ is the scalar field electric charge. The constants $s$, $p$, and $\mu$ are three independent parameters, where $p$ represents the magnetic charge and $\mu$ is the scalar field mass.

The action (1) is invariant under local $U(1)$ gauge transformations $\psi\rightarrow e^{iq\alpha(x^{\theta})}\psi$, where $\alpha(x^{\theta})$ a local gauge function which depends on the spacetime coordinates. As a result, the system possesses a conserved Noether density current, which acts as a source for the electromagnetic field $B^{\mu}$ and is given by:

$$j^{\nu}=ig^{\mu\nu}q\left[\psi^{*}D_{\nu}\psi-\psi(D_{\nu}\psi^{*})\right].$$

(8)

This implies the existence of a total electric charge:

$$Q=\int_{\mathcal{S}}j^{\nu}n_{\nu}dV,$$

(9)

here, $\mathcal{S}$ is a spacelike hypersurface (with the volume element $dV$), and $n_{\nu}$ is the unit normal vector of $\mathcal{S}$. The total scalar particle number $N$ Jetzer:1989av can be obtained through the total charge $Q$, where

$$N=\frac{Q}{q}.$$

(10)

By varying the action (1) with respect to the metric $g_{\mu\nu}$, electromagnetic field $A_{\mu}$, $B_{\mu}$ and scalar field $\psi$, respectively, we can obtain the following equations of motion

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-8\pi G(T^{(1)}_{\mu\nu}+T^{(2)}_{\mu\nu}+T^{(3)}_{\mu\nu})=0,$$

(11)

$$\nabla_{\mu}\left(\frac{\partial\mathcal{L}^{(1)}}{\partial\mathcal{E}}E^{\mu\nu}\right)=0,$$

(12)

$$\nabla_{\mu}F^{\mu\nu}=-j^{\nu},$$

(13)

$$D^{\mu}D_{\mu}\psi-\mu^{2}\psi=0,$$

(14)

with the stress-energy tensor

$$T^{(1)}_{\mu\nu}=-\frac{\partial\mathcal{L}^{(1)}}{\partial\mathcal{E}}E_{\mu\lambda}E_{\mu}{}^{\lambda}+g_{\mu\nu}\mathcal{L}^{(1)},$$

(15)

$$T^{(2)}_{\mu\nu}=-\frac{\partial\mathcal{L}^{(2)}}{\partial\mathcal{F}}F_{\mu\lambda}F_{\mu}{}^{\lambda}+g_{\mu\nu}\mathcal{L}^{(2)},$$

(16)

$$T^{(3)}_{\mu\nu}=\frac{1}{2}\left[\left(D_{\mu}\psi\right)^{*}\left(D_{\nu}\psi\right)+\left(D_{\nu}\psi\right)\left(D_{\mu}\psi\right)^{*}-g_{\mu\nu}\left(\left(D_{\lambda}\psi\right)\left(D^{\lambda}\psi\right)^{*}+m^{2}|\psi|^{2}\right)\right],$$

(17)

where $\mathcal{E}=\frac{1}{4}E^{\mu\nu}E_{\mu\nu}$ and $\mathcal{F}=\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$.

Restricting to static, spherically-symmetric solutions of the field equations, we employ the following ansatz

$$ds^{2}=-n(r)\sigma^{2}(r)dt^{2}+\frac{dr^{2}}{n(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right),$$

(18)

here, $n(r)$ and $\sigma(r)$ are functions of the radial coordinate $r$ only. Then, for the electromagnetic fields Jetzer:1990wr ; Kain:2021bwd and the scalar field Derrick:1964ww ; Wyman:1981bd ; Adib:2002fd ; Schunck:2003kk , we assume

$$A_{\mu}dx^{\mu}=p\cos(\theta)d\varphi,\quad B_{\mu}dx^{\mu}=V(r)dt,\quad\psi=\phi(r)e^{-i\omega t},$$

(19)

where the function $V(r)$ corresponding to the electric potential and the function $\phi(r)$ are two real radial functions, and $\omega>0$ is the real constant corresponding to the frequency of oscillation of the scalar field.

Using the metric (18) and the ansatz (19 ) for the scalar field and electromagnetic fields into the equations (11-14), we can get the following equations about $n(r)$, $\sigma(r)$ $V(r)$, and $\phi(r)$

	$\displaystyle n^{\prime}+n\left(\frac{1}{r}+8\pi Gr\phi^{\prime 2}\right)+\frac{8\pi Gr\phi^{2}}{n\sigma^{2}}\left(\omega^{2}-2q\omega V+q^{2}V^{2}\right)$
	$\displaystyle+\frac{4\pi GrV^{2}}{\sigma}+8\pi Gr\mu^{2}\phi^{2}+\frac{12\pi Grp^{5}}{(p^{2}+r^{2})^{5/2}s}-\frac{1}{r}=0,$		(20)

$$\sigma^{2}-8\pi Gr\phi^{\prime 2}\sigma^{2}-\frac{8\pi Gr\phi^{2}}{n^{2}\sigma}\left(q^{2}V^{2}-2q\omega V+\omega^{2}\right),$$

(21)

$$V^{\prime\prime}+V^{\prime}\left(\frac{2}{r}-\frac{\sigma^{\prime}}{\sigma}\right)+\frac{2q\phi^{2}}{n}\left(qV-\omega\right)=0,$$

(22)

$$\phi^{\prime\prime}+\phi^{\prime}\left(\frac{n^{\prime}}{n}+\frac{\sigma^{\prime}}{\sigma}+\frac{2}{r}\right)+\left(\frac{q^{2}V^{2}}{n\sigma^{2}}+\frac{\omega^{2}}{n\sigma^{2}}-\frac{2q\omega V}{n\sigma^{2}}-\mu^{2}\right)\frac{\phi}{n}=0.$$

(23)

where the prime denotes the differentiation with respect to $r$. And form equations (9), (18), (19), the total electric charge is:

$$Q=8\pi Gq\int^{\infty}_{0}\frac{qV-\omega}{n\sigma}r^{2}\phi^{2}.$$

(24)

Before numerically solving these coupled equations of motion (20-23), we need to impose suitable boundary conditions for $n(r)$, $\sigma(r)$, $\phi(r)$, $V(r)$. Considering the assumption of regularity and asymptotic flatness, the metric functions $n(r)$ and $\sigma(r)$ satisfy:

$$n(0)=1,\quad\sigma(0)=\sigma_{0},\quad n(\infty)=1-\frac{2GM}{r},\quad\sigma(\infty)=1,$$

(25)

For the constants, namely $\sigma_{0}$ and the ADM mass $M$ of the solution, their values can be determined by solving the system of ordinary differential equations. Additionally, for the complex scalar field and the $U(1)$ gauge field $B_{\mu}$, we require the following boundary conditions

$$\left.\frac{d\phi}{dr}\right|_{r=0}=0,\quad\phi(\infty)=0,\quad\left.\frac{dV}{dr}\right|_{r=0}=0,\quad V(\infty)=0.$$

(26)

It is worthwhile to investigate two special solution cases of these equations of motion (20-23). First, when $p=0$ and the scalar field does not vanish, action (1) reduces to Einstein-Maxwell-scalar theory which can describe charged boson star Jetzer:1989av (Further, if $p=q=0$, this model describes pure boson star). Then, in the case of vanishing complex scalar field with $q=0$ and $p\neq 0$, the model is the Bardeen theory and the associated solution describes both spherical solitons or black holes, also referred to as the Bardeen spacetime. The metric is the following form

$$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}(\theta)d\varphi^{2}\right),$$

(27)

here,

$$f(r)=1-\frac{p^{3}r^{2}}{s(r^{2}+p^{2})^{3/2}}.$$

(28)

The asymptotic behavior for this metric function $f(r)$ at infinity is given by

$$f(r)=1-2\frac{p^{3}}{2s}/r+O(1/r^{3}).$$

(29)

From the term $1/r$, one can deduce the ADM mass $M$ of this configuration as $M=p^{3}/2s$ Ayon-Beato:1998hmi . Furthermore, the function $f(r)$ demonstrates a local minimum value at $r=\sqrt{2}p$. In case where $q<3^{3/4}\sqrt{\frac{s}{2}}$, the solutions without event horizons are present. When $q=3^{3/4}\sqrt{\frac{s}{2}}$, degenerate horizons are observed, and for $q>3^{3/4}\sqrt{\frac{s}{2}}$, two distinct horizons exist.

To facilitate numerical computations, we employ the following dimensionless variables introduced by using natural units set by $\mu$ and $G$:

$$r\rightarrow r\mu,\quad\omega\rightarrow\frac{\omega}{\mu},\quad\phi\rightarrow\frac{\sqrt{4\pi}}{M_{Pl}}\phi,\quad V\rightarrow\frac{\sqrt{4\pi}}{M_{Pl}}V,$$

(30)

where $M_{Pl}=1/\sqrt{G}$ denote the Planck mass, Consequently, the dependence on both $G$ and $\mu$ disappears from the equations. Furthermore, the total electric charge and the ADM matter are also can be expressed in units set by $\mu$ and $G$ (Note that, to simplify the output, without loss of generality, we set $s=0.2$ in the following results).

We also introduce a new radial coordinate $x=r/(r+1)$, which maps the semi-infinite region $[0,\infty)$ onto the unit interval $[0,1]$. In addition, the solutions are found by using the finite element method, and the number of grid points in the integration region $[0,1]$ is $1000$. The iterative method we use is the Newton-Raphson method, and the relative error for the numerical solutions in this paper is estimated to be below $10^{-5}$.

In our numerical results, we find that the magnetic charge $p$ can only be less than $3^{3}/4\sqrt{s/2}=3^{3}/4\sqrt{0.1}$ (i.e., $p<3^{3}/4\sqrt{0.1}$). Within this range, two different types of solutions emerge. Based on the electric charge $q$, we refer to these two types of solutions as the “small $q$ solution” and the “large $q$ solution”. The small $q$ solution is a set of solutions with relatively small electric charges, which have an upper limit on $q$. In contrast, the large $q$ solution is a set of solutions with relatively large electric charges, which can tend to infinity but have a lower limit. We present the ranges of $q$ for the small $q$ solution and the large $q$ solution in the second and fourth columns of Tab. 1, and the fourth column of Tab. 1 corresponds to the critical charge $q_{c}$ for the small $q$ solution.

In this paper, we investigated the model of Einstein-Bardeen theory coupled to a charged scalar field and obtained two types of solutions which we call the “small $q$ solution” and the “large $q$ solution”. There exists a limit to the magnetic charge of both solutions and within the limit, we did not find black hole solutions.

For the small $q$ solution, there exists an upper limit to its electric charge. Within this upper limit, there exists a critical charge, and for the same magnetic charge, the curves of the ADM mass $M$ or the number of particles $N$ versus the frequency $\omega$ of the small $q$ solution above the critical charge will be significantly different from those of the small $q$ solution below the critical charge. In addition, we find that after introducing the electromagnetic field, It is possible to obtain the solution for the frozen star without needing $\omega\rightarrow 0$. However, it should be noted that this is only the case when the magnetic charge is relatively small. When the magnetic charge exceeds a certain range, the introduction of the electromagnetic field may result in solutions that fail to form frozen stars.

Instead, for the large $q$ solution, we find that there is no upper limit to its electric charge. In other words, the electric charge of the large $q$ solution can tend towards infinity. And when the electric charge tend to infinity ($q\rightarrow\infty$), the large $q$ solution becomes the pure Bardeen solution. In other words, this solution effectively serves as the probe limit solution. From the curve of the ADM mass $M$ versus the frequency $\omega$, we observe that the minimum frequency of this solution corresponds precisely to the maximum frequency of the small $q$ solution below the critical charge.

There are many interesting extensions of our work. Firstly, this work considers only the ground state solution where the charge scalar field function has no node. In our future work, we plan to consider the excited states solution with more nodes. Secondly, it is also interesting to extend our study to other matter fields including free Dirac fields and Proca fields, etc. Finally, the stability of the small $q$ solution or the large $q$ solution remains unexplored in this paper, one would wonder whether our model can be stable against small perturbations. Therefore, the dynamic stability of the solution is also a crucial subject for further investigation.