Nonsingular black holes and spherically symmetric objects
in nonlinear electrodynamics with a scalar field

General Relativity (GR) is a fundamental pillar for describing gravitational interactions in both strong and weak field regimes. The vacuum solution to the Einstein equation on a static and spherically symmetric (SSS) background is described by a Schwarzschild line metric that contains a mass $M$ of the source. In Einstein-Maxwell theory with the electromagnetic Lagrangian $F=-F_{\mu\nu}F^{\mu\nu}/4$, where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the Maxwell tensor with a vector field $A_{\mu}$, the resulting solution is given by a Reissner-Nordström (RN) metric with electric or magnetic charges. For both Schwarzschild and RN black holes (BHs), there are singularities at the origin ($r=0$) with divergent curvature quantities. This divergent property at $r=0$ also persists for rotating BHs present in the framework of GR.

In GR, Penrose’s singularity theorem [1] establishes that BH singularities at the origin can arise as a natural consequence of gravitational collapse. The validity of this theorem, however, hinges on several assumptions regarding the structure of spacetime and the properties of matter. Among these is the requirement of global hyperbolicity of spacetime. Violating this condition can potentially lead to the existence of nonsingular BHs.

A notable example of such a solution was first introduced by Bardeen [2], who proposed a nonsingular BH with metric components that remain finite as $r\to 0$. Since then, various other regular BH metrics have been proposed in the literature, which offers alternative frameworks for addressing the singularity problem in BH physics [3, 4, 5, 6, 7, 8, 9].

Even though the nonsingular metrics are given apriori in the aforementioned approach, it remains to be seen whether they can be realized in some concrete theories. For this purpose, we need to take into account additional degrees of freedom (DOFs) beyond those appearing in GR. For example, we may consider scalar-tensor theories in which a new scalar DOF is incorporated into the gravitational action [10]. In most general scalar-tensor theories with second-order field equations of motion (Horndeski theories [11, 12, 13, 14]), it is known that the existence of SSS asymptotically-flat hairy BH solutions without ghost/Laplacian instabilities is quite limited [15, 16, 17, 18]. For a radial dependent scalar profile $\phi(r)$, we need a coupling between $\phi$ and a Gauss-Bonnet curvature invariant [19, 20, 21, 22, 23, 24], but the Gauss-Bonnet term diverges at $r=0$. Hence the construction of nonsingular BHs in the context of scalar-tensor theories is generally challenging.

If we consider vector-tensor theories in nonlinear electrodynamics (NED) given by the Lagrangian ${\cal L}(F)$, where ${\cal L}$ is a nonlinear function of $F$, it is possible to realize nonsingular BHs without curvature singularities at $r=0$ [25, 26, 27, 28, 4, 7, 29, 30]. The NED Lagrangian accommodates Euler-Heisenberg theory [31] as well as Born-Infeld theory [32]. In such subclasses of NED theories, the resulting SSS BH solutions possess curvature singularities at $r=0$ [33, 34, 35, 36]. However, there are nonsingular electrically or magnetically charged BHs for some specific choices of the NED Lagrangian. Thus, the vector field with nonlinear Lagrangians of $F$ allows an interesting possibility for realizing regular BHs even at the classical level.

To determine the stability of nonsingular BHs, it is essential to analyze their linear stability by using BH perturbation theory. In Refs. [37, 38, 39, 40, 41], the authors discussed the BH stability by considering the propagation of dynamical perturbations in the region outside the event horizon. Although the conditions for the absence of ghosts and Laplacian instabilities can be satisfied outside the horizon, a recent analysis [42] shows that there is angular Laplacian instability of vector-field perturbations around the regular center. This instability manifests for both electric and magnetic BHs, leading to the rapid enhancement of metric perturbations. Consequently, the nonsingular background metric cannot be maintained as a steady-state solution. This means that nonsingular SSS BHs cannot be realized in the context of NED with the Lagrangian ${\cal L}(F)$.

Motivated by the no-go result in the context of NED, we extend our analysis to explore whether similar properties persist in more general classical field theories. To this end, we incorporate a scalar field $\phi$ with a kinetic term $X$ into the NED framework, considering a Lagrangian of the form ${\cal L}(F,\phi,X)$. For the gravity sector, we consider GR described by the Lagrangian $M_{\rm Pl}^{2}R/2$, where $M_{\rm Pl}$ is the reduced Planck mass and $R$ is the Ricci scalar. Applying such theories to the SSS background, we will show that there are no solutions with mixed electric and magnetic charges (as it happens in NED). Hence we can focus on either electrically or magnetically charged objects, with a nontrivial scalar-field profile.

Theories we will study in this paper belong to a subclass of scalar-vector-tensor theories with second-order field equations of motion. Since they respect $U(1)$ gauge symmetry, there are one scalar, two transverse vectors, and two tensor polarizations as the propagating DOFs. To derive the stability conditions of those five DOFs, we consider linear perturbations in both odd- and even-parity sectors on the SSS background. We expand the corresponding action up to the second order in perturbations by taking into account both electric and magnetic charges. For the electric case, a similar analysis was performed in Refs. [43, 44] as a subclass of Maxwell-Horndeski theories. Since the linear stability of magnetic SSS objects has not been addressed yet, we will do so in this paper.

After deriving conditions for the absence of ghosts and Laplacian instabilities of the five dynamical DOFs, we will apply them to three subclasses of ${\cal L}(F,\phi,X)$ theories: (i) $\tilde{\cal L}(F)+K(\phi,X)$, (ii) ${\cal L}=X+\mu(\phi)F^{n}$, and (iii) ${\cal L}=X\kappa(F)$. We will show that regular SSS objects realized by theories (i) and (iii), which include nonsingular BHs, are excluded by Laplacian instabilities of vector-field perturbations around the origin. In theories (ii), the absence of angular Laplacian instabilities requires the condition $n>1/2$, under which there are no regular BHs with even horizons. Thus, even by extending NED to more general theories with the Lagrangian ${\cal L}(F,\phi,X)$, we do not find even a single example of nonsingular BHs without instabilities. This shows the general difficulty of constructing BHs without singularities in classical field theories. In theories (ii), however, we will show the existence of linearly stable SSS compact objects without event horizons. These regular solutions are present for both electric and magnetic configurations. We will clarify the regions of $n$ in which the regular horizonless compact objects are subject to neither ghosts nor Laplacian instabilities.

This paper is organized as follows. In Sec. II, we derive the SSS background solutions and discuss the properties of them for the electrically and magnetically charged cases. In Sec. III, we expand the action up to quadratic order in perturbations and obtain conditions under which neither ghosts nor Laplacian instabilities are present for five dynamical DOFs. In Sec. IV, we apply the linear stability conditions to theories (i) mentioned above and show that nonsingular SSS objects are prone to angular Laplacian instability. In Sec. V, we show that the absence of angular Laplacian instabilities demands the condition $n>1/2$, under which nonsingular BHs with event horizons do not exist. We also clarify the parameter space of $n$ in which horizonless regular compact objects suffer from neither ghosts nor Laplacian instabilities. In Sec. VI, we show that nonsingular SSS objects in theories (iii) are excluded by Laplacian instabilities either at the origin or at spatial infinity. Sec. VII is devoted to conclusions.

We consider theories in which the Lagrangian ${\cal L}$ in the matter sector depends on a scalar field $\phi$ and the two scalar products

$$X\equiv-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\,,\qquad F\equiv-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$$

(1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength of a covector field $A_{\mu}$. For the gravity sector, we consider GR described by the Einstein-Hilbert Lagrangian $M_{\rm Pl}^{2}R/2$. Then, the total action is given by

$${\cal S}=\int{\rm d}^{4}x\sqrt{-g}\left[\frac{M_{\rm Pl}^{2}}{2}R+{\cal L}(F,\phi,X)\right]\,,$$

(2)

where $g$ is a determinant of the metric tensor $g_{\mu\nu}$.

We study SSS solutions on the background given by the line element

$${\rm d}s^{2}=-f(r){\rm d}t^{2}+h^{-1}(r){\rm d}r^{2}+r^{2}({\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2})\,,$$

(3)

where $f$ and $h$ are functions of the radial distance $r$. For the positivity of $-g=(f/h)r^{4}\sin^{2}\theta$ in Eq. (2), we require that $f/h$ is positive. For the later convenience, we introduce the $r$-dependent function $N(r)$ satisfying

$$f(r)=N(r)h(r)\,,$$

(4)

so that

$$N(r)>0\,.$$

(5)

In the following, we will use the two functions $N(r)$ and $h(r)$ instead of $f(r)$ and $h(r)$.

On the SSS background (3), we consider a radial dependent scalar-field profile $\phi(r)$. For the covector field $A_{\mu}$, the presence of $U(1)$ gauge symmetry in theories given by the action (2) allows us to express the vector-field components in the form

$$A_{\mu}{\rm d}x^{\mu}=A_{0}(r){\rm d}t-q_{M}\cos\theta{\rm d}\varphi\,,$$

(6)

where $A_{0}$ is a function of $r$, and $q_{M}$ is a constant corresponding to a magnetic charge. The scalar products defined in Eq. (1) reduce to

$$X=-\frac{1}{2}h\phi^{\prime 2}\,,\qquad F=\frac{A_{0}^{\prime 2}}{2N}-\frac{q_{M}^{2}}{2r^{4}}\,,$$

(7)

where a prime represents the derivative with respect to $r$.

Varying the action (2) with respect to $N$, $h$, $A_{0}$, and $\phi$, we obtain

	$\displaystyle h^{\prime}-\frac{1-h}{r}-\frac{r}{M_{\rm Pl}^{2}N}\left(N{\cal L}-A_{0}^{\prime 2}{\cal L}_{,F}\right)=0\,,$		(8)
	$\displaystyle\frac{N^{\prime}}{N}=\frac{r\phi^{\prime 2}{\cal L}_{,X}}{M_{\rm Pl}^{2}}\,,$		(9)
	$\displaystyle\left(\frac{r^{2}A_{0}^{\prime}{\cal L}_{,F}}{\sqrt{N}}\right)^{\prime}=0\,,$		(10)
	$\displaystyle{\cal E}_{\phi}\equiv\left(\sqrt{N}hr^{2}\phi^{\prime}{\cal L}_{,X}\right)^{\prime}+\sqrt{N}r^{2}{\cal L}_{,\phi}=0\,,$		(11)

where the notations like ${\cal L}_{,F}\equiv\partial{\cal L}/\partial F$ are used for partial derivatives. If ${\cal L}_{,X}=0$, we require that $N^{\prime}=0$ in general, or $N(r)=1$ by fixing boundary conditions at spatial infinity. If $A_{0}^{\prime}\neq 0$, we can integrate Eq. (10) to give

$${\cal L}_{,F}=\frac{q_{E}\sqrt{N}}{r^{2}A_{0}^{\prime}}\,,$$

(12)

where $q_{E}$ is an integration constant corresponding to an electric charge. From Eqs. (8) and (12), we have

$${\cal L}=\frac{M_{\rm Pl}^{2}}{r^{2}}\left(rh^{\prime}+h-1\right)+\frac{q_{E}A_{0}^{\prime}}{\sqrt{N}r^{2}}\,.$$

(13)

Using Eqs. (9) and (11), we can express ${\cal L}_{,X}$ and ${\cal L}_{,\phi}$, as

	$\displaystyle{\cal L}_{,X}$	$\displaystyle=$	$\displaystyle\frac{M_{\rm Pl}^{2}N^{\prime}}{r\phi^{\prime 2}N}\,,$		(14)
	$\displaystyle{\cal L}_{,\phi}$	$\displaystyle=$	$\displaystyle-\frac{1}{\sqrt{N}r^{2}}\left(\frac{M_{\rm Pl}^{2}N^{\prime}hr}{\sqrt{N}\phi^{\prime}}\right)^{\prime}\,,$		(15)

which are valid for $\phi^{\prime}\neq 0$. Taking the $r$ derivative of Eq. (13), i.e., ${\cal L}^{\prime}(r)={\cal L}_{,F}F^{\prime}+{\cal L}_{,\phi}\phi^{\prime}+{\cal L}_{,X}X^{\prime}$, and employing Eqs. (7), (12), (14), and (15), we obtain

	$\displaystyle 4q_{E}r^{2}A_{0}^{\prime 2}-M_{\rm Pl}^{2}r^{2}N^{-3/2}[2N^{2}(r^{2}h^{\prime\prime}-2h+2)$
	$\displaystyle+rN(2rhN^{\prime\prime}+3rh^{\prime}N^{\prime}+2hN^{\prime})-r^{2}hN^{\prime 2}]A_{0}^{\prime}$
	$\displaystyle+4Nq_{E}q_{M}^{2}/r^{2}=0\,,$		(16)

so that $A_{0}^{\prime}$ is known algebraically in terms of $h$, $N$, and its $r$ derivatives. Interestingly, the $\phi$-dependent terms completely vanish in Eq. (16).

We will consider SSS objects that are regular at $r=0$, including nonsingular BHs with event horizons [2, 3, 4, 5, 6, 7, 8, 9]. In such cases, the Ricci scalar $R$, the squared Ricci tensor $R_{\mu\nu}R^{\mu\nu}$, and the squared Riemann tensor $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ do not diverge at $r=0$. This requires that $h$ and $N$ are expanded around $r=0$, as [45]

	$\displaystyle h(r)$	$\displaystyle=$	$\displaystyle 1+\sum_{n=2}^{\infty}h_{n}r^{n}\,,$		(17)
	$\displaystyle N(r)$	$\displaystyle=$	$\displaystyle N_{0}+\sum_{n=2}^{\infty}N_{n}r^{n}\,,$		(18)

where $h_{n}$, $N_{0}$, and $N_{n}$ are constants. Note that $N_{0}$ is positive due to the condition (5). We substitute Eqs. (17)-(18) and their $r$ derivatives into Eq. (16). Solving the resulting equation for $A_{0}^{\prime}$ and expanding it around $r=0$, we find

$$A_{0}^{\prime}=\pm\frac{\sqrt{N_{0}}\sqrt{-(q_{E}q_{M})^{2}}}{q_{E}r^{2}}+{\cal O}(r^{0})\,.$$

(19)

Then, we have real solutions to $A_{0}^{\prime}$ only if

$$q_{E}q_{M}=0\,,$$

(20)

and hence the dyon BHs with mixed electric and magnetic charges are not allowed¹¹1Equivalently, we may study the lowest order of a discriminant of the second-order algebraic equation and show that it is always negative.. In the following, we will separate the discussion into the electrically and magnetically charged cases.

The linear stability of BHs can be analyzed by considering perturbations on the SSS background (3) [46, 47, 48, 49]. We write the metric tensor in the form $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$, where $\bar{g}_{\mu\nu}$ is the background value and $h_{\mu\nu}$ is the metric perturbation. We expand $h_{\mu\nu}$ in terms of the spherical harmonics $Y_{lm}(\theta,\varphi)$. Without loss of generality, we will focus on the mode $m=0$ and express $Y_{l0}$ as $Y_{l}$ in the following. We also omit the summation for $l$ for each perturbed variable.

We choose the four gauge conditions $h_{t\theta}=0$, $h_{\theta\theta}=0$, $h_{\varphi\varphi}=0$, and $h_{\theta\varphi}=0$. In this case, the four components of $\xi^{\mu}$ under the infinitesimal coordinate transformation $x^{\mu}\to x^{\mu}+\xi^{\mu}$ are fixed. Then, the components of $h_{\mu\nu}$ are given by [50, 51, 52, 44]

	$\displaystyle h_{tt}=f(r)H_{0}(t,r)Y_{l}(\theta),\quad h_{tr}=H_{1}(t,r)Y_{l}(\theta),\quad h_{t\theta}=0,$
	$\displaystyle h_{t\varphi}=-Q(t,r)(\sin\theta)Y_{l,\theta}(\theta),\;\;h_{rr}=f^{-1}(r)H_{2}(t,r)Y_{l}(\theta),$
	$\displaystyle h_{r\theta}=h_{1}(t,r)Y_{l,\theta}(\theta),\quad h_{r\varphi}=-W(t,r)(\sin\theta)Y_{l,\theta}(\theta),$
	$\displaystyle h_{\theta\theta}=0,\quad h_{\varphi\varphi}=0,\quad h_{\theta\varphi}=0\,,$		(28)

where $H_{0}$, $H_{1}$, $H_{2}$, $h_{1}$, $Q$, and $W$ are functions of $t$ and $r$.

We also decompose the scalar and vector fields, as

	$\displaystyle\phi$	$\displaystyle=$	$\displaystyle\bar{\phi}(r)+\delta\phi(t,r)Y_{l}(\theta),$		(29)
	$\displaystyle A_{\mu}$	$\displaystyle=$	$\displaystyle\bar{A}_{\mu}(r)+\delta A_{\mu}\,,$		(30)

where

	$\displaystyle\delta A_{t}=\delta A_{0}(t,r)Y_{l}(\theta),\qquad\delta A_{r}=\delta A_{1}(t,r)Y_{l}(\theta),\qquad$
	$\displaystyle\delta A_{\theta}=0,\qquad\delta A_{\varphi}=-\delta A(t,r)(\sin\theta)Y_{l,\theta}(\theta)\,.$		(31)

Here, we have set $\delta A_{\theta}=0$ by exploiting the fact that the action (2) respects $U(1)$ gauge invariance.

The odd-parity sector contains three perturbations $Q$, $W$, and $\delta A$, while there are seven perturbed fields $H_{0}$, $H_{1}$, $H_{2}$, $h_{1}$, $\delta\phi$, $\delta A_{0}$, and $\delta A_{1}$ in the even-parity sector. After integrating out all nondynamical fields, we have one scalar perturbation $\delta\phi$, two vector modes arising from $\delta A_{\mu}$, and two tensor polarizations arising from the gravity sector. For the electric BH, the linear stability conditions of such five dynamical perturbations were derived in Refs. [43, 44] for more general Maxwell-Horndeski theories. For the magnetic BH, the stability issue has not been addressed yet in theories given by the action (2). In the following, we will obtain the full second-order action of linear perturbations and study the stability of SSS objects with electric and magnetic charges, in turn.