Instability of nonsingular black holes in nonlinear electrodynamics

The vacuum black hole (BH) solutions predicted in General Relativity (GR) possess curvature singularities at their centers ($r=0$). Under several physical assumptions of spacetime and matter, Penrose showed that such singularities arise as an endpoint of the gravitational collapse [1]. However, the existence of singularity-free BHs is not precluded by relaxing some of these assumptions. For example, the nonsingular BH proposed by Bardeen [2] has a regular center due to the absence of global hyperbolicity of spacetime postulated in Penrose’s theorem. Since quantum corrections to GR may manifest themselves in extreme gravity regimes, it is important to investigate whether curvature singularities can be eliminated in extended theories of gravity or matter.

It is known that nonlinear electrodynamics (NED) in the framework of GR allows the existence of spherically symmetric and static (SSS) BHs with regular centers [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The Lagrangian ${\cal L}$ of NED depends on $F=-(1/4)F_{\mu\nu}F^{\mu\nu}$ nonlinearly, where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength of a covector field $A_{\mu}$. The action of Einstein-NED theory is given by

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

(1)

where $g$ is the determinant of a metric tensor $g_{\mu\nu}$, $M_{\rm Pl}$ is the reduced Planck mass, and $R$ is the Ricci scalar. For example, Euler-Heisenberg theory [13] in quantum electrodynamics has a low-energy effective Lagrangian ${\cal L}=F+\alpha F^{2}$, where $\alpha F^{2}$ is a correction to the Maxwell term $F$. NED also accommodates Born-Infeld theory [14] with the Lagrangian ${\cal L}=\mu^{4}[1-\sqrt{1-2F/\mu^{4}}]$, in which the electron’s self-energy is nondivergent by the finiteness of $F$. These subclasses of NED, when coupled with GR, give rise to hairy BH solutions [15, 16, 17, 18], but there are in general curvature singularities at $r=0$ unless the functional form of ${\cal L}(F)$ is further extended.

The common procedure for realizing nonsingular BHs in NED is to assume the existence of regular metrics and reconstruct the Lagrangian ${\cal L}$ from the field equations of motion [3]. In particular, for the magnetic BH, one can directly express ${\cal L}$ as a function of $F$ [5]. Nonsingular electric BHs constructed in this manner have finite values of $F$ and the electric field everywhere. For magnetic BHs, there is the divergence of $F$ at $r=0$, but the force exerted on a charged test particle is finite at any distance $r$ including $r=0$ [6]. Nonsingular BHs can be designed to meet standard energy conditions, but not all regular solutions do [19]. For instance, violation of dominant energy conditions occurs for some regular BHs [2, 20].

Thus, at the background level, there are consistent regular BHs in NED evading Penrose’s theorem. To see whether these BHs do not suffer from theoretical pathologies, we need to address their linear stability by analyzing perturbations on the SSS background. The BH perturbations in NED were studied in Refs. [21, 22, 23, 24, 25] by focusing on the stability outside the outer horizon. It was found that there are viable parameter spaces in which the nonsingular BHs are plagued by neither ghosts nor Laplacian instabilities. However, the BH stability inside the horizon, especially around its center, is still unclear and has not been investigated to our best knowledge. In this letter, we will show that all nonsingular BHs in NED, including both electric and magnetic ones, are unstable due to the angular Laplacian instability around $r=0$.

The line element on the SSS background is given by

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

(2)

where $f$ and $h$ are functions of the radial distance $r$. We consider the following covector-field configuration

$$A_{\mu}=\left[A_{0}(r),0,0,-q_{M}\cos\theta\right]\,,$$

(3)

where $A_{0}$ is a function of $r$, and $q_{M}$ is a constant corresponding to a magnetic charge. Since the theory (1) has $U(1)$ gauge invariance, $A_{\mu}\to A_{\mu}+\partial_{\mu}\chi_{\rm gauge}$, the longitudinal component $A_{1}(r)$ has been eliminated by choosing the gauge field as $\chi_{\rm gauge}(r)=-\int^{r}A_{1}(\rho){\rm d}\rho$.

Varying the action (1) with respect to $A_{0}$, $f$, and $h$, it follows that

	$\displaystyle\left(\sqrt{h/f}\,r^{2}{\cal L}_{,F}A_{0}^{\prime}\right)^{\prime}$	$\displaystyle=$	$\displaystyle 0\,,$		(4)
	$\displaystyle h^{\prime}-\frac{1-h}{r}$	$\displaystyle=$	$\displaystyle\frac{r}{M_{\rm Pl}^{2}f}\left(f{\cal L}-hA_{0}^{\prime 2}{\cal L}_{,F}\right),$		(5)
	$\displaystyle\frac{f^{\prime}}{f}-\frac{h^{\prime}}{h}$	$\displaystyle=$	$\displaystyle 0\,,$		(6)

where a prime represents the derivative with respect to $r$, and ${\cal L}_{,F}\equiv{\rm d}{\cal L}/{\rm d}F$. The explicit form of $F$ is given by

$$F=\frac{hA_{0}^{\prime 2}}{2f}-\frac{q_{M}^{2}}{2r^{4}}\,.$$

(7)

From Eq. (6), we obtain $f=Ch$, where $C$ is a constant. Using time reparametrization invariance, we can impose $f\to 1$ as $r\to\infty$, whereas the asymptotic flatness sets $h\to 1$ at spatial infinity. Then we have $C=1$, so that

$$f=h\,.$$

(8)

With this condition, Eqs. (4) and (5) give

	$\displaystyle A_{0}^{\prime}$	$\displaystyle=$	$\displaystyle\frac{q_{E}}{r^{2}{\cal L}_{,F}}\,,$		(9)
	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle r^{-2}\left[q_{E}A_{0}^{\prime}+M_{\rm Pl}^{2}(rf^{\prime}+f-1)\right]\,,$		(10)

where $q_{E}$ is a constant corresponding to an electric charge. Taking the $r$-derivative of Eq. (10) and combining it with Eq. (9) to eliminate ${\cal L}_{,F}$, we obtain

$$2q_{E}r^{4}A_{0}^{\prime 2}+M_{\rm Pl}^{2}\left(2f-2-r^{2}f^{\prime\prime}\right)r^{4}A_{0}^{\prime}+2q_{E}q_{M}^{2}=0\,,$$

(11)

which algebraically determines $A_{0}^{\prime}$ in terms of $f$ and its second radial derivative.

We are interested in nonsingular BHs with regular centers. To avoid the singularities of Ricci scalar $R$, Ricci squared $R_{\mu\nu}R^{\mu\nu}$, Riemann squared $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ at $r=0$, we require that $f$ is expanded around $r=0$ as [26]

$$f(r)=1+\sum_{n=2}^{\infty}f_{n}r^{n}\,,$$

(12)

where $f_{n}$’s are constants. The deviation of $f(0)$ from 1 results in conical singularities. Any power of $n$ smaller than 2 leads to curvature singularities. Substituting Eq. (12) and its second radial derivative into Eq. (11), two branches of $A_{0}^{\prime}$ have the leading-order terms $\pm\sqrt{-(q_{E}q_{M})^{2}}/(q_{E}r^{2})$. This means that there exist real solutions to $A_{0}^{\prime}$ only if

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

(13)

Thus, the presence of dyon BHs with $q_{E}\neq 0$ and $q_{M}\neq 0$ is forbidden from the regularity of $f$ at $r=0$. From Eq. (13), either $q_{E}$ or $q_{M}$ must be 0. This non-existence of regular dyon BH solutions breaks the electromagnetic duality present in linear electrodynamics, where the property of BHs is determined by their mass and total charge $q_{T}=\sqrt{q_{E}^{2}+q_{M}^{2}}$ (see e.g., [27, 28]).

To study the linear stability of electric and magnetic BHs, we consider metric and vector-field perturbations on the SSS background (2) [29, 30, 31]. For the components of metric perturbations $h_{\mu\nu}$, we choose

	$\displaystyle h_{tt}=f(r)H_{0}(t,r)Y_{l}(\theta),\quad h_{tr}=H_{1}(t,r)Y_{l}(\theta),\quad h_{t\theta}=0,$
	$\displaystyle h_{t\varphi}=-Q(t,r)(\sin\theta)Y_{l,\theta}(\theta),\;\;h_{rr}=f^{-1}(r)H_{2}(t,r)Y_{l}(\theta),$
	$\displaystyle h_{r\theta}=h_{1}(t,r)Y_{l,\theta}(\theta),\quad h_{r\varphi}=-W(t,r)(\sin\theta)Y_{l,\theta}(\theta),$
	$\displaystyle h_{\theta\theta}=0,\quad h_{\varphi\varphi}=0,\quad h_{\theta\varphi}=0,$		(23)

where $Y_{l}(\theta)$ is the $m=0$ component of spherical harmonics $Y_{lm}(\theta,\varphi)$. On the SSS background, we can focus on the axisymmetric modes ($m=0$) without loss of generality. The covector-field perturbation $\delta A_{\mu}$ has the following components

	$\displaystyle\delta A_{t}=\delta A_{0}(t,r)Y_{l}(\theta),\qquad\delta A_{r}=\delta A_{1}(t,r)Y_{l}(\theta),\qquad$
	$\displaystyle\delta A_{\theta}=0,\qquad\delta A_{\varphi}=-\delta A(t,r)(\sin\theta)Y_{l,\theta}(\theta)\,,$		(24)

where the choice $\delta A_{\theta}=0$ is an outcome of the presence of $U(1)$ gauge symmetry.¹¹1Since there is $U(1)$ gauge invariance under the transformation, $A_{\mu}\to A_{\mu}+\partial_{\mu}\chi_{\rm gauge}$, we can eliminate the even-parity mode $\delta A_{\theta}=\delta A_{2}(t,r)Y_{l,\theta}(\theta)$ by choosing the perturbed gauge field $\delta\chi_{\rm gauge}=-\delta A_{2}(t,r)Y_{l}(\theta)$ and making the field redefinitions $\delta A_{0}^{\rm new}=\delta A_{0}-\dot{\delta A}_{2}$, $\delta A_{1}^{\rm new}=\delta A_{1}-\delta A_{2}^{\prime}$. We note that the gauge choice (23) completely fixes the residual gauge degrees of freedom under the infinitesimal transformation $x^{\mu}\to x^{\mu}+\xi^{\mu}$.

The three perturbations $Q$, $W$, $\delta A$ belong to those in the odd-parity sector, while the six perturbations $H_{0}$, $H_{1}$, $H_{2}$, $h_{1}$, $\delta A_{0}$, $\delta A_{1}$ correspond to those in the even-parity sector. We focus on the multiple modes $l\geq 2$ and expand the action (1) up to second order in perturbed fields. The total quadratic-order action can be expressed as ${\cal S}^{(2)}=\int{\rm d}t{\rm d}r\,({\cal L}_{1}+{\cal L}_{2})$, where ${\cal L}_{1}$ and ${\cal L}_{2}$ are given in Appendix A. We introduce the following Lagrange multipliers

	$\displaystyle\chi$	$\displaystyle=$	$\displaystyle r\dot{W}-rQ^{\prime}+2Q-\frac{2{\cal L}_{,F}rA_{0}^{\prime}}{M_{\rm Pl}^{2}}\delta A\,,$		(25)
	$\displaystyle V$	$\displaystyle=$	$\displaystyle\delta A_{0}^{\prime}-\dot{\delta A_{1}}+\frac{A_{0}^{\prime}}{2}(H_{0}-H_{2})\,,$		(26)

where a dot represents the derivative with respect to $t$. The dynamical fields $\chi$ and $V$ correspond to the odd-parity gravitational perturbation and the even-parity electromagnetic perturbation, respectively. We also have the odd-parity electromagnetic mode $\delta A$ and the even-parity gravitational mode $\psi$ defined by

$\displaystyle\psi=rH_{2}-Lh_{1}\,,\quad{\rm where}\quad L=l(l+1)\,.$

(27)

Following the procedure explained in Appendix A, the second-order action, after the elimination of all nondynamical perturbations and the integration by parts, is expressed in the form

$$\tilde{{\cal S}}^{(2)}=\int{\rm d}t{\rm d}r\left(\dot{\vec{\Psi}}^{t}{\bm{K}}\dot{\vec{\Psi}}+\vec{\Psi}^{\prime t}{\bm{G}}\vec{\Psi}^{\prime}+\vec{\Psi}^{t}{\bm{M}}\vec{\Psi}+\vec{\Psi}^{\prime t}{\bm{S}}\vec{\Psi}\right)\,,$$

(28)

where ${\bm{K}},{\bm{G}},{\bm{M}}$ are $4\times 4$ symmetric matrices with components like $K_{11}$, ${\bm{S}}$ is a $4\times 4$ antisymmetric matrix, and

$$\vec{\Psi}^{t}=\left(\chi,\delta A,\psi,V\right)\,.$$

(29)

In the eikonal limit ($l\gg 1$), we will derive the linear stability conditions for electric and magnetic BHs. Unlike past related works [21, 25], our results can be applied to the stability for both $f>0$ and $f<0$.

For the electric BH, we compute Eq. (34) by differentiating Eq. (9) and using the relation $F=A_{0}^{\prime 2}/2$. This gives ${\cal L}_{,FF}=-q_{E}(rA_{0}^{\prime\prime}+2A_{0}^{\prime})/(r^{3}A_{0}^{\prime\prime}A_{0}^{\prime 3})$ and hence $c_{E}^{2}=-rA_{0}^{\prime\prime}/(2A_{0}^{\prime})$. By using Eq. (14), we obtain

$$c_{E}^{2}=c_{f}^{2}\equiv-\frac{r(r^{2}f^{\prime\prime\prime}+2rf^{\prime\prime}-2f^{\prime})}{2(r^{2}f^{\prime\prime}-2f+2)}\,,$$

(38)

which depends on $f$ and its $r$ derivatives alone.

For the magnetic BH, we take the $F$ derivative of Eq. (20) and exploit the relation $F=-q_{M}^{2}/(2r^{4})$. Then, we find that $c_{M}^{2}$ in Eq. (36) reduces to $c_{f}^{2}$ in Eq. (38). Thus, for a given metric function $f(r)$, the squared angular propagation speeds $c_{E}^{2}$ and $c_{M}^{2}$ can be expressed in a unified manner. We have $c_{f}^{2}=1$ at any distance $r$ for the RN metric $f=1-2M/r+q^{2}/(2M_{\rm Pl}^{2}r^{2})$, but this property does not hold for nonsingular BHs.

Let us consider the nonsingular BH with the expansion (12) of $f$ around $r=0$. Since $f>0$ in this regime, the $t$ and $r$ coordinates play the timelike and spacelike roles, respectively. The expansion of $c_{f}^{2}$ leads to

$$c_{f}^{2}=-\frac{3}{2}-\frac{5f_{4}}{4f_{3}}r+\frac{25f_{4}^{2}-36f_{3}f_{5}}{8f_{3}^{2}}r^{2}+{\cal O}(r^{3})\,,$$

(39)

which is valid for $f_{3}\neq 0$. Nonsingular BHs like (18) and (21) correspond to $f_{3}=0$, in which case we have

$$c_{f}^{2}=-2-\frac{9f_{5}}{10f_{4}}r+\frac{81f_{5}^{2}-140f_{4}f_{6}}{50f_{4}^{2}}r^{2}+{\cal O}(r^{3})\,.$$

(40)

Thus, in both cases, the leading-order terms of $c_{f}^{2}$ are negative. For the metric function $f=1+f_{n}r^{n}+\mathcal{O}(r^{n+1})$, we have $c_{f}^{2}=-n/2+{\cal O}(r)$ and hence $c_{f}^{2}\leq-1$ for $n\geq 2$.

We study the behavior of dynamical perturbations $V$ and $\psi$ around $r=0$ for the electric BH. Expressing those fields as $V=\tilde{V}(t)e^{ikr}$ and $\psi=\tilde{\psi}(t)e^{ikr}$ for $f>0$ and taking the limits $l\gg 1$ and $l\gg kr$, the time-dependent parts approximately obey the differential equations

	$\displaystyle\ddot{\tilde{V}}$	$\displaystyle+c_{f}^{2}\,\frac{fl^{2}}{r^{2}}\tilde{V}=\frac{f^{2}\Omega_{f}c_{f}^{4}M_{\rm Pl}^{2}}{r^{3}q_{E}}\tilde{\psi}\,,$		(41)
	$\displaystyle\ddot{\tilde{\psi}}$	$\displaystyle+\frac{fl^{2}}{r^{2}}\tilde{\psi}=\frac{l^{2}q_{E}}{rc_{f}^{2}M_{\rm Pl}^{2}}\tilde{V}\,,$		(42)

where $\Omega_{f}\equiv r^{2}f^{\prime\prime}-2f+2$. If we use the expansion $f=1+f_{n}r_{n}+{\cal O}(r^{n+1})$ around $r=0$, we have that $\Omega_{f}=(n^{2}-n-2)\,f_{n}r^{n}+\mathcal{O}(r^{n+1})$. It is possible to close Eqs. (41) and (42) for one single variable, say $\tilde{\psi}$, finding

$$\ddddot{\tilde{\psi}}+\frac{(1+c_{f}^{2})\,fl^{2}}{r^{2}}\ddot{\tilde{\psi}}+\frac{c_{f}^{2}f^{2}l^{4}}{r^{4}}\tilde{\psi}\simeq 0\,,$$

(43)

where we have neglected the term $-(f^{2}c_{f}^{2}\Omega_{f}l^{2}/r^{4})\tilde{\psi}$. Assuming the solution to Eq. (43) in the form $\tilde{\psi}(t)\propto e^{-i\omega t}$, we obtain

$$\omega^{2}=c_{f}^{2}\,\frac{fl^{2}}{r^{2}}\,,\qquad\omega^{2}=\frac{fl^{2}}{r^{2}}\,.$$

(44)

Since $c_{f}^{2}<0$ in a non-empty set centered around $r=0$, there is always a growing-mode solution ($\omega^{2}=c_{f}^{2}fl^{2}/r^{2}$) besides a stable one ($\omega^{2}=fl^{2}/r^{2}$). However, the presence of the former is enough to make the nonsingular BH unstable. We note that $\tilde{V}$ obeys the same form of a fourth-order differential equation as Eq. (43), so that the two dynamical perturbations $\psi$ and $V$ in the even-parity sector (B) are subject to exponential growth.

The enhancement of $\psi$ and $V$ works as a backreaction to the background BH solution. Then, the background metric is no longer maintained as the steady forms like (18) and (21). For the magnetic BH, the same exponential growth of dynamical perturbations occurs for $\psi$ and $\delta A$ in the sector (D). Such instability is generic for all nonsingular BHs constructed in the framework of NED–including those of Bardeen with metric $f=1-2Mr^{2}/(r^{2}+r_{0}^{2})^{3/2}$ [2] and Hayward with metric $f=1-2Mr^{2}/(r^{3}+2Mr_{0}^{2})$ [20].

A typical time scale of instability arising from the negative value of $c_{f}^{2}$ around $r=0$ is estimated as

$$t_{\rm ins}\simeq\frac{r}{\sqrt{-c_{f}^{2}}l}\,.$$

(45)

We recall that $-c_{f}^{2}$ is of order $c^{2}$, where we restored the speed of light $c$. Since the distance $r$ associated with Laplacian instability is less than the outer horizon radius $r_{h}$, $t_{\rm ins}$ is much shorter than $r_{h}/c$ for $l\gg 1$. If we consider a BH with $r_{h}=10$ km, we have $t_{\rm ins}\lesssim 10^{-5}/l$ sec.

The above results show that nonsingular BHs in NED are always plagued by angular Laplacian instability around $r=0$. For example, the BH solution (21) has the following squared angular propagation speed

$$c_{f}^{2}=\frac{r^{2}-2r_{0}^{2}}{r^{2}+r_{0}^{2}}\,.$$

(46)

As we estimated in Eq. (40), we have $c_{f}^{2}=-2$ at $r=0$. While $c_{f}^{2}$ approaches 1 as $r\to\infty$, $c_{f}^{2}$ is negative in the region $r<\sqrt{2}r_{0}$.

In Fig. 1, we plot $c_{f}^{2}$ and $f$ for the metric (21) with $M=4r_{0}$, in which case there are two horizons at $r_{1}=0.69r_{0}$ and $r_{2}=6.44r_{0}$. Since the expression (46) is valid at any distance $r\geq 0$, there is angular Laplacian instability for $r<\sqrt{2}r_{0}$ (including the region $r_{1}\leq r<\sqrt{2}r_{0}$ with $f<0$). The crucial point is that nonsingular BHs always have a finite range of $r$ where $f$ is expanded as Eq. (12) around $r=0$, in which regime $c_{f}^{2}$ is always negative. In Appendix B, we will confirm that the angular Laplacian instability is robust irrespective of the presence/absence of ghosts and the rescaling of dynamical perturbations.

We have shown that nonsingular BHs in NED are inevitably subject to angular Laplacian instability around $r=0$. This result holds for both electric and magnetic BHs, as the form (38) of $c_{f}^{2}$ is universal to both cases. The Laplacian instability we found is a physical one, in that the even-parity gravitational perturbation $\psi$ is subject to exponential growth through the angular instability of vector-field perturbations ($V$ for the electric BH and $\delta A$ for the magnetic BH). The backreaction of enhanced perturbations to the background would not keep the regular metrics like (18) and (21) as they are.

Our no-go result for the absence of stable static nonsingular BHs is valid for NED, but this is not the case for more general theories. For example, it is of interest to study what happens by incorporating an additional scalar field $\phi$ as the Lagrangian ${\cal L}(\phi,X,F)$ [33, 34, 35], where $X$ is a scalar kinetic term. If such theories with dynamical degrees of freedom still lead to the instability of regular BHs, nonlocal versions of the ultraviolet completion of gravity such as those proposed in Refs. [36, 37, 38, 39] may be the clue to the construction of stable nonsingular BHs.

We thank Valeri Frolov for useful discussions. The work of ADF was supported by the Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 20K03969. ST was supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No. 22K03642 and Waseda University Special Research Project No. 2024C-474.

The second-order action of perturbations, which is obtained after the integration with respect to $\theta$ and $\varphi$, can be written in the form ${\cal S}^{(2)}=\int{\rm d}t{\rm d}r\,({\cal L}_{1}+{\cal L}_{2})$, where

	$\displaystyle{\cal L}_{1}$	$\displaystyle=$	$\displaystyle a_{0}H_{0}^{2}+H_{0}\left[a_{1}H_{2}^{\prime}+La_{2}h_{1}^{\prime}+(a_{3}+La_{4})H_{2}+La_{5}h_{1}+La_{6}\delta A\right]+Lb_{1}H_{1}^{2}+H_{1}(b_{2}\dot{H}_{2}+Lb_{3}\dot{h}_{1})$		(A.1)
			$\displaystyle+c_{0}H_{2}^{2}+LH_{2}(c_{1}h_{1}+c_{2}\delta A)+L(d_{0}\dot{h}_{1}^{2}+d_{1}h_{1}^{2})+Lh_{1}(d_{2}\delta A_{0}+d_{3}\delta A^{\prime})$
			$\displaystyle+s_{1}(\delta A_{0}^{\prime}-\dot{\delta A_{1}})^{2}+(s_{2}H_{0}+s_{3}H_{2}+Ls_{4}\delta A)(\delta A_{0}^{\prime}-\dot{\delta A_{1}})+L(s_{5}\delta A_{0}^{2}+s_{6}\delta A_{1}^{2})\,,$
	$\displaystyle{\cal L}_{2}$	$\displaystyle=$	$\displaystyle L[p_{1}(r\dot{W}-rQ^{\prime}+2Q)^{2}+p_{2}\delta A(r\dot{W}-rQ^{\prime}+2Q)+p_{3}\dot{\delta A^{2}}+p_{4}\delta A^{\prime 2}+Lp_{5}\delta A^{2}+(Lp_{6}+p_{7})W^{2}$		(A.2)
			$\displaystyle~{}~{}+(Lp_{8}+p_{9})Q^{2}+p_{10}Q\delta A_{0}+p_{11}Qh_{1}+p_{12}W\delta A_{1}]\,,$

where we used the condition $h=f$, and

	$\displaystyle a_{0}=\frac{r^{2}}{8}A_{0}^{\prime 2}({\cal L}_{,F}+A_{0}^{\prime 2}{\cal L}_{,FF}),\qquad a_{1}=-\frac{M_{\rm Pl}^{2}rf}{2},\qquad a_{2}=\frac{M_{\rm Pl}^{2}f}{2},\qquad a_{3}=-\frac{M_{\rm Pl}^{2}}{2}-\frac{r^{2}}{4}(2{\cal L}-A_{0}^{\prime 2}{\cal L}_{,F}+A_{0}^{\prime 4}{\cal L}_{,FF}),$
	$\displaystyle a_{4}=-\frac{M_{\rm Pl}^{2}}{4},\qquad a_{5}=\frac{M_{\rm Pl}^{2}(f+1)+r^{2}({\cal L}-A_{0}^{\prime 2}{\cal L}_{,F})}{4r},\qquad a_{6}=\frac{q_{M}}{2r^{2}}({\cal L}_{,F}-A_{0}^{\prime 2}{\cal L}_{,FF}),\qquad b_{1}=-a_{4},$
	$\displaystyle b_{2}=M_{\rm Pl}^{2}r,\qquad b_{3}=2a_{4}\,,\qquad c_{0}=-\frac{a_{3}}{2},\qquad c_{1}=-a_{5},\qquad c_{2}=-a_{6},\qquad d_{0}=-a_{4},\qquad d_{1}=\frac{f}{2r^{4}}(M_{\rm Pl}^{2}r^{2}-q_{M}^{2}{\cal L}_{,F}),$
	$\displaystyle d_{2}=-A_{0}^{\prime}{\cal L}_{,F},\qquad d_{3}=-\frac{q_{M}f{\cal L}_{,F}}{r^{2}},\qquad s_{1}=\frac{r^{2}}{2}({\cal L}_{,F}+A_{0}^{\prime 2}{\cal L}_{,FF}),\qquad s_{2}=A_{0}^{\prime}s_{1},\qquad s_{3}=-A_{0}^{\prime}s_{1},$
	$\displaystyle s_{4}=-\frac{q_{M}A_{0}^{\prime}{\cal L}_{,FF}}{r^{2}},\qquad s_{5}=\frac{{\cal L}_{,F}}{2f},\qquad s_{6}=-\frac{f{\cal L}_{,F}}{2},$		(A.3)
	$\displaystyle p_{1}=\frac{M_{\rm Pl}^{2}}{4r^{2}},\qquad p_{2}=\frac{d_{2}}{r},\qquad p_{3}=s_{5},\qquad p_{4}=s_{6},\qquad p_{5}=\frac{q_{M}^{2}{\cal L}_{,FF}-r^{4}{\cal L}_{,F}}{2r^{6}},\qquad p_{6}=-\frac{M_{\rm Pl}^{2}f}{4r^{2}},$
	$\displaystyle p_{7}=d_{1},\qquad p_{8}=\frac{M_{\rm Pl}^{2}}{4r^{2}f},\qquad p_{9}=-\frac{d_{1}}{f^{2}},\qquad p_{10}=-\frac{d_{3}}{f^{2}},\qquad p_{11}=-A_{0}^{\prime}fp_{10},\qquad p_{12}=-f^{2}p_{10}\,,$		(A.4)

where $s_{4}$ vanishes for both the electric BH ($q_{M}=0$) and the magnetic BH ($A_{0}^{\prime}=0$). The second-order Lagrangians (A.1) and (A.2) with the coefficients (A.3) and (A.4) are valid both for $f>0$ and $f<0$.