Conformal scalar NUT-like dyons in conformal electrodynamics

In the light scattering process, nonlinearities of the electromagnetic interaction are discovered and experimentally proved [1]. The exploration of the nonlinear electrodynamics (NLE) theories in general relativity (GR) theory was inspired by solving the singularity problem of a black hole. Usually the NLE theories are derived from the Lagrangians constructed with quadratic electromagnetic invariants $F_{\mu\nu}F^{\mu\nu}$ and $F_{\mu\nu}{}^{*}F^{\mu\nu}$. There is NLE depending only on the invariant $F_{\mu\nu}F^{\mu\nu}$ proposed by Max Born in [2], albeit the one depending on both invariants proposed in [3] known as the Born–Infeld theory, which preserves Maxwell equations’ electromagnetic duality invariance, seems to be more prevalent. Moreover, a one-loop QED correction to Maxwell’s theory incorporating vacuum polarization effects was raised by Heisenberg and Euler [4]. However, none of them keep the conformal invariance characteristics which is endowed to Maxwell theory. Recently, a nonlinear extension of the source-free Maxwell electrodynamics which preserves both conformal invariance and $SO(2)$ electromagnetic duality was proposed in [5, 6]. Following the convention used in [7], we in this paper call it the conformal electrodynamics, though in [5, 8, 9] it is also named as ModMax theory. The conformal electrodynamics is characterized by a dimensionless parameter $\gamma$, with the Maxwell theory being recovered in the $\gamma\to 0$ limit. For $\gamma>0$ the polarization mode is subluminal and for $\gamma<0$ the polarization mode is superluminal, so in this paper we consider the former case.

The no-hair theorem states that a black hole holds no more hairs other than mass, electric charge, and angular momentum [10]. A minimally coupled scalar cannot be held in the Einstein-scalar theory [11], however, being as a counterexample, the Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole in the Einstein-conformal scalar vacuum leads to a scalar hair [12] that being unbounded at the horizon but not physically troublesome [13]. The Einstein-Maxwell-conformally coupled scalar (EMCS) theory triggered much attention in community [14, 15, 16], which, besides the BBMB black hole, also gives a three-dimensional black hole with the scalar field being regular everywhere [17] and a scalar hairy black hole with a constant scalar field [18]. For the latter one, emission rate of charged particles was investigated in [19], the weak cosmic censorship conjecture was tested in [20], and it was turned out to be stable against full perturbations [21, 22].

Recently, soon after the proposition of the conformal electrodynamics, its applications in the Einstein gravity were put forward [7, 8]. It was shown that there is a screening factor that shields the actual charges of the black hole, and the electric and magnetic fields change qualitatively comparing to the Einstein-Maxwell counterpart. In this paper, we will seek Taub-NUT-like black hole solution in the Einstein-conformal electromagnetic system with a conformally coupled scalar field, which substitutes the Maxwell field in the EMCS with the conformal electrodynamic field. The reason we choose the Taub-NUT-like spacetime is that it provides a representative candidate for which the magnetic fields emergent and the conformal electrodynamics becomes non-trivial. We want to investigate how the conformal electrodynamics and the conformally coupled scalar field interact to modify the Einstein gravity. One may also wonder how the conformal electrodynamics deviates from linear electrodynamics [7], in the situation where the conformally coupled scalar field is added. Thus we will show the strong gravitational effect of the solution, by investigating the innermost stable circular orbits (ISCOs) of charged massive particles around the black hole as well as studying the shadow of the object. To these ends, the NUTty dyons solution together with its thermodynamics will be given in Sec. II. The gravitational effects of the black hole will be elaborated in Sec. III where the ISCOs of charged massive particles will be studied and in Sec. IV where the shadow of the black hole will be explored. Throughout this paper the units are chosen to be ${\rm{c}}={\rm{\hbar}}={\rm{G}}=1$. Notice that in this paper ${\rm{e}}$ denotes the natural constant and $e$ is the electric charge parameter.

In this section, we will study the effects of the NLE dimensionless parameter $\gamma$ and the conformal scalar parameter $\alpha$ on the motion of the charged massive particle, and we will put our emphasis on the circular motion around the NUTty dyons and investigate the ISCO of the particles. Our related investigations in this section benefit from Refs. [39, 40, 41, 42]. The Lagrangian describing the motion of a charged massive particle reads

$$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+qA_{\mu}\dot{x}^{\mu},$$

(59)

where the overhead dot means ordinary derivative with regard to the affine parameter $\lambda$, which is connected to the proper time through the relation $\tau=\mu\lambda$, $q$ is the charge of the particle. The normalizing condition of the charged particle thus can be written as

$$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-\mu^{2},$$

(60)

where $\mu=0,\,1$ for the massless photon and massive particle, respectively. The momenta of the charged massive particle is

$$P_{\mu}=\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}=g_{\mu\nu}\dot{x}^{\nu}+qA_{\mu},$$

(61)

and the Hamiltonian can be obtained as

$$H=P_{\mu}\dot{x}^{\mu}-\mathcal{L}=\frac{1}{2}g^{\mu\nu}\left(P_{\mu}-qA_{\mu}\right)\left(P_{\nu}-qA_{\nu}\right).$$

(62)

To solve the equation of motion for the charged massive particle, we can seek help from the Hamilton-Jacobi method, with the Hamilton-Jacobi equation being written as

$$\frac{\partial S}{\partial\lambda}=H=\frac{1}{2}g^{\mu\nu}\left(P_{\mu}-qA_{\mu}\right)\left(P_{\nu}-qA_{\nu}\right),$$

(63)

where $S$ is the Jacobian action which can be written in the variable-separated form as

$$S=-\frac{1}{2}\lambda+J\phi-Et+S_{r}(r)+S_{\theta}(\theta),$$

(64)

with $J=P_{\phi}\,,E=-P_{t}$ individually the angular momentum and the energy of the charged particle measured at the spatial infinity as constants of motion due to the symmetries of the geometry. $S_{r}(r)$ and $S_{\theta}(\theta)$ are functions of $r$ and $\theta$ to be determined. With the help of Eqs. (63) and (64), we can have the separated equations fulfilled by the functions $S_{r}$ and $S_{\theta}$ as

$$-f(r)[{\rm{d}}S_{r}(r)/{\rm{d}}r]^{2}=\frac{K}{n^{2}+r^{2}}-\frac{[qa(r)+E]^{2}}{f(r)}+\mu^{2},$$

(65)

$$[{\rm{d}}S_{\theta}(\theta)/{\rm{d}}r]^{2}=K-\left[J\csc(\theta)+2En\cot(\theta)\right]^{2},$$

(66)

where $K$ is a separation constant.

According to the relations

$$P_{r}=\frac{\partial S}{\partial r},P_{\theta}=\frac{\partial S}{\partial\theta},$$

(67)

we further have

$$P_{r}=f(r)^{-1}\sqrt{R(r)},$$

(68)

$$P_{\theta}=\sqrt{\Theta(\theta)},$$

(69)

where we have denoted

$$R(r)=[E+qa(r)]^{2}-\left(\frac{K}{n^{2}+r^{2}}+\mu^{2}\right)f(r),$$

(70)

$$\Theta(\theta)=K-(2nE\cot\theta+J\sin^{-1}\theta)^{2},$$

(71)

which are radial and latitudinal effective potentials of the particles. As a result, the Jacobian action as a solution of the Hamilton-Jacobi equation can be written in the form

$\displaystyle S=-\frac{1}{2}\mu^{2}\tau-Et+J\phi+\int^{\theta}\sqrt{\Theta(\theta)}{\rm{d}}\theta+\int^{r}\frac{\sqrt{R(r)}}{f(r)}{\rm{d}}r.$

(72)

After differentiating the Jacobian action relative to the constants of the motion $K,\,\mu,\,E\,,J$ for the particle, we can obtain the integrated forms of the geodesics for the particle, expressed as

$$\int^{\theta}\frac{{\rm{d}}\theta}{\sqrt{\Theta}}=\int^{r}\frac{{\rm{d}}r}{(n^{2}+r^{2})\sqrt{R}},$$

(73)

$$\tau=\int^{r}\frac{{\rm{d}}r}{\sqrt{R}},$$

(74)

	$\displaystyle t=$	$\displaystyle\int^{\theta}\frac{-2n\cot\theta\left(2nE\cot\theta+J\sin^{-1}\theta\right){\rm{d}}\theta}{\sqrt{\Theta(\theta)}}$		(75)
		$\displaystyle+\int^{r}\frac{E+qa(r)}{f(r)\sqrt{R(r)}}{\rm{d}}r,$

$$\phi=\int^{\theta}\frac{2nE\cot\theta+J\sin^{-1}\theta}{\sqrt{\Theta(\theta)}\sin\theta}{\rm{d}}\theta.$$

(76)

Their first-order forms can be explicitly got as

$$\frac{{\rm{d}}t}{{\rm{d}}\tau}=\frac{-2n\cot\theta(2nE\cot\theta+J\sin^{-1}\theta)}{n^{2}+r^{2}}+\frac{E+qa(r)}{f(r)},$$

(77)

$$\frac{{\rm{d}}r}{{\rm{d}}\tau}=\sqrt{R(r)},$$

(78)

$$\frac{{\rm{d}}\theta}{{\rm{d}}\tau}=\frac{\sqrt{\Theta(\theta)}}{n^{2}+r^{2}},$$

(79)

$$\frac{{\rm{d}}\phi}{{\rm{d}}\tau}=\frac{2nE\cot\theta+J\sin^{-1}\theta}{(n^{2}+r^{2})\sin\theta}.$$

(80)

With these equations of motion in hand, we can immediately find the locations of the ISCO for the charged massive particles around the NUTty dyons. To that end, we should let

$$R\left(r_{i}\right)=0,\quad\left.\frac{{\rm{d}}R\left(r\right)}{{\rm{d}}r}\right|_{r=r_{i}}=0,\quad\left.\frac{{\rm{d}}^{2}R\left(r\right)}{{\rm{d}}r^{2}}\right|_{r=r_{i}}=0.$$

(81)

The first one is satisfied by the radial turning point; the second one together with the first one produces the circular orbit with constant radius; the last one restricts that the circular orbit is marginally stable, or in other words, it provides the innermost circular orbit. Besides, the latitudinal conditions

$$\Theta\left(\theta_{i}\right)=0,\quad\left.\frac{{\rm{d}}\Theta\left(\theta\right)}{{\rm{d}}\theta}\right|_{\theta=\theta_{i}}=0,\quad\left.\frac{{\rm{d}}^{2}\Theta\left(\theta\right)}{{\rm{d}}\theta^{2}}\right|_{\theta=\theta_{i}}=0$$

(82)

should also be satisfied, which ensure that the particle is stably located on the position with constant latitude. We numerically calculate the ISCO for the massive particles and obtain the related parameters, of which two representative ones, the radius and the latitude, are shown in Fig. 1. When the NLE parameter is large enough, all parameters tend to be constant. This is easy to understand once we glimpse at the blackening factor Eq. (37) where $\gamma$ penetrates. Besides, from the diagrams, we can also find other important properties. First, the changing tendency of the ISCO radius depends on the value range of the conformal scalar parameter $\alpha$. That is, if $\alpha\geqslant 0$, the ISCO radius increases with respect to the increasing NLE parameter $\gamma$; otherwise, if $\alpha$ belongs to the other branch where $\alpha<-e^{2}$, the ISCO radius behaves contrarily. Secondly, due to the emergence of the NUT parameter $n$, the ISCO will not locate on the equatorial plane. Here we see that how the conformal scalar parameter and the NLE parameter interplay to change the circular plane. With a non-negative conformal scalar parameter, the latitude of the ISCO plane decreases monotonically with respect to the increasing NLE parameter, and this style can also be shared by the case where $\alpha<-e^{2}$. However, if $\alpha$ is small enough, the effect of the NLE parameter $\gamma$ changes. There will be an extreme point where the effect of the NLE parameter $\gamma$ dominates. Lastly, when the NLE parameter is kept unchanged and the conformal scalar parameter increases, the ISCO radius increases while the ISCO latitude decreases. In other words, the closer the ISCO to the equatorial plane, the larger the ISCO radius.

In this section, we will explore the effects of the NLE dimensionless parameter $\gamma$ and the conformal scalar parameter $\alpha$ on the shadow of the conformally scalar NUTty dyons. Refs. [23, 43, 44, 45, 46, 47, 48, 49] are important literature to advance our work here. For simplicity, we set $m=1,E=1$ in what follows. Notice that $\mu=0$ for the photon, whose radial and latitude effective potentials are described by Eqs. (70) and (71) individually. Firstly we should obtain the circular orbits of the photons around the NUTty dyon black hole, which demands

$$R\left(r_{p}\right)=0,\quad\left.\frac{{\rm{d}}R\left(r\right)}{{\rm{d}}r}\right|_{r=r_{p}}=0,\quad\left.\frac{{\rm{d}}^{2}R\left(r\right)}{{\rm{d}}r^{2}}\right|_{r=r_{p}}>0,$$

(83)

where the last condition means that the circular orbit of the photon is radically unstable. We should also have

$$\Theta\left(\theta_{p}\right)=0,\quad\left.\frac{{\rm{d}}\Theta\left(\theta\right)}{{\rm{d}}\theta}\right|_{\theta=\theta_{p}}=0,\quad\left.\frac{{\rm{d}}^{2}\Theta\left(\theta\right)}{{\rm{d}}\theta^{2}}\right|_{\theta=\theta_{p}}<0,$$

(84)

where the last condition denotes that the orbit is latitudinally stable. Using them, we get the related characterized parameters of the photons on the circular orbit as

$$K_{p}=4n^{2}\tan\theta_{p}^{2},$$

(85)

$$J_{p}=-2n\sec\theta_{p},$$

(86)

$$\tan\theta_{p}=\frac{\sqrt{n^{2}+r_{p}^{2}}}{2n\sqrt{f(r_{p})}}.$$

(87)

Besides, the radius of the photon is restricted by the equation

$\displaystyle-f^{\prime}(r_{p})+\frac{2r_{p}f(r_{p})}{n^{2}+r_{p}^{2}}=0.$

(88)

The basis $\left\{\hat{e}_{(t)},\hat{e}_{(r)},\hat{e}_{(\theta)},\hat{e}_{(\varphi)}\right\}$ for the observer can be projected onto basis $\left\{\partial_{t},\partial_{r},\partial_{\theta},\partial_{\varphi}\right\}$ for the spacetime. One usually used orthogonal and normalized tetrad for the observer reads [23]

	$\displaystyle\hat{e}_{(t)}$	$\displaystyle=\sqrt{\frac{g_{\phi\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}\left(\partial_{t}-\frac{g_{t\phi}}{g_{\phi\phi}}\partial_{\phi}\right),$		(89)
	$\displaystyle\hat{e}_{(r)}$	$\displaystyle=\frac{1}{\sqrt{g_{rr}}}\partial_{r},$
	$\displaystyle\hat{e}_{(\theta)}$	$\displaystyle=\frac{1}{\sqrt{g_{\theta\theta}}}\partial_{\theta},$
	$\displaystyle\hat{e}_{(\phi)}$	$\displaystyle=\frac{1}{\sqrt{g_{\phi\phi}}}\partial_{\phi},$

which corresponds to a zero-angular-momentum-observer (ZAMO). An observer in this frame moves with an angular velocity $-g_{t\phi}/g_{\phi\phi}$ relative to spatial infinity, due to the dragging effect of the black hole.

The locally measured four-momentum of the photon can be obtained by its projecting onto $\hat{e}_{(t)}^{\mu}$,

		$\displaystyle p^{(t)}=-p_{\mu}\hat{e}_{(t)}^{\mu},$		(90)
		$\displaystyle p^{(i)}=p_{\mu}\hat{e}_{(i)}^{\mu},$

with $i=r,\theta,\phi$.

For the massless photon, we have

$$\left[p^{(t)}\right]^{2}=\left[p^{(r)}\right]^{2}+\left[p^{(\theta)}\right]^{2}+\left[p^{(\varphi)}\right]^{2}.$$

(91)

So the observation angles can be defined as

		$\displaystyle p^{(r)}=p^{(t)}\cos\tilde{\alpha}\cos\beta,$		(92)
		$\displaystyle p^{(\theta)}=p^{(t)}\sin\tilde{\alpha},$
		$\displaystyle p^{(\phi)}=p^{(t)}\cos\tilde{\alpha}\sin\beta,$

Explicitly, the angular coordinates can be written as

$$\sin\tilde{\alpha}=\frac{p^{(\theta)}}{p^{(t)}},$$

(93)

$$\tan\beta=\frac{p^{(\phi)}}{p^{(r)}}.$$

(94)

The perimeter radius of a circumference at constant $\theta$ and $r$ can be defined by

$$\tilde{r}\equiv\frac{1}{2\pi}\int_{0}^{2\pi}\sqrt{g_{\phi\phi}}{\rm{d}}\phi=\sqrt{g_{\phi\phi}}.$$

(95)

Then the Cartesian coordinate on the sky plane of the observer can be written as

	$\displaystyle x$	$\displaystyle\equiv-\tilde{r}\beta=-\tilde{r}\arctan\left[\frac{p^{(\phi)}}{p^{(r)}}\right]$		(96)
		$\displaystyle=\left.\sqrt{g_{\phi\phi}}\arctan\frac{f(r)\sqrt{g_{rr}}}{\sqrt{R(r)g_{\phi\phi}}}\right|_{(r_{o}\,,\theta_{o})},$

	$\displaystyle y$	$\displaystyle\equiv\tilde{r}\tilde{\alpha}=\tilde{r}\arctan\left[\frac{p^{(\theta)}}{p^{(r)}}\right]$		(97)
		$\displaystyle=\sqrt{\left(n^{2}+r^{2}\right)\sin^{2}\theta-4n^{2}f(r)\cos^{2}\theta}$
		$\displaystyle\left.\quad\times\arcsin\frac{\sqrt{\Theta(\theta)}}{(\zeta-\iota J)\sqrt{n^{2}+r^{2}}}\right|_{(r_{o}\,,\theta_{o})},$

where $r_{o}$ and $\theta_{o}$ are individually radial coordinate and inclination angle of the observer. Here we have denoted $\zeta\equiv\hat{e}_{(t)}^{t},\iota\equiv\hat{e}_{(t)}^{\phi}$, which are evaluated at the photon orbit.