1 Erratum to: Eur. Phys. J. C (2020) 80:761 https://doi.org/10.1140/epjc/s10052-020-8346-3

The sole aim of writing this erratum is to draw the attention of our readers to the mechanism adopted by [1], exclusively section 4, which is possible only if the Hamilton–Jacobi equations with plasma distribution are under special constraints, i.e. the equatorial plane \(\theta =\pi /2\). In Ref. [2], Perlick et al. had broached a similar issue and successfully forged a plasma frequency using the independent coordinate functions f(r) and \(f(\theta )\) as \(\omega _p(x)^2=\frac{f(r)+f(\theta )}{\rho ^2}\). We shall revisit the shadow cast by the rotating charged Kerr–Newman black hole by using the latter plasma distribution in equation 4 of [1] that leads to the separation of the Hamilton–Jacobi equations for a generic unrestrained case, in this regard, figure 4 is precisely re-plotted.

The gravitational field of a Kerr–Newman spacetime in Boyer–Lindquist coordinates reads

$$\begin{aligned} ds^2= & {} -\frac{\Delta }{\rho ^2}(dt-a\sin ^2\theta d\phi )^2+\frac{\rho ^2}{\Delta }dr^2+\rho ^2d\theta ^2\nonumber \\&+\frac{\sin ^2\theta }{\rho ^2}\left( a dt-(r^2+a^2)d\phi \right) ^2, \end{aligned}$$
(1)

with

$$\begin{aligned} \rho ^2= & {} r^2+a^2\cos ^2\theta , \end{aligned}$$
(2)
$$\begin{aligned} \Delta= & {} r^2+a^2-2 M r+Q^2. \end{aligned}$$
(3)

The Hamiltonian that regulates dynamics in a curved spacetime surrounded with a plasma medium is defined as

$$\begin{aligned} {\mathcal {H}}=\frac{1}{2}\Big [g^{\mu \nu }p_{\mu }p_{\nu }+\omega _{\text {p}}(x)^2\Big ], \end{aligned}$$
(4)

where, \(\omega _p\) corresponds to the electron plasma frequency [3]

$$\begin{aligned} \omega _p(x)^2=\frac{4\pi e^2}{m_e}N_e(x). \end{aligned}$$
(5)

Further, the Hamilton–Jacobi equation for the photon motion takes the form

$$\begin{aligned} {\mathcal {H}}\Big (x,\frac{\partial S}{\partial x}\Big )=0\ ~, \end{aligned}$$
(6)

with the Jacobi action ansatz as

$$\begin{aligned} S=-\omega _0 t+p_{\phi }\phi +S_{r}(r)+S_{\theta }(\theta )\ ~, \end{aligned}$$
(7)

where \(p_{\phi }\) and \(\omega _0\) identify the conserved quantities termed respectively as the angular momentum and energy of the test particle. The mathematical expression for plasma frequency is structured below:

$$\begin{aligned} \omega _p(x)^2=\frac{f(r)+f(\theta )}{\rho ^2}\ , \end{aligned}$$
(8)

here, f(r) and \(f(\theta )\) are associated with the radial and angular functions, respectively.

The appropriate choice of plasma distribution brings us to two separate equations by introducing the Carter constant \({{{\mathcal {C}}}}\)

$$\begin{aligned}&(S'_\theta )^2+\big (a\omega _0\sin \theta -\frac{p_\phi }{\sin \theta }\big )^2+f(\theta )={\mathcal {C}}, \end{aligned}$$
(9)
$$\begin{aligned}&\frac{1}{\Delta } -\Delta (S'_r)^2 ((r^2+a^2)\omega _0-ap_\phi )^2-f(r)={\mathcal {C}}~. \end{aligned}$$
(10)
Fig. 1
figure 1

The variation of the spin, charge, plasma and inclination parameters, sequentially, shows the shadow of a Kerr–Newman black hole surrounded by a Shapiro-like plasma distribution \(f(r)=\omega _c^2\sqrt{M^3r}\), when viewed by a local observer

Full size image

Using the Eq. (9) one can easily get the equations for photon trajectories in a plasma medium as

$$\begin{aligned} \rho ^{2}\frac{dt}{d\tau }= & {} a(p_\phi -a\omega _0\sin ^{2}\theta ) +\frac{r^{2}+a^{2}}{\Delta } P(r), \end{aligned}$$
(11)
$$\begin{aligned} \rho ^{2}\frac{dr}{d\tau }= & {} \pm \sqrt{{\mathcal {R}}}, \end{aligned}$$
(12)
$$\begin{aligned} \rho ^{2}\frac{d\theta }{d\tau }= & {} \pm \sqrt{\Theta }, \end{aligned}$$
(13)
$$\begin{aligned} \rho ^{2}\frac{d\phi }{d\tau }= & {} \frac{p_\phi }{\sin ^{2}\theta }-a\omega _0+\frac{a}{\Delta }P(r)\ , \end{aligned}$$
(14)

where P(r) is introduced for simplicity and has the form

$$\begin{aligned} P(r)=(r^2+a^2)\omega _0-ap_\phi \ , \end{aligned}$$
(15)

\({\mathcal {R}}\) and \(\Theta \) are the corresponding radial and angular potentials, defined as

$$\begin{aligned} {\mathcal {R}}= & {} P(r)^2-\Delta \Big [{\mathcal {Q}}+(p_\phi -a\omega _0)^2+f(r)\Big ], \end{aligned}$$
(16)
$$\begin{aligned} \Theta= & {} {\mathcal {Q}}+\cos ^2\theta \left( a^2\omega ^2-p_\phi ^2\sin ^{-2}\theta \right) -f(\theta ), \end{aligned}$$
(17)

where \({\mathcal {Q}}={\mathcal {K}}-(p_\phi -a\omega _0)^2\).

The contour of a black hole shadow is obtained by using the conditions \({\mathcal {R}}=0\) and \({\mathcal {R}}'=0\). Having solved these equations simultaneously provides the constants of motion \({\mathcal {Q}}\) and \(p_\phi \) in terms of the radius r,

$$\begin{aligned} {\mathcal {Q}}= & {} \frac{\left( a p_\phi - \omega _0 \left( a^2+r^2\right) \right) ^2}{\Delta }-(p_\phi -a \omega _0)^2-f(r), \end{aligned}$$
(18)
$$\begin{aligned} p_\phi= & {} \frac{\omega _0}{a}\left[ r^2+a^2{-} \frac{\Delta }{a \Delta '} \left( \sqrt{4 r^2-\frac{f'(r) \Delta '(r)}{\omega _0}}+2 r \right) \right] .\nonumber \\ \end{aligned}$$
(19)

The celestial coordinates (\(\alpha \), \(\beta \)) that determines the silhouette of black hole shadow in the presence of plasma are evaluated as [4]

$$\begin{aligned} \alpha= & {} -\frac{p_\phi }{\omega _0\sin {\theta _0}}, \end{aligned}$$
(20)
$$\begin{aligned} \beta= & {} \pm \frac{1}{\omega _0}\sqrt{{\mathcal {Q}}+\cos ^2{\theta _0}\Big (a^2\omega _0^2-\frac{p_\phi ^2}{\sin ^2{\theta _0}}\Big )-f(\theta _0)}. \end{aligned}$$
(21)

Here, we choose Shapiro-like plasma frequency [4]

$$\begin{aligned} f(r)= & {} \omega _c^2\sqrt{M^3r}, \end{aligned}$$
(22)
$$\begin{aligned} f(\theta )= & {} 0, \end{aligned}$$
(23)

with

$$\begin{aligned} \omega _p^2= & {} \omega _c^2\frac{\sqrt{M^3r}}{r^2+a^2\cos ^2\theta } , \end{aligned}$$
(24)

where \(\omega _c\) is a constant and M is the mass of the black hole.

Now, one may easily get the boundary of a black hole shadow by combining Eqs. (20), (21), (22), and (23). Figure 1 illustrates the evolution of shadow by varying the parameters a, Q, \(\omega _c^2/\omega _0^2\) and \(\theta _0\). Complying with the physical laws, the shadow size decreases by increasing the spin, charge and plasma parameters, also, a shift of the position is observed by changing the inclination parameter.