Black hole shadow in an asymptotically-flat, stationary, and axisymmetric spacetime: The Kerr-Newman and rotating regular black holes
Naoki Tsukamoto





tsukamoto@rikkyo.ac.jp


School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 



The shadow of a black hole can be one of the strong observational evidences for stationary black holes.
If we see shadows at the center of galaxies, 
we would say whether the observed compact objects are black holes.
In this paper, we consider a formula for the contour of a shadow 
in an asymptotically-flat, stationary, and axisymmetric black hole spacetime. 
We show that the formula is useful for obtaining the contour of the shadow of several black holes 
such as the Kerr-Newman black hole and rotating regular black holes.
Using the formula, we can obtain new examples of the contour of the shadow of rotating black holes  
if assumptions are satisfied.   






Black hole shadow in an asymptotically-flat, stationary, and axisymmetric spacetime: 
The Kerr-Newman and rotating regular black holes

    Naoki Tsukamoto
    December 30, 2023
========================================================================================================================================







§ INTRODUCTION

Recently, LIGO detected three gravitational wave events from binary black hole systems <cit.>.
The events showed stellar-mass black holes really exist in our universe.
The physics in strong gravitational field near black holes will be an important topic in not only general relativity but also astronomy.
The black holes are described well by the Kerr black hole solution 
with an Arnowitt-Deser-Misner (ADM) mass M and an angular momentum J
which is an exact solution of the Einstein equations.
There, however, remains some possibility for the other black hole solutions 
because of uncertainty in measuring of gravitational waves <cit.>.

It is believed that there are supermassive black holes in the center of galaxies 
and that isolated stationary black holes would be described well by the Kerr solution.
From observations in weak gravitational fields, 
one can estimate the ADM mass M of suppermassive black holes and the distance between the observer 
and the black holes <cit.>.
To get an evidence that the suppermassive compact objects in the centers of galaxies are black holes 
and that they are not the other exotic compact objects predicted by general relativity,  
we should pay attention to phenomena in strong gravitational fields such as 
an optically thin emission region around a black hole <cit.>,
emissions from a geometrically thin accretion disk around a black hole <cit.>,
and the shadow made by a black hole <cit.>.

In the near future, we may check whether suppermassive black holes in our galaxy and near galaxies can be described by the Kerr black hole.
The Event Horizon Telescope challenges us to measure a shadow made by the suppermassive black hole in the center of our galaxy <cit.>. 
Therefore, the details of the shadows of rotating black holes have been investigated eagerly.  
From an observational viewpoint to check that isolated stationary black holes in nature can be described by the Kerr black hole well, 
the contours of the shadows of dozens rotating black holes have been calculated 
already <cit.>.
Semianalytic calculations <cit.> and
a new numerical method <cit.> of the contour of the shadow of rotating black holes were investigated.
Hioki and Maeda discussed a relation
between the shape of contour of the shadow and the inclination angle and the spin parameter of the Kerr black hole <cit.>.
Tsupko estimated analytically the spin parameter from the shape of the shadow of the Kerr black hole <cit.>.
A method for distinguishing the Kerr black hole from other rotating black holes with the shape of the shadow 
has been investigated by Tsukamoto et al. <cit.> and by Abdujabbarov et al. <cit.>.

The theoretical aspects related to the contour of black hole shadows also have been investigated
since the null geodesics near black holes and other compact objects would determine several important properties of spacetimes.
It was pointed that quasinormal modes <cit.> 
of a static, spherically symmetric, and asymptotically flat black hole in the eikonal limit 
are determined by the parameters of the unstable circular null geodesics <cit.>.
A tight relation between quasinormal modes and gravitational lensing near unstable circular null geodesic <cit.>
was also considered <cit.>.
Very recently, a relation between black hole shadows, spacetime instabilities, and fundamental photon orbits which is a generalization of circular photon orbits 
was discussed in Ref. <cit.>.

These close links between circular null geodesics, quasinormal modes, gravitational lensing, the shadows, and the other phenomena in strong gravitational fields 
in black hole spacetimes might be valid for only some well-known black holes and one might find counterexamples.
For an example, Konoplya and Stuchlik showed that an expected link between the circular null geodesics and quasinormal modes is broken 
in an asymptotically flat black hole spacetime in the Einstein-Lovelock theory <cit.>.
From the theoretical viewpoint, it would be worth to investigate the details of the null geodesic near less-known black holes and of their shadows.

In this paper, 
we investigate an simple analytic formula for the contour of the shadow of rotating black holes
and we apply the formula to two new examples of rotating black holes
and then we examine several known results with the formula.
We emphasize that the formula would help to categorize the shadow of dozens rotating black holes,
and that it can describe new examples of the contours of the shadows of rotating black holes.
 
This paper is organized as follows. 
In Sec. II, we introduce a line element describing a rotating black hole spacetime. 
In Sec. III, we obtain null geodesic equations in the rotating black hole spacetime. 
In Sec. IV, we investigate a formula for the contour of the shadow of the rotating black hole 
and we apply the formula to two examples of rotating black holes.  
In Sec. V, we examine several known results by using the formula.
In Sec. VI, we summarize our result.
In this paper, we use the units in which the light speed and Newton's constant are unity.







§ A LINE ELEMENT IN A ROTATING BLACK HOLE SPACETIME

A Newman-Janis algorithm <cit.> generates a stationary and axisymmetric black hole spacetime  
from a static and spherical black hole spacetime. 
The Newman-Janis algorithm was investigated to find the Kerr-Newman solution 
is an exact solution of the Einstein-Maxwell equations <cit.> 
from the Reissner-Nordström solution.
Recently, the Newman-Janis algorithm has been eagerly applied for regular black hole spacetime to obtain rotating regular black hole metrics.

We consider a line element in an asymptotically-flat, stationary, and axisymmetric black hole spacetime, in the Boyer-Lindquist coordinates,

    ds^2   =    -ρ^2Δ/Σdt^2 +Σsin^2θ/ρ^2[ dϕ-a(r^2+a^2-Δ)/Σdt ]^2
          +ρ^2/Δdr^2 +ρ^2dθ^2
       =   -(1-2m(r)r/ρ^2)dt^2-4m(r)arsin^2θ/ρ^2dϕ dt 
          +( r^2+a^2+ 2m(r)a^2rsin^2θ/ρ^2) sin^2θ dϕ^2
          +ρ^2/Δdr^2 +ρ^2dθ^2,

where

    ρ^2   ≡   r^2+a^2cos^2θ , 
    Δ(r)   ≡   r^2-2m(r)r+a^2, 
    Σ   ≡   (r^2+a^2)^2-a^2Δ(r) sin^2θ

and where a is a spin parameter defined as a≡ J/M 
and M and J are the ADM mass and the angular momentum of the black hole, respectively, 
and m(r) is a function with respect to the radial coordinate satisfying m(r) → M as r →∞.
The line element (<ref>) can be obtained 
when we apply the Newman-Janis algorithm for an asymptotically-flat, static, and spherical black hole spacetime with a line element <cit.>,

    ds^2   =   -(1-2m(r)/r)dt^2 +(1-2m(r)/r)^-1dr^2
          +r^2 ( dθ^2 +sin^2θ dϕ^2).


We assume the existence of an event horizon at r=r_+.
In other words, 
we assume that an equation Δ(r)=0 has one or more positive solutions 
and that r=r_+ is the largest positive solution among them.
We also assume that m(r) is regular in a range r≥ r_+.
In Secs. IV and V, we show that several examples of the function m(r). 

The components of the inverse metric are given by

    g^tt=-Σ/ρ^2Δ,
          g^tϕ=-2m(r)ar/ρ^2Δ,
          g^ϕϕ=Δ -a^2sin^2θ/ρ^2Δsin^2θ,
          g^rr=Δ/ρ^2,
          g^θθ=1/ρ^2.






§ NULL GEODESIC EQUATION

In this section, we investigate the Hamilton-Jacobi method for the motion of a photon in the asymptotically-flat, stationary and axisymmetric black hole spacetime. 
We define the action S=S(λ, x^μ) as a function of the coordinates x^μ and the parameter λ.
The Hamilton-Jacobi equation is obtained by

    ∂ S/∂λ +H =0,

where H≡ g_μνp^μp^ν/2 is the Hamiltonian of the photon motion
and p_μ is the conjugate momentum of the photon given by 

    p_μ≡∂ S/∂ x^μ.

We can write the action S in the following form with the cyclic coordinates t and ϕ:

    S=1/2μ^2λ -Et+L ϕ +S_r(r) +S_θ(θ),

where  the conserved energy E≡ -p_t, the conserved angular momentum L≡ p_ϕ and the mass μ≡ -p_μp^μ=0 of the photon 
are constant along the geodesic
and S_r(r) and S_θ(θ) are functions of the coordinates r and θ, respectively.
We can rewrite the Hamilton-Jacobi equation in

    -Δ( dS_r/dr)^2 +[ (r^2+a^2)E-aL ]^2/Δ
          =( dS_θ/dθ)^2 +(L-aEsin^2θ)^2/sin^2θ.

Both sides of this equation are constant. 
We can divide the Hamilton-Jacobi equation into two equations with respect to r and θ:

    𝒦=-Δ( dS_r/dr)^2 +[ (r^2+a^2)E-aL ]^2/Δ

and

    𝒦=( dS_θ/dθ)^2 +(L-aEsin^2θ)^2/sin^2θ,

where 𝒦 is a constant.

From 

    dx^μ/dλ=p^μ=g^μν p_ν,

we obtain 

    ρ^2dt/dλ=-a(aEsin^2θ-L)+(r^2+a^2)P(r)/Δ(r), 
          ρ^2dr/dλ=σ_r√(R(r)), 
          ρ^2dθ/dλ=σ_θ√(Θ(θ)), 
          ρ^2dϕ/dλ=-( aE-L/sin^2θ)+aP(r)/Δ(r),

where

    P(r)≡ E(r^2+a^2)-aL,
          R(r)≡ P(r)^2-Δ(r) [ (L-aE)^2+𝒬],
          Θ(θ) ≡𝒬+cos^2θ( a^2E^2-L^2/sin^2θ)

and where σ_r=± 1 and σ_θ=± 1 are independent 
and 𝒬 is the Carter constant defined by 𝒬≡𝒦-(L-aE)^2.
R(r) and Θ(θ) should be non-negative for the photon motion.

From Θ(θ) ≥ 0, we obtain

    Θ(θ)/E^2 
    =η +(a-ξ)^2 -( asinθ -ξ/sinθ)^2≥ 0,

where η≡𝒬/E^2 and ξ≡ L/E.

Equation (<ref>) can be rewritten in

    R(r)/E^2=r^4+ ( a^2-ξ^2-η )r^2 +2m(r) [ (ξ-a)^2+η]r -a^2η

and the derivative with respective to the radial coordinate r is given by

    R'/E^2=4r^3+2(a^2-ξ^2-η)r +2m(r) [ (ξ-a)^2 +η] f(r),

where ' denotes the differentiation with respect to the radial coordinate r
and where f(r) is defined as

    f(r)≡ 1+m'r/m.






§ THE SHADOW OF THE ROTATING BLACK HOLE

In this section, we consider a formula for the contour of the shadow of the rotating black hole and apply it for new examples.





 §.§ A formula for the black hole shadow

We give a formula for the contour of the shadow of the rotating black hole.
We assume that the rotating black hole spacetime has unstable circular null orbits satisfying 

    .R(r) |_r=r_0=. R'(r)|_r=r_0=0,

and 

    .R”(r) |_r=r_0 > 0,

where r_0 is the radius of the unstable circular null orbits. 
We also assume 

    r_+≤ r_0.


From Eq. (<ref>), we obtain 

    (4-f_0)r_0^4 +(2-f_0)r_0^2a^2 -[(2-f_0)r_0^2-f_0a^2]η
          =(2-f_0)r_0^2ξ^2

and

    r_0^4-(2-f_0)m_0a^2r_0+[a^2-(2-f_0)m_0r_0]η
          =(2-f_0)m_0r_0 ( ξ^2-2aξ ),

where m_0 and f_0 are defined as m_0≡ m(r_0) and f_0≡ f(r_0), respectively. 
From Eqs. (<ref>) and (<ref>), we obtain a quadratic equation with respective to ξ as 

    a^2(r_0-f_0m_0)ξ^2-2am_0[(2-f_0)r_0^2-f_0a^2]ξ
          -r_0^5 +(4-f_0)m_0r_0^4 -2a^2r_0^3
          +2a^2m_0(2-f_0)r_0^2 -a^4r_0 -a^4m_0f_0=0.

Equation (<ref>) has two real solutions ξ=ξ_± given by

    ξ_±≡m_0[ ( 2-f_0) r_0^2- f_0 a^2] ± r_0Δ_0/a(r_0-f_0m_0),
 
where Δ_0 is defined as Δ_0 ≡Δ(r_0).

We can simplify the solution ξ_+ as 

    ξ_+=r_0^2+a^2/a,

and, by using Eq. (<ref>), we obtain 

    η 
    =η_+≡ -r_0^4/a^2
    =-(a-ξ_+)^2.

We notice that ξ_+ and η_+ are the same as the Kerr black hole case <cit.>.
From Eqs. (<ref>) and (<ref>),
only θ=θ_+, where θ_+ is a constant satisfying ξ_+=asin^2θ_+,  
is permitted in this case.
Thus, the solution ξ=ξ_+ must be rejected for our purpose to describe the black hole shadow. 

We choose the solution ξ=ξ_-, where 

    ξ_-≡4m_0r_0^2-(r+f_0m_0)(r_0^2+a^2)/a(r_0-f_0m_0)
 
since we are interested in the shadow of the black hole.
From Eq. (<ref>), we get η=η_-, where 

    η_-≡r_0^3{  4(2-f_0)a^2m_0-r_0[r_0-(4-f_0)m_0]^2}/a^2(r_0-f_0m_0)^2.


We consider an asymptotically-flat, stationary, and axisymmetric black hole 
seen by an observer at a large distance from the black hole along an inclination angle θ_i.
The contour of the shadow of the black hole can be expressed 
by celestial coordinates α and β in a small part of the celestial sphere of the observer <cit.>.
The celestial coordinates α and β are obtained by 

    α = ξ_-/sinθ_i

and

    β =σ_θ√(η_- +(a-ξ_-)^2 -( asinθ_i -ξ_-/sinθ_i)^2).

See Appendix A for the calculation of the celestial coordinates α and β.
If we obtain a specific form of the function m(r), from Eqs. (<ref>) and (<ref>), 
we can obtain ξ_- and η_-.
Then, by using α and β,
we can draw the contour of the shadow of the black hole with the inclination angle θ_i. 




 §.§ Application

We apply the formula for two new examples of black holes to check that it works well. 





  §.§.§ Rotating black hole with m(r)=2M/(1+e^s/r)

First we consider a rotating black hole applied the Newman-Janis algorithm 
with 

    m(r)=2M/1+e^s/r

suggested in Refs. <cit.>.
Here s is a positive constant.
f_0 is given by

    f_0=r_0+(r_0+s)e^s/r_0/r_0(1+e^s/r_0).

There is a boundary of parameters a and s for existence of the event horizon shown in Fig. 1.
The maximum value of a/M is 1 for s/M = 0, it decreases as s/M increases, and it is 0 for s/M ∼ 1.114.

From Eqs. (<ref>) - (<ref>), 
the shadow of the black hole is obtained as shown in Fig. 2. 






  §.§.§ Rotating black hole with m(r)=4Me^u/√(r)/( 1+e^u/√(r))^2

Next we consider a rotating black hole applied the Newman-Janis algorithm with 

    m(r)=4Me^u/√(r)/( 1+e^u/√(r))^2

where u is a positive constant which is suggested in Ref. <cit.>.
f_0 is obtained as 

    f_0=1-(1-e^u/√(r_0))u/2(1+e^u/√(r_0))√(r_0).

Figure 3 shows a boundary of parameters a and u for existence of the event horizon.

The maximum value of a/M is unity for u/√(M)= 0, decreases as u/√(M) increases, and vanishes for u/√(M)∼ 1.874.
From Eqs. (<ref>) - (<ref>), (<ref>), and (<ref>), 
the contour of the shadow of the black hole is obtained and it is shown in Fig. 4. 








§ EXAMINATION OF KNOWN RESULTS

In this section, we examine that Eqs. (<ref>) and (<ref>) recovers 
several known results of rotating black holes such as the Kerr-Newman black hole and rotating regular black holes. 





 §.§ The Kerr-Newman black hole

The Kerr-Newman solution is an exact solution of the Einstein-Maxwell equations <cit.>.
The function m(r) of the Kerr-Newman spacetime with the electrical charge Q is given by 

    m(r)=M-Q^2/2r.

The Kerr-Newman solution is a black hole solution when M^2-a^2-Q^2 ≥ 0 is satisfied.
The contour of the shadow of the Kerr-Newman black hole was investigated in Ref. <cit.>.
From Eqs. (<ref>) and (<ref>), ξ_- and η_- are given by

    ξ_-=2r_0(2Mr_0-Q^2)-(r_0+M)(r_0^2+a^2)/a(r_0-M)

and

    η_-
    =r_0^2{ 4a^2(Mr_0-Q^2)-[r_0(r_0-3M)+2Q^2 ]^2 }/a^2(r_0-M)^2,

respectively.
Equations (<ref>) and (<ref>) are equal to 
Eqs. (51) and (52) in Ref. <cit.>
and  Eqs. (6) and (7) in Ref. <cit.>.

When the electrical charge Q vanishes, 
the spacetime is the Kerr black hole spacetime <cit.> and 
ξ_- and η_- become

    ξ_-=4Mr_0^2-(r_0+M)(r_0^2+a^2)/a(r_0-M)

and

    η_-
    =r_0^3[ 4a^2M- r_0(r_0-3M)^2 ]/a^2(r_0-M)^2,

respectively.
Equations (<ref>) and (<ref>) are equal to Eqs. (48) and (49) obtained by Bardeen <cit.>.





 §.§ A braneworld black hole with a tidal charge

A rotating black hole localized on a 3-brane in the Randall-Sundrum braneworld <cit.>
was considered by Aliev and Gumrukcuoglu <cit.>.
The function m(r) of the braneworld black hole with a tidal charge b is given by 

    m(r)=M-b/2r.

When M^2-a^2-b ≥ 0 is satisfied, the spacetime is a black hole spacetime.
The line element is the same to the one of the Kerr-Newman black hole spacetime if the tidal charge is nonnegative
while it is not if the tidal charge is negative. 
Notice that the spin parameter a can be larger than the ADM mass M 
when the tidal charge b is negative <cit.>. 

The contour of the shadow of the rotating braneworld black hole with the tidal charge b was investigated in Refs. <cit.>.
From Eqs. (<ref>) and (<ref>), we obtain ξ_- and η_- as

    ξ_-=2r_0(2Mr_0-b)-(r_0+M)(r_0^2+a^2)/a(r_0-M)

and

    η_-
    =r_0^2{ 4a^2(Mr_0-b)-[r_0(r_0-3M)+2b ]^2 }/a^2(r_0-M)^2,

respectively.
We notice that Eqs. (<ref>) and (<ref>) are the same as Eqs. (11) and (12) in Ref. <cit.>.








 §.§ Rotating regular black hole

Recently, rotating regular black hole metrics were suggested eagerly.





  §.§.§ Rotating Bardeen black hole

Applying the Newman-Janis algorithm to the Bardeen black hole <cit.>, 
a rotating regular black hole metric was obtained in Ref. <cit.>.
The function m(r) is given by

    m(r)=M( r^2/r^2+c^2)^3/2

where c is the monopole charge of a self-gravitating magnetic field <cit.>.

The shadow of the rotating Bardeen black hole was investigated in Refs. <cit.>.
From Eqs. (<ref>) and (<ref>), we obtain ξ_- and η_- as

    ξ_-=1/a[(r_0^2+c^2)^5/2-Mr_0^2(r_0^2+4c^2)]{ 4Mr_0^4(r_0^2+c^2) . 
    . -[(r_0^2+c^2)^5/2+Mr^2_0(r_0^2+4c^2) ](r_0^2+a^2) }

and 

    η_-
       =   r_0^4/a^2[(r_0^2+c^2)^5/2-Mr_0^2(r_0^2+4c^2)]^2
          ×{ 4Ma^2(r_0^2+c^2)^5/2(r_0^2-2c^2). 
          .-[ (r_0^2+c^2)^5/2-3Mr_0^4 ]^2},

respectively.
Equations (<ref>) and (<ref>)  are the same as Eq. (2.19) in Ref. <cit.> 
and Eqs. (41) and (42) in Ref <cit.>.
We comment on Ref. <cit.> in Appendix B.





  §.§.§ Rotating Hayward black hole

Applying the Newman-Janis algorithm to a regular black hole considered by Hayward <cit.>, 
a rotating regular black hole metric was obtained in Ref. <cit.>.
The function m(r) is given by

    m(r)=Mr^3/r^3+g^3

where g is a constant.

The contour of shadow of the rotating Hayward black hole was considered in Refs. <cit.>.
From Eqs. (<ref>) and (<ref>), ξ_- and η_- are given by

    ξ_-=1/a[(r_0^3+g^3)^2-Mr_0^2(r_0^3+4g^3)]{ 4Mr_0^4(r_0^3+g^3) . 
    . -[(r_0^3+g^3)^2+Mr_0^2(r_0^3+4g^3) ](r_0^2+a^2) }

and 

    η_-
       =   r_0^4/a^2[(r_0^3+g^3)^2-Mr_0^2(r_0^3+4g^3)]^2
          ×{ 4Ma^2(r_0^3+g^3)^2(r_0^3-2g^3) . 
          . -[ (r_0^3+g^3)^2-3Mr_0^5 ]^2},

respectively.
Equations (<ref>) and (<ref>) are the same as Eqs. (41) and (42) in Ref <cit.>.
See also Appendix B.





  §.§.§ A rotating black hole considered by Ghosh <cit.>

Ghosh applied the Newman-Janis algorithm to one of regular black holes suggested by Balart and Vagenas <cit.>
and obtained a rotating regular black hole metric <cit.>.
The function m(r) in Ref. <cit.> is 

    m(r)=Me^-h/r,

where h is constant.

The contour of the shadow of the black hole was calculated by Amir and Ghosh <cit.>.
From Eqs. (<ref>) and (<ref>), we obtain ξ_- and η_- as

    ξ_-=4Mr_0^3e^-h/r_0-[r_0^2+M(r_0+h)e^-h/r_0](r_0^2+a^2)/a[r_0^2-M(r_0+h)e^-h/r_0]

and

    η_-=r_0^4{4Ma^2(r_0-h)e^-h/r_0-[r_0^2+M(-3r_0+h)e^-h/r_0]^2}/a^2[r_0^2-M(r_0+h)e^-h/r_0]^2,

respectively.
Equations (<ref>) and (<ref>) are the same as Eqs. (22) and (23) in Ref. <cit.>.





  §.§.§ A rotating black hole considered by Tinchev <cit.>

Tinchev suggested a rotating regular black hole with a function

    m(r)=Me^-j/r^2,

where j is a constant, and Tinchev calculated the contour of the shadow of the black hole <cit.>.
From Eqs. (<ref>) and (<ref>), we get 

    ξ_-=4Mr_0^4e^-j/r_0^2-[r_0^3+M(r_0^2+2j)e^-j/r_0^2](r_0^2+a^2)/a[r_0^3-M(r_0^2+2j)e^-j/r_0^2]

and

    η_-
       =   r_0^4/a^2[r_0^3-M(r_0^2+2j)e^-j/r_0^2]^2
          ×{4Ma^2r_0(r_0^2-2j)e^-j/r_0^2. 
          . -[r_0^3-M(3r_0^2-2j)e^-j/r_0^2]^2}.


Equation (<ref>) is equal to ξ_- in Eq. (13) in Ref. <cit.>
while Eq. (<ref>) is not equal to η_- in Eq. (13) in Ref. <cit.>, 
which is given by, in our notation,

    η_-
    =2{ (r_0^2+a^2) [ (r_0m_0)'^2+r_0^2] -4r_0^2m_0(r_0m_0)' }/[(r_0m_0)'-r_0]^2.

When j vanishes, the rotating regular black hole is the Kerr black hole.  
For j=0, Eq. (<ref>) is equal to η_- in the Kerr black hole spacetime obtained as Eq. (<ref>)
while Eq. (<ref>) becomes 

    η_-
    =2 [ (r_0^2+a^2)(r_0^2+M^2)-4M^2r_0^2 ] /(r_0-M)^2

and it is not the same as Eq. (<ref>).





 §.§ A rotating black hole considered by Atamurotov, Ghosh, and Ahmedov <cit.>

Atamurotov et al. investigated the contour of the shadow of a rotating black hole <cit.>
and they claimed that the rotating black hole was a rotating black hole obtained in Ref. <cit.>.
The rotating black hole in Ref. <cit.> has the function m(r) given by

    m(r)=M-K^2(r)/2r,

where K(r) is a function with respective to r. 
Please see Refs. <cit.> and <cit.> carefully.

From Eqs. (<ref>) and (<ref>), we obtain

    ξ_-=2r_0(2Mr_0-K_0^2)-(r_0+M-K_0K_0')(r_0^2+a^2)/a(r_0-M+K_0K_0')

and

    η_-
       =   r_0^2 /a^2(r_0-M+K_0K_0')^2
          ×{ 4a^2 [ r_0(M+K_0K_0')-K_0^2] . 
          .-[r_0(r_0-3M-K_0K_0')+2K_0^2 ]^2 },

where K_0 and K_0' are defined as K_0≡ K(r_0) and K_0'≡ K'(r_0), respectively.
Equation (<ref>) is equal to Eq. (24) in Ref. <cit.>
while Eq. (<ref>) is not equal to η_- obtained by Atamurotov et al. as seen Eq. (25) in Ref. <cit.>.
Atamurotov et al. obtained η_- as, in our notation, 

    η_-
    =16Δ_0 r_0^2a^2-(r_0^2+a^2)Δ_0'+4rΔ_0+Δ_0'a/(Δ_0'a)^2,

where Δ_0' is defined by Δ_0'≡Δ'(r_0).
From a dimensional analysis, we notice that Eq. (25) in Ref. <cit.> or Eq. (<ref>) should be modified. 









 §.§ A rotating black hole considered by Modesto and Nicolini <cit.>

Modesto and Nicolini applied the Newman-Janis algorithm to 
a noncommutative geometry inspired Reissner-Nordström solution obtained in Ref. <cit.>
and obtained a rotating black hole metric <cit.>.
The function m(r) of the rotating black hole spacetime is given by 

    m(r)=n(r)-q^2(r)/2r,

where n(r) and q(r) are functions with respect to r. 
See Eqs. (37), (45) and (46) in Ref. <cit.> or Eq. (1) 
in Ref. <cit.>. [n(r), in our notation, 
is m(r) in a notation in Refs. <cit.>.]

The contour of the shadow of the rotating black hole was investigated by Sharif and Iftikhar <cit.>.
From Eqs. (<ref>) and (<ref>), we obtain ξ_- and η_- as  

    ξ_-
       =   1/a(r_0-n_0-n_0'r_0+q_0q_0')[ 2r_0(2n_0r_0-q_0^2) .  
          . -(r_0+n_0+n_0'r_0-q_0q_0')(r_0^2+a^2) ]

and 

    η_-
       =   r_0^2/a^2(r_0-n_0-n_0'r_0+q_0q_0')^2
          ×{ 4a^2[ r_0(n_0-n_0'r_0+q_0q_0')-q_0^2 ] .  
          . -[ r_0(r_0-3n_0+n_0'r_0-q_0q_0')+2q_0^2 ]^2  },

respectively, where n_0, n'_0, q_0, and q_0' are defined as 
n_0≡ n(r_0), n'_0 ≡ n'(r_0), q_0 ≡ q(r_0), and q_0' ≡ q'(r_0), respectively.
We notice that Eqs. (<ref>) and (<ref>) are not equal to Eqs. (15) and (16) in Ref. <cit.>. 
Equations (15) and (16) in Ref. <cit.> are given by, in our notation, 

    ξ_-
       =   1/a[n_0+r_0(n_0'-1)-q_0q_0'][ n_0(a^2-3r_0^2) .  
          . +r_0(r_0^2+a^2)(n_0'+1)+2q_0^2-q_0q_0'(r_0^2+a^2) ]

and

    η_-
       =   r_0^2/a^2[n_0+r_0(n_0'-1)]^2[ n_0r_0(4a^2-9n_0r_0+6r_0^2) .  
          -2n_0'r_0^2(2a^2+r_0^2-3n_0r_0) -r_0^4(n_0'^2+1) 
          -4q_0^2(a^2+q_0^2-3n_0r_0+n_0'r_0^2+r_0^2) 
          . -q_0'(4a^2+4q_0^3-6n_0q_0'r_0-2n_0'q_0r_0^2-q_0r_0+2q_0r_0^2) ],

respectively.
From a dimensional analysis, 
we notice that Eqs. (15) and (16) in Ref. <cit.> or Eqs. (<ref>) and (<ref>) should be modified.













§ SUMMARY

We have obtained a formula for the contour of the shadow of rotating black holes generated by the Newman-Janis algorithm. 
We have applied the formula to two new examples of the contour of the shadow of rotating black holes.   
We notice the shadows is very similar to the shadow of the Kerr-Newman black hole. 
By using the formula, we have examined ξ_- and η_- of the Kerr-Newman black hole and rotating regular black holes and the other rotating black holes.
The formula would help to categorize the many known results of the shadow of rotating black holes and to obtain new examples of black hole shadows. 





§ ACKNOWLEDGEMENTS

The author thanks R. A. Konoplya for his useful comment. 
This research was supported in part by the National Natural Science Foundation of China under Grant No. 11475065,
the Major Program of the National Natural Science Foundation of China under Grant No. 11690021.






§ CELESTIAL COORDINATES Α AND Β

In this Appendix, we show the calculation for the celestial coordinates α and β <cit.>.
We can express the line element (<ref>) 
in the asymptotically-flat, stationary, and axisymmetric black hole spacetime in the Boyer-Lindquist coordinates as follows:

    ds^2
    = -e^2νdt^2 +e^2ψ( dϕ-ω dt )^2 +e^2χdr^2 +ρ^2dθ^2,

where

    e^ν≡√(ρ^2Δ/Σ), 
          e^ψ≡√(Σsin^2θ/ρ^2), 
          ω≡2m(r)ar/Σ, 
          e^χ≡√(ρ^2/Δ).

The inverse of the metric tensors are obtained as

    g^tt = -e^-2ν, 
          g^t ϕ =-ω e^-2ν, 
          g^ϕϕ =-ω^2e^-2ν +e^-2ψ, 
          g^rr = e^-2χ, 
          g^θθ = ρ^-2.

We use a tetrad frame by the basis-vectors e_(β)αdx^α 
and the contravariant basis-vectors e_(β)^α∂_α:

    e_(0)αdx^α=-e^νdt, 
          e_(1)αdx^α=-ω e^ψdt +e^ψdϕ, 
          e_(2)αdx^α=e^χdr, 
          e_(3)αdx^α=ρ dθ,

and 

    e_(0)^α∂_α = e^-ν∂_t +ω e^-ν∂_ϕ, 
          e_(1)^α∂_α = e^-ψ∂_ϕ, 
          e_(2)^α∂_α = e^-χ∂_r, 
          e_(3)^α∂_α = ρ^-1∂_θ.

The tetrad components of the 4-momentum of the photon  p^(β) are obtained as

    p^(0)   =   -p_(0)=-e^νp^t
       =   -e^ν[ (E-ω L) e^-2ν] = (ω L -E) e^-ν, 
    
    p^(1)   =   p_(1)=e^ψ( -p^tω+ p^ϕ) =Le^-ψ, 
    
    p^(2)   =   p_(2)=e^χp^r,
    
    p^(3)   =   p_(3)=ρ p^θ,


In an asymptotic region r →∞, we obtain

    e^ν→ 1, 
          e^ψ→ r sinθ, 
          ω→2Ma/r^3, 
          e^χ→ 1.

Thus, as r →∞, the tetrad components of the 4-momentum  of the photon p^(β) are given by

    p^(0)→ -E, 
          p^(1)→L/r sinθ, 
          p^(2)→ p^r,
          p^(3)→ r p^θ.


The contour of the shadow of an asymptotically-flat, stationary, and axisymmetric black hole seen by an observer 
at a large distance from the black hole with an inclination angle θ_i is expressed 
by celestial coordinates α and β <cit.>.
The celestial coordinates α and β are defined by

    α≡lim_r→∞( -rp^(1)/p^(0)) 
    = ξ/sinθ_i

and

    β≡lim_r→∞( -rp^(3)/p^(0)) 
    = σ_θ√(Θ/E^2),

respectively.




§ COMMENT ON LI AND BAMBI <CIT.>

In this short Appendix, we comment on Ref. <cit.>.
Li and Bambi claimed that ξ_- and η_- of the rotating Bardeen black hole and the rotating Hayward black hole are complicated 
and they did not show the explicit forms of ξ_- and η_- in Ref. <cit.>. 
ξ_- and η_-, however, are not complicated so much
as we showed them in Eqs. (<ref>) and (<ref>) for the rotating Bardeen black hole 
and in Eqs. (<ref>) and (<ref>) for the rotating Hayward black hole.

We note that we should read Eq. (3.6) in Ref. <cit.>, in our notation,

    . R'(r)|_r=r_0
    =4r_0^3+2(a^2-ξ^2-η)r_0+2m_0[η+(ξ-a)^2]
    =0

as 

    . R'(r)|_r=r_0
    =4r_0^3+2(a^2-ξ^2-η)r_0+2m_0[η+(ξ-a)^2]f_0
    =0

which was obtained as Eq. (2.17) in Tsukamoto et al. <cit.>.



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