$$\mathfrak{a} = -\iint\_{\mathbb{R}^{\mathbb{R}} \underline{\mathbb{T}}\_{\mathbb{S}}^{\infty}} \mathcal{K}dS - \int\_{\mathcal{R}}^{\mathcal{S}} \mathfrak{x}\_{\emptyset} d\ell. \tag{88}$$

Here, *d* is positive when the photon is in the prograde motion, whereas it is negative for the retrograde case. Equation (88) means that *α* is coordinate-invariant for the axisymmetric case. Up until now, we did not use any Equations for gravitational fields. Therefore, the above discussion and results still stand not only in the theory of general relativity but also in a general class of metric theories of gravity only if the light ray in the four-dimensional spacetime is a null geodesic.

#### **8. Weak Deflection of Light in Kerr Spacetime**

#### *8.1. Kerr Spacetime and γij*

In this section, we focus on the weak deflection of light in the Kerr spacetime as an axisymmetric example. Kerr metric in the Boyer–Lindquist form is expressed as:

$$\begin{split} ds^2 &= -\left(1 - \frac{2Mr}{\Sigma}\right)dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}dt d\phi \\ &+ \frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 + \left(r^2 + a^2 + \frac{2a^2Mr\sin^2\theta}{\Sigma}\right)\sin^2\theta d\phi^2, \end{split} \tag{89}$$

where Σ and Δ are defined as:

$$
\Sigma \equiv r^2 + a^2 \cos^2 \theta,\tag{90}
$$

$$
\Delta \equiv r^2 - 2Mr + a^2. \tag{91}
$$

Using the Gauss–Bonnet theorem, the deflection angle of light in the Kerr spacetime was calculated for the asymptotic source and receiver by Werner [67]. However, his method based on the osculating metric is limited within the asymptotic case. Later, Ono et al. developed a different approach using the Gauss–Bonnet theorem that enables the calculation of the deflection angle for the finite distance case in the Kerr spacetime [42].

By using Equations (42) and (43), the generalized optical metric *γij* and the gravitomagnetic term *βi* for the Kerr metric are obtained as:

$$\begin{split} \gamma\_{ij} dx^i dx^j &= \frac{\Sigma^2}{\Delta(\Sigma - 2Mr)} dr^2 + \frac{\Sigma^2}{(\Sigma - 2Mr)} d\theta^2 \\ &+ \left( r^2 + a^2 + \frac{2a^2 Mr \sin^2 \theta}{(\Sigma - 2Mr)} \right) \frac{\Sigma \sin^2 \theta}{(\Sigma - 2Mr)} d\phi^2, \end{split} \tag{92}$$

$$\beta\_i d\mathbf{x}^i = -\frac{2aMr\sin^2\theta}{(\Sigma - 2Mr)}d\phi. \tag{93}$$

Note that *γij* has no linear terms in the Kerr spin parameter *a* because only *g*0*i* in *gμν* has a linear term in *a* and *g*0*i* ∝ *H* contributes to *γij* through a quadratic term *g*0*ig*0*j* ∝ *H*2, as shown by Equation (42).

In order to calculate the Gaussian curvature *K* of the equatorial plane, the geodesic curvature *κg* of the light ray and the geodesic curvature *κ*¯*g* of the circular arc of an infinite radius and of the angles Ψ*R* and Ψ*S*, we use two approximations for the weak field and slow rotation, where *M* and *a* play roles as book-keeping parameters though they are dimensional quantities.

By using Equation (77), we obtain the orbit Equation:

$$\left(\frac{dr}{d\phi}\right)^2 = \frac{b^2 \left\{\frac{a^2}{b^2} + \frac{r}{b}(\frac{r}{b} - \frac{2M}{b})\right\}^2 \left\{\frac{a^2}{b^2}(\frac{2M}{b} + \frac{r}{b}) - \frac{4aM}{b^2} + \frac{2M}{b} - \frac{r}{b} + \frac{r^3}{b^3}\right\}}{\frac{r}{b}\{\frac{2aM}{b^2} + \frac{r}{b} - \frac{2M}{b}\}^2} $$

$$= \frac{r^4}{b^2} - r^2 + 2Mr - \frac{4r^3}{b^3}aM + \mathcal{O}(a^2), \tag{94}$$

where the weak-field and slow-rotation approximations are used in the last line. There are no *M*-squared terms in the last line. The orbit Equation becomes:

$$\left(\frac{du}{d\phi}\right)^2 = F(u) = \frac{1}{b^2} - u^2 + 2Mu^3 - \frac{4u}{b^3}aM + \mathcal{O}(a^2u^4). \tag{95}$$

We solve iteratively Equation (95). In order to find the zeroth-order solution, we solve the truncated Equation (95):

$$\mathcal{O}\left(\frac{du}{d\phi}\right)^2 = \frac{1}{b^2} - u^2 + \mathcal{O}(Mu^3, aMu^4, a^2u^4). \tag{96}$$

The zeroth-order solution for this Equation is:

$$
\mu = \frac{\sin \phi}{b},
\tag{97}
$$

where we use *du dφ φ*=*π*/2 = 0 as the boundary condition. This condition means that the closest approach of the photon orbit is expressed as *r* = *r*0 = 1/*u*0, *φ* = *π*/2. We assume that the linear-order solution with *M* is *u* = sin *φ b* + *u*1(*φ*)*M*. In order to obtain *u*1(*φ*), we substitute this expression of *u* into the Equation (95) with terms linear in *M*:

$$\mathcal{O}\left(\frac{du}{d\phi}\right)^2 = \frac{1}{b^2} - u^2 + 2Mu^3 + \mathcal{O}(aMu^4, a^2u^4). \tag{98}$$

*u*1(*φ*) is thus obtained as:

$$
\mu\_1(\phi) = \frac{1}{b^2} (1 + \cos^2 \phi),
\tag{99}
$$

where we used the boundary condition mentioned above. The solution with *a* is in a form of *u* = sin *φ b* + *M b*2 (1 + cos2 *φ*) + *u*2(*φ*)*a*. Since Equation (95) does not include any linear term in *a*, we find *u*2(*φ*) = 0. The solution with *aM* is *u* = sin *φ b* + *M b*2 (1 + cos2 *φ*) + *u*3(*φ*)*aM*. We substitute this solution into Equation (95):

$$\frac{d^3M}{d\phi}\left\{b^3\frac{d\mu\_3(\phi)}{d\phi}\cos\phi+b^3\mu\_3(\phi)\sin\phi+2\sin\phi\right\}+\mathcal{O}(a^2u^4)=0.\tag{100}$$

Hence, *u*3(*φ*) is obtained as:

$$
\mu\_3(\phi) = -\frac{2}{b^3}.\tag{101}
$$

Bringing the above results together, the iterative solution of Equation (95) is expressed as:

$$u = \frac{\sin\phi}{b} + \frac{M}{b^2} (1 + \cos^2\phi) - \frac{2aM}{b^3} + \mathcal{O}\left(\frac{M^2}{b^3}, \frac{a^2}{b^3}\right). \tag{102}$$

Next, we solve Equation (102) for *φ*. We obtain *φ* as:

$$\phi = \begin{cases} \arcsin(bu) + \frac{-2 + b^2 u^2}{b\sqrt{1 - b^2 u^2}} M + \frac{2aM}{b^2 \sqrt{1 - b^2 u^2}} + \mathcal{O}\left(\frac{M^2}{b^3}, \frac{a^2}{b^3}\right) & \left(|\phi| < \frac{\pi}{2}\right) \\\pi - \arcsin(bu) - \frac{-2 + b^2 u^2}{b\sqrt{1 - b^2 u^2}} M - \frac{2aM}{b^2 \sqrt{1 - b^2 u^2}} + \mathcal{O}\left(\frac{M^2}{b^3}, \frac{a^2}{b^3}\right) & \left(\frac{\pi}{2} < |\phi|\right)' \end{cases} \tag{103}$$

where we can choose the domain of *φ* to be −*π* ≤ *φ* < *π* without loss of generality. In the following, the range of the angular coordinate value *φS* at the source point is −*π* 2 ≤ *φS* < *π* 2 and the range of the angular coordinate value *φR* at the receiver point is |*φR*| > *π* 2 . We find |*bu*| < 1 because the square root in Equation (103) must be real and nonzero, and the values of *b* and *u* are positive. Therefore, *bu* satisfies 0 < *bu* < 1 in our calculation.

#### *8.2. Gaussian Curvature on the Equatorial Plane*

Let us explain how to compute the Gaussian curvature by using Equation (46). In the Kerr case, it becomes:

$$K = \frac{M\left(-6r\left(a^2 + M^2\right) + 6a^2M + 7Mr^2 - 2r^3\right)}{r^5(r - 2M)}$$

$$= -\frac{2M}{r^3} + \mathcal{O}\left(\frac{M^2}{r^4}, \frac{a^2M}{r^5}\right),\tag{104}$$

where the weak-field and slow-rotation approximations are used in the last line.

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Next, we discuss the area element on the equatorial plane by using Equation (47). In the Kerr case, the area element of the equatorial plane is expressed as:

$$dS = [r + \Im M + \mathcal{O}(M^2/r)]dr d\phi. \tag{105}$$

By using Equations (104) and (105), the surface integral of the Gaussian curvature in Equation (88) is performed as:

$$\begin{split} -\int\_{\frac{3}{h}\infty\frac{\varsigma}{\log^{5}s}} \mathrm{K}dS &= \int\_{\phi\_{\mathsf{S}}}^{\Phi\_{\mathsf{R}}} \int\_{\infty}^{\upsilon\_{\mathsf{O}}} (-\frac{2M}{b^{3}}r) dr d\phi + \mathcal{O}\left(\frac{M^{2}}{b^{2}}, \frac{aM^{2}}{b^{3}}, \frac{a^{2}M}{b^{3}}\right) \\ &= 2M \int\_{\phi\_{\mathsf{S}}}^{\Phi\_{\mathsf{R}}} \int\_{0}^{\frac{1}{b}\sin\phi + \frac{M}{b^{2}}(1+\cos^{2}\phi) - \frac{2M}{b^{3}}} dud\phi + \mathcal{O}\left(\frac{M^{2}}{b^{2}}, \frac{aM^{2}}{b^{3}}, \frac{a^{2}M}{b^{3}}\right) \\ &= 2M \int\_{\phi\_{\mathsf{S}}}^{\Phi\_{\mathsf{R}}} \left[\frac{1}{b}\sin\phi\right] d\phi + \mathcal{O}\left(\frac{M^{2}}{b^{2}}, \frac{aM^{2}}{b^{3}}, \frac{a^{2}M}{b^{3}}\right) \\ &= \frac{2M}{b}\left[\cos\phi\_{\mathsf{S}} - \cos\phi\_{\mathsf{R}}\right] + \mathcal{O}\left(\frac{M^{2}}{b^{2}}, \frac{aM^{2}}{b^{3}}, \frac{a^{2}M}{b^{3}}\right) \\ &= \frac{2M}{b}\left[\sqrt{1 - b^{2}u\_{\mathsf{S}}^{2}} + \sqrt{1 - b^{2}u\_{\mathsf{R}}^{2}}\right] + \mathcal{O}\left(\frac{M^{2}}{b^{2}}, \frac{aM^{2}}{b^{3}}, \frac{a^{2}M}{b^{3}}\right), \end{split} \tag{106}$$

where *rOE* in the first line is the solution of Equation (94), we transform the integral variable as *r* = 1/*u* in the second line, and we used cos *φS* = 1 − *b*2*uS*2 + O(*M*/*b*) and cos *φR* = − 1 − *b*2*uR*2 + O(*M*/*b*) from Equation (103) in the last line.

#### *8.3. Path Integral of κg*

Substituting Equation (93) into *βi* in Equation (66) leads to:

$$\begin{split} \kappa\_{\mathcal{S}} &= -\frac{2aM}{r^2(r-2M)} \left( \frac{1-\frac{2M}{r}+\frac{a^2}{r^2}}{1+\frac{a^2}{r^2}+\frac{2a^2M}{r^3}} \right)^{1/2} \\ &= -\frac{2aM}{r^3} + \mathcal{O}\left(\frac{aM^2}{r^4}\right), \end{split} \tag{107}$$

where the weak-field and slow-rotation approximations are used in the last line. We stress that the terms of *anM* (*n* ≥ 2) do not exist in this expression.

The line element for the path integral by Equation (67) becomes:

$$d\ell = \left[\frac{b}{\sin^2 \phi} + \mathcal{O}(M)\right] d\phi,\tag{108}$$

where Equation (102) was used for a relation between *r* and *φ*.

By using Equations (107) and (108), the path integral of *κg* in Equation (88) is performed as:

$$\begin{split}-\int\_{R}^{S} \kappa\_{\mathcal{S}} d\ell &= -\int\_{S}^{R} \frac{2aM}{r^{3}} d\ell + \mathcal{O}\left(\frac{aM^{2}}{r^{4}}\right) \\ &= -\frac{2aM}{b^{2}} \int\_{\phi\_{\rm S}}^{\phi\_{\rm R}} \sin\phi d\phi + \mathcal{O}\left(\frac{aM^{2}}{r^{4}}\right) \\ &= -\frac{2aM}{b^{2}} [\sqrt{1 - b^{2}u\_{R}r^{2}} + \sqrt{1 - b^{2}u\_{S}r^{2}}] + \mathcal{O}\left(\frac{aM^{2}}{b^{3}}\right). \end{split} \tag{109}$$

Here, we assumed *d* > 0, such that the orbital angular momentum can be parallel with the spin of the black hole, and we used a linear approximation of the photon orbit as 1/*r* = *u* = sin *φ*/*b* + O(*M*/*b*2, *aM*/*b*3) from Equation (102). In the retrograde case, *d* becomes negative and the magnitude of the above path integral thus remains the same but the sign of the integral is opposite.

#### *8.4. φRS Part*

The displacement of the angular coordinate *φ* in Equation (87) is computed as:

$$\begin{split} \phi\_{RS} &= \int\_{S}^{R} d\phi \\ &= 2 \int\_{0}^{u\_{0}} \frac{1}{\sqrt{\mathcal{F}(u)}} du + \int\_{u\_{S}}^{0} \frac{1}{\sqrt{\mathcal{F}(u)}} du + \int\_{u\_{R}}^{0} \frac{1}{\sqrt{\mathcal{F}(u)}} du, \end{split} \tag{110}$$

where the orbit equation of Equation (78) was used. We substitute Equation (95) into *F*(*u*) in Equation (110) to obtain:

$$\begin{split} \phi\_{RS} &= \int\_{u\_S}^{u\_0} \left( \frac{1}{\sqrt{u\_0^2 - u^2}} + M \frac{u\_0^{-3} - u^3}{(u\_0^2 - u^2)^{3/2}} - 2aM \frac{u\_0^{-3}(u\_0 - u)}{(u\_0^2 - u^2)^{3/2}} \right) du \\ &+ \int\_{u\_R}^{u\_0} \left( \frac{1}{\sqrt{u\_0^2 - u^2}} + M \frac{u\_0^{-3} - u^3}{(u\_0^2 - u^2)^{3/2}} - 2aM \frac{u\_0^{-3}(u\_0 - u)}{(u\_0^2 - u^2)^{3/2}} \right) du \\ &+ \mathcal{O}(M^2 u\_0^{-2}, d^2 u\_0^{-2}) \\ &= \left( \frac{\pi}{2} - \arcsin\left(\frac{u\_S}{u\_0}\right) + M \frac{(2u\_0 + u\_S)\sqrt{u\_0^{-2} - u\_S^2}}{u\_0 + u\_S} - 2aM \frac{u\_0^{-3}\sqrt{u\_0^{-2} - u\_S^2}}{u\_0^2 + u\_0 u\_S} \right) \\ &+ \left( \frac{\pi}{2} - \arcsin\left(\frac{u\_R}{u\_0}\right) + M \frac{(2u\_0 + u\_R)\sqrt{u\_0^{-2} - u\_R^2}}{u\_0 + u\_R} - 2aM \frac{u\_0^{-3}\sqrt{u\_0^{-2} - u\_R^2}}{u\_0^2 + u\_0 u\_R} \right) \\ &+ \mathcal{O}\left(M^2 u\_0^2, d^2 u\_0^2\right), \end{split} \tag{111}$$

where the prograde case is assumed. In the retrograde motion, the sign of the linear term in *a* is opposite. In Equation (111), the impact parameter *b* is rewritten in terms of the closest approach *u*0 for the integration from *uS*(or *uR*) to *u*0. Namely, Equation (95) tells us the relation between the impact parameter *b* and the inverse of the closest approach *u*0 as *b* = *u*−1 0 + *M* − 2*aMu*0 + O(*M*2*u*0, *a*2*u*0) in the weak-field and slow-rotation approximations. By making use of this relation, Equation (111) is rearranged as:

$$\begin{split} \phi\_{RS} &= \pi - \arcsin(bu\_S) - \arcsin(bu\_R) + \frac{M(2 - b^2 u\_S^2)}{b\sqrt{1 - b^2 u\_S}^2} + \frac{M(2 - b^2 u\_R^2)}{b\sqrt{1 - b^2 u\_R^2}} \\ &- \frac{2aM}{b^2} \left[ \frac{1}{\sqrt{1 - b^2 u\_S^2}^2} + \frac{1}{\sqrt{1 - b^2 u\_R^2}^2} \right] + \mathcal{O}\left(M^2 / b^2, a^2 / b^2\right). \end{split} \tag{112}$$

The first line of this equation recovers Equation (32) of Reference [38].

#### *8.5. Ψ Parts*

In the Kerr spacetime by Equation (89), Equation (85) is:

$$\begin{split} \sin\Psi\_R &= \frac{b}{r\_R} \times \frac{1 - \frac{2M}{r\_R} + \frac{2aM}{br\_R}}{\sqrt{1 - \frac{2M}{r\_R} + \frac{a^2}{r\_R^2}}}\\ &= \frac{b}{r\_R} \left(1 - \frac{M}{r\_R} + \frac{2aM}{br\_R}\right) + \mathcal{O}\left(\frac{M^2}{r\_R^2}, \frac{a^2}{r\_R^2}, \frac{aM^2}{r\_R^3}\right) \\ &= b\mu\_R \left(1 - M\mu\_R + \frac{2aM\mu\_R}{b}\right) + \mathcal{O}\left(M^2\mu\_R^2, a^2\mu\_R^2, aM^2\mu\_R^3\right), \end{split} \tag{113}$$

and Equation (86) is calculated as:

$$\sin(\pi - \Psi\_S) = bu\_S \left( 1 - Mu\_S + \frac{2aMu\_S}{b} \right) + \mathcal{O}\left( M^2 u\_S^2, a^2 u\_S^2, aM^2 u\_S^{-3} \right),\tag{114}$$

where *rR* = 1/*uR*,*rS* = 1/*uS* and we used the weak-field and slow-rotation approximations. By combining Equations (113) and (114), we obtain Ψ*R* and Ψ*S* as:

$$\begin{split} \Psi\_{R} &= \arcsin\left[bu\_{R}\left(1 - Mu\_{R} + \frac{2aMu\_{R}}{b}\right)\right] + \mathcal{O}\left(M^{2}u\_{R}z^{2}u^{2}u\_{R}z^{2}, aM^{2}u\_{R}^{3}\right) \\ &= \arcsin(bu\_{R}) - \frac{Mbu\_{R}}{\sqrt{1 - b^{2}u\_{R}^{2}}} + \frac{2aMu\_{R}}{\sqrt{1 - b^{2}u\_{R}^{2}}} + \mathcal{O}\left(M^{2}u\_{R}z^{2}u\_{R}^{2}u\_{R}^{2}, aM^{2}u\_{R}^{3}\right), \\ \pi-\Psi\_{S} &= \arcsin(bu\_{S}) - \frac{Mbu\_{S}^{2}}{\sqrt{1 - b^{2}u\_{S}^{2}}} + \frac{2aMu\_{S}^{2}}{\sqrt{1 - b^{2}u\_{S}^{2}}} + \mathcal{O}\left(M^{2}u\_{S}^{2}, a^{2}u\_{S}^{2}, aM^{2}u\_{S}^{3}\right). \end{split} \tag{115}$$

By combining these relations, we obtain the Ψ part in Equation (87) as:

$$\begin{split} \Psi\_{R} - \Psi\_{S} &= \arcsin(bu\_{R}) + \arcsin(bu\_{S}) - \pi - \frac{Mbu\_{R}\prescript{2}{}{}{\end{}}{\sqrt{1 - b^{2}u\_{R}\prescript{2}{}{\end{}}{\end{}} - \frac{Mbu\_{S}\prescript{2}{}{\end{}}{\}} \\ &+ \frac{2aMu\_{R}\prescript{2}{}{\frac{2}{\sqrt{1 - b^{2}u\_{R}\prescript{2}{}{\end{}}{\end{}}} + \mathcal{O}\left(M^{2}u\_{R}^{2}, M^{2}u\_{S}^{2}, a^{2}u\_{R}^{2}, a^{2}u\_{S}^{2}, aM^{2}u\_{R}^{3}, aM^{2}u\_{S}^{3}\right)}. \end{split} \tag{116}$$

#### *8.6. Deflection of Light in Kerr Spacetime*

On the equatorial plane in the Kerr spacetime, the deflection angle of light is described by Equations (87) and (88). Let us examine whether the two results agree with each other.

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First, we substitute Equations (112) and (116) into Equation (87). We obtain the deflection angle of light as:

$$\begin{split} \mathfrak{a}\_{\text{prog}} &= \arcsin(bu\_{R}) + \arcsin(bu\_{S}) - \pi - \frac{Mbu\_{R}^{2}}{\sqrt{1 - b^{2}u\_{R}^{2}}^{2}} - \frac{Mbu\_{S}^{2}}{\sqrt{1 - b^{2}u\_{S}^{2}}^{2}} \\ &+ \frac{2aMu\_{R}^{2}}{\sqrt{1 - b^{2}u\_{R}^{2}}^{2}} + \frac{2aMu\_{S}^{2}}{\sqrt{1 - b^{2}u\_{S}^{2}}^{2}} \\ &+ \pi - \arcsin(bu\_{S}) - \arcsin(bu\_{R}) + \frac{M(2 - b^{2}u\_{S})^{2}}{b\sqrt{1 - b^{2}u\_{S}^{2}}^{2}} + \frac{M(2 - b^{2}u\_{R}^{2})}{b\sqrt{1 - b^{2}u\_{R}^{2}}^{2}} \\ &- \frac{2aM}{b^{2}} \left[ \frac{1}{\sqrt{1 - b^{2}u\_{S}^{2}}^{2}} + \frac{1}{\sqrt{1 - b^{2}u\_{R}^{2}}^{2}} \right] + \mathcal{O}\left(\frac{M^{2}}{b^{2}}\right) \\ &= \frac{2M}{b} \left( \sqrt{1 - b^{2}u\_{R}^{2}}^{2} + \sqrt{1 - b^{2}u\_{S}^{2}} \right) \\ &- \frac{2aM}{b^{2}} \left( \sqrt{1 - b^{2}u\_{R}^{2}}^{2} + \sqrt{1 - b^{2}u\_{S}^{2}}^{2} \right) + \mathcal{O}\left(\frac{M^{2}}{b^{2}}\right). \end{split} \tag{117}$$

where the prograde orbit of light is assumed. For the retrograde motion, we obtain:

$$\begin{split} \alpha\_{\text{retro}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 u\_R^2}^2 + \sqrt{1 - b^2 u\_S^2}^2 \right) \\ &+ \frac{2aM}{b^2} \left( \sqrt{1 - b^2 u\_R^2}^2 + \sqrt{1 - b^2 u\_S^2}^2 \right) + \mathcal{O}\left(\frac{M^2}{b^2}\right). \end{split} \tag{118}$$

Next, we substitute Equations (106) and (109) into Equation (88). Then, we obtain the deflection angle of light in the prograde motion as:

$$\begin{split} u\_{\text{prog}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^{-2}} \right) \\ &- \frac{2aM}{b^2} \left( \sqrt{1 - b^2 u\_R^{-2}} + \sqrt{1 - b^2 u\_S^{-2}} \right) + \mathcal{O} \left( \frac{M^2}{b^2} \right), \end{split} \tag{119}$$

and the deflection angle for the retrograde case as:

$$\begin{split} a\_{\text{retro}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 \mu\_R^2} + \sqrt{1 - b^2 \mu\_S^2} \right) \\ &+ \frac{2aM}{b^2} \left( \sqrt{1 - b^2 \mu\_R^2} + \sqrt{1 - b^2 \mu\_S^2} \right) + \mathcal{O} \left( \frac{M^2}{b^2} \right). \end{split} \tag{120}$$

Note that the *a*2 terms in the deflection angle in Equation (87) cancel out thanks to Equation (88). Here, we consider the limit as *uR* → 0 and *uS* → 0. In this limit, we get:

$$
\kappa\_{\infty \text{prog}} \rightarrow \frac{4M}{b} - \frac{4aM}{b^2} + O\left(\frac{M^2}{b^2}\right),
\tag{121}
$$

$$a\_{\infty \, retro} \rightarrow \frac{4M}{b} + \frac{4aM}{b^2} + O\left(\frac{M^2}{b^2}\right). \tag{122}$$

This shows that Equations (117) and (118) agree with the asymptotic deflection angles that are known in earlier works [4,68–70]. Precise analytic treatments of the deflection angle of light were done in a conventional approach, on the equatorial plane of a Kerr black hole [70] and for generic photon orbits in terms of the generalized hypergeometric functions of Appell and Lauricella [71]. They assume that both the source and the receiver are located at the null infinity.

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If we wish to consider the deflection angle of light in a case where the receiver point is closer to the source point than the closest approach point, Equations (117) and (118) become:

$$\begin{split} a\_{\text{prog}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 \boldsymbol{\mu}\_S^2} - \sqrt{1 - b^2 \boldsymbol{\mu}\_R^2}^2 \right) \\ &- \frac{2aM}{b^2} \left( \sqrt{1 - b^2 \boldsymbol{\mu}\_S^2} - \sqrt{1 - b^2 \boldsymbol{\mu}\_R^2}^2 \right) + \mathcal{O} \left( \frac{M^2}{b^2} \right), \\ a\_{\text{retro}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 \boldsymbol{\mu}\_S^2} - \sqrt{1 - b^2 \boldsymbol{\mu}\_R^2}^2 \right) \\ &+ \frac{2aM}{b^2} \left( \sqrt{1 - b^2 \boldsymbol{\mu}\_S^2} - \sqrt{1 - b^2 \boldsymbol{\mu}\_R^2}^2 \right) + \mathcal{O} \left( \frac{M^2}{b^2} \right). \end{split}$$

If we wish to consider the deflection angle of light in such a case that the source point is closer to the receiver than the closest approach point, Equations (117) and (118) become:

$$\begin{split} a\_{\text{prog}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 u\_{\text{R}}^2} - \sqrt{1 - b^2 u\_{\text{S}}^2} \right) \\ &- \frac{2aM}{b^2} \left( \sqrt{1 - b^2 u\_{\text{R}}^2} - \sqrt{1 - b^2 u\_{\text{S}}^2} \right) + \mathcal{O} \left( \frac{M^2}{b^2} \right), \\ a\_{\text{retro}} &= \frac{2M}{b} \left( \sqrt{1 - b^2 u\_{\text{R}}^2} - \sqrt{1 - b^2 u\_{\text{S}}^2} \right) \\ &+ \frac{2aM}{b^2} \left( \sqrt{1 - b^2 u\_{\text{R}}^2} - \sqrt{1 - b^2 u\_{\text{S}}^2} \right) + \mathcal{O} \left( \frac{M^2}{b^2} \right). \end{split}$$

#### *8.7. Finite-Distance Corrections*

In the previous subsections so far, we discussed an effect of the spin of the lens object to the deflection of light. In particular, we do not require the receiver and the source to be located at infinity. The finite-distance correction to the deflection angle of light is defined as *δα*. This is the difference between the asymptotic deflection angle *α*∞ and the deflection angle for the finite distance case. Namely:

$$
\delta \mathfrak{a} \equiv \mathfrak{a} - \mathfrak{a}\_{\infty}.\tag{123}
$$

Equations (117) and (118) tell us the magnitude of the finite-distance correction to the gravitomagnetic bending angle due to the spin. The result is:

$$|\delta \mathfrak{a}\_{GM}| \sim \mathcal{O}\left(\frac{aM}{r\_S^2} + \frac{aM}{r\_R^2}\right)$$

$$\sim \mathcal{O}\left(\frac{J}{r\_S^2} + \frac{J}{r\_R^2}\right),\tag{124}$$

where *buR*, *buS* < 1 is assumed, *J* ≡ *aM* denotes the spin angular momentum of the lens, and the subscript *GM* means the gravitomagnetic part. We introduce the dimensionless spin parameter as: *s* ≡ *a*/*M*. Hence, Equation (124) is rearranged as:

$$|\delta \mathfrak{a}\_{GM}| \sim O\left(\mathrm{s}\left(\frac{M}{r\_S}\right)^2 + \mathrm{s}\left(\frac{M}{r\_R}\right)^2\right). \tag{125}$$

This implies that *δαGM* is of the same order as the second post-Newtonian effect (with the dimensionless spin parameter).

The second-order Schwarzschild contribution to *α* is 15*πM*2/4*b*2. This contribution can be obtained also by using the present method, especially by using a relation between *b* and *r*0 in *M*2 in calculating *φRS*. Appendix A provides detailed calculations at the second order of *M* and *a*. We explain detailed calculations for the integrals of *K* and *κg* in the present formulation. Note that *δαGM* in the above approximations is free from the impact parameter *b*. We can see this fact from Figures 11–13 below.

#### *8.8. Possible Astronomical Applications*

What are possible astronomical applications? As a first example, we consider the Sun, in which its higher multipole moments are ignored for simplicity. Its spin angular momentum denoted as *J* is ∼2 × 1041 m2 kg s−1 [72,73]. This means *G Jc*−2 ∼ 5 × 105 m2, for which the dimensionless spin parameter becomes *s* ∼ 10−1.

Here, our assumption is that a receiver on the Earth observes the light deflected by the Sun, while the distant source is safely in the asymptotic region. For the light ray passing near the Sun, Equation (125) allows us to make an order-of-magnitude estimation of the finite-distance correction. The result is:

$$\begin{aligned} |\delta a\_{GM}| &\sim O\left(\frac{I}{r\_R^2}\right) \\ &\sim 10^{-12} \text{arcsec} \times \left(\frac{I}{f\_\odot}\right) \left(\frac{1 \text{AU}}{r\_R}\right)^2, \end{aligned} \tag{126}$$

where 4*M*/*R* ∼ 1.75 arcsec. ∼ 10−5 rad., where *Modot* means the solar mass and *R* denotes the solar radius. This correction is nearly a pico-arcsecond. Therefore, the correction is beyond the reach of present and near-future technology [74,75].

Figure 11 shows the finite-distance correction to the light deflection. Our numerical calculations are consistent with the above order-of-magnitude estimation. This figure shows also the very weak dependence of *δα* on *b*.

See Figures 12 and 13 for the deflection angle with finite-distance corrections for the prograde motion and retrograde one, respectively, where we choose *rS* ∼ 1.5 × 108 km and *rR* ∼ ∞. The finite-distance correction reduces the deflection angle of light. As the impact parameter *b* increases, the finite-distance correction also increases.

As a second example, we discuss Sgr A∗ that is located at our galactic center. This object is a good candidate for measuring the strong gravitational deflection of light. The distance to the receiver is much larger than the impact parameter of light. On the other hand, some of the source stars may live in our galactic center.

For Sgr A∗, Equation (125) becomes:

$$\begin{split} |\delta \mathfrak{a}\_{GM}| &\sim \mathrm{s} \left( \frac{M}{r\_{\mathrm{S}}} \right)^{2} \\ &\sim 10^{-7} \mathrm{arcsec} \cdot \times \left( \frac{\mathrm{s}}{0.1} \right) \left( \frac{M}{4 \times 10^{6} M\_{\odot}} \right)^{2} \left( \frac{0.1 \mathrm{pc}}{r\_{\mathrm{S}}} \right)^{2} \end{split} \tag{127}$$

where we assume that the mass of the central black hole is *M* ∼ 4 × 106*M*. This correction is nearly at a sub-microarcsecond level. Therefore, it is beyond the capability of present technology (e.g., [31–36]).

See Figure 9 for the finite-distance correction due to the source location. The result in this figure is in agreement with the above order-of-magnitude estimation. This figure suggests the very weak dependence on the impact parameter *b*.

**Figure 11.** *δαGM* for the Sun: The horizontal axis is the distance of the receiver distance *rR*. The vertical axis means the finite-distance correction due to the gravitomagnetic deflection angle of light. The solid curve (blue in color) and dashed one (red in color) denote *b* = *R* and *b* = 105*R*, respectively. The dotted line (black in color) corresponds to the leading term in *δαGM* given by Equation (124). These three curves are overlapped. This implies the very weak dependence of *δαGM* on *b*.

**Figure 12.** *α* in the prograde motion: The horizontal axis is the impact parameter for a photon orbit. The vertical axis means the deflection angle of light. The blue curve is the asymptotic deflection angle by a Kerr black hole. The orange curve means the deflection angle with finite-corrections by a Kerr black hole. The green curve shows the difference between the asymptotic bending angle and the deflection angle with finite-corrections by a Kerr black hole.

**Figure 13.** *α* for light of retrograde motion: The horizontal axis denotes the impact parameter for a photon orbit, and the vertical axis denotes the deflection angle of light. The blue curve is the asymptotic deflection angle by the Kerr black hole. The orange curve is the deflection angle with finite-correction by the Kerr black hole. The green curve shows the difference between the asymptotic bending angle and the deflection angle with finite-correction by the Kerr black hole.

#### **9. Rotating Teo Wormhole: Another Example**

#### *9.1. Rotating Teo Wormhole and Optical Metric*

In this section, we consider a rotating Teo wormhole [76] in order to examine how our method can be applied to a wormhole spacetime. The spacetime metric for this wormhole is:

$$ds^2 = -N^2dt^2 + \frac{dr^2}{1 - \frac{\hbar\_0}{r}} + r^2H^2 \left[d\theta^2 + \sin^2\theta (d\phi - \omega dt)^2\right],\tag{128}$$

where we denote:

$$N = H = 1 + \frac{d(4d\cos\theta)^2}{r},\tag{129}$$

$$
\omega = \frac{2h}{r^3}.\tag{130}
$$

Here, *b*0 means the throat radius of this wormhole, *a*¯ is corresponding to the spin angular momentum, and *d* is a positive constant.

For the rotating Teo wormhole of Equation (128), the components of the generalized optical metric are [43]:

$$\begin{split} \gamma\_{ij}d\mathbf{x}^{i}d\mathbf{x}^{j} &= \frac{r^{7}}{\left(r - b\_{0}\right)\left(r^{4} - 4d^{2}\sin^{2}\theta\right)\left(16d d^{2}\cos^{2}\theta + r\right)^{2}}d\mathbf{r}^{2} \\ &+ \frac{r^{6}}{r^{4} - 4d^{2}\sin^{2}\theta}d\theta^{2} + \frac{r^{10}\sin^{2}\theta}{\left(r^{4} - 4d^{2}\sin^{2}\theta\right)^{2}}d\phi^{2}. \end{split} \tag{131}$$

Here, *γij* is not the induced metric in the Arnowitt-Deser-Misner(ADM) formulation. The components of *βi* are obtained as:

$$
\beta\_i dx^i = -\frac{2\bar{\sigma}r^3 \sin^2 \theta}{r^4 - 4\bar{\sigma}^2 \sin^2 \theta} d\phi. \tag{132}
$$

In this section, we restrict ourselves within the equatorial plane, namely *θ* = *π*/2. On the equatorial plane, the constant *d* in the metric always vanish because *d* is always associated with cos *θ*.

We employ the same way for the Kerr case; we first derive the orbit Equation on the equatorial plane from Equation (77) as:

$$\begin{split} \left(\frac{dr}{d\phi}\right)^{2} &= -\frac{r^{5}(b\_{0}-r)\left(4\bar{a}^{2}b^{2} - 4\bar{a}br^{3} - b^{2}r^{4} + r^{6}\right)}{\left(-4\bar{a}^{2}b + 2\bar{a}r^{3} + br^{4}\right)^{2}} \\ &= \frac{r^{4}}{b^{2}} - r^{2} - \frac{b\_{0}r^{3}}{b^{2}} + b\_{0}r - \frac{4br^{3}}{b^{3}} + \frac{4lb\_{0}r^{2}}{b^{3}} + \mathcal{O}(\hbar^{2}/b^{2}), \end{split} \tag{133}$$

where *b* denotes the impact parameter of the light ray and we use the weak field and slow rotation approximations in the last line. There are no *b*0 squared terms in the last line. The orbit Equation thus becomes:

$$\mathcal{I}\left(\frac{du}{d\phi}\right)^2 = \frac{1}{b^2} - u^2 - \frac{b\_0 u}{b^2} + b\_0 u^3 - \frac{4\hbar u}{b^3} - \frac{4\hbar b\_0 u^2}{b^3} + \mathcal{O}(\hbar^2/b^6). \tag{134}$$

This Equation is iteratively solved as:

$$u = \frac{\sin\phi}{b} + \frac{\cos^2\phi}{2b^2}b\_0 - \frac{2}{b^3}\mathbb{I} + \mathcal{O}\left(\frac{b\_0^2}{b^3}, \frac{\|b\_0\|}{b^4}\right). \tag{135}$$

Solving Equation (135) for *φS* and *φR*, we obtain *φS* and *φR* as:

$$\phi\_S = \arcsin(bu\_S) - \frac{b\sqrt{1 - b^2 u\_S^2}}{2b} + \frac{2\bar{a}}{b^2 \sqrt{1 - b^2 u\_S^2}} + \mathcal{O}\left(\frac{b\rho^2}{b^2}, \frac{\bar{a}b\rho}{b^3}\right),\tag{136}$$

$$\phi\_R = \pi - \arcsin(bu\_R) + \frac{b\_0\sqrt{1 - b^2u\_R}^2}{2b} - \frac{2\hbar}{b^2\sqrt{1 - b^2u\_R}^2} + \mathcal{O}\left(\frac{b\_0^2}{b^2}, \frac{db\_0}{b^3}\right). \tag{137}$$

#### *9.2. Gaussian Curvature*

In the weak-field approximation, the Gaussian curvature of the equatorial plane is:

$$K = -\frac{b\_0}{2r^3} - \frac{56d^2}{r^6} + \mathcal{O}\left(\frac{\hbar^2 b\_0}{r^7}, \frac{d^4}{r^{10}}\right),\tag{138}$$

where *a*¯ and *b*0 play roles as book-keeping parameters in the weak-field approximation. It is not surprising that this Gaussian curvature deviates from Equation (26) in Jusufi and Övgün [77], because their Gaussian curvature describes a different surface that is defined by using the Randers–Finsler metric. The Randers–Finsler metric is quite different from our generalized optical metric *γij*.

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When we perform the surface integral of the Gaussian curvature in Equation (88), we use Equation (135) for a boundary of the integration domain. The surface integral of the Gaussian curvature in Equation (88) is thus calculated as:

$$-\iint\_{\mathcal{S}\subseteq\Box\_{S}^{\otimes}} KdS = \int\_{\phi\_{\mathcal{S}}}^{\phi\_{\mathcal{S}}} \int\_{\infty}^{r(\phi)} \left(-\frac{b\_{0}}{2r^{2}}\right) dr d\phi + \mathcal{O}\left(\frac{b\_{0}^{-2}}{b^{2}}, \frac{db\_{0}}{b^{3}}\right)$$

$$=\frac{b\_{0}}{2} \int\_{\phi\_{\mathcal{S}}}^{\phi\_{\mathcal{S}}} \int\_{0}^{\phi\_{\mathcal{S}}} \int\_{0}^{\frac{\sin\theta}{b^{2}}} \frac{\cos^{2}\theta}{2r^{2}} b\_{0} \cdot \frac{2}{b^{3}} dud\phi + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{db\_{0}}{b^{3}}\right)$$

$$=\frac{b\_{0}}{2} \int\_{\phi\_{\mathcal{S}}}^{\phi\_{\mathcal{S}}} \left[\frac{\sin\phi}{b}\right] d\phi + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{\bar{a}b\_{0}}{b^{3}}\right)$$

$$=\frac{b\_{0}}{2} \left[-\frac{\cos\phi}{b}\right]\_{\phi=\phi\_{\mathcal{S}}}^{\phi\_{\mathcal{S}}} + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{db\_{0}}{b^{3}}\right)$$

$$=\frac{b\_{0}}{2b} \left(\sqrt{1 - b^{2}u\_{R}^{2}} + \sqrt{1 - b^{2}u\_{S}^{2}}\right) + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{db\_{0}}{b^{3}}\right),\tag{139}$$

where we use sin *φR* = *buR* + O(*ab*¯ −2, *b*0*b*−1) and sin *φS* = *buS* + O(*ab*¯ −2, *b*0*b*−1) by Equations (137) and (136) in the last line.

#### *9.3. Geodesic Curvature of Photon Orbit*

We study the geodesic curvature of the photon orbit on the equatorial plane in the stationary and axisymmetric spacetime by using the generalized optical metric. It generally becomes [42]:

$$\kappa\_{\mathcal{S}} = -\sqrt{\frac{1}{\gamma \gamma^{\theta \emptyset}}} \beta\_{\Phi, r} \,. \tag{140}$$

In the Teo wormhole, this expression is rearranged as:

$$\kappa\_{\mathcal{S}} = -\frac{2d}{r^3} + \frac{db\_0}{r^4} + \frac{db\_0}{4r^5}^2 + \frac{db\_0^3}{8r^6}^3 + \mathcal{O}\left(\frac{d^3}{r^7}, \frac{d^3b\_0}{r^8}\right). \tag{141}$$

We compute the path integral of the geodesic curvature of the photon orbit. The detailed calculations and result are:

$$\begin{split} \int\_{S}^{R} \kappa\_{\text{s}}^{\text{R}} d\ell &= \int\_{R}^{\text{S}} \frac{2\bar{a}}{r^{3}} d\ell + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{\bar{a}b\_{0}}{b^{3}}\right) \\ &= \int\_{\pi/2-\phi\_{\text{R}}}^{\pi/2-\phi\_{\text{S}}} \frac{2\bar{a}\cos\theta}{b^{2}} d\theta + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{\bar{a}b\_{0}}{b^{3}}\right) \\ &= \frac{2\bar{a}}{b^{2}} \left[\sin\left(\frac{\pi}{2} - \phi\_{\text{S}}\right) - \sin\left(\frac{\pi}{2} - \phi\_{\text{R}}\right)\right] + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{\bar{a}b\_{0}}{b^{3}}\right) \\ &= \frac{2\bar{a}}{b^{2}} \left(\sqrt{1 - b^{2}u\_{\text{S}}^{2}} + \sqrt{1 - b^{2}u\_{\text{R}}^{2}}\right) + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{\bar{a}b\_{0}}{b^{3}}\right), \end{split} \tag{142}$$

for the retrograde orbit of the photon. In the last line, we used sin *φR* = *buR* + O(*ab*¯ −2, *b*0*b*−1) and sin *φS* = *buS* + O(*ab*¯ −2, *b*0*b*−1) from Equation (135). The above result becomes 4*a*¯/*b*2, as *rR* → ∞ and *rS* → ∞. The sign of the right-hand side in Equation (142) is opposite if the photon is in prograde motion.

#### *9.4. φRS Part*

The rotating Teo wormhole is an asymptotically flat spacetime, as seen from Equation (128). Therefore, the integral of the geodesic curvature of the circular arc segment with an infinite radius can be expressed simply as *φRS*. By using Equations (136) and (137), *φRS* is obtained as:

$$\begin{split} \phi\_{RS} &= \phi\_R - \phi\_S \\ &= n - \arcsin(bu\_R) - \arcsin(bu\_S) + \frac{b\_0\sqrt{1 - b^2u\_R^2}}{2b} + \frac{b\_0\sqrt{1 - b^2u\_S^2}}{2b} \\ &- \frac{2\hbar}{b^2\sqrt{1 - b^2u\_R^2}} - \frac{2\hbar}{b^2\sqrt{1 - b^2u\_S^2}} + \mathcal{O}\left(\frac{b\_0^2}{b^2}, \frac{db\_0}{b^3}\right). \end{split} \tag{143}$$

#### *9.5. Ψ Parts*

For the rotating Teo wormhole in Equation (128), Equation (85) is computed as:

$$
\sin \Psi\_R = bu\_R + 2du\_R^{-2} - 4d^2bu\_R^{-5} \tag{144}
$$

and Equation (86) becomes:

$$\sin(\pi - \Psi\_S) = b\mu\_S + 2\hbar\mu\_S^2 - 4\hbar^2 b\mu\_S \,^5\,. \tag{145}$$

where the slow-rotation approximation is not needed.

Therefore, we obtain Ψ*R* and Ψ*S* as:

$$\Psi\_R = \arcsin(bu\_R) + \frac{2du\_R^2}{\sqrt{1 - b^2u\_R^2}} + \frac{2\bar{a}^2bu\_R^{-5}\left(2b^2u\_R^2 - 1\right)}{\left(b^2u\_R^{-2} - 1\right)^{3/2}} + \mathcal{O}(\bar{a}^3/b^6),\tag{146}$$

$$
\pi - \Psi\_S = \arcsin(bu\_S) + \frac{2hu\_S^2}{\sqrt{1 - b^2u\_S^2}} + \frac{2\bar{u}^2bu\_S^{\frac{5}{2}}\left(2b^2u\_S^2 - 1\right)}{\left(b^2u\_S^2 - 1\right)^{3/2}} + \mathcal{O}(\hbar^3/b^6),
\tag{147}
$$

where we used the slow-rotation approximation.

#### *9.6. Deflection Angle of Light*

We combine Equations (139) and (142) to obtain the deflection angle of light in the prograde orbit as:

$$\begin{split} a\_{\text{prog}} &= \frac{b\_0}{2b} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^2} \right) - \frac{2d}{b^2} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^2} \right) \\ &+ \mathcal{O} \left( \frac{b\_0^{-2}}{b^2}, \frac{db\_0}{b^3} \right). \end{split} \tag{148}$$

The deflection angle of the retrograde light is:

$$\begin{split} a\_{\text{retr0}} &= \frac{b\_0}{2b} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^2} \right) + \frac{2\hbar}{b^2} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^2} \right) \\ &+ \mathcal{O} \left( \frac{b\_0^2}{b^2}, \frac{db\_0}{b^3} \right). \end{split} \tag{149}$$

*Universe* **2019**, *5*, 218

Next, by using Equations (143), (146), and (147), we obtain the deflection angle of the prograde light as:

$$\begin{split} \mathsf{a}\mathsf{prog} &= \pi - \arcsin(bu\_{R}) - \arcsin(bu\_{S}) + \frac{b\sqrt{1-b^{2}u\_{R}^{2}}^{2}}{2b} + \frac{b\sqrt{1-b^{2}u\_{S}^{2}}^{2}}{2b} \\ &- \frac{2\mathsf{il}}{b^{2}\sqrt{1-b^{2}u\_{R}^{2}}^{2}} - \frac{2\mathsf{il}}{b^{2}\sqrt{1-b^{2}u\_{S}^{2}}^{2}} + \arcsin(bu\_{R}) + \frac{2\mathsf{il}u\_{R}^{2}}{\sqrt{1-b^{2}u\_{R}^{2}}^{2}} \\ &- \pi + \arcsin(bu\_{S}) + \frac{2\mathsf{il}u\_{S}^{2}}{\sqrt{1-b^{2}u\_{S}^{2}}^{2}} + \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{db\_{0}}{b^{3}}\right) \\ &= \frac{b\_{0}}{2b}\left(\sqrt{1-b^{2}u\_{R}^{2}} + \sqrt{1-b^{2}u\_{S}^{2}}\right) - \frac{2\mathsf{il}}{b^{2}}\left(\sqrt{1-b^{2}u\_{R}^{2}} + \sqrt{1-b^{2}u\_{S}^{2}}\right) \\ &+ \mathcal{O}\left(\frac{b\_{0}^{2}}{b^{2}}, \frac{db\_{0}}{b^{3}}\right). \end{split} \tag{150}$$

The deflection angle of light in the retrograde orbit is:

$$\begin{split} a\_{\text{retro}} &= \frac{b\_0}{2b} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^2} \right) + \frac{2\bar{a}}{b^2} \left( \sqrt{1 - b^2 u\_R^2} + \sqrt{1 - b^2 u\_S^2} \right) \\ &+ \mathcal{O} \left( \frac{b\bar{b}^2}{b^2}, \frac{\bar{a}b\_0}{b^3} \right). \end{split} \tag{151}$$

The deflection of light in the prograde (retrograde) orbit is weaker (stronger) when increasing the angular momentum of the Teo wormhole. The reason is as follows. The local inertial frame in which the light travels at the light speed *c* in general relativity moves faster (slower). Hence, the time-of-flight of light becomes shorter (longer). On light propagation, a similar explanation is done by using the dragging of the inertial frame also by Laguna and Wolsczan [78]. They discussed the Shapiro time delay. The expression of the deflection angle of light by a rotating Teo wormhole is similar to that by Kerr black hole.

In Equations (150) and (151), the source and receiver can be located at finite distances from the wormhole. In the limit as *rR* → ∞ and *rS* → ∞, Equations (148) and (149) become:

$$a \text{prog} \rightarrow \frac{b\_0}{b} - \frac{4\overline{a}}{b^2} + \mathcal{O}\left(\frac{b\_0^2}{b^2}, \frac{db\_0}{b^3}\right),$$

$$a\_{\text{retro}} \rightarrow \frac{b\_0}{b} + \frac{4\overline{a}}{b^2} + \mathcal{O}\left(\frac{b\_0^2}{b^2}, \frac{db\_0}{b^3}\right). \tag{152}$$

They are in complete agreement with Equations (39) and (56) in Jusufi and Övgün [77], where they restrict themselves within the asymptotic source and receiver (*rR* → ∞ and *rS* → ∞).

#### *9.7. Finite-Distance Corrections in the Teo Wormhole Spacetime*

To be precise, we define the finite-distance correction to the deflection angle of light as the difference between the asymptotic deflection angle *α*∞ and the deflection angle for the finite distance case. It is denoted as *δα*.

We consider the following situation. An observer on the Earth sees the light deflected by the solar mass. The source of light is located in a practically asymptotic region. In other words, we choose *b*0 = *M*, *a*¯ = *J*, *rR* ∼ 1.5 × 108 km, *rS* ∼ ∞. See Figure 14 for the finite-distance correction due to the impact parameter *b*. In Figure 14, the green curve means the difference between the asymptotic bending angle and the deflection angle with finite-distance corrections, the blue curve denotes the asymptotic deflection angle, and the orange curve is the deflection angle with finite-distance corrections by the rotating Teo wormhole. The deflection angle is decreased by the finite-distance correction. If the impact parameter *b* increases, the finite-distance correction also increases.

See also Figure 15 for numerical calculations of the finite-distance correction due to the impact parameter *b*.  The deflection of light is stronger in a Kerr black hole case for the chosen values.

**Figure 14.** *α* in the Teo wormhole: The blue curve is the asymptotic deflection angle by the rotating Teo wormhole. The orange curve is the deflection angle with finite-distance corrections by the rotating Teo wormhole. The blue curve shows the difference between the asymptotic deflection angle and the deflection angle with finite-distance corrections by the rotating Teo wormhole.

**Figure 15.** *α* for prograde motion of light: The horizontal axis is the impact parameter of photon orbit. The vertical axis means the deflection angle of light. The blue curve means the deflection angle with finite-distance corrections by the Kerr black hole. The red curve corresponds to that by the rotating Teo wormhole. For the purpose of this comparison, the mass of a Kerr black hole *M* and the throat radius of a rotating Teo wormhole *b*0 are chosen as *M* = *b*0 = *M*. The spin angular momentum of a Kerr black hole and that of a rotating Teo wormhole are chosen as the same as that the Sun for simplicity.

#### **10. Summary**

In this paper, we provided a brief review of a series of works on the deflection angle of light for a light source and receiver in a non-asymptotic region. [38,39,42,43]. The validity and usefulness of the new formulation come from the GB theorem in differential geometry. First, we discussed how to define the gravitational deflection angle of light in a static, spherically symmetric, and asymptotically flat spacetime, for which we assume the finite-distance source and receiver. We examined whether our definition is invariant geometrically by using the GB theorem. By using our definition, we carefully computed finite-distance corrections to the light deflection in Schwarzschild spacetime. We considered both the cases of weak deflection and the strong one. Next, we extended the definition to stationary and axisymmetric spacetimes. This extension allows us to compute finite-distance corrections for Kerr black holes and rotating Teo wormholes. We verified that these results are consistent with previous works in the infinite-distance limit. We mentioned also the finite-distance corrections to the light deflection by Sagittarius A∗. It is left as future work to apply the present formulation to other interesting spacetime models and also to extend it to a more general spacetime structure.

**Funding:** This work was supported in part by JSPS research fellowship for young researchers (T.O.); in part by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research, No. 18J14865 (T.O.) and No. 17K05431 (H.A.); and in part by by the Ministry of Education, Culture, Sports, Science, and Technology, No. 17H06359 (H.A.).

**Acknowledgments:** We are grateful to Marcus Werner for the stimulating and very fruitful discussions. We thank Takao Kitamura, Asahi Ishihara, and Yusuke Suzuki for useful conversations. We would like to thank Yuuiti Sendouda, Ryuichi Takahashi, Yuya Nakamura, and Naoki Tsukamoto for useful conversations.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Detailed Calculations at** *O***(***M***2/***b***2) and** *O***(***a***2/***b***2) in Kerr Spacetime**

First, we investigate the Gaussian curvature *K* of the equatorial plane in the Kerr spacetime. Here, we assume the weak-field and slow-rotation approximations. Up to the second order, *K* is expanded as:

$$\begin{split} K &= \frac{R\_{r\phi r\phi}}{\gamma} \\ &= -\frac{2M}{r^3} + \frac{3M^2}{r^4} + O\left(\frac{a^2M}{r^5}\right), \end{split} \tag{A1}$$

where *γ* denotes det(*γij*). There are no *a*2 terms in *K*. More interestingly, only the *a*2*M* term at the third-order level exists in *K*. By noting that *K* begins with *O*(*M*), what we need for the second-order calculations is only the linear-order term in the area element on the equatorial plane. This is obtained as:

$$\begin{split} dS & \equiv \sqrt{\gamma} dr d\phi \\ &= \left[ r + 3M + O\left(\frac{M^2}{r}\right) \right] dr d\phi \end{split} \tag{A2}$$

where terms at *O*(*a*) and at *O*(*a*2) do not exist in *dS*. This is because all terms including the spin parameter cancel out in *γ* for *θ* = *π*/2 and *γ* thus depends only on *M*, as shown by direct calculations. *Universe* **2019**, *5*, 218

By using Equations (A1) and (A2), the surface integration of the Gaussian curvature is performed as:

− -- *KdS* = *rOE* ∞ *dr φR φS dφ* − 2*M r*3 + 3*M*2 *r*4 (*r* + 3*M*) + *O M*3 *b*3 , *aM*2 *b*3 , *a*2*M b*3 = - 1 *b* sin *φ*+ *M b*2 (1+cos2 *φ*) 0 *du φR φS dφ* (2*M* + 3*uM*2) + *O M*3 *b*3 , *aM*2 *b*3 , *a*2*M b*3 = *φR φS* + 2*M b* sin *φ* + *M*2 2*b*2 (7 + cos2 *φ*) , *dφ* + *O M*3 *b*3 , *aM*2 *b*3 , *a*2*M b*3 =2*M b* + cos *φ* ,*φS φR* + *M*2 2*b*2 + 30*φ* + sin(2*φ*) 4 ,*φR φS* + *O M*3 *b*3 , *aM*2 *b*3 , *a*2*M b*3 =2*M b* + 1 − *b*2*uS*2 + 1 − *b*2*uR*2 , + 15*M*2 4*b*2 [*π* − arcsin(*buS*) − arcsin(*buR*)] + *M*2 4*b*2 [ *buS*(15 − 7*b*2*uS* 2) 1 − *b*2*uS*2 + *buR*(15 − 7*b*2*uR*2) 1 − *b*2*uR*2 ] + *O M*3 *b*3 , *aM*2 *b*3 , *a*2*M b*3 , (A3)

where we use, in the second line, an iterative solution for the orbit equation of Equation (77) in the Kerr spacetime.

Next, we study the geodesic curvature. On the equatorial plane, we find:

$$\begin{split} \mathbb{K}\_{\mathcal{S}} &= -\frac{1}{\sqrt{\frac{\Sigma^2}{\Delta(\Sigma - 2Mr)} \left(r^2 + a^2 + \frac{2a^2Mr\sin^2\theta}{\Sigma}\right) \frac{\Sigma\sin^2\theta}{(\Sigma - 2Mr)}}} \beta\_{\Phi,r} \\ &= -\frac{2aM}{r^3} + O\left(\frac{aM^2}{r^3}\right). \end{split} \tag{A4}$$

Note that *a*2 terms do not exist. Therefore, we obtain:

$$\begin{split} \int\_{\varepsilon\_p} \kappa\_{\mathbb{S}} d\ell &= -\int\_{\mathbb{S}}^R d\ell \left[ \frac{2aM}{r^2} + O\left(\frac{aM^2}{r^3}\right) \right] \\ &= -\frac{2aM}{b^2} \int\_{\mathbb{S}^3}^{\Phi\_R} \cos\theta d\theta + O\left(\frac{aM^2}{b^3}\right) \\ &= \frac{2aM}{b^2} [\sqrt{1 - b^2 \mu\_R^2} + \sqrt{1 - b^2 \mu\_S^2}] + O\left(\frac{aM^2}{b^3}\right), \end{split} \tag{A5}$$

where we use sin *φS* = *rS*2 − *b*2/*rS* + *O*(*M*/*rS*) and sin *φR* = − *rR*2 − *b*2/*rR* + *O*(*M*/*rR*). By combining Equations (A3) and (A5), we obtain:

$$\begin{split} a &= -\int\_{\frac{\pi}{R}\frac{\pi a}{\sqrt{\pi}}} \mathrm{K}dS - \int\_{R}^{S} \kappa\_{S} d\ell \\ &= \frac{2M}{b} \left[ \sqrt{1 - b^{2}u\_{S}{}^{2}} + \sqrt{1 - b^{2}u\_{R}{}^{2}} \right] \\ &+ \frac{15M^{2}}{4b^{2}} \left[ \pi - \arcsin(bu\_{S}) - \arcsin(bu\_{R}) \right] \\ &+ \frac{M^{2}}{4b^{2}} \left[ \frac{bu\_{S}(15 - 7b^{2}u\_{S}{}^{2})}{\sqrt{1 - b^{2}u\_{S}{}^{2}}} + \frac{bu\_{R}(15 - 7b^{2}u\_{R}{}^{2})}{\sqrt{1 - b^{2}u\_{R}{}^{2}}} \right] \\ &- \frac{2aM}{b^{2}} \left[ \sqrt{1 - b^{2}u\_{R}{}^{2}} + \sqrt{1 - b^{2}u\_{S}{}^{2}} \right] + O\left(\frac{M^{3}}{b^{3}}, \frac{aM^{2}}{b^{3}}, \frac{a^{2}M}{b^{3}}\right). \end{split} \tag{A6}$$

Note that *a*2 terms and *a*3 ones do not appear in *α* for the finite distance situation as well as in the infinite distance limit. If we assume the infinite distance limit *uR*, *uS* → 0, Equation (A6) becomes:

$$
\alpha \to \frac{4M}{b} + \frac{15\pi M^2}{4b^2} - \frac{4aM}{b^2}.\tag{A7}
$$

This agrees with the known results, especially on the numerical coefficients at the order of *M*2 and *aM*.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Deflection Angle of Photons through Dark Matter by Black Holes and Wormholes Using Gauss–Bonnet Theorem**

#### **Ali Övgün 1,2**

Received: 17 March 2019; Accepted: 9 May 2019; Published: 14 May 2019

**Abstract:** In this research, we used the Gibbons–Werner method (Gauss–Bonnet theorem) on the optical geometry of a black hole and wormhole, extending the calculation of weak gravitational lensing within the Maxwell's fish eye-like profile and dark-matter medium. The angle is seen as a partially topological effect, and the Gibbons–Werner method can be used on any asymptotically flat Riemannian optical geometry of compact objects in a dark-matter medium.

**Keywords:** gravitational lensing; weak deflection; dark matter; Gauss–Bonnet theorem; black hole; wormhole

**PACS:** 95.30.Sf; 98.62.Sb; 97.60.Lf

#### **1. Introduction**

Black holes are an essential component of our universe, and one of the most important discoveries in astrophysics is that, when stars die, they can collapse to become extremely small objects. Black holes provide an important opportunity for probing and testing the fundamental laws of the universe. For example, gravitational waves from black holes and neutron star mergers have been recently detected [1]. Black holes may also hint at the nature of quantum gravity at small scales that change the area law of entropy. Quantum gravity is far from understood, though theoretically it has seen tremendous progress, and, in a few years, the Event Horizon Telescope may provide some more information about it [2–5].

In 1854, Maxwell presented the solution to a mathematical problem related to the passage of rays through a sphere of variable refractive index, and he noted that the potential existence of a medium of this kind would possess exceptional optical properties [6]. This is similar to the reflection of the crystalline lens in fish. This optical tool is Maxwell's fish eye (MFE), the condition in which all light rays form circular trajectories. It was a remarkable accomplishment to visualize that a lens whose refractive index increases toward a point could form perfect images [7].

Luneburg discovered that the ray propagation of MFE is equivalent to ray propagation on a homogeneous sphere with a unit radius and a unit refractive index within geometrical optics [8]. This showed that the imaging of variants that have been applied to microwave devices and the fish-eye lens in photography form an extremely wide-angled image, almost hemispherical in coverage. In 2009, Leonhardt showed that MFE is also good for waves, and it enables the production of super-resolution imaging with perfect lensing, which requires negative refractive-index materials. This began a debate and offered a rich area of research to explore [9–11]. It was shown that perfect imaging in a homogeneous three-dimensional region is also possible [12]. MFE happens when all light rays

arising from any point within converge at its conjugate, which means that power released from a source can only be fully absorbed at its image point, resulting in perfect imaging. There has been a rapid increase in the importance of perfect imaging in theoretical and experimental optics [13–15].

Fermat's principle says that light rays always follow extremal optical paths, with path length being measured by refractive index *n*. The formula for MFE indicates the interesting possibility that rays generate a perfect image in a black hole. The refractive index depends only on distance *r* from the origin [13,14]. In this paper, we try to understand the effect of an MFE-like profile on the deflection angle. For simplicity, we used the uniform MFE-like profile, which is different from a nonuniform MFE profile.

Gravitational lensing is a useful tool of astronomy and astrophysics [16], in which light rays from distant stars and galaxies are deflected by a planet, a black hole, or dark matter [17,18]. The detection of dark-matter filaments [19] using weak deflection is a very relevant topic because it can help in understanding the large-scale structure of the universe [20]. To build a sky map (the refractive index of the entire visible universe), there is ongoing research on the observation of the effect of cosmological weak deflection on temperature fluctuations in the cosmic microwave background (CMB) [21]. From a theoretical point of view, new methods have been proposed to calculate deflection angle. One of them is the Gauss–Bonnet theorem (GBT), which was first proposed by Gibbons and Werner using optical geometry [22–24]. The deflection angle is seen as a partially topological effect that can be calculated by integrating the Gaussian curvature of the optical metric outward from the light ray by using the following equation: [22,23]

$$\hbar = -\int \int\_{D\_{\text{eq}}} \mathbf{K} dA. \tag{1}$$

Since Gibbons and Werner's paper on weak deflection angles by GBT provided a unique perspective, this method has been applied in various cases [25–47].

Dark matter makes up to 27% of the total mass–energy of the universe [48]. We can only detect dark matter from its gravitational interactions, and we only know that dark matter is nonbaryonic, nonrelativistic (or cold), and it has weak nongravitational interactions. There are many dark-matter candidates, such as weakly interacting massive particles (WIMPs), super-WIMPs, axions, and sterile neutrinos [49]. It has been proposed that dark matter is a composite, such as the dark-atom model, which we investigate here using the deflection of light through it. Dark matter, although suppressed, generally has electromagnetic interactions [50], such that the medium of dark matter should have some optical properties that a traveling photon can sense because of the frequency-dependent refractive index. The refractive index regulates the speed at which a wave is propagated via a medium. The particles of dark matter do not get electrically charged, but they can couple to other particles that have a virtual electromagnetic charge, and can also couple to photons [51–54]. To find the amplitude of dark-matter annihilation into two photons, we must first calculate the scattering amplitude. One can obtain the index of the refractive of light, where the real part is related to the speed of propagation.

To investigate weak deflection through dark matter, we consider the propagation effects for the case that particles of dark matter (warm thermal relics or axionlike particles) have low mass whose number density is larger than ordinary matter. Put simply, dark matter interacts with photons (if only through quantum fluctuations), resulting in a refractive index. The relationship between refractive index and the forward Compton amplitude at relatively low photon energies [50] (Mfwd ∼ −*ε*2*e*2) is

$$m = 1 + \frac{\rho}{4m\_{dm}^2 \omega^2} \mathcal{M}\_{\text{fwd}\prime} \tag{2}$$

where *ω* is the measured photon frequency, and *ρ* = 1.1 × 10−6 GeV/cm3 is the present-day dark-matter density [50]. Neglecting spin, the amplitude is a real and even function of *ω* (for photon energies below the inelastic threshold); additionally, the coefficients of the *O*(*ω*2*n*) terms are positive

and spin-dependent interactions can lead to odd powers in the expansion about *ω*. Their presence could give information on the spin of dark matter. Hence, the refractive index becomes [50]:

$$m = 1 + \frac{\rho}{4m\_{dm}^2} \left[ \frac{A}{\omega^2} + B + \mathbb{C}\omega^2 + \mathcal{O}\left(\omega^4\right) \right]. \tag{3}$$

To do so, we suppose that photons can be deflected through dark matter due to dispersive effects. We used the index of refractive *n*(*ω*) that is manipulated by the scattering amplitude of the light and dark matter in the forward [50].

Gravitational lensing in plasma has been studied in various cases [45,46,55–57]. For the first time, Bisnovatyi-Kogan and Latimer showed that, due to the dispersive properties of plasma, even in homogeneous plasma, gravitational deflection differs from a vacuum deflection angle [56,57]. Moreover, it was shown that the deflection angle is increased due to the presence of plasma [55]. Afterward, Crisnejo and Gallo calculated weak lensing in a plasma medium using the GBT [45].

The main motivation of this research is to shed light on the unexpected features of spacetimes in regards to an MFE-like profile, and to derive the deflection angle of black holes using the Gauss–Bonnet theorem in weak limits for a dark-matter medium. We suppose that the refractive index of the medium is spatially nonuniform but it is uniform at large distances. We also investigated the effect of various parameters on the refractive index of the medium, which has not been covered in previous studies.

#### **2. Effect of Medium on Deflection Angle of Schwarzschild Black Hole Using Gauss–Bonnet Theorem**

In this section, we first describe the black-hole solution in a static and spherically symmetric spacetime. Then, we apply the MFE-like profile within the GBT to calculate the weak deflection angle.

The Schwarzschild black-hole spacetime reads

$$ds^2 = -f(r)dt^2 + g(r)dr^2 + r^2(d\theta^2 + \sin^2\theta d\varphi^2),\tag{4}$$

with metric functions

$$f(r) \quad = g(r)^{-1} = \quad 1 - \frac{2M}{r}.\tag{5}$$

Analysis of the geodesics equation, the ray equation, is calculated by

$$\varphi = \int \frac{b\sqrt{g(r)}\,\mathrm{d}r}{r^2\sqrt{\frac{1}{f(r)} - \frac{b^2}{r^2}}} \,\mathrm{}\tag{6}$$

where *b* is the impact parameter of the unperturbed photon.

Our universe is homogeneous and isotropic on large scales. Now, we consider isotropic coordinates that are nonsingular at the horizon and time direction is a Killing vector. Moreover, time slices become Euclidean with a conformal factor, and one can calculate refractive index *n* of light rays around the black hole. Another important feature of isotropic coordinates is that they satisfy Landau's condition of coordinate clock synchronization:

$$\frac{\partial}{\partial \mathbf{x}\_j} \left( -\frac{\mathcal{G}\_{00}}{\mathcal{g}\_{00}} \right) = \frac{\partial}{\partial \mathbf{x}\_i} \left( -\frac{\mathcal{G}\_{0j}}{\mathcal{g}\_{00}} \right) (i, j = 1, 2, 3). \tag{7}$$

Using transformation

$$r = \rho \left( 1 + \frac{M}{2\rho} \right)^2,\tag{8}$$

the Schwarzschild black hole is rewritten in isotropic coordinates (where *ρ* is an isotropic radial coordinate) [45]

$$ds^2 = -F(\rho)dt^2 + G(\rho)(d\rho^2 + \rho^2 d\Omega^2),\tag{9}$$

with

$$F(\rho) = \left(\frac{\rho - \frac{M}{2}}{\rho + \frac{M}{2}}\right)^2, \text{ and } G(\rho) = \left(\frac{\rho + \frac{M}{2}}{\rho}\right)^4. \tag{10}$$

The metric becomes nonsingular at horizon *r* = 2*M*. It can also be written in Fermat form of the metric:

$$ds^2 = F(\rho)[-dt^2 + n(\rho)^2(d\rho^2 + \rho^2 d\Omega^2)],\tag{11}$$

with index of refractive *n*(*ρ*) = *c v*(*ρ*). For the Schwarzschild black-hole medium, the refractive index reads

$$m = \frac{\left(1 + \frac{M}{2\rho}\right)^3}{\left(1 - \frac{M}{2\rho}\right)^2} \tag{12}$$

and it can be approximated for large *ρ M*

$$m \approx 1 + \frac{2M}{\rho}.\tag{13}$$

Now, the ray equation becomes

$$
\varphi = \int \frac{bd\rho}{\rho^2 \sqrt{n^2 - \frac{b^2}{\rho^2}}}.\tag{14}
$$

To discuss the deflection angle and extract information of the MFE-like profile, the GBT was used instead of the null geodesics method. The GBT is calculated using the negative Gauss curvature of the optical metric.

#### *2.1. Case 1*

Let us start from the constant case for medium *nm* as refractive index:

$$n\_m = n\_{0\prime} \tag{15}$$

where *n*0 is a constant refractive index of the medium; here, we consider the GBT to obtain the deflection angle in a medium in weak field limits.

Let us write the optical Schwarzschild spacetime in an equatorial plane [45]:

$$d\sigma^2 = \frac{n\_m^2}{f(\rho)}[g(\rho)d\rho^2 + \rho^2 d\rho^2].\tag{16}$$

Then, we calculate the Gaussian optical curvature:

$$K = -2\frac{M}{n\_0 \rho^2 \rho^3} + 3\frac{M^2}{n\_0 \rho^2 \rho^4} + O(M^3),\tag{17}$$

which is negative everywhere that gives a universal property of black-hole metrics [23].

It reduces to this form at a linear order of *M*:

$$K \approx -2\frac{M}{n\_0 \epsilon^2 \rho^3}.\tag{18}$$

*Universe* **2019**, *5*, 115

This result is used to evaluate the deflection angle using a nonsingular domain outside the light ray (*Dρ*, with boundary *∂Dρ* = *γ* ∪ *Cρ*) [22]:

$$\int\_{D\_{\rho}} \mathcal{K} \, dS + \oint\_{\partial D\_{\rho}} \kappa \, dt + \sum\_{i} \theta\_{i} = 2\pi \chi(D\_{\rho}), \tag{19}$$

where *κ* stands for the geodesic curvature, and *K* is the Gaussian optical curvature, with exterior angles *θi* = (*θO*, *θS*) and Euler characteristic number *χ*(*Dρ*) = 1. At weak limits, (*ρ* → ∞), *θO* + *θS* → *π*. Then, the GBT reduces to

$$\int\_{D\_{\rho}} \mathbf{K} \, dS + \oint\_{\mathbb{C}\_{\rho}} \mathbf{x} \, dt \stackrel{\rho \to \infty}{=} \iint\_{D\_{\infty}} \mathbf{K} \, dS + \int\_{0}^{\pi + \mathbb{A}} d\varphi = \pi. \tag{20}$$

For geodesics *γ*, geodesic curvature vanishes *κ*(*γ*) = 0, and we have

$$\kappa(\mathbb{C}\_{\rho}) = |\nabla\_{\mathbb{C}\_{\rho}} \dot{\mathbb{C}}\_{\rho}| \, \tag{21}$$

with *Cρ* = *ρ* = constant. The GBT becomes

$$\lim\_{\rho \to \infty} \int\_0^{\pi + \hat{\alpha}} \left[ \kappa \frac{d\sigma}{d\rho} \right] \Big|\_{C\_{\rho}} d\rho = \pi - \lim\_{\rho \to \infty} \int \int\_{D\_{\rho}} \mathcal{K} dS\_{\prime} \tag{22}$$

and one can calculate

$$\frac{d\sigma}{d\rho}\Big|\_{C\_{\rho}} = n\_{\text{m}} \left(\frac{\rho^3}{\rho - 2M}\right)^{1/2},\tag{23}$$

where, for very large radial distance,

$$
\kappa(\mathbb{C}\_{\rho})dt = d\,\boldsymbol{\varrho}.\tag{24}
$$

Therefore, as expected, for this number density profile and physical metric (which imply that the optical metric is asymptotically Euclidean), we corroborate that

$$\lim\_{\rho \to \infty} \kappa \frac{d\sigma}{d\rho} \Big|\_{\mathcal{C}\_{\rho}} = 1. \tag{25}$$

At a linear order in *M*, it follows to use Equation (22) in limit *ρ* → ∞, and taking geodesic curve *γ*, approximated by its flat Euclidean version parametrized as *ρ* = *b*/ sin *ϕ*, with *b* representing the impact parameter in the physical spacetime that

$$\hbar = -\lim\_{\rho \to \infty} \int\_0^\pi \int\_{\frac{b}{\sin \rho}}^\rho \mathcal{K} dS. \tag{26}$$

After nontrivial calculation, we calculate that the deflection angle of the Schwarzschild black hole in medium for the leading order terms is

$$
\hat{\kappa} = 4 \frac{M}{n\_0 b'} \tag{27}
$$

which agrees with the well-known results in the limit at which its presence is negligible (*n*0 = 1); this expression reduces to known vacuum formula *α*ˆ = 4 *M b* ., so that GBT exhibits a partially topological effect. This method can be used in any asymptotically flat Riemannian optical metrics.

#### *2.2. Case 2*

Now, we apply the different model of the MFE-like medium [10]

$$m = \frac{z\_0}{1 + z^2},\tag{28}$$

where *z*0 and *z* are a constant.

The Gaussian curvature of the optical metric approximating in leading orders is negative everywhere and found as:

$$K = -2\frac{\left(z^2 + 1\right)^2 M}{z\_0 \rho^2 \rho^3} + O(M^3),\tag{29}$$

Then, using the same method, we calculate the deflection angle as follows:

$$
\hbar \simeq 4 \, \frac{Mz^2}{z\_0 \, b} + 4 \, \frac{M}{z\_0 \, b} \,. \tag{30}
$$

At *z* = 0 and *z*0 = 1, it reduces to the exact Schwarzschild case.

*2.3. Case 3*

The refractive index for the dark-matter medium [50]

$$n(\omega) = 1 + \beta A\_0 + A\_2 \omega^2 \tag{31}$$

where *β* = *ρ*0 4*m*2*ω*2 and *ρ*0 are the mass density of the scattered dark-matter particles, and *A*0 = −2*ε*2*e*2 and *A*2*j* ≥ 0.

The terms in O *ω*2 and higher are related to the polarizability of the dark-matter candidate. Note that the order of *ω*−2 is due to the charged dark-matter candidate and *ω*2 for a neutral dark-matter candidate. Moreover, there may be a linear term in *ω* when parity and charge-parity asymmetries are present.

The Gaussian curvature is obtained as:

$$K \approx -2\frac{M}{\left(A\_2\omega^2 + \beta A\_0 + 1\right)^2 \rho^3} + O(M^3) \tag{32}$$

The deflection angle is found as follows:

$$\hat{\mathfrak{A}} = 4 \frac{M}{\left(A\_2 \,\omega^2 + 1\right)b} - 4 \frac{MA\_0}{\left(A\_2 \,\omega^2 + 1\right)^2 b} \beta + O\left(\beta^2\right) \tag{33}$$

The effect of dark matter can be seen by comparison with the above deflection angle by the Schwarzschild black hole. Hence, dark matter gives a small deflection angle compared to the standard case.

#### **3. Effect of Medium on Deflection Angle of Schwarzschild-Like Wormhole Using Gauss–Bonnet Theorem**

In this section, we consider the static Schwarzschild-like wormhole solution [58] with metric

$$ds^2 = -(1 - 2M/r + \lambda^2)dt^2 + \frac{dr^2}{1 - 2M/r} + r^2d\Omega\_{(2)}^2,\tag{34}$$

which reduces to the black-hole metric in Equation (4) at *λ* = 0. Using the transformation of *t* → *t*/ √ 1 + *λ*2 and *M* → *M*(1 + *λ*2), the metric functions of the Schwarzschild-like wormhole spacetime become:

$$f(r) = 1 - \frac{2M}{r}, \quad g(r)^{-1} = 1 - \frac{2M(1 + \lambda^2)}{r}.\tag{35}$$

*3.1. Case 1*

We first use the constant profile as refractive *nm* = *n*0 to calculate the deflection angle in the medium in weak field limits. Using the same procedure, we obtain the optical metric, and calculate the Gaussian optical curvature for the Schwarzschild-like wormhole at a linear order of M as follows:

$$K \approx -\frac{\left(\lambda^2 + 2\right)M}{\rho^3 n\_0^2} + O(M^3),\tag{36}$$

and after similar calculations, the corresponding deflection angle in the leading order terms is

$$\dot{\kappa} = 2\frac{M\lambda^2}{n\_0 \, b} + 4\frac{M}{n\_0 \, b'} \, \tag{37}$$

which agrees with the well-known results in the limit in which the medium is negligible (*n*0 = 1) [42].

*3.2. Case 2*

To see the effect of the MFE-like medium, we use [10]

$$m = \frac{z\_0}{1 + z^2},\tag{38}$$

where *z*0 and *z* are a constant. The Gaussian curvature of the optical metric approximating in leading orders is negative everywhere and found as:

$$K \approx -\frac{\left(z^2 + 1\right)^2 \left(\lambda^2 + 2\right)M}{\rho^3 z\_0 r^2} \tag{39}$$

Using the GBT, the deflection angle is calculated as follows:

$$\hbar \simeq 2\frac{M\lambda^2 z^2}{z\_0 \, b} + 2\frac{M\lambda^2}{z\_0 \, b} + 4\frac{Mz^2}{z\_0 \, b} + 4\frac{M}{z\_0 \, b} \,. \tag{40}$$

At *z* = 0 and *z*0 = 1, it reduces to the previous result [42].

#### *3.3. Case 3*

Finally, we use the refractive index for dark matter given in Equation (31) to calculate the deflection angle of a wormhole in a medium. The Gaussian curvature is obtained as:

$$K \approx -\frac{\left(\lambda^2 + 2\right)M}{\rho^3 \left(A\_2 \omega^2 + A\_0 \beta + 1\right)^2} \tag{41}$$

The deflection angle is found as follows:

$$\hat{\alpha} = 2 \frac{M\lambda^2}{\left(A\_2 \,\,\omega^2 + A\_0 \,\,\beta + 1\right)b} + 4 \frac{M}{\left(A\_2 \,\,\omega^2 + A\_0 \,\,\beta + 1\right)b} \tag{42}$$

We find that the deflected photon through dark matter around the Schwarzschild-like wormhole has a large deflection angle compared to the standard case.

#### **4. Conclusions**

We calculated the deflection angle of black holes and wormholes in a dark-matter medium using the GBT. This was achieved by constructing optical metrics. In summary, we investigated that GBT is a partially topological effect. We demonstrated this by using three different cases. In the first case, we used the constant profile as a refractive index. Then, by constructing the optical geometry and using the GBT, we obtained the deflection angle in the weak field limit. The deflection angle of

the Schwarzschild black hole was correctly calculated in a medium that has a constant *n*0 refractive index. In the second case, we used the MFE-like model (but uniform in large distances), repeated the calculation, and showed that it produces a similar effect.

In Section 2, we repeated our method on the Schwarzschild-like wormhole to see the effect of the dark-matter medium when light is propagated through it. Note that we supposed that the refractive index is spatially nonuniform as long as it is uniform at large distances. In the first case, we again used the constant refractive index, we considered the MFE-like profile, and, finally, the medium for the dark matter was taken to find the deflection angle in the weak field limit. We concluded that the deflection angle by a black hole decreases in a medium of dark matter, as seen in Equation (33). On the other hand, deflection angle by a wormhole increases, as seen in Equation (42).

These results suggest that weak deflection within a dark-matter medium or MFE-like model (perfect imaging), can be calculated using the Gibbons and Werner method, which gives us hints to understanding the nature of dark matter.

**Funding:** This work was supported by Comisión Nacional de Ciencias y Tecnología of Chile through FONDECYT Grant No. 3170035 (A.Ö.)

**Acknowledgments:** A.Ö. is grateful to the Institute for Advanced Study, Princeton for their hospitality.

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Shadow Images of a Rotating Dyonic Black Hole with a Global Monopole Surrounded by Perfect Fluid**

**Sumarna Haroon 1, Kimet Jusufi 2,3,\* and Mubasher Jamil 1,4**

Received: 29 November 2019; Accepted: 21 January 2020; Published: 24 January 2020

**Abstract:** In this paper, we revisit the rotating global monopole metric and extend the metric to a rotating dyonic global monopole in the presence of a perfect fluid. We then show that the surface topology at the event horizon, related to the metric computed, is a 2-sphere using the Gauss-Bonnet theorem. By choosing *ω* = −1/3, 0, 1/3 we investigate the effect of dark matter, dust and radiation on the silhouette of a black hole. The presence of the global monopole parameter *γ* and the perfect fluid parameters *υ* also deform the shape of a black hole's shadow, which has been depicted through graphical illustrations. Finally, we analyse the energy emission rate of a rotating dyonic global monopole surrounded by perfect fluid with respect to parameters.

**Keywords:** rotating black hole; global monopole; perfect fluid; scalar field; shadows

#### **1. Introduction**

Black holes are fascinating objects predicted to exist by Einstein's theory of general relativity. Recent astrophysical observation shows that such objects may exist at the center of almost every galaxy [1,2]. By studying the light-like geodesics around black holes it is shown that photons can be absorbed by the black hole or can escape from black holes [3]. That is to say a boundary is defined between these two categories of light-like geodesics, giving rise to a dark region known as the shadow. Very recently, a project known as the Event Horizon Telescope (EHT) Collaboration announced the first image concerning the detection of an event horizon of a supermassive black hole at the center of a giant elliptical galaxy, M87 [4,5]. That being said, the black hole shadow has recently become a hot topic among researchers for the simple fact it is best to evaluate the soon-expected observational data. Historically, Synge was the first to propose the apparent shape of a spherically symmetric black hole [6]. After that Luminet [7] discussed the appearance of a Schwarzschild black hole, the shadow of a Kerr black hole was studied by Bardeen [8], the shadow of Kerr-Newman black holes [9], naked singularities with deformation parameters [10], Kerr-Nut spacetimes [11], while shadows of black holes in Chern-Simons modified gravity, Randall-Sundrum braneworlds, and Kaluza-Klein rotating black holes have been studied in References [12–14], and many other interesting studies concerning the effect of dark matter and cosmological constant on the shadow images [15–22], Kerr-like wormholes as well as traversable wormholes and many other interesting studies [23–31]. Some authors have also tried to test theories of gravity by using the observations obtained from the shadow of Sgr A\* [32–35]. Note that the new general approach for shadow calculation for axially symmetric black holes was

developed in Reference [36], while the intrinsic curvature and topology of shadows in Kerr spacetime was developed in References [37,38].

Global monopoles are topological defects which may have been produced during the phase transitions in the early universe. In fact, global monopoles are just one type of topological defects. Other types of topological objects are expected to exist including domain walls and cosmic strings (e.g., [39]). More precisely, a global monopole is a heavy object characterized by spherically symmetry and divergent mass. Such objects, which may have been formed during the phase transition of a system composed of a self-coupling triplet of scalar fields *φa* which undergoes a spontaneous breaking of global *O*(3), gauge symmetry down to *U*(1). The gravitational field of a static global monopole was found by Barriola and Vilenkin for the first time and are expected to be stable against spherical as well as polar perturbations [40]. According to their model, global monopoles are configurations whose energy density decreases with the distance *r*−2 and whose spacetimes exhibit a solid angle deficit given by Δ = 8*π*2*γ*2, where *γ* is the scale of gauge-symmetry breaking. The rotating metric of a global monopole was investigated by Filho and Bezerra [41] Gravitational lensing by rotating global monopoles has been investigated in Reference [42] and more recently in Reference [43]. Among other things, global monopoles are expected to rotate and to carry magnetic charges.

 In Section 2, we consider the gravitational field of a static dyonic black hole (SDBH) with a global monopole surrounded by perfect fluid. In Section 3, by applying a complex coordinate transformation known as the Newman-Janis method [44] we find the spacetimes of a rotating dyonic black hole (RDBH) with a global monopole surrounded by perfect fluid. In Section 4, we consider the null geodesics using Hamilton-Jacobi equation. In Section 5, we study the impact of dark matter, dust and radiation on the shape of global monopole shadow. In Section 6, we study the energy emission rate. Finally in Section 7, we comment on our results.

#### **2. An SDBH with a Global Monopole in Perfect Fluid**

The action, *S*(*EM*), for Einstein Maxwell gravity along with actions *S*(*D*) and S respectively defining presence of a global monopole and matter distribution, can be altogether written as:

$$\mathcal{S} = \mathcal{S}^{(EM)} + \mathcal{S}^{(D)} + \mathcal{S}.\tag{1}$$

The Einstein-Maxwell action *S*(*EM*) is given by:

$$S^{(EM)} = \int \sqrt{-g} d^4 \mathbf{x} \left(\frac{\mathcal{R}}{2\kappa} - \frac{1}{4} F\_{\mu\nu} F^{\mu\nu}\right) ,\tag{2}$$

where *κ* = 8*π*. The quantities *g*, R and *Fμν* are, respectively, the determinant of the metric *gμν* associated to the gravitational field, the scalar invariant and the electromagnetic tensor. Also *μ*, *ν* = 0, 1, 2, 3.

The corresponding Einstein field equations read:

$$\mathcal{R}\_{\mu\nu} - \frac{1}{2} \mathcal{g}\_{\mu\nu} \mathcal{R} = 8\pi T\_{\mu\nu}. \tag{3}$$

While the corresponding Maxwell equations are:

$$
\nabla\_{\mu} \mathcal{F}^{\mu \nu} = 0. \tag{4}
$$

Here *Tμν* is the total stress energy tensor which we discuss later in this section. Since we are considering a dyonic black hole, which means that it is comprised of both electric charge *QE* and magnetic charge *QM*, the electromagnetic potential has two non zero terms, that is, [45,46]:

$$\mathbf{A} = \frac{Q\_E}{r} dt - Q\_M \cos\theta d\varphi. \tag{5}$$

The only non-vanishing components of the electromagnetic tensor:

$$F\_{tr} = -F\_{rt} = \frac{Q\_E}{r^2},\ \ F\_{\theta\varphi} = -F\_{q\theta} = Q\_M \sin \theta. \tag{6}$$

Now the action *S*(*D*) corresponds to the matter having a defect– a global monopole which is a heavy object formed in the phase transition of a system composed by a self-coupling scalar triplet field Φ**s**, where *s* runs from 1 to 3. Thus the action in presence of a matter field Φ**s** coupled to gravity that characterizes a global monopole [40]:

$$S^{(D)} = \int \sqrt{-g}d^4x \left(\frac{1}{2}g^{\mu\nu}\partial\_{\mu}\Phi^{\ast}\partial\_{\nu}\Phi^{\ast} - \frac{\lambda}{4}\left(\Phi^2 - \gamma^2\right)^2\right),\tag{7}$$

where Φ2 = Φ**s**Φ**s**, while *λ* is the self-interaction term and *γ* is the scale of a gauge-symmetry breaking. The monopole can be described through the field configuration Φ**s** = *γh*(*r*)*x***s** |**x**| , in which *x***s** = {*r* sin *θ* cos *ϕ*,*r* sin *θ* sin *ϕ*,*r* cos *θ* }, such that |**x**| = *r*2, and *h*(*r*) is a function of radial coordinate *r*.

The field equations for the scalar field Φ**s** reduces to a single equation for *h*(*r*) given as:

$$f(r)h''(r) + \left[\frac{2f(r)}{r} + \frac{1}{2f(r)}(f^2(r))'\right]h'(r) - \frac{2h(r)}{r^2} - \lambda\gamma^2 h(r)\left(h^2(r) - 1\right) = 0. \tag{8}$$

With these equations in mind, and without loss of generality we can choose a spherically symmetric metric written as follows:

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2d\theta^2 + r^2\sin^2\theta d\varphi^2. \tag{9}$$

In our case the total stress-energy momentum reads:

$$T\_{\mu\nu} = T\_{\mu\nu}^{(EM)} + T\_{\mu\nu}^{(D)} + \mathcal{T}\_{\mu\nu} \tag{10}$$

in which:

$$T^{(EM)}\_{\mu\nu} = \frac{1}{4\pi} \left( F\_{\mu\sigma} F\_{\nu}{}^{\sigma} - \frac{1}{4} \mathcal{g}\_{\mu\nu} F\_{\rho\sigma} F^{\rho\sigma} \right) , \tag{11}$$

$$T^{(D)}\_{\mu\nu} = \partial\_{\mu}\phi^{a}\partial\_{\nu}\phi^{a} - \frac{1}{2}g\_{\mu\nu}g^{\rho\sigma}\partial\_{\rho}\phi^{a}\partial\_{\sigma}\phi^{a} - \frac{g\_{\mu\nu}\lambda}{4}\left(\phi^{2} - \gamma^{2}\right)^{2},\tag{12}$$

and T*μν* is the energy-momentum tensor of the surrounding matter. The energy momentum-tensor of the surrounding fluid has the following components [47]:

$$
\mathcal{T}^t{}\_t = \mathcal{T}^r{}\_r = -\rho,\tag{13}
$$

and:

$$
\mathcal{T}^{\theta}{}\_{\theta} = \mathcal{T}^{\varphi}{}\_{\varphi} = \frac{1}{2} (1 + 3\omega) \rho. \tag{14}
$$

*Universe* **2020**, *6*, 23 4 of 18

Outside the core *h* → 1 and the energy-momentum tensor of the monopole has the following components [40]:

$$T^{(D)^{I}\_{\
u}}\_{\
u} = T^{(D)^{I}\_{\
u}}\_{\
u} = -\gamma^2 \left[ \frac{h^2}{r^2} + f(r) \frac{(h')^2}{2} + \frac{\lambda \gamma^2}{4} (h^2 - 1)^2 \right] \to -\frac{\gamma^2}{r^2} \tag{15}$$

$$T^{(D)\stackrel{\theta}{\theta}}{}\_{\theta} = T^{(D)}{}\_{\theta}{}^{\theta} = -\gamma^2 \left[ f(r) \frac{(h'(r))^2}{2} + \frac{\lambda \gamma^2}{4} (h^2(r) - 1)^2 \right] \to 0. \tag{16}$$

The surrounding matter, whose action is denoted by S in Equation (1), can generally be a dust, radiation, quintessence, cosmological constant, phantom field or even any combination of them. Thus, the Einstein's field equations yield:

$$\frac{rf'(r) + f(r) - 1}{r^2} + \frac{8\pi\gamma^2}{r^2} + \frac{Q\_E^2}{r^4} + \frac{Q\_M^2}{r^4} + 8\pi\rho = 0,\tag{17}$$

$$\frac{r f'''(r) + 2f'(r)}{2r} - \frac{Q\_E^2}{r^4} - \frac{Q\_M^2}{r^4} - 4\pi\rho(3\omega + 1) = 0. \tag{18}$$

Now by solving the set of differential equations (18) and (19) one obtains the following general solution for the metric:

$$f(r) = 1 - 8\pi\gamma^2 - \frac{2M}{r} + \frac{Q\_E^2}{r^2} + \frac{Q\_M^2}{r^2} - \frac{\upsilon}{r^{1+3\omega\prime}}\tag{19}$$

with the energy density in the form:

$$
\rho = -\frac{3\,\omega\,\upsilon}{8\pi r^{3(1+\omega)}}.\tag{20}
$$

Note that, *υ* is the perfect fluid parameter. From the weak energy condition it follows the positivity of the energy density of the surrounding field, *ρ* ≥ 0, which should satisfy the following constraint *ωυ* ≤ 0.

#### **3. An RDBH with a Global Monopole in Perfect Fluid**

We now extend the study of static global monopole solution and obtain its rotating counterpart. For this we apply Newman-Janis formalism to the metric (9) along with (19). As a first step to this formalism, we transform Boyer-Lindquist (BL) coordinates (*t*,*r*, *θ*, *φ*) to Eddington-Finkelstein (EF) coordinates (*u*,*r*, *θ*, *φ*). This can be achieved by using the coordinate transformation:

$$dt^{\prime} = \,^t du + \frac{dr}{1 - 8\pi\gamma^2 - \frac{2M}{r} + \frac{Q\_F^2}{r^2} + \frac{Q\_M^2}{r^2} - \frac{\upsilon}{r^{3\omega + 1}}},\tag{21}$$

which yields line element in the form:

$$ds^2 = -\left(1 - 8\pi\gamma^2 - \frac{2M}{r} + \frac{Q\_E^2}{r^2} + \frac{Q\_M^2}{r^2} - \frac{\upsilon}{r^{3\omega+1}}\right) du^2 - 2dudr + r^2 d\Omega^2,\tag{22}$$

where *d*Ω2 = *dθ*2 + sin2 *θdφ*2. It is worth noting that, compared to the previous work in Reference [41], we shall use the metric form (9) along with *f*(*r*) given by Equation (19) to obtain a simple metric for the rotating black hole with a global monopole. This metric can be decomposed in terms of null tetrads as:

$$\mathcal{g}^{\mu\nu} = -l^{\mu}n^{\nu} - l^{\nu}n^{\mu} + m^{\mu}\overline{m}^{\nu} + m^{\nu}\overline{m}^{\mu},\tag{23}$$

where the null vectors are defined as:

$$l^{\mu}{}\_{\;\;\;\;\nu} = \delta^{\mu}\_{\;\;\nu}{}\_{\;\;\;\nu} \tag{24}$$

$$n^{\mu} \quad = \quad \delta^{\mu}\_{\mu} - \frac{1}{2} \left( 1 - 8\pi\gamma^2 - \frac{2M}{r} + \frac{Q\_E^2}{r^2} + \frac{Q\_M^2}{r^2} - \frac{\upsilon}{r^{3\omega+1}} \right) \delta^{\mu}\_r,\tag{25}$$

$$m^{\mu} \quad = \frac{1}{\sqrt{2}r} \left( \delta^{\mu}\_{\theta} + \frac{i}{\sin \theta} \delta^{\mu}\_{\phi} \right), \tag{26}$$

$$
\overline{m}^{\mu} \quad = \frac{1}{\sqrt{2}r} \left( \delta^{\mu}\_{\theta} - \frac{i}{\sin \theta} \delta^{\mu}\_{\phi} \right). \tag{27}
$$

It is obvious from the notation that *m*¯ *μ* is a complex conjugate of *mμ*. These vectors further satisfy the conditions for normalization, orthogonality and isotropy as:

$$l^{\mu}l\_{\mu} = n^{\mu}n\_{\mu} = m^{\mu}m\_{\mu} = \bar{m}^{\mu}\bar{m}\_{\mu} = 0,\tag{28}$$

$$l^{\mu}m\_{\mu} = l^{\mu}\vec{m}\_{\mu} = n^{\mu}m\_{\mu} = n^{\mu}\vec{m}\_{\mu} = 0,\tag{29}$$

$$-l^{\mu}n\_{\mu} = m^{\mu}\vec{m}\_{\mu} = 1.\tag{30}$$

Following the Newman–Janis prescription we write:

$$\mathbf{x}'^{\mu} = \mathbf{x}^{\mu} + ia(\delta\_r^{\mu} - \delta\_u^{\mu})\cos\theta \to \begin{cases} u' = u - ia\cos\theta, \\ r' = r + ia\cos\theta, \\ \theta' = \theta, \\ \phi' = \phi. \end{cases} \tag{31}$$

in which *a* stands for the rotation parameter. Next, let the null tetrad vectors *Za* = (*l a*, *na*, *ma*, *m*¯ *a*) undergo a transformation given by *Zμ* = (*∂xμ*/*∂xν* )*Zν* , following:

$$l^{\prime \mu}\_{\;\;\;\;\nu} = \;\delta^{\mu}\_{\;\;\;\nu} \tag{32}$$

$$m^{\prime \mu} \quad = \quad \delta^{\mu}\_{\mu} - \frac{1}{2} \mathcal{F} \delta^{\mu}\_{r}, \tag{33}$$

$$m^{\prime \mu} \quad = \frac{1}{\sqrt{2\,\Sigma}} \left[ (\delta^{\mu}\_{\mu} - \delta^{\mu}\_{r}) \dot{a} \sin \theta + \delta^{\mu}\_{\theta} + \frac{i}{\sin \theta} \delta^{\mu}\_{\phi} \right],\tag{34}$$

$$\overline{m}^{\prime \mu} \quad = \frac{1}{\sqrt{2\,\Sigma}} \left[ (\delta^{\mu}\_{\mu} - \delta^{\mu}\_{r}) \dot{a} \sin \theta + \delta^{\mu}\_{\theta} + \frac{i}{\sin \theta} \delta^{\mu}\_{\phi} \right],\tag{35}$$

where we replaced *f*(*r*) to F(*r*, *a*, *θ*) and *h*(*r*) = *r*2 to Σ(*r*, *a*, *θ*). With the help of the above equations the contravariant components of new metric are computed as:

$$\begin{aligned} \text{g}^{\mu\mu} &= \quad \frac{a^2 \sin^2 \theta}{\Sigma}, \text{ g}^{\mu\phi} = \frac{a}{\Sigma'} \text{ g}^{\mu r} = 1 - \frac{a^2 \sin^2 \theta}{\Sigma}, \\\text{g}^{rr} &= \quad \mathcal{F} + \frac{a^2 \sin^2 \theta}{\Sigma}, \text{ g}^{r\phi} = -\frac{a}{\Sigma'} \text{ g}^{\mu\theta} = \frac{1}{\Sigma'}, \\\text{g}^{\mu\phi} &= \quad \frac{1}{\Sigma \sin^2 \theta}. \end{aligned} \tag{36}$$

The new metric is found as follows:

$$\begin{split} ds^2 &= -\mathcal{F} du^2 - 2 du dr + 2a \sin^2 \theta \left( \mathcal{F} - 1 \right) du d\phi + 2a \sin^2 dr d\phi + \Sigma d\theta^2 \\ &+ \sin^2 \theta \left[ \Sigma + a^2 \left( 2 - \mathcal{F} \right) \sin^2 \theta \right] d\phi^2. \end{split} \tag{37}$$

*Universe* **2020**, *6*, 23 6 of 18

Using the method without a complexification introduced in Reference [48] we revert the EF coordinates back to BL coordinates by using the following transformation:

$$du = dt + \lambda(r)dr, \quad d\phi = d\varphi + \chi(r)dr,\tag{38}$$

where

$$
\lambda(r) = \frac{-a^2 - k(r)}{f(r)h(r) + a^2}, \ \chi(r) = \frac{-a}{f(r)h(r) + a^2}, \ k(r) = h(r) = r^2,\tag{39}
$$

with

$$\mathcal{F}\_{-}=\frac{f(r)h(r)+a^{2}\cos^{2}\theta}{\left(k(r)+a^{2}\cos^{2}\theta\right)^{2}}\Sigma.\tag{40}$$

Hence the rotating black hole solution in Boyer-Lindquist coordinates turns out to be [49]:

$$\begin{split} ds^2 &= -\frac{f(r)h(r) + a^2 \cos^2 \theta}{\left(k(r) + a^2 \cos^2 \theta\right)^2} \Sigma dt^2 + 2a \sin^2 \theta \frac{f(r)h(r) - k(r)}{\left(k(r) + a^2 \cos^2 \theta\right)^2} \Sigma dt d\phi + \frac{\Sigma}{f(r)h(r) + a^2} dr^2 + \Sigma d\theta^2 \\ &+ \Sigma \sin^2 \theta \left[1 + a^2 \sin^2 \theta \frac{2k(r) - f(r)h(r) + a^2 \cos^2 \theta}{\left(k(r) + a^2 \cos^2 \theta\right)^2}\right] d\phi^2. \end{split}$$

Following Reference [48] using the condition *k*(*r*) = *h*(*r*) = *r*2 one can find Σ = *r*2 + *a*2 cos2 *θ*. Finally the metric can also be written as:

$$\begin{split} ds^2 &= -\left(1 - \frac{r^2(1 - f(r))}{\Sigma}\right) dt^2 - 2a\sin^2\theta \left(\frac{r^2(1 - f(r))}{\Sigma}\right) dt d\varphi + \frac{\Sigma}{\Delta} dr^2 + \Sigma d\theta^2 \\ &+ \sin^2\theta \left[\frac{(r^2 + a^2)^2 - a^2\Delta\sin^2\theta}{\Sigma}\right] d\varphi^2, \end{split} \tag{41}$$

where in order to simplify the notation we introduce the following quantities:

$$
\Delta \quad = \quad r^2 f(r) + a^2 = r^2 + a^2 - 2Mr - 8\pi r^2 \gamma^2 + Q\_E^2 + Q\_M^2 - \frac{\upsilon}{r^{3\omega - 1}} \tag{42}
$$

where *f*(*r*) is given by Equation (19). In this work, we consider three different cases of *ω* = −1/3 dark matter dominant, 0 (dust dominant) and 1/3 (radiation dominant). For spin *a* = 0, perfect fluid parameter *υ* = 0 and no charges, the above metric reduces to Schwarzschild black hole with global monopole [50]. It would certainly be interesting to generalize our solution by including the cosmological constant as well. As a particular example, we point out the Kerr-Newman-NUT black hole which is a subclass of the Plebanski-Demianski class [51], but our solution it seems not to be the case. The shadows of Kerr-Newman-NUT black holes with cosmological constant has been investigated by Grenzebach et al. [52]. The electromagnetic field of a black hole is defined by its vector potential. As already mentioned, in case of a static black hole the vector potential is given by Equation (5). For the rotating case, the Newman-Janis method can also be applied on Equation (5) using a gauge transformation such that *grr* = 0 and *Ar* = 0. For the detailed procedure the authors refer the readers to Reference [46]. The vector potential computed through Newman-Janis formalism for a rotating dyonic black hole is given by [46]:

$$\mathbf{A} = \left(\frac{rQ\_E - aQ\_M\cos\theta}{\Sigma}\right)dt + \left(-\frac{ra}{\Sigma}Q\_E\sin^2\theta + \frac{r^2 + a^2}{\Sigma}Q\_M\cos\theta\right)d\varphi. \tag{43}$$

*Universe* **2020**, *6*, 23 7 of 18

It has been shown in Reference [48] that metric similar to (41) satisfies the Einstein field equations. For the Einstein tensor *Gμν* and energy-momentum tensor *Tμν*, the Einstein field equations are given by *Gμν* = *Rμν* − 1/2*gμν* R = 8*πTμν*. For simplicity, let *f*(*r*) = 1 − 2*F*(*r*)/*r*2, where *F*(*r*) = 4*πγ*2*r*2 + *Mr* − (*Q*2 *E* + *Q*2 *M*)/2 + *υ r*1−3*ω*/2, then the nonvanishing components of *Gμν* are:

$$\begin{aligned} \mathbf{G}\_{ll} &= \begin{array}{ll} \frac{2}{\Sigma^{3}} \left( 2F(r) - \left( (r^{2} + a^{2}) + a^{2} \sin^{2} \theta \right) \right) \left( F(r) - rF'(r) \right) - \frac{a^{2} \sin^{2} \theta}{\Sigma^{2}} F''(r), \\ \mathbf{G}\_{rr} &= \begin{array}{ll} \frac{2}{\Sigma \Delta} \left( F(r) - rF'(r) \right), \\\\ \mathbf{G}\_{\theta \theta} &= \begin{array}{ll} -\frac{2}{\Sigma} \left( F(r) - rF'(r) \right) - F''(r), \\\\ \frac{4a \sin^{2} \theta}{\Sigma^{3}} \left( \left( r^{2} + a^{2} \right) - F(r) \right) \left( F(r) - rF'(r) \right) + \frac{a}{\Sigma^{2}} \left( r^{2} + a^{2} \right) \sin^{2} \theta F''(r), \\\\ \mathbf{G}\_{\theta \theta'} &= \begin{array}{ll} \frac{\sin^{2} \theta}{\Sigma^{3}} \left( 4a^{2} \sin^{2} \theta F(r) - \left( r^{2} + a^{2} \right) \left( 2(r^{2} + a^{2}) + a^{2} \sin^{2} \theta \right) \right) \left( F(r) - rF'(r) \right) - \frac{(r^{2} + a^{2}) \sin^{2} \theta}{\Sigma^{2}} F''(r). \end{array} \end{aligned} \tag{44}$$

In terms of the orthogonal basis, for the metric (41):

$$\begin{aligned} \varepsilon\_t^\mu &= \quad \frac{1}{\sqrt{\Sigma \Delta}} \left( r^2 + a^2 , 0, 0, a \right), & \varepsilon\_r^\mu &= \frac{1}{\sqrt{\Sigma}} \left( 0, 1, 0, 0 \right), \\\varepsilon\_\theta^\mu &= \quad \frac{1}{\sqrt{\Sigma}} \left( 0, 0, 1, 0 \right), & \varepsilon\_\varphi^\mu &= \frac{1}{\sqrt{\Sigma \Delta}} \left( a \sin^2 \theta , 0, 0, 1 \right). \end{aligned} \tag{45}$$

and the Einstein tensor *Gμν*, the energy momentum tensor is expressed as:

$$\begin{array}{rcl}p\_{\varGamma} &=& \frac{1}{8\pi\pi}e\_{t}^{\mu}\,\mathbf{e}\_{t}^{\nu}\,\mathbf{G}\_{\mu\nu} & p\_{r} = \frac{1}{8\pi}\mathbf{e}\_{r}^{\mu}\,\mathbf{e}\_{r}^{\nu}\,\mathbf{G}\_{\mu\nu},\\p\_{\varGamma} &=& \frac{1}{8\pi\pi}\mathbf{e}\_{\theta}^{\mu}\,\mathbf{e}\_{\theta}^{\nu}\,\mathbf{G}\_{\mu\nu}, & p\_{\varGamma} = \frac{1}{8\pi\pi}\mathbf{e}\_{\theta}^{\mu}\,\mathbf{e}\_{\varphi}^{\nu}\,\mathbf{G}\_{\mu\nu}.\end{array} \tag{46}$$

Equations (41)–(46) gives the components for energy momentum tensor as:

$$\begin{aligned} p\_t &=& \frac{1}{8\pi\Sigma^2} \left( 8\pi\gamma^2 r^2 - 3v\omega r^{1-3\omega} + (Q\_E^2 + Q\_M^2) \right) = -p\_{r\prime} \\ p\_\theta &=& -p\_r - \frac{1}{8\pi\Sigma} \left( 8\pi\gamma^2 - \frac{3v\omega (1-3\omega)}{2r^{1+3\omega}} \right) = p\_\theta. \end{aligned} \tag{47}$$

Analogous to a Kerr black hole, a ring singularity resides inside the black hole defined by metric (41). This can be demonstrated by computing the points at which the Kretschmann scalar K*s* = *RμνσρRμνσρ* turns to infinity. For the metric (41), the Kretschmann scalar has the value:

$$\mathcal{K}\_s = \frac{Z(r, a, \theta, Q\_{\mathbb{E}\prime} Q\_{M\prime} \omega, \upsilon, \gamma)}{r^{2(1+3\omega)}(r^2 + a^2 \cos^2 \theta)^2}. \tag{48}$$

where *Z*(*r*, *a*, *θ*, *QE*, *QM*, *ω*, *υ*, *γ*) is a tedious function. From the above expression, we observe that for *ω* = −1/3, 0, 1/3 the poles lie *r*2 + *a* cos2 *θ* = 0 or when *r* = 0 and *θ* = *π*/2. This leads us to the interpretation that a test particle moving in an equatorial plane *θ* = *π*/2 will hit the singularity at *r* = 0.

#### *3.1. Surface Topology*

It is interesting to determine the surface topology of the global monopole spacetime at the event horizon. At a fixed moment in time *t*, and a constant *r* = *r*+, the metric (41) reduces to:

$$ds^2 = \Sigma(r\_+, \theta)d\theta^2 + \left(2Mr\_+ + 8\pi r\_+^2 \gamma^2 - Q\_E^2 - Q\_M^2 + \frac{v}{r\_+^{3\omega - 1}}\right)^2 \frac{\sin^2 \theta}{\Sigma(r\_+, \theta)}d\varphi^2,\tag{49}$$

The above metric has the following determinant:

$$\det \mathbf{g}^{(2)} = \begin{pmatrix} 2Mr\_+ + 8\pi r\_+^2 \gamma^2 - Q\_E^2 - Q\_M^2 + \frac{\upsilon}{r\_+^{2\omega - 1}} \end{pmatrix}^2 \sin^2 \theta. \tag{50}$$

**Theorem 1.** *Let* M *be a compact orientable surface with metric g*(2)*, and let K be the Gaussian curvature with respect to g*(2) *on* M*. Then, the Gauss-Bonnet theorem states that:*

$$\int\_{\mathcal{M}} \int\_{\mathcal{M}} \mathcal{K} \, dA = 2\pi \chi(\mathcal{M}).\tag{51}$$

Note that *dA* is the surface line element of the 2-dimensional surface and *χ*(M) is the Euler characteristic number. It is convenient to sometimes express the above theorem in terms of the Ricci scalar, in particular for the 2-dimensional surface there is a simple relation between the Gaussian curvature and Ricci scalar given by:

$$K = \frac{\mathcal{R}}{2}.\tag{52}$$

Yielding the following from:

$$\frac{1}{4\pi} \iint\_{\mathcal{M}} \mathcal{R}dA = \chi(\mathcal{M}).\tag{53}$$

A straightforward calculation using the metric (49) yields the following result for the Ricci scalar:

$$\mathcal{R} = \frac{2(r\_+^2 + a^2)(r\_+^2 - 3a^2 \cos^2 \theta)}{\left(r\_+^2 + a^2 \cos^2 \theta\right)^3} \tag{54}$$

From the GBT we find:

$$\chi(\mathcal{M}) = \frac{1}{4\pi} \int\_0^{2\pi} \int\_0^{\pi} \left[ \frac{2(r\_+^2 + a^2)(r\_+^2 - 3a^2 \cos^2 \theta)}{\left(r\_+^2 + a^2 \cos^2 \theta\right)^3} \right] \sqrt{\det g^{(2)}} d\theta d\varphi. \tag{55}$$

Finally, solving the integral we find:

$$
\chi(\mathcal{M}) = 2.\tag{56}
$$

Hence the surface topology is a 2-sphere at the event horizon, since we know that *χ*(M)*sphere* = 2.

#### *3.2. Shape of Ergoregion*

Let us now proceed to study the shape of the ergoregion of an RDBH with a global monopole. In particular we shall be interested to plot the shape of the ergoregion in the *xz*-plane. Recall that the horizons of the RDG can be found by solving Δ = 0:

$$r^2 + a^2 - 2Mr - 8\pi r^2 \gamma^2 + Q\_E^2 + Q\_M^2 - \frac{\upsilon}{r^{3\omega - 1}} = 0,\tag{57}$$

on the other hand, the limit surfaces or inner and outer ergosurface is given by *gtt* = 0, i.e.,

$$r^2 + a^2 \cos^2 \theta - 2Mr - 8\pi r^2 \gamma^2 + Q\_E^2 + Q\_M^2 - \frac{\upsilon}{r^{3\omega - 1}} = 0. \tag{58}$$

There is an interesting process which relies on the presence of an ergoregion, namely from such a rotating black hole energy can be extracted, and is known as the Penrose process. In Figure 1 we plot the shape of ergoregion for different values of *a*, *ω*, *γ*, and *υ*. One can observe that the event horizon and static limit surface meet at the poles while the region between them is the ergoregion

which supports negative energy orbits. Furthermore, the shape of ergoregion depends on the spin *a*, however due to the small values of *υ* we observe small changes related to the value of *ω*.

**Figure 1.** Plots showing the shape of ergoregion in *xz*-plane for different values of *a*, *ω*, and *υ*. We have chosen *QE* = *QM* = 0.1 in all plots. The blue and the red lines correspond to horizons and static limit surfaces, respectively. The outer red line corresponds to the static limit surface, whereas the two blue lines correspond to the two horizons. Due to the small values of *υ* and a arbitrary value of *a* we observe almost indistinguishable plots for the shape of the ergoregion.

#### **4. Null Geodesics**

Our main objective is to study the shadow cast by the black hole defined by metric (41). To do so, we first need to analyze the geodesics structure of photons moving around the compact gravitational source. This will enable us to detect the unstable photon orbits which in turn defines the boundary of the shadow.

To observe the null geodesics around the RDGM present in perfect fluid, we consider the Hamilton-Jacobi method. The Hamilton-Jacobi equation is given by:

$$
\partial\_{\mathsf{T}} \mathcal{J} = -\mathcal{H}.\tag{59}
$$

In the above equation:

**On Left Side** J is the Jacobi action, defined as the function of affine parameter *τ* and coordinates *xμ* i.e., J = J (*τ*, *xμ*).

**On Right Side** H is the Hamiltonian of test particle's motion and is equivalent to *gμν∂μ*J *∂ν*J .

In the spacetime under consideration, along the photon geodesics the energy *E* and momentum *L*, defined respectively by Killing fields *ξt* = *∂t* and *ξφ* = *∂φ*, are conserved. The mass *m* = 0 of the photon is also constant. Using these constants of motion we can thus separate the Jacobi function as:

$$\mathcal{J} = \frac{1}{2}m^2\tau - Et + L\phi + \mathcal{J}\_r(r) + \mathcal{J}\_\theta(\theta),\tag{60}$$

where the functions J*r*(*r*) and J*θ* (*θ*) respectively depends on coordinates *r* and *θ*.

Combining Equations (59) and (60) yields a set of equations, which describes the dynamics of a test particle around the rotating black hole in perfect fluid matter, as:

$$
\Sigma \frac{dt}{d\tau} = \frac{r^2 + a^2}{\Delta} [E(r^2 + a^2) - aL] - a(aE \sin^2 \theta - L),
\tag{61}
$$

$$
\Sigma \frac{dr}{d\tau} = \sqrt{\mathcal{R}(r)},
\tag{62}
$$

$$
\Sigma \frac{d\theta}{d\tau} = \sqrt{\Theta(\theta)},
\tag{63}
$$

$$
\Sigma \frac{d\varphi}{d\tau} = \frac{a}{\Delta} [E(r^2 + a^2) - aL] - \left( aE - \frac{L}{\sin^2 \theta} \right),
\tag{64}
$$

where R(*r*) and Θ(*θ*) read as:

$$\mathcal{R}(r) = \left[E(r^2 + a^2) - aL\right]^2 - \Delta[m^2r^2 + (aE - L)^2 + \mathcal{K}].\tag{65}$$

$$\Theta(\theta) = \mathcal{K} - \left(\frac{L^2}{\sin^2 \theta} - a^2 E^2\right) \cos^2 \theta,\tag{66}$$

with K the Carter constant.

#### **5. Circular Orbits**

Now we consider a gravitational source placed between a light emitting source and an observer at infinity. The photons emitted from the light source will form two kinds of trajectories—the ones which eventually fall into the black hole and the ones which scatter away from it. The region separating these trajectories contains unstable circular orbits. These unstable circular orbits form a dark region in sky thus forming the contour of the shadow. In this section we intend to discuss the presence of unstable circular orbits around the black hole under consideration. For this we consider photon as a test particle and hence take *m* = 0. We can express the radial geodesic equation in terms of effective potential *V*eff of photon's radial motion as:

$$
\Sigma^2 \left( \frac{dr}{d\tau} \right)^2 + V\_{\rm eff} = 0.
$$

For our convenience we introduce two independent parameters *ξ* and *η* [53] as:

$$
\xi = \mathcal{L}/\mathcal{E}, \qquad \eta = \mathcal{K}/\mathcal{E}^2. \tag{67}
$$

The effective potential in terms of these two parameters is then expressed as:

$$V\_{\rm eff} = \Delta((a-\xi)^2 + \eta) - (r^2 + a^2 - a\,\xi)^2,\tag{68}$$

at some constant *r* = *rc* the conditions:

where we have replaced *V*eff/*E*2 by *V*eff. Figure 2 shows the variation in effective potential associated with the radial motion of photons. From the figure we observe that in all three cases the value of

effective potential decreases with increase in parameter *γ*. Now the circular photon orbits exists when

$$V\_{\rm eff}(r) = 0, \qquad \frac{dV\_{\rm eff}(r)}{dr} = 0 \tag{69}$$

are satisfied. We then use Equation (68) in Equation (69) and thus obtain:

$$[\eta + (\mathfrak{f} - a)^2] \Delta - (r^2 + a^2 - a\mathfrak{g})^2 = 0,\tag{70}$$

$$4(r^2 + a^2 - a\xi) - [\eta + (\xi - a)^2] \mathcal{A}(r) = 0,\tag{71}$$

where:

$$\mathcal{A}(r) = 1 - 8\pi\gamma^2 - \frac{M}{r} + (\frac{3\omega - 1}{2})\frac{\upsilon}{r^{1+3\omega}}.\tag{72}$$

**Figure 2.** The effective potential of photon moving in equatorial plane, with respect to its radial motion: *ω* = 1/3 for radiation, *ω* = 0 for dust and *ω* = −1/3 for dark matter.

Combining Equations (70) and (71) results in:

$$a \; \zeta = r^2 + a^2 - \frac{2 \; \Delta}{\mathcal{A}(r)} \; , \tag{73}$$

$$\eta = \frac{4\Lambda}{\mathcal{A}(r)^2} - \frac{1}{a^2} \left( r^2 - \frac{\Lambda}{\mathcal{A}(r)^2} \right)^2 \tag{74}$$

It is worth mentioning here that impact parameters, *ξ* and *η*, will be affected not just by radial coordinate *r*, spin parameter *a* and mass of black hole *M* but also by electric charge *QE*, magnetic charge *QM*, monopole parameter *γ* and perfect fluid parameter *υ*. The unstable circular orbits are located at local maxima of the potential curves, that is, when *V* eff < 0 or:

$$\left(\Delta'^2 + 2\Delta\Lambda'^{\prime}\right)r + 2\Delta\Lambda' > 0\tag{75}$$

#### **6. Silhoutte of Black Holes**

In this section, we extend our calculations to observe shadow of RDGM surrounded by perfect fluid. To gain the optical image we specify the observer at position (*ro*, *θo*), where *ro* = *r* → ∞ and *θo* is the angular coordinate at infinity, on observer's sky. The new coordinates, also widely known as celestial coordinates, *α* and *β*, are then introduced. These coordinates are selected such that *α* and *β* correspond to the apparent perpendicular distance of the image from axis of symmetry and its projection on the equatorial plane, respectively.

*Universe* **2020**, *6*, 23 12 of 18

Due to the presence of global monopole, we have asymptotically non flat solutions due to the global nontrivial topology. Now we obtain the proper celestial coordinates for the asymptotically non-flat solution by adopting [10]:

$$\kappa = \lim\_{r \to \infty} -r \frac{p^{(\phi)}}{p^{(t)}} \tag{76}$$

$$\beta = \lim\_{r \to \infty} r \frac{p^{(\theta)}}{p^{(t)}} \tag{77}$$

where (*p*(*t*), *p*(*r*), *p*(*θ*), *p*(*φ*)) are the tetrad components of the photon momentum with respect to locally nonrotating reference frame. So basically one can define the observer's sky as the usual cases in which the observer bases *e μ* (*ν*) can be expanded as a form in the coordinate bases. In the limit *r* → ∞ can relate the above coordinates to parameters *ξ* and *η*, which then yield:

$$\alpha\_{\perp} = \sqrt{1 - 8\pi\gamma^2} \frac{\xi}{\sin\theta} \tag{78}$$

$$\mathcal{J}\_{\pm} = \pm \sqrt{1 - 8\pi\gamma^2} \sqrt{\eta + a^2 \cos^2 \theta - \xi^2 \cot^2 \theta},\tag{79}$$

for the case *ω* = 0 and *ω* = 1/3. And similarly:

$$a\_{\parallel} = -\sqrt{1 - 8\pi\gamma^2 - \upsilon} \frac{\xi}{\sin\theta} \tag{80}$$

$$\mathcal{S}\_{\pm} = \pm \sqrt{1 - 8\pi\gamma^2 - \upsilon} \sqrt{\eta + a^2 \cos^2 \theta - \xi^2 \cot^2 \theta} \tag{81}$$

for the case of quintessence, that is, *ω* = −1/3. We observe that in the dark matter case there is a similar contribution term compared to the global monopole. In the limit *γ* → 0 and *υ* → 0 we obtain the usual relations for celestial coordinates for the asymptotically flat solution. We expect that the parameters involved in RDGM in the presence of a perfect fluid will affect the shape of its shadow. This can be clearly confirmed through Equation (81) as it depends not only on spin parameter *a* and angular coordinate *θo* but also on *γ*, *ω* and perfect fluid parameter *υ*. Later, we will justify our results also through graphical interpretations.

As our observer is placed in the equatorial plane (*θ* = *π*/2), *α* and *β* reduce to:

$$\mathfrak{a}\_{\shortparallel} = -\sqrt{1 - 8\pi\gamma^2} \text{ } \mathfrak{F}\tag{82}$$

$$
\beta\_{\perp} = \pm \sqrt{1 - 8\pi\gamma^2} \sqrt{\eta\_{\prime}} \tag{83}
$$

for the case *ω* = 0 and *ω* = 1/3. And

$$
\alpha \quad = \quad -\sqrt{1 - 8\pi\gamma^2 - \upsilon} \quad \text{(} \text{\textquotedblleft} \text{\textquotedblright} \text{\textquotedblright} \text{\textquotedblright} \tag{84}
$$

$$
\beta \quad = \pm \sqrt{1 - 8\pi\gamma^2 - \upsilon} \sqrt{\eta} \tag{85}
$$

for the case *ω* = −1/3. Figures 3 and 4 show deformation in the shapes of the shadow with respect to monopole parameter *γ* and perfect fluid parameter *υ*, respectively. It is a well known observation now that the rotational effect in a black hole distorts its shape. That being said, we notice in Figure 3 that for small spin parameter, *a*, the shadow of the black hole maintains a circular shape along with the increase in its size with the inclination of *γ*. As for larger spin values, the shadow is clearly distorted and matches with its Kerr counterpart in perfect fluid [19] for *γ* = 0. Figure 4 shows the effect of parameter *υ* on the rotating dyonic black hole with a global monopole present in perfect fluid. It is noticed in Figure 4 that as perfect fluid parameter, *υ*, increases the size of the shadow also increases. A distortion is noticed in shape of the shadow when the spin parameter *a* is increased. Also, in the case of dark matter and dust, there is a significant change in the size of the shadow with respect to *υ*. On the other hand, in the case of radiation we do not observe any significant effect of perfect fluid parameter *υ*, in fact the effect is negligibly small.

In [10], the authors introduce two observables, radius *Rs* and distortion *δs*, to analyze the size and form of the shadow. The first observable *Rs* is the approximate radius of the shadow. It is defined by considering a reference circle passing through three points on the boundary of the shadow, such that (*α*tp, *β*tp) is the top most point on the shadow, (*α*bm, *β*bm) is the bottom most point on the shadow and (*α*r, 0) is the point corresponding to unstable circular orbit seen by an observer on reference frame. Thus:

$$R\_s = \frac{(\alpha\_{\rm tp} - \alpha\_r)^2 + \beta\_{\rm tp}^2}{2|\alpha\_{\rm tp} - \alpha\_r|}. \tag{86}$$

**Figure 3.** Variation in shape of a rotating dyonic global monopole surrounded by a perfect fluid. Magnetic and electric charges are kept constant such that *QE* = 10−2 = *QM*. In each graph the Kerr case, that is, *γ* = 0 and *υ* = 0, is represented by a solid line, *γ* = 0.05 by dotdashed and *γ* = 0.08 by dashed lines. For dark matter (*ω* = −1/3) and dust (*ω* = 0) case *υ* = 0.01, whereas *υ* = −0.01 in the case of radiation (*ω* = 1/3).

**Figure 4.** Variation in the shape of a rotating dyonic black hole with global monopole surrounded by a perfect fluid, for different values of perfect fluid parameter *υ*. Magnetic and electric charges along with the global monopole parameter are kept constant such that *QE* = 10−2 = *QM* and *γ* = 0.08. For dark matter and dust case *υ* = 0 (Solid), 0.05 (DotDashed) and 0.1 (Dashed). In the case of radiation *υ* = 0 (Solid), −0.01 (DotDashed) and −0.05 (Dashed).

The second observable *δs* is the distortion parameter. Let *DCS* be the difference between the contour of shadow and the reference circle. Then for the point (*α*˜ *p*, 0) lying on the reference circle and the point (*αp*, 0) lying on the contour of the shadow, *DCS* = |*α*˜ *p* − *αp*|. Thus:

$$
\delta\_{\mathfrak{k}} = \frac{\tilde{\alpha}\_p - \alpha\_p}{\mathcal{R}\_{\mathfrak{s}}}.
$$

For our case, we consider the points (*α*˜ *p*, 0) and (*αp*, 0) to be on the equatorial plane, opposite to the point (*α*r, 0). The variations in these observables with respect to monopole parameter *γ* are graphically presented in Figure 5. The dependence of *Rs* on parameter *γ* is such that as *γ* increases the radius *Rs* also increases. Thus the size of the shadow increases with increase in monopole parameter *γ*. Whereas the distortion *δs* decreases monotonically with an increase in *γ*. This tells us that with respect to circumference of reference circle, the shadow of the rotating black hole is significantly distorted for *γ* ∈ [0, 0.1] but for *γ* > 0 it may not show any distortion and thus we may obtain a perfect circle.

**Figure 5.** The quantities *Rs* and *δs* with respect to parameter *γ*.

As we have considered our observer to be at infinity so in this case the area of the black hole shadow will be approximately equal to high energy absorption cross section as discussed in Reference [18]. For a spherically symmetric black hole the absorption cross section oscillates around Π*ilm*, a limiting constant value. For a black hole shadow with radius *Rs*, we adopt the value of Π*ilm* as calculated by [18]:

$$
\Pi\_{ilm} \cong \,\,\pi R\_s^2.
$$

The energy emission rate of the black hole is thus defined by:

$$\frac{d^2E(\sigma)}{d\sigma dt} = 2\pi^2 \frac{\Pi\_{ilm}}{e^{\sigma/T} - 1} \sigma^3 \,\,\,\,$$

where *σ* is the frequency of the photon and *T* represents the temperature of the black hole at outer horizon, that is, *r*+, given by:

$$\begin{aligned} T(r\_+) &= \lim\_{r \to r\_+} \frac{\partial\_r \sqrt{\mathcal{g}\_{tt}}}{2\pi \sqrt{\mathcal{g}\_{rr}}} \\ &= \left( 2a^2 \left( f(r) - 1 \right) + r(r^2 + a^2) f'(r) \right) \frac{r}{4\pi \left( r^2 + a^2 \right)^2} \end{aligned}$$

For all three cases, radiation, dust and dark matter, the energy emission rate is graphically presented in Figure 6 where we notice that the energy emission rate decreases with an increase in parameter *γ*. A slight shift to the lower frequency is also observed while *γ* increases. The spin parameter *a* also effects the shape of the energy emission rate as an abrupt decrease in energy emission rate is noticed for higher spin value.

Recent studies have pointed out a connection between the topology of the shadow shape by introducing a new quantity such as the local curvature radius of the shadow. In our paper, we have applied the Gauss-Bonnet theorem to the horizon surface area to prove that the topology is a 2 sphere. It will be interesting to see if one can find the shadow radius by means of Gauss-Bonnet theorem applied directly to metric by using a relation between the horizon radius and the photon sphere. We are planning to work on such a project in the near future.

**Figure 6.** The figure shows the energy emission rate when *a* = 0.46 (**upper panel**) and *a* = 0.92 (**lower panel**).

#### **7. Conclusions**

In this paper, we have used the complex transformations pointed out by Newman and Janis to obtain an RDGM solution in the presence of a perfect fluid matter. Using the Gauss-Bonnet theorem we have shown that the surface topology of an RDGM is indeed a 2-sphere. Furthermore by choosing *ω* = −1/3, 0, 1/3 we have explored the impact of dark matter, dust, radiation, as well as the global monopole parameter *γ*, and perfect fluid parameters *υ*, on the silhouette of a black hole. We have found that a rotating dyonic black hole with a global monopole retains a circular shape for small spin parameter. Whereas for high spin like *a* = 0.98*M* the shadow of RDGM is distorted. Also as monopole parameter *γ* increases, a slight shift towards the right is also noticed in the shape of the shadow of the black hole under consideration. The two observables, *Rs* and *δs*, are also discussed. Finally, we analyze the energy emission rate of a rotating dyonic global monopole surrounded by perfect fluid with respect to parameters.

**Author Contributions:** All three authors contributed equally to the conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing–original draft preparation, writing–review and editing, visualization, supervision, project administration, funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We would like to thank the editor and Marcus Werner for helpful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **EHT Constraint on the Ultralight Scalar Hair of the M87 Supermassive Black Hole**

#### **Pedro V. P. Cunha 1,2,3,\*,†, Carlos A. R. Herdeiro 1,2,3,† and Eugen Radu 1,2,†**

Received: 20 September 2019; Accepted: 23 November 2019; Published: 27 November 2019

**Abstract:** Hypothetical ultralight bosonic fields will spontaneously form macroscopic bosonic halos around Kerr black holes, via superradiance, transferring part of the mass and angular momentum of the black hole into the halo. Such a process, however, is only efficient if resonant—when the Compton wavelength of the field approximately matches the gravitational scale of the black hole. For a complex-valued field, the process can form a stationary, bosonic field black hole equilibrium state—a black hole with synchronised hair. For sufficiently massive black holes, such as the one at the centre of the M87 supergiant elliptic galaxy, the hairy black hole can be robust against its own superradiant instabilities, within a Hubble time. Studying the shadows of such scalar hairy black holes, we constrain the amount of hair which is compatible with the Event Horizon Telescope (EHT) observations of the M87 supermassive black hole, assuming the hair is a condensate of ultralight scalar particles of mass *μ* ∼ 10−20 eV, as to be dynamically viable. We show the EHT observations set a weak constraint, in the sense that typical hairy black holes that could develop their hair dynamically, are compatible with the observations, when taking into account the EHT error bars and the black hole mass/distance uncertainty.

**Keywords:** ultralight particles; black hole shadow; event horizon telescope

#### **1. Introduction**

The hypothesis that all astrophysical black holes (BHs) when near equilibrium are well described by the Kerr metric [1]—*the Kerr hypothesis*—yields a remarkable scenario. It means that throughout the whole mass spectrum, ranging from solar mass BHs, with *M* ∼ *M*, all the way until the most supermassive black holes known, with *M* ∼ 1010*M*, the immense population of astrophysical BHs correspond to the very same object, with only two macroscopic degrees of freedom. One of these is the mass, which merely rescales the BH, leaving a single degree of freedom with impact on the BH phenomenology—the spin. The Kerr hypothesis, therefore, encodes an economical natural order: The landscape of gravitational atoms (BHs) that compose the dark Universe is made up of a single species, varying only in size (by, at least, ten orders of magnitude!) and spin, but otherwise indistinguishable. Such uniformity is a trademark of the microscopic world, where all elementary particles of a given species are indistinguishable, but not of the macroscopic world, where variety is ubiquitous.

Despite the current lack of tension between observations and the Kerr hypothesis, there are reasons to consider the latter but a fair approximation, within current precision, rather than a fundamental truth. Both fundamental problems—such as a quantum version of the laws of gravity, and how it impacts on the physics of horizons and classical singularities—and the phenomenological problem of accounting for the elusive dark matter and dark energy, suggest our current understanding of gravity is incomplete. It may therefore be that the Kerr hypothesis is strictly false for astrophysical BHs at all scales. But an alternative possibility is that the Kerr hypothesis is violated at a higher degree at some narrow interval of scales only, wherein new physics exists, remaining an excellent approximation outside this interval.

A concrete realisation of the latter possibility is provided by scenarios of hypothetical ultralight bosonic particles that could constitute part of the dark matter population [2–4]. Inspired by the QCD axion [5], and with theoretical support in string theory [6], these scenarios suggest that a landscape of such particles, negligibly interacting with standard model constituents, might exist, leaving their gravitational interactions as the only possible smoking gun for their identification. Amongst these, an exciting possibility, is their interaction with BHs, which, could single out a scale (or a range of scales) wherein BHs deviate from the Kerr paradigm.

Spinning BHs (in particular Kerr BHs) are energy and angular momentum reservoirs that can be classically mined. A very well suited tool for such mining is an ultralight bosonic field. Then, through the phenomenon of superradiance [7,8] an appropriate small seed of such field (provided, say, as a quantum fluctuation) will grow into a macroscopic condensate of bosonic particles—a Bose–Einstein condensate—storing a non-negligible fraction of the original BH mass and angular momentum. When the energy/angular momentum transfer from the BH to the bosonic halo saturates, the corresponding BH-halo state may or may not be stationary. If the bosonic field is real, the BH-halo system emits gravitational radiation and slowly decays back to a Kerr BH [9]. But if the bosonic field is complex, the BH-halo system is stationary, which is, in fact, a hairy BH—dubbed BHs with synchronised hair [10].

BHs with synchronised hair are not absolutely stable. They are themselves prone to their own superradiant instabilities [11–13]. However, the timescale of these instabilities is larger than the one of the initial Kerr superradiant instability that formed the hair, and, in the right mass range it becomes cosmologically large. A suggestive possibility is then the following. A Kerr BH develops ultralight bosonic hair in an astrophysical time scale; the hairy BH is then effectively stable, since it is superradiantly unstable only in a cosmological timescale. This turns out to be a realisable scenario for supermassive BHs, such as the one recently observed by the Event Horizon Telescope (EHT) collaboration [14–16] at the centre of the supergiant elliptic galaxy M87, henceforth referred to as the M87 BH.

The scenario in this paper is therefore that the M87 BH is hairy, due to an appropriate ultralight scalar field. Appropriate means its mass is in the right range to make the superradiant instability of the original Kerr BH grow in a sufficiently small fraction of the Hubble time, yielding a hairy BH that is stable in a cosmological time scale. Since the shadows of a Kerr and a hairy BH with the same total mass and angular momentum differ [17], and since the EHT observation is compatible with the M87 BH being of Kerr type, we shall then inquire how much the EHT observations constrain the hair. As we shall see, for the most interesting mass ranges, as to make the hairy BH dynamically viable, the EHT constraint is weak, and it is essentially compatible with a hairy BH that could have dynamically formed from superradiance and it is in a long lived, albeit not absolutely stable, state.

This paper is organised as follows. In Section 2 we discuss the physical scenario under which a BH with synchronised scalar hair could form from superradiance and be effectively stable within a cosmological timescale. In Section 3 we describe the part of the domain of existence of hairy BHs that is dynamically viable, according to the criteria in Section 2 and that we shall study in the remaining of the paper. In Section 4 we consider the Kerr BH shadow and we obtain an approximate expression for the shadow areal radius, valid for arbitrary observation angle and dimensionless spin value. In Section 5 we analyse the shadows of the hairy BHs in the relevant domain and obtain an approximate expression for the areal shadow radius, in terms of that of a comparable Kerr BH, i.e., with the same mass, and a parameter measuring the hairiness of the BH. In Section 6 the expression obtained in Section 5 is applied to the case of the M87 supermassive BH. Then, considering the EHT observational errors, together with the errors in the mass estimate, we constrain the hairiness compatible with the observations. Brief final remarks are given in Section 7. Unless otherwise stated, natural units *c* = 1 = *G* = *h* are used.

#### **2. The Hair Formation and Hair Instability Timescales**

The timescale of BH superradiance is extremely sensitive to the occurrence, or not, of a resonance between the gravitational scale of the BH and the Compton wavelength of the ultralight particles. Consider a massive, complex, scalar field, Φ, described by the Klein–Gordon equation, ✷Φ = *μ*2Φ, with mass *μ*, on the background of a Kerr BH with mass *M*. Maximal efficiency occurs for *Mμ* 0.4 [18] and for a spin close to extremality; for the M87 BH, for which we take *M*M87 ∼ 6 × 109*M*1 this resonant scalar field mass is:

$$
\mu\_{\rm r} = \frac{0.4}{M\_{\rm M87}} \simeq 4 \times 10^{-20} \,\text{eV} \,\text{.}\tag{1}
$$

At maximal efficiency, the e-folding time of the superradiant instability's leading mode is [18]:

$$
\Delta t \sim 10^{\text{V}} \tau\_{\text{LC}} \,\tag{2}
$$

where *τLC* is the light crossing time of the BH. For the M87 BH, Δ*t* ∼ 104 years. We call this the *hair formation timescale*. This means that if an ultralight boson of mass *μ*r exists, the M87 BH would develop scalar hair in an astrophysically short time scale.

If the resonance *μ* = *μ*r is missed, however, this timescale grows extremely fast: as (*Mμ*)−8 [23] for the leading mode and *Mμ* 1; or as 107*e*3.7*Mμ* for *Mμ* 1 [9,24]. In other words, if *Mμ* misses the resonant sweet spot by one order of magnitude, either above or below, the timescale of the leading mode of the superradiant instability of Kerr becomes considerably larger than the Hubble time, and the Kerr BH does not become hairy. On the lower end, the hair formation timescale becomes larger than one tenth of the Hubble time for *Mμ* < 0.05 [13], for the M87 BH mass. Thus, a conservative bound on the formation timescale is to take *Mμ* > 0.1, when the formation timescale becomes around a thousandth of the Hubble time.

Supermassive BHs in matter rich environments, such as galactic centres, are expected to grow in time. Thus, a supermassive BH with ∼ 109*M*, such as the one at the centre of M87, will have evolved, due to accretion and mergers, from one (or many) BHs with mass several orders of magnitude lower, see e.g., [25]. Only when the BH grows to the size that resonates with ∼ *μ*r does the superradiant energy/angular momentum extraction becomes efficient producing a sufficiently non-Kerr BH. At all other scales BHs remain Kerr-like.

Once the hairy BH forms, one must consider its leading superradiant instability. The leading instability has an e-folding time, dubbed *hair instability timescale*, larger than the Hubble time if *Mμ* 0.25 [13]. Thus, a Kerr BH with mass *M* = *M*M87 becomes hairy in an astrophysical timescale and the hair is stable in a cosmological timescale if (conservatively rounding off 0.25 to 0.3):

$$
\mu M\_{\text{MS7}} \in [0.1, 0.3] \quad \Rightarrow \quad \mu \in [1, 3] \times 10^{-20} \text{ eV} \,. \tag{3}
$$

We remark that there is a dependence on the BH dimensionless spin parameter *a* in determining the resonant scalar field mass *μ*r, although the instability is considerably more sensitive to *Mμ* than *a*. If the spin is not near extremal, this changes the ideal value of *Mμ* given in Equation (1), which could

1 For the considerations in this section this approximate value suffices. More accurate values will be considered in Section 6. This value is suggested from stellar dynamics [19] and favoured by the EHT observations [14]. A value half of this is suggested by gas dynamics [20]. The spin of the M87 BH is largely unknown, with different claims in the literature, see e.g., [21,22].

push down slightly, but not significantly, the lower end value of the interesting mass range given in Equation (3) (see also discussion in Section 7).