Generalized Gibbons-Werner method for stationary spacetimes

Let $D$ be a compact and connected region on a two-dimensional Riemannian manifold (as depicted in Fig. 1).

The boundary of $D$, denoted as $\partial D$, consists of piecewise smooth components $\partial D_{i}$ ($i=1,2,\cdots$), and the jump angles at each vertex are represented by $\eta_{i}$ in the positive sense. Then the GBT can be expressed as [73]

$$\iint_{D}K\mathrm{d}S+\sum_{i}\int_{\partial D_{i}}\kappa\mathrm{d}l+\sum_{i}\eta_{i}=2\pi\chi(D),$$

(2.1)

where $K$ and $\mathrm{d}S$ are the Gaussian curvature and area element of $D$, respectively; $\kappa$ and $\mathrm{d}l$ represent the geodesic curvature and line element of $\partial D$, respectively; $\chi(D)$ denotes the Euler characteristic number of $D$.

Eq. (2.1) establishes a fundamental relation between the integral of curvature quantities and the Euler characteristic of the region $D$. The GBT serves as a powerful tool for analyzing the geometric property of surfaces and their topological characteristics.

The JMRF metric, constructed by Chanda $et\ al.$ in 2019 [74], is of great significance in geometrodynamics and provides a framework for studying the particle’s motion in stationary spacetimes. Consider a coordinate $(t,\boldsymbol{x})$ where $\left(\partial/\partial t\right)^{a}$ is the Killing vector corresponding to the stationary property of the spacetime, the metric of the stationary spacetime states

$$\mathrm{d}s^{2}=g_{00}(\boldsymbol{x})\mathrm{d}t^{2}+2g_{0i}(\boldsymbol{x})\mathrm{d}t\mathrm{d}x^{i}+g_{ij}(\boldsymbol{x})\mathrm{d}x^{i}\mathrm{d}x^{j}.$$

(2.2)

Then the corresponding JMRF metric reads [74]

$$\mathrm{d}\tilde{s}=\sqrt{\alpha_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}}+\beta_{i}\mathrm{d}x^{i},$$

(2.3)

where the components of the JMRF metric are given by

	$\displaystyle\alpha_{ij}$	$\displaystyle=\frac{E^{2}+m^{2}g_{00}}{-g_{00}}\left(g_{ij}-\frac{g_{0i}g_{0j}}{g_{00}}\right),$		(2.4)
	$\displaystyle\beta_{i}$	$\displaystyle=-E\frac{g_{0i}}{g_{00}}.$		(2.5)

Here, $m$ and $E$ stand for the rest mass and relativistic energy of the particle, respectively, and we dropped ’$(\boldsymbol{x})$’ from ’$g_{00}(\boldsymbol{x})$’, ’$g_{0i}(\boldsymbol{x})$’, and ’$g_{ij}(\boldsymbol{x})$’ . The Riemannian metric $\alpha_{ij}$ and one-form $\beta_{i}$ must satisfy the inequality $\sqrt{\alpha^{ij}\beta_{i}\beta_{j}}<1$. We denote the three-dimensional space determined by $\alpha_{ij}$, i.e. the Riemannian part of the JMRF metric, as $M^{\left(\alpha 3\right)}$. For a geodesic in the four-dimensional stationary spacetime Eq. (2.2), its spatial projection $\gamma$ is a geodesic in the three-dimensional JMRF space Eq. (2.3) [74]. Additionally, the JMRF metric encompasses various special cases, including the Jacobi metric applicable to massive particles in SSS spacetimes [75], the optical Randers-Finsler metric applicable to photons in SAS spacetimes [49], and the optical metric applicable to photons in SSS spacetimes [48]. Researchers have utilized the JMRF metric, as well as its specific forms, to investigate the gravitational deflection of particles [48, 49, 76, 77, 50, 78, 75, 79, 80, 51, 81, 82, 83, 58, 52, 53, 84, 85, 63, 86, 54, 87, 88, 89], the Kepler orbit [90], the motion of charged particles [91], and Hawking radiation [92, 93]. A comprehensive discussion on the JMRF metric can be found in [94].

For SAS spacetimes, the metric in the Boyer-Lindquist coordinates can be expressed as

$$\mathrm{d}s^{2}=g_{tt}\left(r,\theta\right)\mathrm{d}t^{2}+g_{rr}\left(r,\theta\right)\mathrm{d}r^{2}+g_{\theta\theta}\left(r,\theta\right)\mathrm{d}\theta^{2}+g_{\phi\phi}\left(r,\theta\right)\mathrm{d}\phi^{2}+2g_{t\phi}\left(r,\theta\right)\mathrm{d}t\mathrm{d}\phi,$$

(2.6)

and the metric of the corresponding $M^{\left(\alpha 3\right)}$ can be written as

$$\mathrm{d}{\hat{l}}^{2}=\alpha_{rr}\left(r,\theta\right)\mathrm{d}r^{2}+\alpha_{\theta\theta}\left(r,\theta\right)\mathrm{d}\theta^{2}+\alpha_{\phi\phi}\left(r,\theta\right)\mathrm{d}\phi^{2}.$$

(2.7)

We focus on the motion in the equatorial plane ($\theta=\pi/2$), and $M^{\left(\alpha 3\right)}$ reduces to a two-dimensional Riemannian space (denoted as $M^{\left(\alpha 2\right)}$ for simplicity). The metric of $M^{\left(\alpha 2\right)}$ reads

$$\mathrm{d}l^{2}=\alpha_{rr}\left(r\right)\mathrm{d}r^{2}+\alpha_{\phi\phi}\left(r\right)\mathrm{d}\phi^{2},$$

(2.8)

in which

$$\alpha_{rr}\left(r\right)=\frac{E^{2}+m^{2}g_{tt}}{-g_{tt}}g_{rr},\qquad\alpha_{\phi\phi}\left(r\right)=\frac{E^{2}+m^{2}g_{tt}}{-g_{tt}}g_{\phi\phi}.$$

(2.9)

The corresponding $\beta_{i}$ states

$$\beta_{\phi}\left(r\right)=-E\frac{g_{t\phi}}{g_{tt}}.$$

(2.10)

Here, we dropped ’$\left(r,\theta=\pi/2\right)$’ from ’$g_{tt}\left(r,\theta=\pi/2\right)$’, ’$g_{rr}\left(r,\theta=\pi/2\right)$’, ’$g_{\phi\phi}\left(r,\theta=\pi/2\right)$’, and ’$g_{t\phi}\left(r,\theta=\pi/2\right)$’ for simplicity.

Furthermore, the equation of motion for unbound particles moving in the equatorial plane of the SAS spacetime equipped with metric (2.6) can be written as [67]

$$\left(\frac{\mathrm{d}u}{\mathrm{d}\phi}\right)^{2}=\frac{u^{4}\left(g_{t\phi}^{2}-g_{tt}g_{\phi\phi}\right)\left[g_{tt}b^{2}v^{2}+2g_{t\phi}bv+g_{\phi\phi}\left(1+g_{tt}-g_{tt}v^{2}\right)+g_{t\phi}^{2}\left(v^{2}-1\right)\right]}{g_{rr}(g_{tt}bv+g_{t\phi})^{2}},$$

(2.11)

where $u=1/r$, $b$ symbolizes the impact parameter, $v$ is the velocity of particles.

In 2017, by taking into account the finite distance from a lens object to a light source and a observer, Ishihara $et\ al.$ investigate the finite-distance corrections for the deflection angle of photons [50]. With the GW method, they defined a finite-distance deflection angle and prove it is geometric invariant, namely well-defined.

As shown in Fig. 2,

in a two-dimensional Riemannian manifold, $L$ represents the lens, $S$ and $R$ are the source and the receiver of particles, respectively, $\gamma=\overset{\curvearrowright}{SR}$ is the trajectory from $S$ to $R$, then the deflection angle is defined by [50]

$$\delta=\Psi_{R}-\Psi_{S}+\phi_{RS},$$

(2.12)

in which $\phi_{RS}=\phi_{R}-\phi_{S}$ is the increment of the azimuthal coordinate, $\Psi_{S}$ and $\Psi_{R}$ are the angle between the tangent vector along $\gamma$ and the radial outward vector at $S$ and $R$, respectively. In the limit as $S$ and $R$ tend to infinity, $\Psi_{S}=\pi$ and $\Psi_{R}=0$, the above formula reduces to the usual infinite-distance deflection angle $\delta=\phi_{RS}-\pi$. The definition Eq. (2.12) has been widely employed in the investigation of the deflection of particles [80, 51, 55, 59, 58, 61, 95, 60, 63, 54, 96, 87, 64, 97, 98, 65, 66, 99].

As shown in Fig. 3,

in the $M^{\left(\alpha 2\right)}$ corresponding to the SAS spacetime equipped with metric (2.6), $L$, $S$, $R$, $\gamma$, $\Psi_{S}$, and $\Psi_{R}$ have the same meaning as that in Fig. 2, $C_{\infty}$ is an infinite circular arc, intersecting the outgoing radial curve $\overrightarrow{LS}$ and $\overrightarrow{LR}$ at $S_{\infty}$ and $R_{\infty}$, respectively. Then the integral region is constructed as $D_{\infty}=_{R}^{R_{\infty}}\square_{S}^{S_{\infty}}$. Applying the GBT Eq. (2.1) to $D_{\infty}$ leads to

$$\iint_{D_{\infty}}K\mathrm{d}S+\int_{\overrightarrow{SS}_{\infty}}\kappa\mathrm{d}l+\int_{C_{\infty}}\kappa\mathrm{d}l+\int_{\overrightarrow{R_{\infty}R}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{RS}}\kappa\mathrm{d}l+\eta_{S}+\eta_{S_{\infty}}+\eta_{R_{\infty}}+\eta_{R}=2\pi\chi\left(D_{\infty}\right).$$

(3.3)

$\int_{\overrightarrow{SS}_{\infty}}\kappa\mathrm{d}l=\int_{\overrightarrow{R_{\infty}R}}\kappa\mathrm{d}l=0$, since $\overrightarrow{SS_{\infty}}$ and $\overrightarrow{R_{\infty}R}$ are geodesics (see Appendix A of Ref. [67] for the proof). $\int_{\overset{\curvearrowright}{RS}}\kappa\mathrm{d}l=-\int_{\gamma}\kappa\mathrm{d}l$, $\eta_{S}=\pi-\Psi_{S}$, $\eta_{R}=\Psi_{R}$, $\eta_{S_{\infty}}=\eta_{R_{\infty}}=\pi/2$, and $\chi\left(D_{\infty}\right)=1$ as $D_{\infty}$ is simply connected.

In the scenario where the spacetime is asymptotically flat, $C_{\infty}$ can be treated as a circular arc in the flat space, i.e. $\int_{C_{\infty}}\kappa\cdot\mathrm{d}l=\int_{\phi_{S}}^{\phi_{R}}\left(1/r_{\infty}\right)\cdot r_{\infty}\mathrm{d}\phi=\phi_{RS}$. According to Eq. (3.3) and the related results, we can express the deflection angle as

$$\delta=-\iint_{D_{\infty}}K\mathrm{d}S+\int_{\gamma}\kappa\mathrm{d}l,$$

(3.4)

in which the definition Eq. (2.12) is adopted. The above formula has been used to calculate the deflection angle of particles in Refs. [55, 56, 57, 58, 59, 60, 61, 62, 54, 64, 65, 66, 68], where the spacetime is asymptotically flat.

Based on the presence or absence of the singularity as the radial coordinate approaches infinity, asymptotically nonflat SAS spacetimes can be divided into two categories: the infinity-reachable and the infinity-unreachable. Examples of the infinity-reachable include the Kerr-like black hole in bumblebee gravity [100, 101], while the Kerr-de Sitter spacetime [69] and the rotating solution in conformal Weyl gravity [70, 71] fall under the category of the infinity-unreachable.

In the case of infinity-reachable asymptotically nonflat SAS spacetimes, the construction of an infinite integral region is still allowed. However, the calculation formula of the deflection angle becomes more intricate and relies on the specific metric. To illustrate this, we briefly review the work in [63], where the Kerr-like black hole in bumblebee gravity is considered. The metric for such black hole is given by

$$\mathrm{d}s^{2}=-\left(1-\frac{2Mr}{\rho^{2}}\right)\mathrm{d}t^{2}-\frac{4Mar\lambda\sin^{2}\theta}{\rho^{2}}\mathrm{d}\phi\mathrm{d}t+\frac{\rho^{2}}{\Delta}\mathrm{d}r^{2}+\rho^{2}\mathrm{d}\theta^{2}+\frac{A\sin^{2}\theta}{\rho^{2}}\mathrm{d}\phi^{2},$$

(3.5)

in which $\lambda=\sqrt{1+l}$, $\rho^{2}=r^{2}+\lambda^{2}a^{2}\cos^{2}\theta$, $\Delta=\left(r^{2}-2Mr\right)/\lambda^{2}+a^{2}$, $A=\left(r^{2}+\lambda^{2}a^{2}\right)^{2}-\Delta\lambda^{4}a^{2}\sin^{2}\theta$. Here, $M$ and $a$ are the mass and rotation parameter of the black hole, respectively, and $l$ is the Lorentz violation parameter. In this scenario, Fig. 3 can still serve as the illustration, and the relevant constructions and results, including Eq. (3.3), remain valid except for the integral of the geodesic curvature along $C_{\infty}$. The author derived $\int_{C_{\infty}}\kappa\mathrm{d}l=\left(1/\lambda\right)\phi_{RS}$ instead of the result obtained in asymptotically flat spacetime. Consequently, the deflection angle is expressed as

$$\delta=-\iint_{D_{\infty}}K\mathrm{d}S+\int_{\gamma}\kappa\mathrm{d}l+\left(1-\frac{1}{\lambda}\right)\phi_{RS},$$

(3.6)

in which an additional item, the change in the coordinate angle, appears compared to Eq. (3.4) due to the existence of the bumblebee vector field. It is important to note that the formula for the deflection angle will vary depending on the specific metrics employed.

As for the infinity-unreachable asymptotically nonflat SAS spacetime, the infinite integral region $D_{\infty}$ is ill-defined, rendering the GWOIA method invalid.

As shown in Fig. 4,

in the $M^{\left(\alpha 2\right)}$, $L$, $S$, $R$, and $\gamma$ have the same meaning as that in Fig. 2. $C_{a}=\overset{\curvearrowright}{S_{a}R_{a}}$ is the auxiliary circular arc with $r_{c}>r_{\gamma}^{max}$, and intersects with $\overrightarrow{LS}$ and $\overrightarrow{LR}$ at $S_{a}$ and $R_{a}$, respectively. Thus we obtain a quadrilateral region $D_{a}=^{R_{a}}_{R}\square^{S_{a}}_{S}$. Here, the subscript $a$ indicates "arbitrary". The application of the GBT to $D_{a}$ yields

$$\iint_{D_{a}}K\mathrm{d}S+\int_{\overrightarrow{SS}_{a}}\kappa\mathrm{d}l+\int_{C_{a}}\kappa\mathrm{d}l+\int_{\overrightarrow{R_{a}R}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{RS}}\kappa\mathrm{d}l+\eta_{S}+\eta_{S_{a}}+\eta_{R_{a}}+\eta_{R}=2\pi\chi(D_{a}).$$

(4.1)

Substituting $\kappa(\overrightarrow{SS}_{a})=\kappa(\overrightarrow{R_{a}R})=0$, $\int_{\overset{\curvearrowright}{RS}}\kappa\mathrm{d}l=-\int_{\gamma}\kappa\mathrm{d}l$, $\eta_{S}=\pi-\Psi_{S}$, $\eta_{S_{a}}=\eta_{R_{a}}=\pi/2$, $\eta_{R}=\Psi_{R}$, and $\chi\left(D_{a}\right)=1$ into the above equation leads to

$$\iint_{D_{a}}K\mathrm{d}S+\int_{C_{a}}\kappa\mathrm{d}l-\int_{\gamma}\kappa\mathrm{d}l+\Psi_{R}-\Psi_{S}=0.$$

(4.2)

With the definition Eq. (2.12), the deflection angle can be written as

$$\delta=-\iint_{D_{a}}K\mathrm{d}S-\int_{C_{a}}\kappa\mathrm{d}l+\phi_{RS}+\int_{\gamma}\kappa\mathrm{d}l.$$

(4.3)

Firstly, we analyze $\iint_{D_{a}}K\mathrm{d}S$. According to Eqs. (2.8) and (3.2), we have

$$\int K\sqrt{\alpha}\mathrm{d}r=-\frac{\alpha_{\phi\phi,r}}{2\sqrt{\alpha}}+Const.$$

(4.4)

Introducing $H(r)$ to denote the indefinite integral of $K\sqrt{\alpha}$ with respect to the radial coordinate up to a constant, namely,

$$H(r)=-\frac{\alpha_{\phi\phi,r}}{2\sqrt{\alpha}},$$

(4.5)

we derive

$$\iint_{D_{a}}K\mathrm{d}S=\int_{\phi_{S}}^{\phi_{R}}\int_{r_{\gamma}}^{r_{c}}K\sqrt{\alpha}\mathrm{d}r\mathrm{d}\phi=\int_{\phi_{S}}^{\phi_{R}}\left[H(r_{c})-H(r_{\gamma})\right]\mathrm{d}\phi.$$

(4.6)

Secondly, we analyze $\int_{C_{a}}\kappa\mathrm{d}l$. According to Liouville’s formula for geodesic curvature, the geodesic curvature of the circular arc in $M^{(\alpha 2)}$ can be expressed as (Chapter 4 of Ref. [102])

$$\kappa^{(c)}=-\Gamma^{r}_{\phi\phi}\frac{\alpha^{1/2}}{\alpha^{3/2}_{\phi\phi}}=\frac{\alpha_{\phi\phi,r}}{2\alpha_{\phi\phi}\sqrt{\alpha_{rr}}}.$$

(4.7)

Introducing $G(r)=\kappa^{(c)}\mathrm{d}l/\mathrm{d}\phi$ to denote the integrand of the integral of geodesic curvature along the auxiliary circular arc with respect the azimuthal coordinate, we derive

$$\int_{C_{a}}\kappa\mathrm{d}l=\int_{\phi_{S}}^{\phi_{R}}\left[\kappa^{(c)}\frac{\mathrm{d}l}{\mathrm{d}\phi}\right]_{r=r_{c}}\mathrm{d}\phi=\int_{\phi_{S}}^{\phi_{R}}G(r_{c})\mathrm{d}\phi.$$

(4.8)

According to Eqs. (2.8) and (4.7) we obtain

$$G(r)=\frac{\alpha_{\phi\phi,r}}{2\sqrt{\alpha}}.$$

(4.9)

Finally, substituting Eqs. (4.6) and (4.8) into Eq. (4.3) yields

$$\delta=-\int_{\phi_{S}}^{\phi_{R}}\left[H(r_{c})-H(r_{\gamma})+G(r_{c})-1\right]\mathrm{d}\phi+\int_{\gamma}\kappa\mathrm{d}l.$$

(4.10)

We find that $H(r)=-G(r)$ regardless of the value of $r$ according to Eqs. (4.5) and (4.9), thus the above formula becomes

$$\delta=\int_{\phi_{S}}^{\phi_{R}}\left[1+H(r_{\gamma})\right]\mathrm{d}\phi+\int_{\gamma}\kappa\mathrm{d}l.$$

(4.11)

Now, we show that Eq. (4.11) also holds for the situation $r_{\gamma}^{max}\geq r_{c}\geq r_{\gamma}^{min}$. As depicted in Fig. 5,

the auxiliary circular arc $C_{a}$ satisfies $r_{\gamma}^{max}\geq r_{c}\geq r_{\gamma}^{min}$, and intersects with $\gamma$ at $P$ and $Q$. Thus we obtain two triangle regions $D_{a1}=_{P}\triangle_{S_{a}}^{S}$ and $D_{a3}=_{R}\triangle^{Q}_{R_{a}}$, and a digon $D_{a2}$ with vertexes $P$ and $Q$. $\xi$ and $\omega$ are the angle between $C_{a}$ and $\gamma$ at $P$ and $Q$, respectively.

Applying the GBT to $D_{a1}$ leads to

$$\iint_{D_{a1}}K\mathrm{d}S+\int_{\overrightarrow{S_{a}S}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{SP}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{PS_{a}}}\kappa\mathrm{d}l+\eta_{S_{a}}+\eta_{S}+\eta_{P}^{(1)}=2\pi\chi(D_{a1}).$$

(4.12)

Using $\kappa(\overrightarrow{S_{a}S})=0$, $\eta_{S_{a}}=\pi/2$, $\eta_{S}=\Psi_{S}$, $\eta_{P}^{(1)}=\pi-\xi$, and $\chi(D_{a1})=1$, we derive

$$\iint_{D_{a1}}K\mathrm{d}S+\int_{\overset{\curvearrowright}{SP}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{PS_{a}}}\kappa\mathrm{d}l+\Psi_{S}-\xi=\frac{\pi}{2}.$$

(4.13)

Applying the GBT to $D_{a2}$ leads to

$$\iint_{D_{a2}}K\mathrm{d}S+\int_{\overset{\curvearrowright}{QP}}\kappa\mathrm{d}l+\int_{\overset{\Large\curvearrowright}{PQ}}\kappa\mathrm{d}l+\eta_{P}^{\left(2\right)}+\eta_{Q}^{\left(2\right)}=2\pi\chi\left(D_{a2}\right),$$

(4.14)

where $\overset{\curvearrowright}{QP}$ denotes the curve followed from $Q$ to $P$ along the trajectory $\gamma$, $\overset{\curvearrowright}{PQ}$ represents the curve followed from $P$ to $Q$ along the circular arc $C_{a}$. Using $\eta_{P}^{\left(2\right)}=\pi-\xi$, $\eta_{Q}^{\left(2\right)}=\pi-\omega$, and $\chi(D_{a2})=1$, we derive

$$\iint_{D_{a2}}K\mathrm{d}S+\int_{\overset{\curvearrowright}{QP}}\kappa\mathrm{d}l+\int_{\overset{\Large\curvearrowright}{PQ}}\kappa\mathrm{d}l-\xi-\omega=0.$$

(4.15)

Applying the GBT to $D_{a3}$ leads to

$$\iint_{D_{a3}}K\mathrm{d}S+\int_{\overset{\curvearrowright}{QR}}\kappa\mathrm{d}l+\int_{\overrightarrow{RR_{a}}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{R_{a}Q}}\kappa\mathrm{d}l+\eta_{Q}^{(3)}+\eta_{R}+\eta_{R_{a}}=2\pi\chi\left(D_{a3}\right).$$

(4.16)

Using $\kappa\left(\overrightarrow{RR_{a}}\right)=0$, $\eta_{Q}^{(3)}=\pi-\omega$, $\eta_{R}=\pi-\Psi_{R}$, $\eta_{R_{a}}=\pi/2$, and $\chi\left(D_{a3}\right)=1$, we derive

$$\iint_{D_{a3}}K\mathrm{d}S+\int_{\overset{\curvearrowright}{QR}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{R_{a}Q}}\kappa\mathrm{d}l-\omega-\Psi_{R}=-\frac{\pi}{2}.$$

(4.17)

Acoording to Eq. (2.12), Eq. (4.13)$+$Eq. (4.17)$-$Eq. (4.15) results in

$$\delta=\iint_{D_{a1}}K\mathrm{d}S-\iint_{D_{a2}}K\mathrm{d}S+\iint_{D_{a3}}K\mathrm{d}S-\int_{C_{a}}\kappa\mathrm{d}l+\phi_{RS}+\int_{\gamma}\kappa\mathrm{d}l.$$

(4.18)

With the help of Eqs. (4.5) and (4.9), the deflection angle is expressed as

	$\displaystyle\delta=$	$\displaystyle\int_{\phi_{S}}^{\phi_{P}}\left[H(r_{\gamma})-H(r_{c})\right]\mathrm{d}\phi-\int_{\phi_{P}}^{\phi_{Q}}\left[H(r_{c})-H(r_{\gamma})\right]\mathrm{d}\phi+\int_{\phi_{Q}}^{\phi_{R}}\left[H(r_{\gamma})-H(r_{c})\right]\mathrm{d}\phi$		(4.19)
		$\displaystyle-\int_{\phi_{S}}^{\phi_{R}}G(r_{c})\mathrm{d}\phi+\phi_{RS}+\int_{\gamma}\kappa\mathrm{d}l$
	$\displaystyle=$	$\displaystyle\int_{\phi_{S}}^{\phi_{R}}\left[H(r_{\gamma})-H(r_{c})\right]\mathrm{d}\phi-\int_{\phi_{S}}^{\phi_{R}}G(r_{c})\mathrm{d}\phi+\phi_{RS}+\int_{\gamma}\kappa\mathrm{d}l$
	$\displaystyle=$	$\displaystyle\int_{\phi_{S}}^{\phi_{R}}\left[1+H(r_{\gamma})\right]\mathrm{d}\phi+\int_{\gamma}\kappa\mathrm{d}l,$

which is the same as Eq. (4.11). Additionally, if $C_{a}$ intersects with $\gamma$ at more or fewer points, one can also obtain the above formula by applying the GBT to more or fewer quadrilateral, triangle, or digon regions.

As shown in Fig. 6,

$r_{c}<r_{\gamma}^{min}$, applying the GBT to $D_{a}=^{R}_{R_{a}}\square_{S_{a}}^{S}$ yields

$$\iint_{D_{a}}K\mathrm{d}S+\int_{\overrightarrow{S_{a}S}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{SR}}\kappa\mathrm{d}l+\int_{\overrightarrow{RR}_{a}}\kappa\mathrm{d}l+\int_{\overset{\curvearrowright}{R_{a}S_{a}}}\kappa\mathrm{d}l+\eta_{S_{a}}+\eta_{S}+\eta_{R}+\eta_{R_{a}}=2\pi\chi(D_{a}).$$

(4.20)

Using $\kappa(\overrightarrow{S_{a}S})=\kappa(\overrightarrow{RR}_{a})=0$, $\eta_{S_{a}}=\eta_{R_{a}}=\pi/2$, $\eta_{S}=\Psi_{S}$, $\eta_{R}=\pi-\Psi_{R}$, $\chi\left(D_{a}\right)=1$ and Eq. (2.12), we derive

$$\delta=\iint_{D_{a}}K\mathrm{d}S+\int_{\overset{\curvearrowright}{R_{a}S_{a}}}\kappa\mathrm{d}l+\phi_{RS}+\int_{\gamma}\kappa\mathrm{d}l.$$

(4.21)

Namely,

	$\displaystyle\delta=$	$\displaystyle\int_{\phi_{S}}^{\phi_{R}}\left[H(r_{\gamma})-H(r_{c})\right]\mathrm{d}\phi-\int_{\phi_{S}}^{\phi_{R}}G(r_{c})\mathrm{d}\phi+\phi_{RS}+\int_{\gamma}\kappa\mathrm{d}l$		(4.22)
	$\displaystyle=$	$\displaystyle\int_{\phi_{S}}^{\phi_{R}}\left[1+H(r_{\gamma})\right]\mathrm{d}\phi+\int_{\gamma}\kappa\mathrm{d}l,$

which is the same as Eq. (4.11).

Although Eq. (4.11) is simpler than that in the conventional GWOIA method (Eqs. (3.4) and (3.6)), calculating the integral of geodesic curvature along the trajectory is still not straightforward. Here, we recast it into a form that further simplifies the calculation process for practical computations. We denote the integrand of the integral of geodesic curvature along the trajectory with respect to the azimuthal coordinate by $T\left(r\right)=\kappa_{\gamma}\mathrm{d}l/\mathrm{d}\phi$. Then using Eqs. (2.7), (2.8) and (3.1), we obtain

$$T\left(r\right)=-\beta_{\phi,r}\sqrt{\frac{1}{\alpha_{\phi\phi}}\left(\frac{\mathrm{d}r}{\mathrm{d}\phi}\right)^{2}+\frac{1}{\alpha_{rr}}},\qquad\text{or}\qquad T\left(r\right)=-\beta_{\phi,r}\sqrt{\frac{r^{4}}{\alpha_{\phi\phi}}\left(\frac{\mathrm{d}u}{\mathrm{d}\phi}\right)^{2}+\frac{1}{\alpha_{rr}}}.$$

(4.23)

As a consequence, we express the integral of geodesic curvature along $\gamma$ as

$$\int_{\gamma}\kappa\mathrm{d}l=\int_{\phi_{S}}^{\phi_{R}}\left.\left[\kappa_{\gamma}\frac{\mathrm{d}l}{\mathrm{d}\phi}\right]\right|_{r=r_{\gamma}}\mathrm{d}\phi=\int_{\phi_{S}}^{\phi_{R}}T(r_{\gamma})\mathrm{d}\phi,$$

(4.24)

with which Eq. (4.11) is finally simplified as

$$\delta=\int_{\phi_{S}}^{\phi_{R}}\left[1+H(r_{\gamma})+T(r_{\gamma})\right]\mathrm{d}\phi.$$

(4.25)

This formula is applicable to all kinds of SAS spacetime, including the asymptotically flat, the infinity-reachable asymptotically nonflat, and the infinity-unreachable asymptotically nonflat. It provides an unified description for the finite-distance deflection angle of particles moving in the equatorial plane of SAS spacetimes, the calculation process using the simplified formula Eq. (4.25) is simpler and more straightforward. It can be summarized as follows:

• •

  Firstly, substitute the metric into Eq. (2.11) to obtain the $\left(\mathrm{d}u/\mathrm{d}\phi\right)^{2}$, and the orbit solution $u\left(\phi\right)=1/r(\phi)$ and $\phi\left(u\right)$.

• •

  Secondly, substitute the metric into Eqs. (2.9) and (2.10) to get $\alpha_{rr}$, $\alpha_{\phi\phi}$ and $\beta_{\phi}$.

• •

  Thirdly, substitute $\alpha_{rr}$, $\alpha_{\phi\phi}$, $\beta_{\phi}$, $\left(\mathrm{d}u/\mathrm{d}\phi\right)^{2}$ and $r_{\gamma}=1/u(\phi)$ into the integrand of Eq. (4.25) to derive $f(\phi)$, where

  $$f\left(\phi\right)=1+H(r_{\gamma})+T(r_{\gamma}).$$

  (4.26)

• •

  Finally, obtain the deflection angle in terms of $u_{R}$ and $u_{S}$ with

  $$\delta=F\left(\phi_{R}\right)-F\left(\phi_{S}\right),$$

  (4.27)

  where $\phi_{R}=\phi(u_{R})$ and $\phi_{S}=\phi(u_{S})$, $\phi_{R}>\pi/2$ and $\phi_{S}<\pi/2$ are usually assumed, $u_{R}$ and $u_{S}$ are respectively the reciprocal of the radial coordinate of the observer (receiver) and source, and $F\left(\phi\right)$ denotes the indefinite integral of $f\left(\phi\right)$.

The scheme of the generalized GW encompasses various special cases, such as the deflection angle of photons ($v=1$), the infinite-distance deflection angle ($u_{S}=u_{R}=0$, only valid for spacetimes where the source and observer can reach infinity), and the deflection angle of SSS counterparts (the rotation parameter vanishes). Furthermore, for the GW method, numerous examples demonstrate that if the spacetime can be approximated as Minkowski spacetime ($\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}$) in terms of the quantities of interest under the zeroth-order approximation, the $n$-th order deflection angle can be obtained using the $(n-1)$-th order orbit solution.

Our discovery of the expression $H(r_{c})=-G(r_{c})$ is the cornerstone of the generalized GW method. It neutralizes the $G(r_{c})$ in the integral of geodesic curvature and the $H(r_{c})$ in the integral of Gaussian curvature regardless of the chosen value of $r_{c}$. The resulting Eq. (4.25) does not depend on the location of the auxiliary circular arc. Our method encompasses the conventional GWOIA method as a specific case with $r_{c}=\infty$. To illustrate the relationship between the two methods, we employ the generalized GW method with $r_{c}=\infty$ to reproduce the calculation formula in the conventional GWOIA method. When $r_{c}=\infty$ the deflection angle can be expressed by Eq. (4.10) since $\infty>r_{\gamma}^{max}$. Firstly, we reproduce Eq. (3.4), which corresponds to asymptotically flat SAS spacetimes. Given the asymptotic flatness of spacetimes, we have $\alpha_{rr}\left(\infty\right)=1$ and $\alpha_{\phi\phi}\left(\infty\right)=r^{2}$ according to Eq. (2.8). Substituting these values into Eq. (4.9) yields $G\left(r_{c}=\infty\right)=1$, with which Eq. (4.10) can reduce to Eq. (3.4). Secondly we reproduce Eq. (3.6), which corresponds to an infinity-reachable nonflat SAS spacetime (Kerr-like black hole in bumblebee gravity). Using the result $G(r_{c}=\infty)=1/\lambda$ from Ref. [63], Eq. (4.10) directly reduces to Eq. (3.6).

The generalized GW and the corresponding simplified calculation formula holds significant meaning in five aspects:

• (a)

  The GW method can be extended to infinity-unreachable spacetimes by constructing an appropriate finite integral region, since the auxiliary circular arc can be chosen arbitrarily instead of taking the limit $r_{c}=\infty$ in previous works.

• (b)

  For infinity-reachable asymptotically nonflat SAS spacetimes, our calculation formula Eq. (4.25) is universally applicable, unlike that in the conventional GWOIA method which varies depending on specific metrics.

• (c)

  Our method can be viewed as an unified description to the GW method for calculating the finite-distance and infinite-distance deflection angle of massive and massless particles in SSS and SAS spacetimes with or without asymptotical flatness. This is the reason why we call it the generalized GW method.

• (d)

  Compared to calculation formulas in the conventional GWOIA method (Eqs. (3.4) and (3.6)), our formula Eq. (4.25) is easier to apply in practical computation processes. Since it involves only a single integral of an fully simplified integrand, rather than a double integral of the intricate Gaussian curvature and a single integral of the geodesic curvature required by Eqs. (3.4) and (3.6).

• (e)

  Since the presence of the surface integral over $D_{\infty}$ in the calculation formula in conventional GWOIA method (Eqs. (3.4) and (3.6)), the deflection angle seems depend on the nature of the region between the trajectory and infinity. While our result, Eq. (4.25), clearly demonstrates that the deflection angle is solely determined by the properties of the trajectory itself, which is closer to our intuitive understanding.

The Kerr metric with Boyer-Lindquist coordinates states [103]

$$\mathrm{d}s^{2}=-\left(1-\frac{2Mr}{\Sigma}\right)\mathrm{d}t^{2}-\frac{4aMr\sin^{2}\theta}{\Sigma}\mathrm{d}t\mathrm{d}\phi+\frac{\Sigma}{\Delta}\mathrm{d}r^{2}+\Sigma\mathrm{d}\theta^{2}+\left[\Delta+\frac{2Mr\left(r^{2}+a^{2}\right)}{\Sigma}\right]\sin^{2}\theta\mathrm{d}\phi^{2},$$

(5.1)

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $\Delta=r^{2}-2Mr+a^{2}$. We adopt the calculation steps in Sec.4.4 to calculate the deflection angle of particles moving in the equatorial plane of Kerr spacetime. Firstly, according to Eq. (2.11), we obtain the equation of motion

$$\left(\frac{\mathrm{d}u}{\mathrm{d}\phi}\right)^{2}=\frac{1-b^{2}u^{2}}{b^{2}}+M\cdot\frac{2u\left(b^{2}u^{2}v^{2}-v^{2}+1\right)}{b^{2}v^{2}}+\mathcal{O}\left(M^{2},Ma,a^{2}\right).$$

(5.2)

Then the orbit solution can be derived with the perturbation method ²²2When $M=a=0$, Kerr metric becomes the Minkowski spacetime. Here we calculate the second order deflection angle with respect to $M$ and $a$, thus the orbit solution up to first order is enough.

$$u\left(\phi\right)=\frac{\sin\phi}{b}+M\cdot\frac{1+v^{2}\cos^{2}\phi}{b^{2}v^{2}}+\mathcal{O}\left(M^{2},Ma,a^{2}\right).$$

(5.3)

In addition, the iterative solution of $\phi$ can be obtained by using the above formula

$$\phi(u)=\begin{cases}\Phi(u),&\text{if }\left|\phi\right|<\frac{\pi}{2},\\ \pi-\Phi(u),&\text{if }\left|\phi\right|>\frac{\pi}{2},\end{cases}$$

(5.4)

where

$$\Phi\left(u\right)=\arcsin\left(bu\right)-M\cdot\frac{1+v^{2}-b^{2}u^{2}v^{2}}{bv^{2}\sqrt{1-b^{2}u^{2}}}+\mathcal{O}\left(M^{2},Ma,a^{2}\right).$$

(5.5)

Secondly, we substitute the metric (5.1) into Eqs. (2.9) and (2.10) to obtain $\alpha_{rr}$, $\alpha_{\phi\phi}$, and $\beta_{\phi}$. Thirdly, substituting $\alpha_{rr}$, $\alpha_{\phi\phi}$, $\beta_{\phi}$, and Eq. (5.2) into Eqs. (4.5) and (4.23), and using Eq. (4.26), we derive

	$\displaystyle f(\phi)=$	$\displaystyle M\cdot\frac{\left(v^{2}+1\right)\sin\phi}{bv^{2}}+M^{2}\cdot\frac{1}{4b^{2}v^{4}}\left[\left(v^{2}-2\right)^{2}\cos(2\phi)+3\left(v^{2}+4\right)v^{2}\right]$		(5.6)
		$\displaystyle-Ma\cdot\frac{2\sin\phi}{b^{2}v}+\mathcal{O}\left(M^{3},M^{2}a,Ma^{2},a^{3}\right),$

in which $r_{\gamma}$ is expressed as the reciprocal of Eq. (5.3). Finally, substituting $F\left(\phi\right)$ (derived by integrating $f(\phi)$), $\phi_{R}=\pi-\Phi\left(u_{R}\right)$, and $\phi_{S}=\Phi\left(u_{S}\right)$ into Eq. (4.27) yields the deflection angle of particles moving in the equatorial plane

	$\displaystyle\delta=$	$\displaystyle M\cdot\frac{\left(v^{2}+1\right)\left(\sqrt{1-b^{2}u{{}_{R}}^{2}}+\sqrt{1-b^{2}u{{}_{S}}^{2}}\right)}{bv^{2}}+M^{2}\cdot\left\{\frac{3\left(v^{2}+4\right)[\arccos(bu_{R})+\arccos(bu_{S})]}{4b^{2}v^{2}}\right.$		(5.7)
		$\displaystyle+\frac{u_{S}\left[3v^{2}\left(v^{2}+4\right)-b^{2}u{{}_{S}}^{2}\left(3v^{4}+8v^{2}-4\right)\right]}{4bv^{4}\sqrt{1-b^{2}u{{}_{S}}^{2}}}\left.+\frac{u_{R}\left[3v^{2}\left(v^{2}+4\right)-b^{2}u{{}_{R}}^{2}\left(3v^{4}+8v^{2}-4\right)\right]}{4bv^{4}\sqrt{1-b^{2}u{{}_{R}}^{2}}}\right\}$
		$\displaystyle-Ma\cdot\frac{2\left(\sqrt{1-b^{2}u_{R}^{2}}+\sqrt{1-b^{2}u_{S}^{2}}\right)}{b^{2}v}+\mathcal{O}\left(M^{3},M^{2}a,Ma^{2},a^{3}\right).$

Li and Jia also obtained Eq. (5.7) with the conventional GWOIA method in Ref. [54], where the calculation process is much more complicated than ours and the redundant second order term $Ma$ is retained in the orbit solution.