Appendix A: The geodesic equations and determination of the deflection angle
In this appendix, we mainly focus on the derivation of the geodesic equations and the usage of these equations to establish the relationship between constants of motion and the deflection angles. We start from the Hamilton-Jacobi equations for KNTN space-time and use separation of variables to derive ODEs of the coordinates. The case \(\displaystyle \theta \equiv \frac{\pi }{2}\) is treated separately. Then we rewrite these equations in integral form and calculate the integrals over r and \(\theta\) respectively. Finally, we use Taylor expansion with respect to \(\Delta \theta\) for the evaluation of these integrals, in order to solve the value of \(\Delta \theta\) and \(\Delta \phi\).
Following [52], the Hamilton-Jacobi equation governing geodesic motion of KNTN metric
$$\begin{aligned} 2 \frac{\partial S}{\partial \tau }&=g^{\mu \nu }\frac{\partial S}{\partial x^{\mu }} \frac{\partial S}{\partial x^{\nu }}\\&=-\frac{\Sigma }{\Delta }\left( (1+\frac{a p}{\Sigma }) \frac{\partial S}{\partial t} +\frac{a}{\Sigma }\frac{\partial S}{\partial \phi }\right) ^2 +\frac{\Sigma }{\sin ^2\theta } \left( \frac{p}{\Sigma }\frac{\partial S}{\partial t} +\frac{1}{\Sigma }\frac{\partial S}{\partial \phi }\right) ^2\\&\quad +\frac{\Delta }{\Sigma }\left( \frac{\partial S}{\partial r}\right) ^2 +\frac{1}{\Sigma }\left( \frac{\partial S}{\partial \theta }\right) ^2 \end{aligned}$$
(A.1)
could be evaluated by imposing the assumption of the principal function for null geodesics
$$\begin{aligned} S = -E t + L_{z} \phi + S_{r}(r) + S_{\theta }(\theta ). \end{aligned}$$
(A.2)
Here E and \(L_{z}\) are the conserved quantities corresponding to the Killing vectors \(\displaystyle \frac{\partial }{\partial t}\) and \(\displaystyle \frac{\partial }{\partial \phi }\) of the KNTN space-time:
$$\begin{aligned} E&= \frac{1}{\Sigma } \left( (\Delta -a^2\sin ^2\theta )\dot{t} +(\Delta p-a(\Sigma +a p)\sin ^2\theta )\dot{\phi }\right) \\ L_{z}&= \frac{1}{\Sigma } \left( (\Delta p-a(\Sigma +a p)\sin ^2\theta )\dot{t} +((\Sigma +a p)^2\sin ^2\theta - p^2 \Delta )\dot{\phi }\right) , \end{aligned}$$
(A.3)
and the principal function S does not explicitly include the affine parameter \(\tau\) since the geodesics in consideration are null [53].
The resulting equation
$$\begin{aligned} \frac{1}{\Delta }\left( -E(\Sigma +a p)+a L_{z}\right) ^2 -\frac{1}{\sin ^2\theta }(-E p + L_{z})^2 -\Delta \left( \frac{{\text{d}}S_{r}}{{\text{d}}r}\right) ^2 -\left( \frac{{\text{d}}S_{\theta }}{{\text{d}}\theta }\right) ^2 =0 \end{aligned}$$
(A.4)
is separated with respect to the variables r and \(\theta\) into
$$\begin{aligned} -\mathcal {Q}&=\Delta \left( \frac{{\text{d}}S_{r}}{{\text{d}}r}\right) ^2 -\frac{1}{\Delta }\left( -E (r^2+l^2+a^2)+a L_{z}\right) ^2 +(L_{z}-a E)^2,\\ \mathcal {Q}&=\left( \frac{{\text{d}}S_{\theta }}{{\text{d}}\theta }\right) ^2 +\frac{1}{\sin ^2\theta } (-p E+L_{z})^2-(L_{z}-a E)^2 \end{aligned}$$
(A.5)
by introducing the Carter constant \(\mathcal {Q}\).
Then since
$$\begin{aligned} \frac{{\text{d}}S_{r}}{{\text{d}}r}&\equiv p_{r} =\frac{\partial \mathscr {L}}{\partial \dot{r}} =\frac{\Sigma }{\Delta }\dot{r},\\ \frac{{\text{d}}S_{\theta }}{{\text{d}}\theta }&\equiv p_{\theta } =\frac{\partial \mathscr {L}}{\partial \dot{\theta }} =\Sigma \dot{\theta }, \end{aligned}$$
(A.6)
where
$$\begin{aligned} \mathscr {L} = \frac{1}{2}g_{\mu \nu }\dot{x}^{\mu }\dot{x}^{\nu } \end{aligned}$$
(A.7)
is the Lagrangian of KNTN space-time satisfying
$$\begin{aligned} S = \int \mathscr {L} d\tau , \end{aligned}$$
(A.8)
the r and \(\theta\) components of geodesic equations turn out to be
$$\begin{aligned} \Sigma \dot{r}&= \pm \sqrt{R(r)}\\ \Sigma \dot{\theta }&= \pm \sqrt{\Theta (\theta )}, \end{aligned}$$
(A.9)
where
$$\begin{aligned} R( r)&=-\Delta \left( \mathcal {Q} +( L_{z} -a E)^{2}\right) +\left( a L_{z} -E \left( r^{2} +l^{2} +a^{2}\right) \right) ^{2},\\ \Theta ( \theta )&=\mathcal {Q} +( L_{z} -a E)^{2} -\frac{1}{\sin ^{2} \theta }\left( - E (a\sin ^{2} \theta -2l\cos \theta )+ L_{z} \right) ^{2}. \end{aligned}$$
(A.10)
In the following part we will take \(E=1\), \(\displaystyle \lambda = \frac{L_{z}}{E}=L_{z}\) and \(\displaystyle \eta = \frac{\mathcal {Q}}{E^2}=\mathcal {Q}\) for simplicity.
However, the above procedure of variable separation does not apply to the geodesics with \(\displaystyle \theta \equiv \frac{\pi }{2}\), i.e., those lying in the orbital plane. Instead, by inserting the assumption
$$\begin{aligned} S = -E t + L_{z} \phi + S_{r}(r) \end{aligned}$$
(A.11)
into
$$\begin{aligned} 2 \frac{\partial S}{\partial \tau }&=-\frac{\Sigma }{\Delta }\left( (1+\frac{a p}{\Sigma }) \frac{\partial S}{\partial t} +\frac{a}{\Sigma }\frac{\partial S}{\partial \phi }\right) ^2 +\Sigma \left( \frac{p}{\Sigma }\frac{\partial S}{\partial t} +\frac{1}{\Sigma }\frac{\partial S}{\partial \phi }\right) ^2 +\frac{\Delta }{\Sigma }\left( \frac{\partial S}{\partial r}\right) ^2, \end{aligned}$$
(A.12)
we derive
$$\begin{aligned} \frac{1}{\Delta }\left( -E(\Sigma +a p)+a L_{z}\right) ^2 -(-E p + L_{z})^2 -\Delta \left( \frac{{\text{d}}S_{r}}{{\text{d}}r}\right) ^2 =0, \end{aligned}$$
(A.13)
which is the r component of geodesic equations. Note that (A.5) and (A.6) still hold in this case as long as we set \(\mathcal {Q}=0\) and \(S_{\theta }\) being any constant. Therefore the general geodesic equations (A.9) are not violated, with \(\eta = 0\) and \(\Theta \equiv 0\).
The remaining two equations are transformed from (A.3):
$$\begin{aligned} \dot{t}&=\frac{1}{\Sigma \Delta }\left( r^{2} +a^{2} +l^{2}\right) \left( r^{2} +a^{2} +l^{2} -a\lambda \right) \\&\quad +\frac{1}{\Sigma \sin ^{2} \theta }\left( a\sin ^{2} \theta -2l\cos \theta \right) \left( \lambda -a\sin ^{2} \theta +2l\cos \theta \right) ,\\ \dot{\phi }&=\frac{a}{\Sigma \Delta }\left( r^{2} +a^{2} +l^{2} -a\lambda \right) +\frac{1}{\Sigma \sin ^{2} \theta }\left( \lambda -a\sin ^{2} \theta +2l\cos \theta \right) . \end{aligned}$$
(A.14)
In the following we will provide some details in the treatment of the spatial geodesic equations in the integral form. In order to evaluate the integral
$$\begin{aligned} \int _{r_{\min }}^{\infty }\frac{{\text{d}} r}{\sqrt{R( r)}}, \end{aligned}$$
(A.15)
we utilise the techniques in [54], introducing the new dimensionless variable
$$\begin{aligned} \textbf{x} = \frac{r_{\min }}{r} \end{aligned}$$
(A.16)
such that
$$\begin{aligned}&2\int _{r_{\min }}^{\infty }\frac{{\text{d}} r}{\sqrt{R( r)}}\\&=2\int _{0}^{1}\Big ( r_{\min }^2 +(a^2+2l^2-\lambda ^2-\eta )\textbf{x}^2 +2m\left( \eta +(\lambda -a)^2\right) \textbf{x}^3/r_{\min }\\&\quad +\left( (a^2+l^2-a \lambda )^2+(l^2-a^2-Q^2)(\eta +(\lambda -a)^2)\right) \textbf{x}^4/r_{\min }^2 \Big )^{-1/2}{\text{d}}\textbf{x}\\&=2\int _{0}^{1}\Big ( (a^2+2l^2-\lambda ^2-\eta )(\textbf{x}^2-1) +2m\left( \eta +(\lambda -a)^2\right) (\textbf{x}^3-1)/r_{\min }\\&\quad +\left( (a^2+l^2-a \lambda )^2+(l^2-a^2-Q^2)(\eta +(\lambda -a)^2)\right) (\textbf{x}^4-1)/r_{\min }^2 \Big )^{-1/2}{\text{d}}\textbf{x}\\&=2\int _{0}^{1}(r_{\min }^{(0)})^{-1} (1-\textbf{x}^2)^{-1/2} \Bigg ( 1-\tilde{a}^2-2\tilde{l}^2 -2\tilde{m}\big (\tilde{\eta }+ (\tilde{\lambda }-\tilde{a})^2\big ) \frac{r_{\min }^{(0)}}{r_{\min }} \frac{\textbf{x}^2+\textbf{x}+1}{\textbf{x}+1}\\&\quad -\Big ((\tilde{a}^2+\tilde{l}^2-\tilde{a} \tilde{\lambda })^2+(\tilde{l}^2-\tilde{a}^2-\tilde{Q}^2) \big (\tilde{\eta }+(\tilde{\lambda }-\tilde{a})^2\big )\Big ) \Big (\frac{r_{\min }^{(0)}}{r_{\min }}\Big )^2 (\textbf{x}^2+1) \Bigg )^{-1/2}{\text{d}}\textbf{x}. \end{aligned}$$
(A.17)
After Taylor expansion of the integrand up to the third order of \(\tilde{a}\), \(\tilde{m}\), \(\tilde{Q}\) and \(\tilde{l}\), the resulting function can be directly integrated over \(\textbf{x}\) as in [54], such that we have
$$\begin{aligned} \int ^{r}\frac{\mathrm {| d} r\mathrm {| }}{\sqrt{R( r)}}&= r{_{\min }^{( 0)}}^{-1} \Big ( \pi +4\tilde{m}+32\tilde{m}\tilde{l}^{2} -16\tilde{m}\tilde{Q}^{2} -8\tilde{m}\tilde{a}\tilde{\lambda } +4\tilde{m}\tilde{a}^{2}( 4\tilde{\lambda }^{2} -1)\\&\quad +\frac{7}{4}\pi \tilde{l}^{2} -\frac{3}{4} \pi \tilde{Q}^{2} -3\pi \tilde{a}\tilde{l}^{2}\tilde{\lambda } +\frac{3}{2}\pi \tilde{a}\tilde{Q}^{2}\tilde{\lambda } -\frac{1}{4}\pi \tilde{a}^{2} +\frac{3}{4}\pi \tilde{\lambda }^{2}\tilde{a}^{2}\\&\quad +\frac{128}{3}\tilde{m}^{3} +\frac{15}{4} \pi \tilde{m}^{2} -15\pi \tilde{m}^{2}\tilde{a}\tilde{\lambda }\Big ), \end{aligned}$$
(A.18)
and the other integral involving r can be treated in the same manner.
For the integration involving \(\theta\), we have derived
$$\begin{aligned} \int ^{\theta }\frac{|\mathrm {d\theta } |}{\sqrt{\Theta ( \theta )}} =\left( \int _{0}^{\sigma _{{\rm s}}} +\int _{0}^{\sigma_{{\rm o}}}\right) H(\sigma ) {\text{d}}\sigma , \end{aligned}$$
(A.19)
where
$$\begin{aligned} H(\sigma )&= {r_{\min }^{(0)}}^{-1} \left( 1-\frac{2\tilde{\lambda }\tilde{\eta }^{-1/2}}{\cos \sigma +1}\tilde{l} +2\tilde{\eta }^{-1/2}\big (\frac{\tilde{\lambda }^{2}}{\cos \sigma +1} -\tilde{\eta }\cos \sigma \big )\tilde{a}\tilde{l}\right. \\&\quad \left. +\frac{1}{2}(\tilde{\lambda }^{2} -\tilde{\eta }\cos ^{2} \sigma )\tilde{a}^{2} -\frac{2\big ( 1-\tilde{\lambda }^{2}\tilde{\eta }^{-1} +2( 1+\tilde{\lambda }^{2}\tilde{\eta }^{-1})\cos \sigma +\cos ^{2} \sigma \big )}{(\cos \sigma +1)^{2}}\tilde{l}^{2}\right. \\&\quad \left. +\frac{\tilde{\lambda }\tilde{\eta }^{-1/2}\big ( 2(\tilde{\eta } -\tilde{\lambda }^{2}) +\tilde{\eta }\cos \sigma (5\cos \sigma +\cos 2\sigma +3)\big )}{\cos \sigma +1}\tilde{a}^{2}\tilde{l}\right. \\&\quad \left. +\frac{\tilde{\lambda }\big ( 18-4\tilde{\lambda }^{2}\tilde{\eta }^{-1} +( 27+8\tilde{\lambda }^{2}\tilde{\eta }^{-1})\cos \sigma +10\cos 2\sigma +\cos 3\sigma \big )}{(\cos \sigma +1)^{2}}\tilde{a}\tilde{l}^{2}\right. \\&\quad \left. +\frac{2\tilde{\lambda }\tilde{\eta }^{-1/2}\big ( 6-\tilde{\lambda }^{2}\tilde{\eta }^{-1} +8( 1+\tilde{\lambda }^{2}\tilde{\eta }^{-1})\cos \sigma +( 2-\tilde{\lambda }^{2}\tilde{\eta }^{-1})\cos 2\sigma \big )}{(\cos \sigma +1)^{3}}\tilde{l}^{3} \right) . \end{aligned}$$
(A.20)
After denoting
$$\begin{aligned} J(\sigma ) = \int H(\sigma ) {\text{d}} \sigma , \end{aligned}$$
(A.21)
it leads to
$$\begin{aligned} J(\pi - \sigma_{{\rm o}} )+J(\sigma_{{\rm o}})&= r{_{{\rm min}}^{( 0)}}^{-1} \left( \pi -4\tilde{\lambda }\tilde{\eta }^{-1/2}\csc \sigma_{{\rm o}}\tilde{l} -\frac{1}{4} \pi (\tilde{\eta } -2\tilde{\lambda }^{2})\tilde{a}^{2}\right. \\&\quad \left. +4\tilde{\eta }^{-1/2}(\tilde{\lambda }^{2}\csc \sigma_{{\rm o}} -4\tilde{\eta }\sin \sigma_{{\rm o}})\tilde{a}\tilde{l}\right. \\&\quad \left. -2\tilde{\eta }^{-1}( \pi \tilde{\eta } -4\tilde{\lambda }^{2}\cot ^{2} \sigma_{{\rm o}}\csc \sigma_{{\rm o}})\tilde{l}^{2}\right. \\&\quad \left. +\tilde{\lambda }\tilde{\eta }^{-1/2}\csc \sigma_{{\rm o}}( 9\tilde{\eta } -4\tilde{\lambda }^{2} -3\tilde{\eta }\cos 2\sigma_{{\rm o}})\tilde{a}^{2}\tilde{l}\right. \\&\quad \left. +4\tilde{\lambda }\tilde{\eta }^{-1}\big ( 3\pi \tilde{\eta } +\cot ^{2} \sigma_{{\rm o}}\csc \sigma_{{\rm o}}( -\tilde{\eta } -4\tilde{\lambda }^{2} +\tilde{\eta }\cos 2\sigma_{{\rm o}})\big )\tilde{a}\tilde{l}^{2}\right. \\&\quad \left. +\frac{8}{3}\tilde{\lambda }\tilde{\eta }^{-3/2}\csc \sigma_{{\rm o}}\big ( 6\tilde{\eta } -3\tilde{\lambda }^{2} +\tilde{\lambda }^{2}\csc ^{2} \sigma_{{\rm o}}( 17-12\csc ^{2} \sigma_{{\rm o}})\big )\tilde{l}^{3} \right) ,\\ J'(\pi - \sigma_{{\rm o}})&= r{_{{\rm min}}^{( 0)}}^{-1} \left( 1-\tilde{\lambda }\tilde{\eta }^{-1/2}\csc ^{2}\frac{\sigma_{{\rm o}}}{2}\tilde{l} -\frac{1}{4}(\tilde{\eta } -2\tilde{\lambda }^{2} +\tilde{\eta }\cos 2\sigma_{{\rm o}})\tilde{a}^{2}\right. \\&\quad \left. +\tilde{\eta }^{-1/2}( 2\tilde{\eta }\cos \sigma_{{\rm o}} +\tilde{\lambda }^{2}\csc ^{2}\frac{\sigma_{{\rm o}}}{2})\tilde{a}\tilde{l}\right. \\&\quad \left. +\frac{1}{2}\tilde{\eta }^{-1}\big (\tilde{\lambda }^{2} (2\cos \sigma_{{\rm o}} +1)\csc ^{4}\frac{\sigma_{{\rm o}}}{2} -4\tilde{\eta }\big )\tilde{l}^{2} \right) ,\\ \frac{1}{2}J''(\pi - \sigma_{{\rm o}})&= r{_{{\rm min}}^{( 0)}}^{-1} \Big (-\frac{1}{4}\tilde{\lambda }\tilde{\eta }^{-1/2}\sin \sigma_{{\rm o}}\csc ^{4}\frac{\sigma_{{\rm o}}}{2}\tilde{l}\Big ),\\ \frac{1}{6}J'''(\pi - \sigma_{{\rm o}})&= 0, \end{aligned}$$
(A.22)
where we have kept the terms up to the third order for \(\displaystyle J(\pi - \sigma_{{\rm o}} )+J(\sigma_{{\rm o}})\), second order for \(\displaystyle J'(\pi - \sigma_{{\rm o}})\), first order for \(\displaystyle \frac{1}{2}J''(\pi - \sigma_{{\rm o}})\) and zeroth order for \(\displaystyle \frac{1}{6}J'''(\pi - \sigma_{{\rm o}})\), such that Taylor expansion up to the third order are considered overall in (44).
After the calculation of \(\Delta \sigma\) through (48), the value of \(\Delta \theta\) can be derived through (53), up to the third order of \(\tilde{m}\), \(\tilde{a}\), \(\tilde{Q}\) and \(\tilde{l}\). The result can be expressed in the form
$$\begin{aligned} \Delta \theta&=C_{m}\tilde{m} +C_{a}\tilde{a} +C_{Q}\tilde{Q} +C_{l}\tilde{l} +C_{mm}\tilde{m}^{2} +C_{ma}\tilde{m}\tilde{a} +C_{mQ}\tilde{m}\tilde{Q} +C_{ml}\tilde{m}\tilde{l}\\&\quad +C_{aa}\tilde{a}^{2} +C_{aQ}\tilde{a}\tilde{Q} +C_{al}\tilde{a}\tilde{l} +C_{QQ}\tilde{Q}^{2} +C_{Ql}\tilde{Q}\tilde{l} +C_{ll}\tilde{l}^{2}\\&\quad +C_{mmm}\tilde{m}^{3} +C_{mma}\tilde{m}^{2}\tilde{a} +C_{mmQ}\tilde{m}^{2}\tilde{Q} +C_{mml}\tilde{m}^{2}\tilde{l}\\&\quad +C_{maa}\tilde{m}\tilde{a}^{2} +C_{maQ}\tilde{m}\tilde{a}\tilde{Q} +C_{mal}\tilde{m}\tilde{a}\tilde{l} +C_{mQQ}\tilde{m}\tilde{Q}^{2} +C_{mQl}\tilde{m}\tilde{Q}\tilde{l} +C_{mll}\tilde{m}\tilde{l}^{2}\\&\quad +C_{aaa}\tilde{a}^{3} +C_{aaQ}\tilde{a}^{2}\tilde{Q} +C_{aal}\tilde{a}^{2}\tilde{l} +C_{aQQ}\tilde{a}\tilde{Q}^{2} +C_{aQl}\tilde{a}\tilde{Q}\tilde{l} +C_{all}\tilde{a}\tilde{l}^{2}\\&\quad +C_{QQQ}\tilde{Q}^{3} +C_{QQl}\tilde{Q}^{2}\tilde{l} +C_{Qll}\tilde{Q}\tilde{l}^{2} +C_{lll}\tilde{l}^{2}, \end{aligned}$$
(A.23)
with the following coefficients for the first order:
$$\begin{aligned} C_{m} = 4\csc \theta_{{\rm o}} \mu ,\ C_{l} = 4\tilde{\lambda }\csc \theta_{{\rm o}},\ C_{a} = C_{Q}=0, \end{aligned}$$
(A.24)
the second order:
$$\begin{aligned}&C_{mm} = \frac{15}{4} \pi \csc \theta_{{\rm o}} \mu -8\tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc ^{2} \theta_{{\rm o}},\ C_{ma} = -8\tilde{\lambda }\csc \theta_{{\rm o}} \mu ,\\&C_{ml} = 8\tilde{\lambda }\cot \theta_{{\rm o}}(1 -2\csc ^{2} \theta_{{\rm o}}\tilde{\lambda }^{2} ) \mu ^{-1},\ C_{al} = 4\sin \theta_{{\rm o}} -8\tilde{\lambda }^{2}\csc \theta_{{\rm o}},\\&C_{QQ} = -\frac{3}{4} \pi \csc \theta_{{\rm o}} \mu ,\ C_{ll} = 8\tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc ^{2} \theta_{{\rm o}},\\&C_{mQ} = C_{aa} = C_{aQ} = C_{Ql} = 0, \end{aligned}$$
(A.25)
and the third order:
$$\begin{aligned}C_{mmm} & = -\frac{32}{3}\tilde{\lambda }^{2} (\cos 2\theta_{{\rm o}} +2)\csc ^{5} \theta_{{\rm o}} \mu -15\pi \tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc ^{2} \theta_{{\rm o}} +\frac{128}{3}\csc \theta_{{\rm o}} \mu ,\\ C_{mma} & = \tilde{\lambda }\csc \theta_{{\rm o}}( 32\tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc \theta_{{\rm o}} -15\pi \mu ),\\ C_{mml} & = 16 \tilde{\lambda }\cos \theta_{{\rm o}}\cot \theta_{{\rm o}} + 96\tilde{\lambda }^{3}\csc \theta_{{\rm o}}\cot ^{4} \theta_{{\rm o}} \mu ^{-2}\\ &\quad -\frac{15}{2}\pi \tilde{\lambda }\csc \theta_{{\rm o}}(\cos \theta_{{\rm o}}-2\tilde{\lambda }^2\cot \theta_{{\rm o}}\csc \theta_{{\rm o}}) \mu ^{-1} \\&\quad +128\tilde{\eta }\tilde{\lambda }\csc \theta_{{\rm o}}\big ( -2\tilde{\lambda }^{2} (\cos 2\theta_{{\rm o}} +2)\csc ^{4} \theta_{{\rm o}} +1\big )\mu ^{-2},\\ C_{maa} & = 2\tilde{\eta }\csc \theta_{{\rm o}}\tilde{\lambda }^{2} \mu ^{-1} +( 11\tilde{\lambda }^{2} -3\tilde{\eta } +\cos 2\theta_{{\rm o}})\csc \theta_{{\rm o}} \mu ,\\ C_{mal} & =\big (\sin 4\theta_{{\rm o}} -2( 7\tilde{\lambda }^{2} +\tilde{\eta })\sin 2\theta_{{\rm o}}\big )\mu ^{-3} +32\cot \theta_{{\rm o}}( 3\tilde{\lambda }^{4} -2\tilde{\lambda }^{6}\csc ^{2} \theta_{{\rm o}}) \mu ^{-3},\\ C_{mQQ} & = ( 3\pi \tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc \theta_{{\rm o}} -16\mu )\csc \theta_{{\rm o}} ,\\ C_{mll} & = -32\tilde{\lambda }^{4} (\cos 2\theta_{{\rm o}} +2)\csc ^{5} \theta_{{\rm o}} \mu ^{-1} +40\csc \theta_{{\rm o}} \mu -64\csc ^{3} \theta_{{\rm o}}\tilde{\lambda }^{4}\mu ^{-3}\\ &\quad -8( 11\tilde{\lambda }^{2} +8\tilde{\eta })\csc \theta_{{\rm o}}\tilde{\lambda }^{2}\mu ^{-3} +8( 3\tilde{\lambda }^{2} +\tilde{\eta })\sin \theta_{{\rm o}}\mu ^{-3},\\ C_{aal} & = 4\tilde{\lambda }( 2\tilde{\lambda }^{2} -2\tilde{\eta } +\cos 2\theta_{{\rm o}})\csc \theta_{{\rm o}},\\ C_{aQQ} & = \frac{3}{2} \pi \tilde{\lambda }\csc \theta_{{\rm o}} \mu ,\\ C_{all} & = -\tilde{\lambda }\csc \theta_{{\rm o}}( 32\cot \theta_{{\rm o}}\csc \theta_{{\rm o}}\tilde{\lambda }^{2} +15\pi \mu ) ,\\ C_{QQl} & = -\frac{3}{4}\pi \tilde{\lambda }( 4\cot \theta_{{\rm o}}\csc ^{2} \theta_{{\rm o}}\tilde{\lambda }^{4} -6\cot \theta_{{\rm o}}\tilde{\lambda }^{2} +\sin 2\theta_{{\rm o}}) \mu ^{-3},\\ C_{lll} & = -\frac{16}{3}\tilde{\lambda }\big ( 3-2\tilde{\lambda }^{2} (\cos 2\theta_{{\rm o}} +2)\csc ^{4} \theta_{{\rm o}}\big )\csc \theta_{{\rm o}} \\ &\quad +\frac{15}{2} \pi \tilde{\lambda }(\sqrt{\tilde{\eta }} +\cos \theta_{{\rm o}}) \mu ^{-1}\csc \theta_{{\rm o}} +\frac{15}{4} \pi \csc \theta_{{\rm o}} \mu ,\\ C_{mmQ} & = C_{maQ} = C_{mQl} = C_{aaa} = C_{aaQ} = C_{aQl} = C_{QQQ} = C_{Qll} = 0. \end{aligned}$$
(A.26)
Note that in the above expressions we have \(\displaystyle \tilde{\eta }= 1-\tilde{\lambda }^2\) by definition (29). Thus by taking the sum of all these terms, the expression of \(\Delta \theta\) is derived.
In the treatment of the second integral over theta (55), we introduced the primitive function \(P(\sigma )\). The following values are calculated where we kept the terms up to the third order for \(\displaystyle P(\pi - \sigma_{{\rm o}} )+P(\sigma_{{\rm o}})\), second order for \(\displaystyle P'(\pi - \sigma_{{\rm o}})\), first order for \(\displaystyle \frac{1}{2}P''(\pi - \sigma_{{\rm o}})\) and zeroth order for \(\displaystyle \frac{1}{6}P'''(\pi - \sigma_{{\rm o}})\) with respect to the small quantities:
$$\begin{aligned} P(\pi - \sigma_{{\rm o}} )+P(\sigma_{{\rm o}})&=-4\mu ^{-1}\tilde{l} +\frac{1}{2} \pi \tilde{\lambda }\tilde{a}^{2} +4\tilde{\lambda } \mu ^{-1}\tilde{a}\tilde{l}\\&\quad +2\left( \cos ^{2} \theta_{{\rm o}}\tilde{\lambda }^{2} \mu ^{-3} +\left( 2\tilde{\eta } -2\tilde{\lambda }^{2} -\cos ^{2} \theta_{{\rm o}}\right) \mu ^{-1}\right) \tilde{a}^{2}\tilde{l}\\&\quad +4\pi \tilde{a}\tilde{l}^{2} +\frac{8}{3}\left( 2\tilde{\eta } -\cos ^{2} \theta_{{\rm o}}\left( 2+3\sin ^{2} \theta_{{\rm o}}\right) \right) \mu ^{-5}\tilde{l}^{3},\\ P'(\pi - \sigma_{{\rm o}})&=\tilde{\lambda }\csc ^{2} \theta_{{\rm o}} -2\left( \cos \theta_{{\rm o}} +\tilde{\lambda }^{2}\tilde{\eta }^{1/2}\csc ^{2} \theta_{{\rm o}}\right) \mu ^{-2}\tilde{l}\\&\quad +\left( 4\tilde{\eta }^{1/2}\cos \theta_{{\rm o}} +3\cos 2\theta_{{\rm o}} +\left( \tilde{\eta } +7\tilde{\lambda }^{2}\right) -4\tilde{\lambda }^{4}\csc ^{2} \theta_{{\rm o}}\right) \tilde{\lambda } \mu ^{-4}\tilde{l}^{2}\\&\quad +\frac{1}{2}\left( -\tilde{\eta }\csc ^{2} \theta_{{\rm o}} +1\right) \tilde{\lambda }\tilde{a}^{2} +2\left( \tilde{\lambda }^{2}\tilde{\eta }^{1/2}\csc ^{2} \theta_{{\rm o}} +\cos \theta_{{\rm o}}\right) \mu ^{-2}\tilde{\lambda }\tilde{a}\tilde{l},\\ \frac{1}{2}P''(\pi - \sigma_{{\rm o}})&=\tilde{\lambda }\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}} \mu -\Big ((\cos \theta_{{\rm o}} +\tilde{\eta }^{1/2})^{2} \mu ^{-3} \\&\quad +\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}}(\cos 2\theta_{{\rm o}} -\tilde{\eta } +3\tilde{\lambda }^{2})\tilde{\eta }^{1/2} \mu ^{-1}\Big )\tilde{l},\\ \frac{1}{6}P'''(\pi - \sigma_{{\rm o}})&=\frac{1}{3}\tilde{\lambda }\csc ^{2} \theta_{{\rm o}}\left( 3\left( \tilde{\eta } +2\tilde{\lambda }^{2}\right) \csc ^{2} \theta_{{\rm o}} -2-4\tilde{\lambda }^{2}\csc ^{4} \theta_{{\rm o}}\right) . \end{aligned}$$
(A.27)
With these coefficients and other quantities already obtained, the value of \(\Delta \phi\) can be determined by plugging in the expression of \(\Delta \sigma\) into (32), and keeping the terms up to the third order of \(\tilde{a}\), \(\tilde{m}\), \(\tilde{Q}\) and \(\tilde{l}\). Finally we arrive at
$$\begin{aligned} \Delta \phi&=F_{m}\tilde{m} +F_{a}\tilde{a} +F_{Q}\tilde{Q} +F_{l}\tilde{l} +F_{mm}\tilde{m}^{2} +F_{ma}\tilde{m}\tilde{a} +F_{mQ}\tilde{m}\tilde{Q} +F_{ml}\tilde{m}\tilde{l}\\&\quad +F_{aa}\tilde{a}^{2} +F_{aQ}\tilde{a}\tilde{Q} +F_{al}\tilde{a}\tilde{l} +F_{QQ}\tilde{Q}^{2} +F_{Ql}\tilde{Q}\tilde{l} +F_{ll}\tilde{l}^{2}\\&\quad +F_{mmm}\tilde{m}^{3} +F_{mma}\tilde{m}^{2}\tilde{a} +F_{mmQ}\tilde{m}^{2}\tilde{Q} +F_{mml}\tilde{m}^{2}\tilde{l}\\&\quad +F_{maa}\tilde{m}\tilde{a}^{2} +F_{maQ}\tilde{m}\tilde{a}\tilde{Q} +F_{mal}\tilde{m}\tilde{a}\tilde{l} +F_{mQQ}\tilde{m}\tilde{Q}^{2} +F_{mQl}\tilde{m}\tilde{Q}\tilde{l} +F_{mll}\tilde{m}\tilde{l}^{2}\\&\quad +F_{aaa}\tilde{a}^{3} +F_{aaQ}\tilde{a}^{2}\tilde{Q} +F_{aal}\tilde{a}^{2}\tilde{l} +F_{aQQ}\tilde{a}\tilde{Q}^{2} +F_{aQl}\tilde{a}\tilde{Q}\tilde{l} +F_{all}\tilde{a}\tilde{l}^{2}\\&\quad +F_{QQQ}\tilde{Q}^{3} +F_{QQl}\tilde{Q}^{2}\tilde{l} +F_{Qll}\tilde{Q}\tilde{l}^{2} +F_{lll}\tilde{l}^{2}, \end{aligned}$$
(A.28)
with the first-order coefficients:
$$\begin{aligned} F_{m} = 4\tilde{\lambda }\csc ^{2} \theta_{{\rm o}} ,\ F_{l} = -4(1 -\csc ^{2} \theta_{{\rm o}}\tilde{\lambda }^{2} ) \mu ^{-1},\ F_{a} = F_{Q} = 0, \end{aligned}$$
(A.29)
the second-order coefficients:
$$\begin{aligned}&F_{mm} = 16\tilde{\lambda }\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}} \mu +\frac{15}{4} \pi \tilde{\lambda }\csc ^{2} \theta_{{\rm o}},\ F_{ma} = 4-8\tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}},\\&F_{ml} = 8\cot \theta_{{\rm o}}\csc \theta_{{\rm o}}( 4\tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}} -1 ) ,\ F_{al} = 8\tilde{\lambda }(1 -\csc ^{2} \theta_{{\rm o}}\tilde{\lambda }^{2} ) \mu ^{-1},\\&F_{QQ} = -\frac{3}{4}\pi \tilde{\lambda }\csc ^{2} \theta_{{\rm o}},\ F_{ll} = -8\tilde{\lambda }\big (\cot \theta_{{\rm o}}\csc \theta_{{\rm o}}( 2\tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}} -3)\tilde{\lambda }^{2} +\cos \theta_{{\rm o}}\big ) \mu ^{-3},\\&F_{mQ} = F_{aa} = F_{aQ} = F_{Ql} = 0, \end{aligned}$$
(A.30)
and the third-order coefficients:
$$\begin{aligned} F_{mmm} & = 30\pi \tilde{\lambda }\csc ^{3} \theta_{{\rm o}}\cot \theta_{{\rm o}} \mu -\frac{256}{3}\tilde{\lambda }^{3}\csc ^{6} \theta_{{\rm o}} +64\tilde{\lambda }( 2\tilde{\lambda }^{2} +\tilde{\eta })\csc ^{4}\theta_{{\rm o}}, \\ F_{mma} & = 5\pi -64\tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}} \mu -15\pi \tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}}, \\ F_{mml} & = 30\pi \tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}} -\frac{15}{2} \pi \cot \theta_{{\rm o}}\csc \theta_{{\rm o}} +16\big ( -16\tilde{\lambda }^{4}\csc ^{6} \theta_{{\rm o}} \\&\quad +4\tilde{\lambda }^{2}( 7\tilde{\lambda }^{2} +4\tilde{\eta })\csc ^{4} \theta_{{\rm o}} -( 13\tilde{\lambda }^{2} +2\tilde{\eta })\csc ^{2} \theta_{{\rm o}} +1\big ) \mu ^{-1},\\ F_{maa} & = -4\tilde{\lambda }\big ( (\tilde{\eta } -3\tilde{\lambda }^{2})\csc ^{2} \theta_{{\rm o}} +2\big ),\\ F_{mal} & = 16\tilde{\lambda }\cot \theta_{{\rm o}}\csc \theta_{{\rm o}}( 3-8\tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}}),\\ F_{mQQ} & = - 6\pi \tilde{\lambda }\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}} \mu -16\tilde{\lambda }\csc ^{2} \theta_{{\rm o}},\\ F_{mll} & = 8 \tilde{\lambda }\csc ^{6} \theta_{{\rm o}}( 12\cos 2\theta_{{\rm o}}\tilde{\lambda }^{2} +19\tilde{\lambda }^{2} -\tilde{\eta } +\cos 4\theta_{{\rm o}}),\\ F_{aal} & = \big ( -4\tilde{\lambda }^{2}(\tilde{\eta } -3\tilde{\lambda }^{2})\csc ^{2} \theta_{{\rm o}} -( 13\tilde{\lambda }^{2} -3\tilde{\eta } +\cos 2\theta_{{\rm o}})\big ) \mu ^{-1} ,\\ F_{aQQ} & = \frac{3}{2} \pi \tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}} -\frac{\pi }{2},\\ F_{all} & = 5\pi -15\pi \tilde{\lambda }^{2}\csc ^{2} \theta_{{\rm o}} -8( 8\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}}\tilde{\lambda }^{4} -6\cot \theta_{{\rm o}}\csc \theta_{{\rm o}}\tilde{\lambda }^{2} +\cos \theta_{{\rm o}}) \mu ^{-1},\\ F_{QQl} & = \frac{3}{2} \pi \cot \theta_{{\rm o}}\csc \theta_{{\rm o}} -6\pi \tilde{\lambda }^{2}\cot \theta_{{\rm o}}\csc ^{3} \theta_{{\rm o}}, \\ F_{lll} & = \frac{15}{4} \pi \csc ^{2} \theta_{{\rm o}}\tilde{\lambda } +\frac{15}{2} \pi (\cos \theta_{{\rm o}} +\sqrt{\tilde{\eta }})\csc ^{2} \theta_{{\rm o}}\tilde{\lambda }^{2} \mu ^{-2} \\ & \quad +\frac{64}{3}\csc ^{2} \theta_{{\rm o}}\big ( -4\tilde{\lambda }^{2}\csc ^{4} \theta_{{\rm o}} +3( 2\tilde{\lambda }^{2} +\tilde{\eta })\csc ^{2} \theta_{{\rm o}}-2\Big )\tilde{\lambda }^{4} \mu ^{-3} \\ &\quad +\frac{1}{3}\big ( 16\csc ^{2} \theta_{{\rm o}} ( -12\csc ^{2} \theta_{{\rm o}}\tilde{\lambda }^{2} +25\tilde{\lambda }^{2} +18\tilde{\eta })\tilde{\lambda }^{4}\\ &\quad +( -207\tilde{\lambda }^{4} -90\tilde{\eta }\tilde{\lambda }^{2} +5\tilde{\eta }^{2}) -4( 2\tilde{\eta } -\tilde{\lambda }^{2})\cos 2\theta_{{\rm o}} +3\cos 4\theta_{{\rm o}}\big ) \mu ^{-5},\\ F_{mmQ} & = F_{maQ} = F_{mQl} = F_{aaa} = F_{aaQ} = F_{aQl} = F_{QQQ} = 0. \end{aligned}$$
(A.31)
Therefore, we have deduced the expressions of \(\Delta \theta\) and \(\Delta \phi\) using the geodesic equations. By inserting these expressions back into (26) and (28), we find that \(X-x\) is a third order small quantity with respect to \(\tilde{m}\), \(\tilde{a}\), \(\tilde{Q}\) and \(\tilde{l}\). Thus by (30), \(X-x\) is equal to the Faraday rotation angle \(\chi\) up to the third order.
