A Propagation Vector of Light in Kerr Spacetime: Reference Needed

Haorong Wu
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The asymptotic behavior of a propagation vector is given in a paper. Need suggestions of reference to understand it.
Hi, there. I am currently reading the paper, Gravitational Faraday rotation induced by a Kerr black hole (https://doi.org/10.1103/PhysRevD.38.472). After Eq. (2.4), it reads that
From the equation of motion for a light ray, the asymptotic behavior of ##k_i## near the position of the source or of the observer is given by \begin{align}
k^t\rightarrow& 1,\\
k^r\rightarrow& k^r/|k^r| ,\\
k^\theta\rightarrow &\beta /r^2,\\
k^\phi \rightarrow &\lambda/(r^2\sin^2\theta),
\end{align}
where ##\beta=(\eta-\lambda^2\cot^2\theta +a^2\cos^\theta)^{1/2}k^\theta/|k^\theta|##, and ##\lambda## and ##\eta## are constants of motion.

The paper does not provide the derivation of the equations and no related reference is listed. Also, ##k^i## is not clearly defined in the paper, so I assume it takes the form as ##k^i=dx^i/d\tau## where ##\tau## is some affine parameter. But the concepts of the two constants of motion, ##\lambda## and ##\eta##, are also unfamiliar. I know there are four constants of motion in Kerr spacetime, i.e., the mass, the energy, the ##z## component of angular momentum, and the Carter constant. I could not find the definitions of the two constants of motion, ##\lambda## and ##\eta##, in the paper.

I would be grateful if anyone could share some insights or opinions.

Thanks ahead.
 
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I think I have solved it partially.

From the EOM of photons in Kerr spacetime, \begin{align}
\rho^2 k^r=&\pm \sqrt{R(r)},\\
\rho^2 k^\theta=&\pm \sqrt{\Theta(\theta)},\\
\rho^2 k^\phi=&-(aE-\frac{L_z}{\sin^2 \theta})+\frac a \Delta P(r),
\end{align} where at large ##r##, ##R(r)\rightarrow r^4##, ##\Theta(\theta)=\eta+a^2 \cos^2 \theta-\lambda^2 \cot^2 \theta##, ##P(r)\rightarrow Er^2##, and the ##k^\mu## is normalized such that ##k^t=E=1##, and ##\eta## is the Carter constant ##Q##, ##\lambda## is ##L_z##. The signs in the first two equations are defined as ##\pm 1=\frac {k^r}{\left | k^r \right |}=\frac {k^\theta}{\left | k^\theta \right |}##.

Then I can derive the equations in the paper.

The only left question is the normalization of ##k^\mu##. Is it a convention to normalize it by ##k^t=E=1##?
 
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