Graduate Propagation Vector of Light in Kerr Spacetime: Reference Needed

[Haorong Wu]

TL;DR

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.

 

[Haorong Wu]

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 $k^t=E=1$?