# Asymmetric Light Bending in the Equatorial Kerr Metric

Arthur B. Congdon  Savitri V. Iyer  Department of Physics and Astronomy, State
University of New York at Geneseo, 1 College Circle, Geneseo, New York 14454,
USA  Charles R. Keeton  Department of Physics and Astronomy, Rutgers, the
State University of New Jersey, 136 Frelinghuysen Road, Piscataway, New Jersey
08854, USA

###### Abstract

The observation of the bending of light by mass, now known as gravitational
lensing, was key in establishing general relativity as one of the pillars of
modern physics. In the past couple of decades, there has been increasing
interest in using gravitational lensing to test general relativity beyond the
weak deflection limit. Black holes and neutron stars produce the strong
gravitational fields needed for such tests. For a rotating compact object, the
distinction between prograde and retrograde photon trajectories becomes
important. In this paper, we explore subtleties that arise in interpreting the
bending angle in this context and address the origin of seemingly
contradictory results in the literature. We argue that analogies that cannot
be precisely quantified present a source of confusion.

##  I Introduction

One of the simplest formulas to emerge from general relativity (GR) is the
expression for the bending angle $\hat{\alpha}$ of a light ray that passes by
an object of mass $M$ at a minimum distance $r_{0}$:

$\hat{\alpha}=\frac{4GM}{c^{2}r_{0}}\equiv 2\,\frac{R_{S}}{r_{0}}\,,$ (1)

where $R_{S}=2GM/c^{2}$ is the object’s Schwarzschild radius. The
straightforward interpretation of Eq. (1) is that the amount of bending is
proportional to the mass of the deflector, and inversely proportional to the
distance of closest approach of the light ray to the deflector. The appearance
of $c$ and $G$ is not surprising since light and gravity are involved. Apart
from a factor of two, this result can be derived from Newton’s law of
universal gravitation, provided that we treat a light ray as a stream of
discrete particles. However, the Newtonian derivation relies on canceling out
the mass of the photon, which we know to be zero. Even so, one of the great
advantages of the Newtonian picture of gravity is that light bending can be
understood as just one more example of classical orbital mechanics. This
formulation has led to a rich literature in both astrophysics and mathematical
physics (see the books [1, 2, 3, 4, 5] and references therein).

Yet explaining gravitational lensing in purely classical terms misses the
essentially relativistic character of the phenomenon. Consider Eq. (1), for
example. In order for a Newtonian description to be physically meaningful, we
would expect, at the very least, that the relativistic expression for the
bending angle would reduce to Eq. (1) in some appropriate limit, say,
$r_{0}\gg R_{S}$. Instead, we find a weak-field bending angle that is twice
the Newtonian prediction. Explaining this discrepancy requires us to write
down the exact relativistic bending angle, and expand the result in a Taylor
series. The first-order term in $R_{S}/r_{0}$ yields Eq. (1).

A fully relativistic treatment of light bending shows that, in addition to the
two lensed images that arise from the Newtonian bending angle, there is an
infinite sequence of fainter images (see the review by Bozza [6] and
references therein, especially the original discovery by Darwin [7]). If the
deflector is compact enough to have a shadow border, which can be thought of
as an event horizon for light, a photon that passes close to this region
without falling into it can execute any number of orbits before continuing on
to the observer. For a rotating compact object such as a black hole or ultra-
dense neutron star, an asymmetry develops between prograde photons, which co-
rotate with the deflector, and retrograde photons, which counter-rotate. The
upshot for lensing is that the bending angle for prograde and retrograde
photons is not the same. Both exact and approximate expressions for the
bending angle in the Kerr metric have appeared in the literature, but
questions of interpretation remain. After summarizing what is known about the
bending angle in the Kerr metric, we identify and seek to resolve a subtle but
important discrepancy that has appeared in the literature.

##  II Light Bending in the Equatorial Kerr Metric

Figure 1: Lensing geometry for prograde motion in the equatorial plane of the
Kerr metric. Arrows indicate the direction of propagation of the lensed photon
and sense of rotation of the deflector. Retrograde motion is obtained by
reflecting the photon’s trajectory about the horizontal axis ($b\to-b$).
Adapted from Fig. 3.1 of [5].

The spacetime around a rotating compact object is described by the Kerr metric
[8], which is typically written in Boyer-Lindquist [9] coordinates
($t,r,\theta,\phi$), where $\theta$ is measured with respect to the axis of
rotation. For simplicity, we work in the equatorial plane ($\theta=\pi/2$),
where the line element takes the form

$ds^{2}=g_{tt}\,c^{2}\,dt^{2}+g_{rr}\,dr^{2}\ +\
g_{\phi\phi}\,d\phi^{2}+2g_{t\phi}\,c\,dt\,d\phi.$ (2)

The metric coefficients are given by

$\displaystyle g_{tt}(r)=-\left(1-\frac{2m}{r}\right)$ (3a) $\displaystyle
g_{rr}(r)=\left(1-\frac{2m}{r}+\frac{a^{2}}{r^{2}}\right)^{-1}$ (3b)
$\displaystyle g_{\phi\phi}(r)=r^{2}+a^{2}+\frac{2ma^{2}}{r}$ (3c)
$\displaystyle g_{t\phi}(r)=-\frac{2ma}{r}$ (3d)

where $m\equiv GM/c^{2}$ is the mass scaled by constants to have dimensions of
length. The spin parameter, which has the dimension of length, is defined by
$a\equiv j/c$, where $j$ is the angular momentum per unit mass of the compact
object. Without loss of generality, we take $j\geq 0$. It is convenient to
define the dimensionless spin parameter $\hat{a}\equiv a/m$, which is
restricted to the interval $0\leq\hat{a}\leq 1$. The Schwarzschild limit
corresponds to $\hat{a}=0$, while maximal rotation corresponds to $\hat{a}=1$.
The case $\hat{a}>1$, where an event horizon with some $r>0$ gives way to a
naked singularity at $r=0$, is not considered in this Note.

###  II.1 Equations of Motion

The equations of motion for a particle subject to a given metric $ds$ follow
from the Lagrangian $L=(ds/d\lambda)^{2}/2$, where $\lambda$ is an affine
parameter. For a massive particle, we can identify $\lambda$ as the proper
time. For a massless particle such as a photon, where the proper time
vanishes, we can define $\lambda$ to be any parameter that specifies the
spacetime coordinates everywhere along the trajectory.

The Lagrangian for a particle in the equatorial plane of the Kerr metric is

$L(r,\dot{t},\dot{r},\dot{\phi})=\frac{1}{2}\left(c^{2}g_{tt}\dot{t}^{2}+g_{rr}\dot{r}^{2}+g_{\phi\phi}\dot{\phi}^{2}+2g_{t\phi}c\dot{t}\dot{\phi}\right)\,,$
(4)

where an overdot denotes differentiation with respect to $\lambda$. We can
identify two constants of motion:

$\displaystyle\varepsilon$ $\displaystyle\equiv-\frac{\partial
L}{\partial\dot{t}}=-c^{2}g_{tt}\dot{t}-cg_{t\phi}\dot{\phi}$ (5a)
$\displaystyle\ell$ $\displaystyle\equiv\frac{\partial
L}{\partial\dot{\phi}}=cg_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}\,\,.$ (5b)

For a massive particle, $\varepsilon$ and $\ell$ reduce to the energy per unit
mass and angular momentum per unit mass, respectively, when $r\gg m$. Since
these quantities do not depend on the mass of the particle, they can be
straightforwardly extended to include the massless case. It follows from our
sign convention for $j$ that $\ell>0$ for a prograde trajectory and $\ell<0$
for a retrograde trajectory.

Solving for $\dot{t}$ and $\dot{\phi}$ in terms of $\varepsilon$ and $\ell$
yields

$\dot{t}=\frac{1}{c^{2}}\left(\frac{c\ell g_{t\phi}+\varepsilon
g_{\phi\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\right)=\frac{\varepsilon}{c^{2}}\frac{\left[r^{2}+a^{2}\left(1+\frac{2m}{r}\right)-\frac{2ma}{r}\frac{c\ell}{\varepsilon}\right]}{\left(r^{2}-2mr+a^{2}\right)}$
(6)

and

$\dot{\phi}=-\,\frac{1}{c}\left(\frac{c\ell g_{tt}+\varepsilon
g_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\right)=\ell\frac{\left(1-\frac{2m}{r}+\frac{2ma}{r}\frac{\varepsilon}{c\ell}\right)}{\left(r^{2}-2mr+a^{2}\right)}\,.$
(7)

Substituting these expressions into Eq. (4), and setting $L=0$ along a null
geodesic, we obtain

$\dot{r}^{2}=\frac{\varepsilon^{2}}{c^{2}}+\frac{1}{r^{2}}\left(\frac{\varepsilon^{2}a^{2}}{c^{2}}-\ell^{2}\right)+\frac{2m}{r^{3}}\left(\frac{\varepsilon^{2}a^{2}}{c^{2}}-\frac{2a\varepsilon\ell}{c}+\ell^{2}\right)\,.$
(8)

###  II.2 Bending Angle

In terms of the change in azimuthal angle $\phi$ as the light ray travels from
the source to the observer, the bending angle is given by

$\displaystyle\hat{\alpha}$
$\displaystyle=-\pi+\left(\int_{\rm{r_{src}}}^{r_{0}}+\int_{r_{0}}^{{\rm
r_{obs}}}\right)\frac{d\phi}{dr}\,dr$
$\displaystyle=-\pi+\left(-\int_{\rm{r_{src}}}^{r_{0}}+\int_{r_{0}}^{{\rm
r_{obs}}}\right)\left|{\frac{d\phi}{dr}}\right|dr\,.$ (9)

The signs of the integrals correspond to a photon traveling inward from the
source at distance $r_{\rm{}_{src}}$ to the distance of closest approach
$r_{0}$, and then outward from $r_{0}$ to the observer at $r_{\rm{}_{obs}}$.
Since the observer and source are assumed to be far from the lens, we let
$r_{\rm{}_{src}},r_{\rm{}_{obs}}\to\infty$. (See Fig. 1.) This leads to

$\hat{\alpha}=-\pi+2\int_{r_{0}}^{\infty}\left|{\frac{d\phi}{dr}}\right|\,dr=-\pi+2\int_{r_{0}}^{\infty}\left|{\frac{\dot{\phi}}{\dot{r}}}\right|\,dr\,.$
(10)

To obtain the bending angle in the Kerr metric, we rewrite Eqs. (7) and (8) as

$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{\ell}{br}\left[\frac{br-2m(b-a)}{r^{2}-2mr+a^{2}}\right]$
(11a) $\displaystyle\dot{r}^{2}$ $\displaystyle=$
$\displaystyle\frac{\ell^{2}}{b^{2}r^{3}}P(r)\,,$ (11b)

where

$P(r)\equiv r^{3}-(b^{2}-a^{2})r+2m(b-a)^{2}\,,$ (12)

and $b\equiv c\ell/\varepsilon$ is the impact parameter. Since a lensed photon
follows an unbound orbit, its energy is positive ($\varepsilon>0$). Thus,
$b>0$ for a prograde trajectory ($\ell>0$), and $b<0$ for a retrograde
trajectory ($\ell<0$). Substituting Eqs. (11) into Eq. (10) yields the Kerr
bending angle,

$\hat{\alpha}=-\pi+2\int_{r_{0}}^{\infty}\sqrt{\frac{r}{P(r)}}\,\frac{br-2m(b-a)}{r^{2}-2mr+a^{2}}dr\,.$
(13)

The integral above can be expressed in terms of elliptic integrals of the
third kind [10]. In the Schwarzschild case ($a=0$), these reduce to elliptic
integrals of the first kind [7, 11].

We can express $r_{0}$ in terms of $a$, $b$, and $m$ by noting that
$\dot{r}=0$ when $r=r_{0}$. At such a turning point, $P(r)$ in Eq. (11b)
vanishes for $r\neq 0$. Depending on the value of $b$, the equation $P(r)=0$
has either two or zero solutions with $r>0$. First, consider the case of two
solutions, denoted by $r_{1}$ and $r_{2}$ ($r_{1}<r_{2}$). As long as $r_{1}$
and $r_{2}$ lie outside the event horizon at $r_{H}\equiv
m+\sqrt{m^{2}-a^{2}}$, we interpret $r_{1}$ as the maximum distance of an
outward-moving photon, and $r_{2}$ as the minimum distance of an inward-moving
photon. In other words, $r_{0}\equiv r_{2}$. Since $P(r)<0$ for
$r_{1}<r<r_{2}$, $\dot{r}$ is imaginary in this region. Thus motion between
$r_{1}$ and $r_{2}$ is not possible. If $P(r)=0$ has no solutions for
$r>r_{H}$, an incoming photon will fall inside the event horizon.

The critical impact parameter $b_{c}$ separates the cases when $P(r)$ admits
two ($\left|{b}\right|>\left|{b_{c}}\right|$) or zero
($\left|{b}\right|<\left|{b_{c}}\right|$) turning points. As
$\left|{b}\right|\to\left|{b_{c}}\right|$ from above, the two positive zeros
of $P(r)$ approach a common value $r_{c}$. Thus $P^{\prime}(r_{c})=0$ when
$b=b_{c}$. Solving for $r_{c}$ yields

$r_{c}=\sqrt{\frac{b_{c}^{2}-a^{2}}{3}}\,.$ (14)

To understand what $r_{c}$ means physically, we differentiate Eq. (11b), and
solve for $\ddot{r}$:

$\ddot{r}=\frac{\ell^{2}}{2b^{2}}\left[-\frac{3}{r^{4}}P(r)+\frac{1}{r^{3}}P^{\prime}(r)\right]\,.$
(15)

Since $P(r_{c})=P^{\prime}(r_{c})=0$, we conclude that $\ddot{r}=0$ when
$r=r_{c}$. Thus a photon with $b=b_{c}$ approaches a circular orbit at
$r_{c}$. The circle of radius $r_{c}$ is known as the shadow border because an
observer at $r>r_{c}$ cannot detect a photon emitted from any $r\leq r_{c}$.
If we allow for photon trajectories that are not restricted to the equatorial
plane, the shadow border becomes a two-dimensional surface.

Figure 2: Paths of prograde (upper curve) and retrograde (lower curve) photons
for a deflector with spin parameter $\hat{a}=0.9$. The rays start out far to
the right of the figure with the same unsigned impact parameter $|b|=7.5m$.
The axes are labeled in units of $m$.

To solve for $b_{c}$ in terms of $a$ and $m$, we substitute Eq. (14) into Eq.
(12), and set $P(r_{c})=0$. This leads to the cubic equation

$(b_{c}+a)^{3}=27m^{2}(b_{c}-a)\,,$ (16)

which has a doubly degenerate solution with $b_{c}>0$, and a single solution
with $b_{c}<0$. The respective critical impact parameters for prograde and
retrograde motion are denoted by $b^{+}_{c}$ and $b^{-}_{c}$, and are given by

$b^{\pm}_{c}=-a\pm
6m\cos{\left[\frac{1}{3}\cos^{-1}{\left(\mp\frac{a}{m}\right)}\right]}\,.$
(17)

In the non-rotating case ($a=0$), we recover $b_{c}=3m$. In the case of
maximal rotation ($a=m$), $b_{c}^{+}=2m$ and $b_{c}^{-}=-7m$.

##  III Asymmetry between Prograde and Retrograde Motion

It is not obvious from Eq. (13), or from the representation of the bending
angle in terms of elliptic integrals, that prograde and retrograde photons are
deflected by different amounts. This asymmetry becomes apparent when we
consider the bending angle in two asymptotic cases: the weak deflection limit
(WDL) and the strong deflection limit (SDL). In the WDL, $\left|{b}\right|\gg
m$, so that the bending angle can be written as a power series in
$m/\left|{b}\right|$. Starting respectively from the elliptic integral
representation of the bending angle and from Eq. (13), Iyer and Hansen [12]
and Aazami  _et al._ [13] find

$\begin{split}\hat{\alpha}_{\pm}&=4\left(\frac{m}{|b|}\right)+\left(\frac{15\pi}{4}\mp
4\hat{a}\right)\left(\frac{m}{|b|}\right)^{2}\\\
&\quad\quad+\left(\frac{128}{3}\mp
10\pi\hat{a}+4\hat{a}^{2}\right)\left(\frac{m}{|b|}\right)^{3}\\\
&\quad\quad\quad+\left(\frac{3465\pi}{64}\mp
192\hat{a}+\frac{285\pi}{16}\hat{a}^{2}\mp
4\hat{a}^{3}\right)\left(\frac{m}{|b|}\right)^{4}\\\
&\quad\quad\quad\quad+O\left[\left(\frac{m}{|b|}\right)^{5}\right]\,\end{split}$
(18)

in the equatorial plane of the Kerr metric for a fixed value of $\hat{a}$.
Apart from the first-order term in $m/\left|{b}\right|$, which is independent
of spin, all other terms increase (decrease) the bending angle of a retrograde
(prograde) photon relative to the non-rotational case.

In the SDL, where $\left|{b}\right|\sim m$, a power series representation of
the bending angle would require one to identify a different expansion
parameter. However, this would prove to be an exercise in futility, since the
bending angle diverges logarithmically [14] as
$\left|{b}\right|\to\left|{b_{c}}\right|\sim m$. This same behavior is seen
for the Schwarzschild bending angle [7]. Thus, we examine the bending angle
numerically in this case. Figure 2 shows the trajectories of two photons with
the same value of $\left|{b}\right|$, but on opposite sides of the horizontal
axis. Notice that the bending occurs mostly within a few gravitational radii
of the deflector. To understand why the retrograde photon is bent more than
its prograde counterpart, there is an additional asymmetry in Kerr lensing
that warrants discussion. The critical radius for photon capture takes on the
single value $r_{c}=3m$ in the Schwarzschild metric, but has two different
values, $r_{c}^{+}$ (prograde) and $r_{c}^{-}$ (retrograde) in the Kerr
metric, which depend on the mass and spin of the deflector. They approach the
Schwarzschild value $r_{c}=3m$ as $\hat{a}\to 0$. At maximal rotation
($\hat{a}=1$), $r_{c}^{+}=m$ and $r_{c}^{-}=4m$. Since $r_{c}^{+}<r_{c}^{-}$,
a prograde photon can get closer to the deflector than a retrograde photon
without being captured. A less intuitive but more useful equivalent statement
is that, for a fixed value of $\left|{b}\right|$, a retrograde photon gets
closer to its critical radius than a prograde photon gets to its critical
radius. Thus, the bending angle of the retrograde photon is the larger of the
two.

For the sake of plotting the bending angle in the Kerr and Schwarzschild
metrics for both prograde and retrograde motion over the full range
$\left|{b_{c}^{\pm}}\right|<\left|{b}\right|<\infty$, Iyer and Hansen [10]
introduce the parameter $b^{\prime}\equiv 1-\left|{b_{c}^{\pm}/b}\right|$,
which ranges from zero in the extreme SDL to unity in the extreme WDL. They
plot the bending angle against $b^{\prime}$ in their Figs. 4 and 5, finding
greater deflection for prograde photons than for retrograde photons with the
same value of $b^{\prime}$. Yet this conclusion runs counter to our discussion
above. This is a direct consequence of the dependence of $b^{\prime}$ on
$b_{c}^{\pm}$, which has distinct values for prograde and retrograde motion.
In other words, prograde and retrograde photons with the same value of
$b^{\prime}$ have different values of $b$, and vice-versa. The most natural
way around this difficulty is to plot the bending angle against the physical
impact parameter separately for the WDL and SDL (see Fig. 3). In both limits,
we see that the magnitude of the bending angle is indeed larger for retrograde
photons than prograde photons, with the Schwarzschild value lying somewhere in
between.

It is tempting to interpret the bending angle in the prograde and retrograde
cases by means of analogy with a rotating fluid. A retrograde photon, which
moves “upstream,” has to overcome the “gravitational current” induced by the
deflector on its way to the observer, while a prograde photon, which moves
“downstream,” is swept along with the current. This results in a larger
bending angle for a retrograde photon than for a prograde photon.
Unfortunately, the same analogy can lead to the opposite conclusion [10].
Aazami  _et al._ [13], who find a larger bending angle for retrograde photons,
use their analysis of time delays to argue that retrograde photons “spend more
time” in the gravitational field of the deflector than prograde photons. The
apparent tension between Iyer and Hansen [10] and Aazami  _et al._ [13]
disappears when the mathematical results of the two papers are compared,
without reference to analogies or figures. Thus, analogies that are intended
to explain physical results can become an accidental exercise in confirmation
bias instead. Finding intuitive explanations for results in a field as non-
intuitive as GR is a particularly risky undertaking.

Figure 3:  Exact bending angle $\hat{\alpha}$ as a function of
$\left|{b}\right|/m$ for different values of the spin parameter $\hat{a}$. For
Kerr cases, prograde rays are shown as solid lines and retrograde rays are
shown as dashed lines. The left panel shows the strong deflection limit; the
vertical dotted lines indicate the critical impact parameter
$b_{c}/m=3\sqrt{3}$ for the Schwarzschild case (black) as well as the limits
$b_{c}^{+}/m=2$ and $\left|{b_{c}^{-}}\right|/m=7$ that correspond to prograde
and retrograde photons, respectively, for $\hat{a}=1$ (gray). The right panel
shows the weak deflection limit. (The colors and line styles are the same in
both panels.)

## Acknowledgments

ABC would like to thank Ted Burkhardt, Jerod Caligiuri, Carl Droms, Tim Jones,
Allan Moser, and Erik Nordgren for their helpful comments on the manuscript.
SVI gratefully acknowledges Robert Sinesi, who prepared Figure 2 and was
supported by the NASA NY State Space Grant.

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