# Optical Polarization Holonomy in the Kerr Metric

Mark T. Lusk<EMAIL_ADDRESS>Department of Physics, Colorado School of Mines,
Golden, CO 80401, USA

###### Abstract

Polarization holonomy is analytically determined for a class of closed,
spherical trajectories of light transiting a black hole in the Kerr metric.
The leading order geometric optics approximation admits a closed-form
expression of such paths, and sets of source/receiver locations are quantified
for a spectrum of black hole angular momenta. A conserved, conformal Yano-
Killing scalar is then exploited to determine the evolving polarization.
Polarization holonomy, the angle between outgoing and incoming polarizations,
is quantified for the spectrum of admissible direct and retrograde
trajectories.

##  I Introduction

The fabric of modern physics is woven from holonomies that characterize the
structure of spacetime and parameter manifolds[1, 2, 3]. These might be as
conceptually immediate as the angle between initial and final orientations of
a vector field parallel transported around a loop on a curved surface[4] or
the geometric phase of a closed circuit on the Poincaré sphere[5]. However,
they also manifest as Wilson loops for the non-Abelian gauge fields of the
Standard Model[6], underpin Loop Quantum Gravity[7], and provide the requisite
structure for Topological Quantum Computing[8].

Within the setting of stationary spacetimes, the polarization of light
exhibits a helicity dependent evolution that results in the rotation of
linearly polarized planewaves. Such Gravitational Faraday Rotation has been
considered for decades[9, 10, 11, 12], and care is taken to construct a co-
evolving reference frame from which to measure the polarization between two
observers[11, 13]. The consideration of closed trajectories would make this
unnecessary, though, since the polarization can be self-referenced. In
particular, a holonomy can be quantified by comparing the outgoing and
incoming polarizations at the common 3-space position of source and receiver.
Circuits of this sort can be engineered from self-intersecting trajectories.

Recent work has made it straightforward to construct several morphologically
distinct classes of trajectories using Jacobi elliptic functions[14, 15]. One
category is associated with non-trivial geodesics of constant radius, referred
to as spherical trajectories. These self-intersecting paths can be cropped so
that they start and end at the same point in 3-space, where the wave vector
may be discontinuous. The initial and final polarizations may also be
misaligned at such locations.

For these closed circuits, the Walker-Penrose Theorem[16, 17] can be used to
analytically quantify polarization holonomy as a function of source position
and black hole angular momentum. This allows holonomy to be quantified over
the entire range of admissible direct and retrograde transits. The analytical
approach and focus on holonomy distinguishes this work from that of earlier
studies of light emanating from accretion disks[18] and the exchange of
quantum information between a pair of observers near a Kerr black hole[13].

##  II Approach

Because it can be exasperating to reconcile disparate notations, a brief
review of the geometric optics approximation is provided along with a summary
of the key scalars conserved in the Kerr metric due to isometries and hidden
symmetries. Geometrized units are adopted to convert all physical quantities
to length equivalents using black hole mass, M, speed of light, c, and
gravitational constant, G[10]. All such quantitaties are then non-
dimensionalized using the mass length equivalent, $MG/c^{2}$. With black hole
(Komar) angular momentum $J$, the singularity rotation rate is thus
characterized by the non-dimensional parameter, $a=Jc/(M^{2}G)$[19], which has
a magnitude less than one in non-extremal settings[20].

###  II.1 Propagation of Light and Polarization within a Geometric Optics
Approximation

The source-free Maxwell equation with metric $g_{\mu\nu}$ is [10]

$\nabla_{\nu}F_{\mu\nu}=0.$ (1)

Here $\nabla_{\nu}$ represents the components of the covariant gradient, while
$F_{\mu\nu}$ is the skew-symmetric electromagnetic tensor, constrained by the
Bianchi Identity[21] to be representable in terms of magnetic vector potential
$A_{\nu}$:

$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}.$ (2)

Applying the Lorenz gauge condition,

$\nabla\cdot{\bf A}=0,$ (3)

and noting that the Ricci tensor, $R^{\mu}_{\,\,\nu}$, can be introduced
using[10]

$\nabla^{\nu}\nabla_{\nu}A^{\mu}=\nabla^{\mu}\nabla_{\nu}A^{\nu}+R^{\mu}_{\,\,\nu}A^{\nu},$
(4)

we arrive at the de Rham form of the Maxwell equation:

$-\nabla^{\nu}\nabla_{\nu}A^{\mu}+R^{\mu}_{\,\,\nu}A^{\nu}=0.$ (5)

This representation lends itself to an Eikonal expansion of the vector
potential, $\bf A$, in terms of the perturbation parameter
$\epsilon:=\lambda/L$, the ratio of wavelength $\lambda$ and characteristic
length $L$:

$A_{\mu}={\rm Re}\left[(a_{\mu}+\epsilon
b_{\mu}+\epsilon^{2}c_{\mu}+...)e^{iS/\epsilon}\right].$ (6)

The wave vector is defined as the phase gradient ${\bf p}:=\nabla S$.
Coefficients $a_{\mu}$, $b_{\mu}$, etc. are functions of position, as is the
phase accumulation $S$. Substitution of this expansion into Eq. 5, and
separation by powers of $\epsilon$, gives that the highest-order relation is
independent of the de Rham tensor and is simply a statement that the wave
vector has null character:

${\bf p}\cdot{\bf p}=0.$ (7)

Of course, this is equivalent to saying that the wave vector is parallel
transported,

$p^{\nu}\nabla_{\nu}p_{\mu}=0,$ (8)

the geodesic equation that governs the trajectory of light within the lowest-
order geometric optics approximation[10].

Application of the Eikonal expansion, Eq. 6, to the Lorenz condition of Eq. 3,
implies that the polarization of light, ${\bf f}:={\bf a}/|{\bf a}|$, is
orthogonal to the trajectory at leading order:

${\bf f}\cdot{\bf p}=0.$ (9)

Polarization is therefore parallel transported as well:

$p^{\nu}\nabla_{\nu}f_{\mu}=0.$ (10)

The transport equations for wave vector and polarization each constitute a set
of coupled, first-order ordinary differential equations (ODEs) that can be
solved numerically. While that is certainly one approach to determining a
given trajectory and associated polarization, there is a better way to
proceed.

The evolution equation for trajectory can be re-expressed, using three
conserved scalars, as four decoupled, first-order ODEs for spacetime position.
These, in turn, can be solved in terms of Jacobi elliptic functions. The
polarization along such paths can then be determined in closed form as well,
using a _hidden symmetry_ associated with a conformal Killing-Yano tensor.
Once the Killing symmetries are explained, this is carried out in Kerr
spacetime to engineer closed geodesics that exhibit polarization holonomy.

###  II.2 Killing Fields

####  II.2.1 Isometries

The strategy outlined above is made possible through the identification of
operations for which the associated spacetime metric is invariant—i.e.
isometries. This is equivalent to saying that the Lie derivatives of the
metric are zero with respect to Killing vectors, $\bf v$, or Killing tensors,
$\hat{K}$[22, 16, 23, 19, 24]:

${\cal L}_{v}g_{\mu\nu}=0,\quad{\cal L}_{K}g_{\mu\nu}=0.$ (11)

A more operationally useful form of these is

$\nabla_{(\mu}v_{\nu)}=0,\quad\nabla_{(\mu}K_{\nu\gamma)}=0,$ (12)

where parentheses indicate the symmetry operator. Because the wave vector is
also parallel transported, there is a conserved scalar, an integral of motion,
associated with each such equation.

In this work, attention is restricted to spacetime with a Kerr metric,
expressed using oblate spheroidal Boyer-Lindquist coordinates[25]
$\\{t,r,\theta,\phi\\}$ with covariant components of

$[g]=\left(\begin{array}[]{cccc}\frac{2r}{\Sigma}-1&0&0&-\frac{2ar\mathbb{s}^{2}}{\Sigma}\\\
0&\frac{\Sigma}{\Delta}&0&0\\\ 0&0&\Sigma&0\\\
-\frac{2ar\mathbb{s}^{2}}{\Sigma}&0&0&\frac{\mathbb{s}^{2}}{\Sigma}\left((a^{2}+r^{2})^{2}-\mathbb{s}^{2}\Delta
a^{2}\right)\\\ \end{array}\right).$ (13)

Here $\mathbb{c}:=\cos\theta,\quad\mathbb{s}:=\sin\theta$, while
$\Sigma:=r^{2}+a^{2}\mathbb{c}^{2}$ and $\Delta:=a^{2}-2r+r^{2}$ are standard
in this setting.

Since neither time, $t$, nor azimuthal angle, $\phi$, appear in the Kerr
metric, it has two Killing vectors, ${\bf v}_{t}$ and ${\bf v}_{\phi}$, with
Boyer-Lindquist coordinates of $\\{1,0,0,0\\}$ and $\\{0,0,0,1\\}$,
respectively. The associated conserved scalars are the spacetime counterparts
to conserved scalars for energy and angular momentum:

$\epsilon:=-{\bf v}_{t}\cdot{\bf p},\quad\ell:={\bf v}_{\phi}\cdot{\bf p}.$
(14)

####  II.2.2 Hidden Symmetries

There are also symmetries not associated with configurational isometries but,
rather, symmetry operations in the dynamical state space. These are referred
to as _hidden symmetries_ , and the associated operations are manifested in a
generalization of Eqs. 12, the conformal Killing-Yano equation[26]:

$\nabla_{\mu}H_{\nu\lambda}=\frac{1}{3}g_{\mu\nu}\nabla^{\gamma}H_{\gamma\lambda}-\frac{1}{3}g_{\mu\lambda}\nabla^{\gamma}H_{\gamma\nu}.$
(15)

The skew-symmetric 2-form, $\hat{H}$, is the _Principal Tensor_[26], and its
Hodge dual, $\hat{F}:={}^{*}\\!\hat{H}$, is also a conformal Killing-Yano
tensor. These fields are considered more fundamental than Killing tensors
since their square always generates a symmetric Killing tensor,

$K_{\mu\nu}=F_{\mu\gamma}F_{\nu\alpha}g^{\alpha\gamma},$ (16)

while the reverse is not necessarily true.

In Boyer-Lindquist coordinates, the covariant components are:

$[H]=\left(\begin{array}[]{cccc}0&r&a^{2}\mathbb{c}\mathbb{s}&0\\\
-r&0&0&ar\mathbb{s}^{2}\\\
-a^{2}\mathbb{c}\mathbb{s}&0&0&a\mathbb{c}\mathbb{s}\left(a^{2}+r^{2}\right)\\\
0&-ar\mathbb{s}^{2}&-a\mathbb{c}\mathbb{s}\left(a^{2}+r^{2}\right)&0\\\
\end{array}\right)$ (17)

and

$[F]=\left(\begin{array}[]{cccc}0&-a\mathbb{c}&ar\mathbb{s}&0\\\
a\mathbb{c}&0&0&-a^{2}\mathbb{c}\mathbb{s}^{2}\\\
-ar\mathbb{s}&0&0&r\mathbb{s}\left(a^{2}+r^{2}\right)\\\
0&a^{2}\mathbb{c}\mathbb{s}^{2}&-r\mathbb{s}\left(a^{2}+r^{2}\right)&0\\\
\end{array}\right).$ (18)

Eqs 16 and 18 allow the Killing tensor components to be constructed as well:

$[K]=\left(\begin{array}[]{cccc}a^{2}\left(1-\frac{2\mathbb{c}^{2}r}{\Sigma}\right)&0&0&K_{t\phi}\\\
0&-\frac{a^{2}\mathbb{c}^{2}\Sigma}{\Delta}&0&0\\\ 0&0&r^{2}\Sigma&0\\\
K_{t\phi}&0&0&K_{\phi\phi}\\\ \end{array}\right),$ (19)

where

$\displaystyle
K_{t\phi}=-\frac{a\mathbb{s}^{2}\left(a^{2}\mathbb{c}^{2}\Delta+r^{2}\left(a^{2}+r^{2}\right)\right)}{\Sigma}$
(20) $\displaystyle
K_{\phi\phi}=\frac{\mathbb{s}^{2}\left(a^{4}\Delta\mathbb{c}^{2}\mathbb{s}^{2}+r^{2}\left(a^{2}+r^{2}\right)^{2}\right)}{\Sigma}.$
(21)

Finally, it is useful to construct a third conformal Killing-Yano 2-form,
$\hat{Z}:=\hat{H}+\imath\hat{F}$, with a matrix representation obtained
immediately from Eqs. 17 and 18.

For any parallel propagated vector $\bf q$, tensors $\hat{H}$ and $\hat{F}$
exhibit scalar conservations of

$\bf q\cdot\hat{H}\cdot{\bf p}=0,\quad\bf q\cdot\hat{F}\cdot{\bf p}=0.$ (22)

More useful, though, is the conserved quantity, $\mathbb{k}$, associated with
polarizations $\bf f$ and the Killing-Yano tensor $\hat{Z}$:

$\mathbb{k}:={\bf f}\cdot\hat{Z}\cdot{\bf p}.$ (23)

This is the complex scalar of the Walker-Penrose Theorem[16, 17]. Its
magnitude is related to Carter’s constant, $Q$,[22, 23] by

$Q=|\mathbb{k}|-(\ell-a\varepsilon)^{2}.$ (24)

For the Kerr metric in the Boyer-Linquist chart, $\mathbb{k}$ can be
represented as[18, 23]:

$\mathbb{k}=r\alpha-a\beta\mathbb{k}-\imath(r\beta+a\alpha\mathbb{c}),$ (25)

where

$\displaystyle\alpha:=f^{1}p^{0}+a(f^{3}p^{1}-f^{1}p^{3})\mathbb{s}^{2}$ (26)
$\displaystyle\beta:=-af^{2}p^{0}\mathbb{s}+(-f^{3}p^{2}+f^{2}p^{3})(a^{2}+r^{2})\mathbb{s}.$
(27)

###  II.3 Trajectories of Light

The most familiar application of Killing fields is associated with the
construction of trajectories by identifying the wave vector as the derivative
of position: ${\bf p}=d{\bf x}/dt$. The Killing tensor, $\hat{K}$, satisfies
Eq. 122, and the conserved scalar is the magnitude of $\mathbb{k}$–i.e.

$|\mathbb{k}|={\bf p}\cdot\hat{K}\cdot{\bf p}.$ (28)

In Boyer-Lindquist coordinates, this is a first-order ODE in affine parameter
$\tau$, for $dr/d\tau\equiv p^{1}$:

$|\mathbb{k}|=\frac{\left(\varepsilon\left(a^{2}+r^{2}\right)-a\ell\right)^{2}}{\Delta}-\frac{(p^{1})^{2}\Sigma^{2}}{\Delta}.$
(29)

Likewise, the Boyer-Lindquist expression for Eq. 7 amounts to a first-order
ODE in terms of both $p^{1}$ and $d\theta/d\tau\equiv p^{2}$:

$\displaystyle 4\varepsilon^{2}r(a^{2}+r^{2})$
$\displaystyle+2a^{2}\ell^{2}-8a\varepsilon
r\ell+2\Delta\varepsilon^{2}\Sigma$ $\displaystyle=\,\ell^{2}(a^{2}+\Delta)$
$\displaystyle\csc^{2}\theta+2\Sigma^{2}((p^{1})^{2}+(p^{2})^{2}\Delta).$ (30)

Two additional ODEs are supplied by the integrals of motion of Eq. 14 in terms
of $dt/d\tau\equiv p^{0}$ and $d\phi/d\tau\equiv p^{3}$:

$\displaystyle\varepsilon$
$\displaystyle=\frac{2ap^{3}r\mathbb{s}^{2}}{\Sigma}-p^{0}\left(\frac{2r}{\Sigma}-1\right)$
$\displaystyle\ell$
$\displaystyle=\frac{\mathbb{s}^{2}\left(p^{3}\left(\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta\mathbb{s}^{2}\right)-2ap^{0}r\right)}{\Sigma}.$
(31)

Eqs. 29, II.3, and II.3a,b, comprise four first-order ordinary differential
equations for the four trajectory components[27]. While they can be written as
separate ODEs for each position coordinate, the equations are still coupled
because $\Sigma=r^{2}+a^{2}\mathbb{c}$. This can be removed with a simple re-
scaling of parametrization from $\tau$ to the Mino parameter, $s$[28]:

$\frac{dx^{\mu}}{ds}:=\frac{\Sigma}{\epsilon}p^{\mu}\equiv\frac{\Sigma}{\epsilon}\frac{dx^{\mu}}{d\tau}.$
(32)

It is also useful to observe that there are only two independent scalars in
these equations, impact parameter $\lambda:=\ell/\varepsilon$ and Carter ratio
$\eta:=Q/\varepsilon^{2}$. The resulting equations of motion are

$\displaystyle\left(\frac{dr}{ds}\right)^{2}$ $\displaystyle={\mathcal{R}}(r)$
(33) $\displaystyle\left(\frac{d\theta}{ds}\right)^{2}$
$\displaystyle=\Theta(\theta)$ (34) $\displaystyle\frac{d\phi}{ds}$
$\displaystyle=\frac{a}{\Delta}(r^{2}+a^{2}-a\lambda)+\frac{\lambda}{\mathbb{s}^{2}}-a$
(35) $\displaystyle\frac{dt}{ds}$
$\displaystyle=\frac{(r^{2}+a^{2})}{\Delta}(r^{2}+a^{2}-a\lambda)+a(\lambda-a\mathbb{s}^{2}),$
(36)

where

$\displaystyle{\mathcal{R}}(r)$
$\displaystyle=\left(a^{2}-a\lambda+r^{2}\right)^{2}-\Delta\left((\lambda-a)^{2}+\eta\right)$
(37) $\displaystyle\Theta(\theta)$
$\displaystyle=\eta-(a\mathbb{c})^{2}-\left(\lambda\mathbb{c}/\mathbb{s}\right)^{2}.$
(38)

Given a position for the light source, and characterizing its direction with
parameters $\lambda$ and $\eta$, the first two equations can be solved
independently to obtain the radial and polar trajectories. The results can
then be substituted into the second pair to determine the azimuthal and
temporal trajectories.

###  II.4 Evolution of Polarization

The equations of motion and their solution are agnostic with regard to
evolving polarization. The utility of the complex-valued Fermi-Walker constant
of Eq. 25, $\mathbb{k}$, is that it allows the polarization of light to be
determined along any such path. Since it is complex valued, its real and
imaginary components provide two equations for the four components of the
polarization vector $\bf f$ provided that the initial polarization is known. A
third equation is generated by the orthogonality of polarization and the wave
vector–i.e. $\bf f\cdot\bf p=0$. Finally, tangents to the trajectory are null
vectors, so the polarization is only unique modulo a factor of the wave
vector. The factor can always be chosen so that the temporal component of the
polarization is equal to zero. It is therefore possible to solve for the
polarization at any point along a trajectory, a primary application of the
Walker-Penrose theorem.

##  III Application

###  III.1 Closed Spherical Trajectories

As detailed in a recent study, algebraic expressions for trajectories of
constant radius can be constructed using Jacobi elliptic functions[15]. This
reduces the number of 3-space components to two, making it easier to identify
closed circuits that exhibit polarization holonomy. For such transits, the
impact parameter $\lambda$ and Carter ratio $\eta$ are

$\displaystyle\lambda\equiv\frac{\ell}{\varepsilon}:=a+\frac{r}{a}\biggl{(}r-\frac{2\Delta}{r-1}\biggr{)}$
(39)
$\displaystyle\eta\equiv\frac{Q}{\varepsilon^{2}}:=\frac{r^{3}}{a^{2}}\biggl{(}\frac{4\Delta}{(r-1)^{2}}-r\biggr{)}.$
(40)

This implies that a path is completely fixed once an initial position and
rotation parameter, $a$, are prescribed. The trajectories are given by the
following equations[14, 15], with requisite functions defined in the Appendix:

$\displaystyle\theta(s)$
$\displaystyle=\cos^{-1}\left[-\nu_{\theta}\sqrt{u_{+}}\,\,{\rm
sn}\left(\sqrt{-a^{2}u_{-}}(s+\nu_{\theta}\mathcal{G}_{\theta}),\frac{u_{+}}{u_{-}}\right)\right]$
$\displaystyle\phi(s)$ $\displaystyle=\lambda G_{\phi}$ $\displaystyle+$
$\displaystyle\frac{2a}{r_{+}-r_{-}}\left[\left(r_{+}-\frac{a\lambda}{2}\right)\frac{s}{r-r_{+}}-\left(r_{-}-\frac{a\lambda}{2}\right)\frac{s}{r-r_{-}}\right]$
$\displaystyle t(s)$ $\displaystyle=I(s)+a^{2}G(s).$ (41)

The travel of light, on a path of constant radius, will eventually overlap
with itself. This feature can be exploited to engineer a class of trajectories
for which each member starts and ends at a specific radius and polar angle.
Suppose that these and the rotation parameter are specified. Set the initial
azimuthal angle to zero, and solve Eq. III.12 for the value of Mino parameter,
$s$, at which $\phi=2\pi$. For a given value of $a$, the points for which this
is possible comprise an _initial position arc_ in spacetime. For instance, the
set of all send/receive points with $a=0.99$ is shown in Fig. 1.

Figure 1:  _Send/receive points resulting in closed circuits._ Sets of
retrograde (red) and direct (green) start/end points are shown for $a=0.99$.
Green and red curves that are semi-transparent (top) and dashed (bottom)
identify admissible complementary sets ($\theta\rightarrow
180^{\circ}-\theta$). The top plot gives a 3-space projection of the
positions, with the outer ergosphere shown in gray and the outer event horizon
in black. Note that, for the direct case, it is possible to start/end within
the ergosphere even though the light may escape it before returning.

The methodology was then applied to construct initial position arcs for direct
and retrograde paths over a range of black hole rotation values. These are
shown in Fig. 2, where complementary sets ($\theta\rightarrow
180^{\circ}-\theta$) are not plotted for the sake of clarity. Notice that the
range of admissible initial radii decreases with decreasing rotation rate,
with an asymptote at the Schwarzschild black hole radius for unstable circular
orbits $r=3$.

Figure 2:  _Source/Observation Points_. Admissible initial/final states are
plotted for several values of rotation parameter $a$. Positions resulting in
closed retrograde trajectories are shown with solid curves while positions
that produce closed direct trajectories are dashed with a light gray outline
to guide the eye. Complementary sets ($\theta\rightarrow 180^{\circ}-\theta$)
are not plotted for the sake of clarity.

Eqs. III.1 were then used, with the source/receiver positions of Fig. 2, to
generate closed trajectories for representative send/receive points and values
of rotation parameter $a$. These are shown in Figs. 3 and 4 for direct and
retrograde orbits, respectively. Each trajectory is characterized by tangent
vectors that are discontinuous at the common location of the source and
receiver. The associated misorientation of polarization constitutes a holonomy
that can be engineered to take on a wide range of values.

Figure 3:  _3-space projection of direct trajectories._ Views are from front
(left) and below (right). Arrows highlighted in green indicate direction of
propagation, while the direction of black hole rotation is identified in the
top panel of each column. The starting point of each trajectory is highlighted
with a green sphere. Each row is associated with a particular value of
rotation parameter, $a$. Panels include the outer ergosphere (light gray) and
the outer event horizon (dark gray). Figure 4:  _3-space projection of
retrograde trajectories._ Views are from front (left) and below (right).
Arrows highlighted in green indicate direction of propagation, while the
direction of black hole rotation is identified in the top panel of each
column. The starting point of each trajectory is highlighted with a green
sphere. Each row is associated with a particular value of black hole rotation
parameter, $a$. Panels include the outer ergosphere (light gray) and the outer
event horizon (dark gray).

###  III.2 Evolution of Polarization for Spherical Trajectories

For a specific trajectory, the evolution of polarization holonomy can be
determined in two ways, both requiring only an initial polarization. The first
method was detailed in Approach subsection D. It employs the null character of
tangent vectors, polarization orthogonality with respect to propagation, and
the Walker-Penrose constant $\mathbb{k}$, of Eq. 25, to obtain an closed-form
expression for the polarization at any point along a given trajectory. The
second method, utilized here as a check on the analytical approach,
numerically solves the parallel transport equation, Eq. 10. A subsequent gauge
transformation is then implemented so that the resulting field has no temporal
component at any point along the trajectory. Both methods were used to produce
evolving polarization states for a number of trajectories, and it was verified
that the numerical results matched the analytical predictions.

A given trajectory can support two orthogonal polarization vectors.
Interestingly though, it is possible to construct initial polarization states
such that only one vector exhibits a polarization holonomy. In particular,
polarizations that are initially tangent to the local radial coordinate line
never accumulate a polarization holonomy. Because of this, we need only
construct the initial polarization to be orthogonal to such a radial
polarization. The result can then be viewed as _the_ polarization holonomy of
the trajectory.

A representative evolution of polarization is shown in Fig. 5. The angle
between the initial (blue) and final (red) polarizations is the holonomy.

Figure 5:  _Evolution of Polarization_. The 3-space projection of a retrograde
trajectory is shown in yellow for rotation parameter $a=0.99$ and start/end
polar angle $\theta_{0}=177.3^{\circ}$ . Along this path, the polarization of
light is plotted as colored sticks with color starting blue and ending red.
The view is from below. Since the temporal component is engineered to always
be zero, the angle between the red and blue sticks is the polarization
holonomy. The figure includes the outer ergosphere (light gray) and the outer
event horizon (dark gray).

Polarization holonomy can now be quantified for the entire class of closed
trajectories. The procedure is to determine the angle of mismatch between
between initial and final polarizations of each trajectory:

$\chi=\cos^{-1}\left({\bf f}_{\scriptstyle final}\cdot{\bf f}_{\scriptstyle
init}\right).$ (42)

Results are plotted in Fig. 6 for a spectrum of rotation parameters. In each
case, the holonomy is calculated over the entire set of radii and polar angles
for which a closed trajectory (of our identified class) is possible. The
figure is intended to serve as a concise visual guide for understanding and
engineering polarization holonomies.

Figure 6:  _Polarization Holonomy_. The angle, $\chi$, between initial and
final polarizations as a function of rotation parameter, $a$, and start/end
polar angle, $\theta_{0}$. Results for retrograde trajectories are shown with
solid curves while results for direct trajectories are dashed curves with a
light gray outline to guide the eye.

##  IV Conclusions

A holonomy is exhibited when closed, three-space circuits of light have
distinct outgoing and incoming polarizations. This is not simply an artifact
of the non-unique description of polarization associated with null-valued wave
vectors. Within the lowest-order geometric optics approximation in Kerr
spacetime, a class of such circuits has been used to construct closed-form
expressions for both the path and evolving polarization. Outgoing and incoming
wave vectors are not aligned, a necessary but not sufficient condition for
holonomy. The results have been verified, for specific cases, by numerically
solving the parallel transport equation for polarization.

The Walker-Penrose Theorem was used to quantify polarization holonomy as a
function of source/receiver radius and polar angle along with the black hole
angular momentum per unit mass. For a given rotation parameter, $a$, the
trajectories and associated holonomy can be smoothly changed by moving along
well-defined initial position arcs. Both direct and retrograde paths were
considered, and it was found that the latter are always associated with radii
that are less than their direct counterparts. In addition, direct trajectories
allow for holonomic geodesics that start and end within the ergosphere but
emerge from it over an intermediate interval.

The approach of this work is in contrast to studies of Gravitational Faraday
Rotation on open trajectories, in which a co-evolving frame must be used as a
reference. In fact, the net rotation of polarization relative to a Fermi-
Walker frame[11] or a frame composed of principal null directions of the Weyl
tensor[13] would be zero, verified in both cases. This is because the
orientation of the frames themselves are discontinuous at the common location
of source and receiver.

It is tempting to draw analogies between the polarization holonomies
considered here and those associated with light propagating through
inhomogeneous dielectrics[29, 30, 31, 32]. The rotating singularities of the
present work, though, correspond to three-dimensionally inhomogeneous,
_active_ dielectrics[33], making it challenging to produce a laboratory
surrogate[34].

There are a number of natural extensions to the present work. Perhaps the most
straightforward of these would be to extend the results to include charged
black holes. It would also be interesting to consider a higher-order geometric
optics approximation[12], allowing the spin dependence of polarization
holonomy to be determined. For quantized radiation, pairs of entangled photons
can be used to elucidate a black hole version of the Geometric Phase of
Entanglement[35, 36]. Finally, it may be possible to identify holonomies
associated with spatial modes of light as in, for instance, Laguerre-Gaussian,
Hermite-Gaussian, and Bessel beams[37, 38].

##  V Acknowledgments

It is a pleasure to acknowledge useful discussions with Brendan McLane, Alex
Tinguely, Andrew Voitiv, and Mark Siemens.

##  Appendix A Equations of Motion

Closed-form, spherical solutions can be constructed that satisfy the Equations
of Motion, Eq 33. The process is tedious, but the ultimate result, Eqs. III.1,
is very practical and easy to use. Only the essential equations are provided
below, since a detailed development is given elsewhere[14, 15].

Define $A$, $B$, and $C$.

$\displaystyle A$ $\displaystyle:=a^{2}-\eta-\lambda^{2}$ $\displaystyle B$
$\displaystyle:=2(\eta+(\lambda-a)^{2})$ $\displaystyle C$
$\displaystyle:=-a^{2}\eta$ (43)

Define $P$, $Q$, $w_{\pm}$ and $z$.

$\displaystyle P$ $\displaystyle:=-\frac{A^{2}}{12}-C$ $\displaystyle Q$
$\displaystyle:=-\frac{A}{3}\left(\left(\frac{A}{6}\right)^{2}-C\right)-\frac{B^{2}}{8}$
(44) $\displaystyle w_{\pm}$
$\displaystyle:=\left(\pm\sqrt{\frac{P^{3}}{27}+\frac{Q^{2}}{4}}-\frac{Q}{2}\right)^{1/3}$
$\displaystyle z$ $\displaystyle:=\sqrt{\frac{w_{+}+w_{-}}{2}-\frac{A}{6}}$

Define key radii, $r_{1}$, $r_{2}$, $r_{3}$, and $r_{4}$.

$\displaystyle r_{1}$
$\displaystyle:=-\sqrt{-\frac{A}{2}+\frac{B}{4z}-z^{2}}-z$ $\displaystyle
r_{2}$ $\displaystyle:=\sqrt{-\frac{A}{2}+\frac{B}{4z}-z^{2}}-z$
$\displaystyle r_{3}$
$\displaystyle:=-\sqrt{-\frac{A}{2}+\frac{B}{4z}-z^{2}}+z$ (45) $\displaystyle
r_{4}$ $\displaystyle:=\sqrt{-\frac{A}{2}+\frac{B}{4z}-z^{2}}+z$
$\displaystyle r_{\pm}$ $\displaystyle:=1\pm\sqrt{1-a^{2}}$

Define $x_{1}$ and $k_{1}$.

$\displaystyle x_{1}$
$\displaystyle:=\sqrt{\left(\frac{r-r_{2}}{r-r_{1}}\right)\left(\frac{r_{3}-r_{1}}{r_{3}-r_{2}}\right)}$
$\displaystyle k_{1}$
$\displaystyle:=\frac{\left(r_{3}-r_{2}\right)\left(r_{4}-r_{1}\right)}{\left(r_{3}-r_{1}\right)\left(r_{4}-r_{2}\right)}$
(46)

Define $\Delta_{\theta}$ and $u_{\pm}$.

$\displaystyle\Delta_{\theta}$
$\displaystyle:=\frac{1}{2}\left(1-\frac{\eta+\lambda^{2}}{a^{2}}\right)$
$\displaystyle u_{\pm}$
$\displaystyle:=\Delta_{\theta}\pm\sqrt{\Delta_{\theta}^{2}+\eta/a^{2}}$ (47)

Define $F_{1}$, $E_{1}$, $\Pi_{1}$, and $\Pi_{\pm}$. These are expressed in
terms of Jacobi elliptic functions $E$, $F$, and $\Pi$.

$\displaystyle F_{1}$
$\displaystyle:=\frac{2F\left(\sin^{-1}\left(x_{1}\right),k_{1}\right)}{\sqrt{\left(r_{3}-r_{1}\right)\left(r_{4}-r_{2}\right)}}$
$\displaystyle E_{1}$
$\displaystyle:=\sqrt{\left(r_{3}-r_{1}\right)\left(r_{4}-r_{2}\right)}E\left(\sin^{-1}\left(x_{1}\right),k_{1}\right)$
$\displaystyle\Pi_{1}$
$\displaystyle:=\frac{2\Pi\left(\frac{r_{3}-r_{2}}{r_{3}-r_{1}},\sin^{-1}\left(x_{1}\right),k_{1}\right)}{\sqrt{\left(r_{3}-r_{1}\right)\left(r_{4}-r_{2}\right)}}$
(48) $\displaystyle\Pi_{\pm}$
$\displaystyle:=\frac{2}{\sqrt{\left(r_{3}-r_{1}\right)\left(r_{4}-r_{2}\right)}}\frac{r_{2}-r_{1}}{(r_{\pm}-r_{1})(r_{\pm}-r_{2})}$
$\displaystyle\quad\times\Pi\left[\left(\frac{r_{\pm}-r_{1}}{r_{\pm}-r_{2}}\right)\left(\frac{r_{3}-r_{2}}{r_{3}-r_{1}}\right),\sin^{-1}\left(x_{1}\right),k_{1}\right]$

Define $\mathcal{I}_{0}$, $\mathcal{I}_{1}$, $\mathcal{I}_{2}$, and
$\mathcal{I}_{\pm}$.

$\displaystyle\mathcal{I}_{0}$ $\displaystyle:=F_{1}$
$\displaystyle\mathcal{I}_{1}$
$\displaystyle:=r_{1}F_{1}+(r_{2}-r_{1})\Pi_{1}$
$\displaystyle\mathcal{I}_{2}$
$\displaystyle:=\frac{\sqrt{\mathcal{R}}}{r-r_{1}}-\frac{1}{2}(r_{1}r_{4}+r_{2}r_{3})F_{1}-E_{1}$
(49) $\displaystyle\mathcal{I}_{\pm}$
$\displaystyle:=-\Pi_{\pm}-\frac{F_{1}}{r_{\pm}-r_{1}}$

Define $\mathcal{G}_{\theta}$ and $\nu_{\theta}$.

$\displaystyle\mathcal{G}_{\theta}$
$\displaystyle:=\frac{-1}{\sqrt{-a^{2}u_{-}}}F\left(\sin^{-1}\left(\frac{\cos\theta_{\scriptstyle
init}}{\sqrt{u_{+}}}\right),\frac{u_{+}}{u_{-}}\right)$ (50)
$\displaystyle\nu_{\theta}$ $\displaystyle:={\rm Sign(p^{3}_{\scriptstyle
init})}\equiv{\rm Sign(p^{\theta}_{\scriptstyle init})}$

Define $\mathcal{G}_{\phi}$ and $G_{\phi}$.

$\displaystyle\mathcal{G}_{\phi}$
$\displaystyle:=\frac{-1}{\sqrt{-a^{2}u_{-}}}\Pi\left[u_{+},\sin^{-1}\left(\frac{\cos\theta_{\scriptstyle
init}}{\sqrt{u_{+}}}\right),\frac{u_{+}}{u_{-}}\right]$ $\displaystyle
G_{\phi}$ $\displaystyle:=-\nu_{\theta}\mathcal{G}_{\phi}$ (51)
$\displaystyle-$ $\displaystyle\frac{1}{\sqrt{-a^{2}u_{-}}}\Pi\left[u_{+},{\rm
am}\left(\sqrt{-a^{2}u_{-}}(s+\nu_{\theta}\mathcal{G}_{\theta}),\frac{u_{+}}{u_{-}}\right),\frac{u_{+}}{u_{-}}\right]$

Define functions $I_{\pm}(s)$ and $I(s)$.

$\displaystyle I_{\pm}(s)$ $\displaystyle:=\frac{s}{r-r_{\pm}}$ $\displaystyle
I(s)$
$\displaystyle:=\frac{4}{r_{+}-r_{-}}\left[r_{+}\left(r_{+}-\frac{a\lambda}{2}\right)I_{+}(s)\right]$
$\displaystyle\,\,-\frac{4}{r_{+}-r_{-}}\left[r_{-}\left(r_{-}-\frac{a\lambda}{2}\right)I_{-}(s)\right]$
(52) $\displaystyle\,\,+2rs+r^{2}s+4s$

Define function $G(s)$ in terms of Jacobi elliptic functions, $E$ and $F$, as
well as the Jacobi amplitude function, ${\rm am}$.

$\displaystyle G(s)$
$\displaystyle:=\frac{-u_{+}}{\sqrt{-a^{2}u_{-}}}\left(\frac{u_{-}}{u_{+}}\right)$
$\displaystyle\quad\times
E\left[\rm{am}\left(\sqrt{-a^{2}u_{-}}(s+\nu_{\theta}\mathcal{G}_{\theta}),\frac{u_{+}}{u_{-}}\right),\frac{u_{+}}{u_{-}}\right]$
$\displaystyle\,\,+\frac{u_{+}}{\sqrt{-a^{2}u_{-}}}\left(\frac{u_{-}}{u_{+}}\right)$
(53) $\displaystyle\quad\times
F\left[\rm{am}\left(\sqrt{-a^{2}u_{-}}(s+\nu_{\theta}\mathcal{G}_{\theta}),\frac{u_{+}}{u_{-}}\right),\frac{u_{+}}{u_{-}}\right]$

The expressions for spherical trajectories, r, can now be written out as a
function of the Mino parameter, $s$:

$\displaystyle\theta(s)$
$\displaystyle=\cos^{-1}\left[-\nu_{\theta}\sqrt{u_{+}}\,\,{\rm
sn}\left(\sqrt{-a^{2}u_{-}}(s+\nu_{\theta}\mathcal{G}_{\theta}),\frac{u_{+}}{u_{-}}\right)\right]$
$\displaystyle\phi(s)$ $\displaystyle=\lambda G_{\phi}$ $\displaystyle+$
$\displaystyle\frac{2a}{r_{+}-r_{-}}\left[\left(r_{+}-\frac{a\lambda}{2}\right)\frac{s}{r-r_{+}}-\left(r_{-}-\frac{a\lambda}{2}\right)\frac{s}{r-r_{-}}\right]$
$\displaystyle t(s)$ $\displaystyle=I(s)+a^{2}G(s).$ (54)

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