# Snell’s law for the Schwarzschild black hole

X. H. Zheng1<EMAIL_ADDRESS>J. X. Zheng2 1Department of Physics, Queen’s
University of Belfast, BT7 1NN, N. Ireland  2Department of Electrical and
Electronic Engineering, Imperial College London, SW7 2AZ, England

###### Abstract

The Wheeler equation, for electromagnetic disturbances in a gravitational
field, was found by Fiziev to have exact solutions both above and below the
event horizon, in the form of waves propagating both inwardly and outwardly.
This observation can be interpreted and verified from the optical point of
view, entirely on the basis of the Schwarzschild metric for length contraction
and time dilation, in order to derive a differential version of Snell’s law
for the Schwarzschild black hole. It reveals interesting physics, including
the correct amount of light deflection by the Sun, internal and external
Oppenheimer-Snyder cones of the black hole, properties of its phonon sphere
and the conclusion that light-rays are kept below the horizon by length
contraction and time dilation rather than deflection.

##  I Introduction

Electromagnetic fields in the Schwarzschild metric have been investigated for
a long time Bonnor ; Lovelock ; Papapetrou ; Linet ; Bachelot ; Bachelot2 ;
Gutsunaev ; Choquet ; Kroon ; Blue ; Ovsiyuk ; Fernandes  and Fiziev secured
an unusual advance in 2006 Fiziev . With an explicit substitution he
transformed the Wheeler equation Wheeler  into Heun’s equation Ronveaux  which
is known to have exact solutions, both above and below the event horizon of a
black hole, in the form of waves propagating both inwardly and outwardly. He
presented a number of such solutions and commented on their eventual physical
significance and some open problems Fiziev .

The Wheeler equation was derived in 1955, from a system of fundamental
equations for coupled electromagnetic and gravitational fields, to study
strong electromagnetic disturbances capable of distorting spacetime to trap
themselves Wheeler . Sometimes the Wheeler equation is referred to as the
Regge-Wheeler equation probably because, in 1957, the two authors jointly
published a paper on a related topic Regge . In the case of a black hole and
weak electromagnetic disturbances the Wheeler equation assumes a specific form
in the so-called tortoise coordinate in association with the Schwarzschild
metric. With this specific Wheeler equation Fiziev applied his substitution
and linked it to Heun’s equation.

Heun’s equation was studied by Karl Heun in the late 19th century and still is
an active topic of contemporary investigation. It is a second order ordinary
differential equation with 4 singularities in its general form. By the process
of confluence the number of singularities can be reduced (to 3 in the case of
the Wheeler equation) for a rich variety of useful and important applications.
Solutions to Heun’s equation include local solutions (power series), Heun
functions, Heun polynomials and path-multiplicative solutions Ronveaux . The
solutions Fiziev found are mostly local solutions, where a countable set
exists below the event horizon, simultaneously finite at the horizon and
centre, and obviously square integrable. In his opinion, if these are proven
to form a complete set in the corresponding functional space, they will
present a natural basis of normal modes for a well-defined expansion of any
small perturbation of the metric in the interior of the black hole Fiziev .

The perspective of Fiziev amounts to a sophisticated scheme to analyse waves,
essentially via Fourier expansion in terms of the so-called normal modes,
which are solutions to the Wheeler equation, with a boundary value (zero for
example) imposed upon the event horizon by some physics which, to the best of
our knowledge, has not yet been identified.

Here we tackle the problem from the optical point of view so that, as long as
light-rays exist in the void below the event horizon, in addition to their
presence above, it is sufficient for us to know length contraction and time
dilation from the Schwarzschild metric to determine the speed of the waves in
order to trace the rays, with no need to identify additional physics. The
result is a differential version of Snell’s law which, like its counterpart in
classical optics, reveals interesting physics.

For example, above the event horizon, our version of Snell’s law enables us to
find the correct amount of light deflection by the Sun via straightforward
evaluation. It enables us to define quantitatively the external Oppenheimer-
Snyder cone, together with properties of the so-called phonon sphere, to
determine whether or not a ray launched above the event horizon can escape
from the gravitational field. It also tells us clearly that, if incapable of
escaping, the ray must encounter the event horizon at a right angle.

Below the event horizon, we find from our version of Snell’s law the internal
Oppenheimer-Snyder cone to determine the allowed angles for light-rays to
propagate in both the outward and inward directions. Historically Einstein
started with optics to derive the general theory of relativity Einstein ,
adding length contraction and time dilation to the arsenal of optics as
devices to manipulate light-rays. We conclude that deflection, the traditional
device in optics to manipulate light, on its own cannot retain light-rays
below the event horizon.

We arrange this article as follows. In Section II we explain the motions of
light-rays in a gravitational field. In Section III we specify relations
between the speed of light and gravity. In Section IV we derive a differential
version of Snell’s law for black holes. In Section V we apply Snell’s law to
find the correct amount of light deflection by the Sun. In Sections VI and VII
we study analytical and numerical trajectories of the rays. In Section VIII we
discuss the Wheeler equation. Brief discussions and conclusions are in Section
IX.

##  II Huygens principle and gravity

Figure 1: Illustrations of the Huygens principle originally from Einstein,
where $P_{1}$ and $P_{2}$ are centres on the first wave front to emit
spherical wavelets whose envelope becomes the next front to emit further
wavelets, $C_{1}$ and $C_{2}$ are visual speeds of light (as defined in
Section II) which become slower the stronger the gravity, and $dt$
differential time of wavelet expansion. The ray is launched either parallel to
(upper part) or towards (lower part) the event horizon (location indicated by
$N$).

In 1678 the Dutch physicist Huygens proposed that every point to which a
luminous disturbance reaches becomes a source of a spherical wavelet, and the
sum of these wavelets determines the form of the luminous disturbance at any
subsequent time Huygens . He applied this principle to explain linear and
spherical wave propagation, and to derive the laws of reflection and
refraction. In 1911 Einstein applied the same principle to explain deflection
of light-rays in a gravitational field, and to outline the general theory of
relativity Einstein . In FIG. 1 we redraw the historic illustration by
Einstein with some modifications. We follow Einstein to show how the wavelets
emitted from two exemplary points on the first wave front, $P_{1}$ and
$P_{2}$, expand and determine the second wave front. In addition we show how
the same process repeats itself to determine the third front.

In FIG. 1 the wave fronts can be seen as the floors and roof of a two level
building, with the wavelets for pillars to support the first floor and roof.
Einstein stated: “the velocity of light in the gravitational field is a
function of the place” Einstein . From now on we will refer to Einstein’s
“velocity of light” in the quotation as the speed of light seen by a distant
observer, or the visual speed of light. We also will refer to the velocity of
light seen by a local observer the local speed of light, or simply the speed
of light, which is a relativistic invariant. In the upper part of the figure
gravity is uneven at $P_{1}$ and $P_{2}$ so that the building tilts towards
$P_{2}$ where the visual speed of light is slower due to stronger gravity. In
the lower part of FIG. 1 gravity is even at $P_{1}$ and $P_{2}$ so that the
building does not tilt but with its levels piled closer to each other the
stronger the gravity.

Now we have a problem: what would happen if a ray were launched from below the
event horizon to approach the horizon at a right angle, as is shown in the
lower part of FIG. 1? It definitely will not make a U-turn and fall back to
the centre, because it is well known the wavelets in the Huygens principle
cannot propagate in the backward direction. From the optical point of view,
what could stop the ray from escaping across the event horizon?

We will show that light-rays launched outwardly from below the event horizon
indeed encounter the horizon at right angles. They are kept below the horizon
not by deflection but by length contraction and time dilation, which in our
view are new devices in the arsenal of optics to manipulate light, capable of
lending deflection a helping hand to impede and eventually stop the escaping
rays on their outward trajectories.

##  III visual speed of light and gravity

Above the event horizon the Schwarzschild metric can be written as

$\displaystyle(ds)^{2}=(d\ell^{\prime})^{2}-c^{2}(dt^{\prime})^{2},\;\;\;\;\;r>r_{s}$
(1)

where $c$ is the local speed of light, or speed of light, invariant under all
circumstances,

$\displaystyle(d\ell^{\prime})^{2}=\bigg{(}1-\frac{r_{s}}{r}\bigg{)}^{\\!-1}\\!\\!\\!(dr)^{2}+r^{2}(d\Omega)^{2},$
(2) $\displaystyle(dt^{\prime})^{2}=\bigg{(}1-\frac{r_{s}}{r}\bigg{)}(dt)^{2}$

measure length contraction and time dilation, $r_{s}$ being the Schwarzschild
radius, $(d\Omega)^{2}=(d\theta)^{2}+\sin^{2}(\theta)(d\phi)^{2}$. We are
reminded the Schwarzschild metric is presented in the coordinate system
$(t,r,\theta,\phi)$ accommodating a distant observer. It is valid only in the
presence of the factor $(1-r_{s}/r)^{-1}$ for length contraction, and
$1-r_{s}/r$ for time dilation. It tells us $d\ell^{\prime}/dt^{\prime}=c$,
because $ds=0$ for light-rays, that is the speed of light is invariant in the
eyes of a local observer experiencing both length contraction and time
dilation. But this is not true in the eyes of the distant observer. Letting

$\displaystyle\cos^{2}(\alpha)\,(d\ell)^{2}=(dr)^{2},$ (3)
$\displaystyle\sin^{2}(\alpha)\,(d\ell)^{2}=r^{2}(d\Omega)^{2}$

where $\alpha$ is the angle of incidence made by the ray and the radial
direction, explicitly specified as $\alpha_{1}$ and $\alpha_{2}$ at $B$ and
$C$ in FIG. 2, we find from Eqs. (1) nand (III)

$\displaystyle\left(\frac{d\ell^{\prime}}{dt^{\prime}}\right)^{\\!2}=\frac{r^{2}}{(r-r_{s})^{2}}\bigg{[}1-\frac{r_{s}}{r}\sin^{2}(\alpha)\bigg{]}\bigg{(}\frac{d\ell}{dt}\bigg{)}^{\\!2}$
(4)

or $(d\ell^{\prime}/dt^{\prime})^{2}=V^{2}(d\ell/dt)^{2}$ to relate the local
and visual speed of light. We should take notice that in Eq. (4)
$d\ell^{\prime}$ and $dt^{\prime}$ are local measurements, whereas $r$,
$r_{s}$, $\alpha$ and $d\ell$ are from the distant observer, comparable with
say an object and its optical image. Consequently $V$ can be compared with the
refractive index that defines images in classical optics. Indeed Eq. (4) tells
us clearly that, when $\alpha=0$, we have $v=d\ell/dt=(1-r/r_{s})\,c$ which is
the well known formula for the visual speed of a ray in the radial direction
of Schwarzschild black hole from the eyes of the distant observer.

We follow the same procedure to investigate light-rays below the event
horizon. In this case the Schwarzschild metric is written as

$\displaystyle-(ds)^{2}=(d\ell^{\prime})^{2}-c^{2}(dt^{\prime})^{2},\;\;\;\;\;r<r_{s}$
(5)

where

$\displaystyle(d\ell^{\prime})^{2}=\bigg{(}\frac{r_{s}}{r}-1\bigg{)}^{\\!-1}\\!\\!\\!(dr)^{2}-r^{2}(d\Omega)^{2},$
(6) $\displaystyle(dt^{\prime})^{2}=\bigg{(}\frac{r_{s}}{r}-1\bigg{)}(dt)^{2}$

which too leads to $d\ell^{\prime}/dt^{\prime}=c$ when $ds=0$. Eqs. (III) and
(III), like the Schwarzschild metric, are presented in the coordinate system
$(t,r,\theta,\phi)$, valid only in the presence of the length contraction and
time dilation factors. Eq. (4) still applies but under the condition

$\displaystyle\sin^{2}(\alpha)\leq\frac{r}{r_{s}}$ (7)

in order to avoid $V$, defined below Eq. (4), from becoming imaginary, a new
issue arising from the term $-r^{2}(d\Omega)^{2}$ in Eq. (III).

To understand Eqs. (4) and (7) it is worth noting that Eqs. (III) and (III)
are our choice, or ansatz, to test if we are correct to anticipate light-rays
below the event horizon. Our choice assumes consistency between Eqs. (1) and
(5), true because they differ just by the sign of $(ds)^{2}$, which vanishes
in the case of light-rays. In addition our choice is evidently supported by
the observation of Fiziev that the Wheeler equation has exact solutions both
above and below the horizon Fiziev . In fact in the Wheeler equation the time
factor, $\exp(i\omega t)$, (declared to be an ansatz by Fiziev) also is a
choice to test if the equation has similar solutions both above and below the
event horizon Fiziev ; Wheeler . The test might fail, or become reduction to
absurdity, in case the ansatz leads to a contradiction in general. Since Eq.
(III) leads to a contradiction only when Eq. (7) is violated, not in general,
we might be correct to expect light-rays below the event horizon, provided we
can find the physical reason for Eq. (7).

In the Schwarzschild metric length contraction takes place only in the
$r$-direction. Above the event horizon it has less of an effect in other
directions due to the term $r^{2}(d\Omega)^{2}$ in Eq. (III). Below the event
horizon, however, length contraction is always enhanced by the term
$-r^{2}(d\Omega)^{2}$ in Eq. (III). When Eq. (7) has been violated, the
enhancement to length contraction turns out to be so significant that
$(d\ell^{\prime})^{2}$ in Eq. (III) becomes negative, indicating the wavelets
in the Huygens principle can no longer be in phase to direct their energy to
form a thin pencil around a single direction, similar to what happens to the
refractive ray when total reflection occurs. Indeed the criterion for total
reflection in classical optics does bear a fair resemblance to Eq. (7).

Eq. (7) defines a cone about the radial direction, with aperture $\rightarrow
0$ when $r\rightarrow 0$. In a landmark paper in 1939 Oppenheimer and Snyder
revealed that, by the gravitational deflection of light, all energy emitted
from the surface of a collapsing star will almost all be reduced to within a
cone about the outward normal, of progressively shrinking aperture as the star
contracts Oppenheimer . From now on we will refer to the cone defined in Eq.
(7) as the internal Oppenheimer-Snyder cone, because it also is from
gravitational deflection, located below the event horizon, aperture
progressively shrinking with decreasing values of $r$, similar to what happens
on the surface of a contracting star.

##  IV Snell’s law for black holes

Figure 2: Passage of a ray defined as $r(\phi)$ in the $(r,\phi)$ plane of the
spherical coordinates, $(r,\phi,\theta)$, with $r_{1}=r(\phi_{1})$,
$r_{2}=r(\phi_{2})$, $\delta\phi=\phi_{2}-\phi_{1}$. The ray is shown to start
from $A$, be deflected at $B$ with an angle of incidence, $\alpha_{1}$, and
refraction, $\alpha_{2}+\delta\phi$, both against the $r$-direction.
Eventually it reaches $C$ with an angle of incidence, $\alpha_{2}$, also
measured against $r$. The visual speed of the ray is determined by $V_{1}$ or
$V_{2}$ when $r<r_{1}$ or otherwise.

We apply Eq. (4) to specify the rule to determine the direction of light-rays
in the Schwarzschild metric. On account of the principle of phase matching we
must have

$\displaystyle V_{1}\sin(\alpha_{1})=V_{2}\sin(\alpha_{2}+\delta\phi)$ (8)

for the ray in FIG. 2. Here $V_{1}=V(r_{1},\alpha_{1})$ and
$V_{2}=V(r_{2},\alpha_{2})$ are values of $V$ defined in Eq. (4), $\alpha_{1}$
and $\alpha_{2}$ ray incidence against the outward normal, and $\delta\phi$
incremental angle made by $r$ when it rotates to take the values $r_{1}$ and
$r_{2}$. Eq. (8) can be written as

$\displaystyle
V_{1}\frac{\sin(\alpha_{2}+\delta\phi)-\sin(\alpha_{1})}{\alpha_{2}+\delta\phi-\alpha_{1}}\,(\alpha_{2}+\delta\phi-\alpha_{1})$
$\displaystyle=-\frac{V_{2}-V_{1}}{\delta\phi}\sin(\alpha_{2}+\delta\phi)\,\delta\phi$

which can readily be transformed into a differential equation. To this end we
divide the above equation with $\delta\phi$ and let $\delta\phi\rightarrow 0$
so that $(\alpha_{2}+\delta\phi-\alpha_{1})/\delta\phi\rightarrow
d\alpha/d\phi+1$, assuming $\alpha_{2}\rightarrow\alpha_{1}$ when
$\delta\phi\rightarrow 0$. At the same time we find $\cos(\alpha_{1})$ on the
left hand side of the above equation. We also find $dV/d\phi$ and
$\sin(\alpha_{1})$ on the right hand side of the equation, assuming
$V_{2}\rightarrow V_{1}$ when $\delta\phi\rightarrow 0$. Letting
$\alpha_{1}=\alpha$ for generality, we find via simple rearrangement of the
terms that

$\displaystyle\frac{d\alpha}{d\phi}=-1-\frac{r}{V}\frac{dV}{dr}\frac{\tan(\alpha)\,dr}{rd\phi}$

where

$\displaystyle\frac{\tan(\alpha)\,dr}{rd\phi}=\lim_{\delta\phi\rightarrow
0}\frac{\tan(\alpha_{2})\,(r_{2}-r_{1})}{r_{1}\delta\phi}=1$

from the geometry in FIG. 2, giving

$\displaystyle\frac{d\alpha}{d\phi}=-1-\frac{r}{V}\frac{dV}{dr},\;\;\;\;\;\;\frac{dr}{d\phi}=r\cot(\alpha)$
(9)

as a differential version of Snell’s law for Schwarzschild black holes. With
the help of the expression of $V$ in Eq. (4), and some rather involved
algebra, we find

$\displaystyle\left[1-\frac{2r_{s}}{r}\sin^{2}(\alpha)\right]\frac{d\alpha}{d\phi}=-\left[1-\frac{r_{s}}{2r}\sin^{2}(\alpha)\right]$
$\displaystyle+\bigg{[}1-\frac{r_{s}}{r}\sin^{2}(\alpha)\bigg{]}\frac{r_{s}}{r-r_{s}}$
(10)

to evaluate $d\alpha/d\phi$ in Eq. (9).

##  V light deflection by the sun

Letting $r_{1}=6.96\times 10^{8}$ and $r_{s}=2.95\times 10^{3}$ be the radius
and Schwarzschild radius of the Sun in meters respectively, we have
$r_{s}/r<r_{s}/r_{1}=4.24\times 10^{-6}$ for a ray passing the Sun in a
grazing angle. Since in Eq. (4) the term $(r_{s}/r)\sin^{2}(\alpha)$
represents a high order correction, due to inhomogeneous length contraction in
different directions, we neglect similar terms in Eq. (IV) (radial gravity
approximation) and find

$\displaystyle\frac{d\alpha}{d\phi}=-1+\frac{r_{s}}{r-r_{s}}$ (11)

where the effect of gravitational deflection is represented by the last term
on the right hand side. To a very good approximation $r-r_{s}\simeq
r_{1}/\cos(\phi)$ holds to give

$\displaystyle\alpha=\bigintsss_{-0.5\pi}^{\;0.5\pi}\\!\\!\\!\\!\frac{r_{s}}{r-r_{s}}\,d\phi\simeq\frac{2r_{s}}{r_{1}}=1.75\mbox{
arc seconds}$ (12)

as the amount of light deflection by the Sun, consistent with the well-known
astronomical observation.

##  VI analytical trajectories: radial gravity approximation

Figure 3: Event horizon (interior shaded, radius = $r_{s}$) and analytic
trajectories of rays in radial gravity approximation from Eq. (14) and
detailed discussion in Section VI. The rays are launched at $r_{0}=0.5$ or
$2r_{s}$ with $0.05\leq\alpha_{0}\leq 0.48\pi$ until $r\sim r_{s}$. Afterwards
the rays are illustrated to continue their journeys along straight lines,
leaving a schematic gap behind indicating length contraction and time
dilation.

To illustrate the physics we integrate Eq. (11) analytically for simplicity
and clarity. By using the second formula in Eq. (9) to eliminate $d\phi$ we
find

$\displaystyle\frac{r}{dr}\;\frac{d\sin(\alpha)}{\sin(\alpha)}=-1+\frac{r_{s}}{r-r_{s}}$
(13)

which is a closed relationship between $\alpha$ and $r$ ready to be
integrated. The result is

$\displaystyle\frac{r/r_{s}}{1-r_{s}/r}\sin(\alpha)=\frac{r_{0}/r_{s}}{1-r_{s}/r_{0}}\sin(\alpha_{0})$
(14)

for trajectories of light-rays in radial gravity approximation, where $r_{0}$
and $\alpha_{0}$ are the initial conditions of the ray. Letting $r$ and
$r_{s}$ be the radius and Schwarzschild radius of the Earth for example, we
have $r/r_{s}\simeq r_{0}/r_{s}=7.18\times 10^{8}$ and Eq. (14) tells us
$\sin(\alpha)\simeq\sin(\alpha_{0})$, that is a ray around us must travel
along a straight path in its initial direction. If the ray is launched to
ascend and pulled back by gravity afterwards, then $\alpha=\pi/2$ must occur
at some stage, giving via Eq. (14)

$\displaystyle\frac{r_{s}-r_{0}}{r_{0}^{2}}\,r^{2}+\sin(\alpha_{0})\,r-\sin(\alpha_{0})\,r_{s}=0$
(15)

to determine the maximum value of $r$ the ray can reach. Eq. (15) has no real
solution if

$\displaystyle\sin(\alpha_{0})\leq
2\,\sqrt{\;\,\frac{r_{s}}{r_{0}}-\bigg{(}\frac{r_{s}}{r_{0}}\bigg{)}^{\\!2}}$
(16)

which means the initial angle is too steep for the ray to be pulled back. Eq.
(16) can be compared with Eq. (7) and can be seen as an extended criteria for
total reflection. From now on we will refer to the cone defined by Eq. (16) as
the external Oppenheimer-Snyder cone, because the two authors discussed a
similar cone above the surface of a collapsing star Oppenheimer .

Eq. (16) tells us that $\alpha_{0}=0$ when $r_{0}=r_{s}$, that is, immediately
above the event horizon, a ray still can defy gravity and escapt if it is
launched in the radial direction. On the other hand we have $\alpha_{0}=\pi/2$
when $r_{0}=2r_{s}$ where lies the so-called phonon sphere Nitta  made from
circular orbits of rays launched horizontally.

Data from Eq. (14) are ready to be presented graphically to illustrate the
physics. All we have to do is to find $\phi$ from the second equation in Eq.
(9) with a simple numerical procedure. To this end we need $\phi_{0}$, in
addition to $r_{0}$ and $\alpha_{0}$, for the initial conditions to launch the
rays. Above the event horizon in FIG. 3 we launch rays at $r_{0}=2r_{s}$ and
$\phi_{0}=0.25\pi$ (north-east direction) with $0.05\leq\alpha_{0}\leq
0.48\pi$ to descend and ascend in the clockwise and anticlockwise directions
respectively. The ascending rays are traced until $r=2.5r_{s}$ whereas the
descending rays are traced to encounter the event horizon ($r\simeq r_{s}$
with $\alpha=0$). Afterwards the rays are shown to continue their journey to
illustrate their straight trajectories ($\alpha\equiv 0$) until $r=0.5r_{s}$.

We also launch rays below the event horizon at $r_{0}=0.5r_{s}$ and
$\phi_{0}=1.25\pi$ (south-west direction) with $0.05\leq\alpha_{0}\leq
0.48\pi$ to ascend in anticlockwise directions. The rays are traced to
encounter the event horizon ($r\simeq r_{s}$ with $\alpha=0$) and continue
their journey along straight lines ($\alpha\equiv 0$) until $r=2.5r_{s}$. We
are reminded that $r=r_{s}$ is a singular point where $\alpha$ in Eq. (14) has
no definition. Each of the trajectories in FIG. 3 is made from two pieces
separated by a gap to indicate infinite length contraction and time dilation.

##  VII numerical trajectories

Figure 4: Numerical trajectories of light-rays from Eq. (VII) and detailed
discussion in Section VII, with conventions in FIG. 3, the schematic gaps
indicating infinite length contraction and time dilation in particular.

To solve Eq. (IV) we use the second equation in Eq. (9) to replace $d\phi$
with $d\alpha$ and apply the following numerical scheme

$\displaystyle\frac{u_{n+1}}{u_{n}}=1+\frac{\Delta\xi_{n}}{\xi_{n}-2u_{n}^{2}}\left[\frac{2-\xi_{n}}{\xi_{n}-1}\;\;\;\;\;\;\;\;\;\right.$
$\displaystyle+\left.\left(\frac{3}{2\xi_{n}}-\frac{1}{\xi_{n}-1}\right)u_{n}^{2}\right]$
(17)

where $u_{n}=\sin(\alpha_{n})$, $\xi_{n}=r_{n}/r_{s}$,
$\Delta\xi_{n}=\xi_{n+1}-\xi_{n}$, $n=0,1,2,...$, iteratively for rays above
the event horizon, with the initial condition $\xi_{0}=2$ and
$\phi_{0}=0.25\pi$, $0.05\leq\alpha_{0}\leq 0.3\pi$, and let the ray ascend
until $\xi=2.5$, or descend until $\xi=1+\Delta\xi$, that is one numerical
step above the event horizon. Afterwards we let the rays continue their
journeys along straight trajectories until $\xi=0.5$. We also apply Eq. (VII)
with the initial condition $\xi_{0}=0.5$ and $\phi_{0}=1.25\pi$,
$0.05\leq\alpha_{0}\leq 0.16\pi$, and let the ray ascend until
$\xi=1-\Delta\xi$, that is just one numerical step short below the event
horizon. Afterwards we let the rays continue their journeys along straight
trajectories until $\xi=2.5$.

We evaluate Eq. (VII) numerically with a first-order Runge-Kutta procedure
written in house, always with 8000 steps when $\xi$ varies between 0.5 and
2.5. We use the analytical solution in Eq. (14) as a reference to test
accuracy of the outcome of Eq. (VII), with terms in $\sin^{2}(\alpha)$ removed
from Eq. (IV). We find that, over the range of $\xi$, the r.m.s. deviation in
$\alpha$ is $1.15\times 10^{-4}$ on average when $\xi_{0}=0.5$ and $1.38\times
10^{-4}$ when $\xi_{0}=2$.

Our numerical trajectories from Eq. (VII) are shown graphically in FIG. 4
ready to be compared with the analytical trajectories in FIG. 3. We find that,
if we do not assume $\sin(\alpha)=0$ for radial gravity approximation, then we
have to let $\alpha\leq 0.16\pi$ to restrict the initial incidence when the
ray is launched at $\xi=0.5$. The reason apparently lies in the internal
Oppenheimer cone in Eq. (7) demanding $\alpha<0.25\pi$ when $\xi=0.5$. Above
the event horizon we find we have to let $\alpha\leq 0.3\pi$ initially when
the ray is launched at $\xi=2$.

It is remarkable that, in FIG. 3, the ray launched with $r_{0}=2r_{s}$ and
$\alpha_{0}=0.48\pi$ follows a nearly circular trajectory to orbit the event
horizon. Furthermore, with $r_{0}=2r_{s}$ and $\alpha_{0}=\pi/2$, we can
balance Eq. (14) with $r\equiv 2r_{0}$ and $\alpha\equiv\pi/2$, which
represents a perfect circular trajectory. However it is clear from FIG. 4
that, numerically, we cannot launch a ray to follow this trajectory. The
reason lies in that, if we let $\sin^{2}(\alpha)=1$ in Eq. (IV), Eq. (13) will
be replaced by

$\displaystyle\frac{r}{dr}\;\frac{d\sin(\alpha)}{\sin(\alpha)}=-1+\frac{r_{s}}{r-r_{s}}+\frac{r_{s}}{r-2r_{s}}$

for transverse gravity approximation. It tells us $r\equiv 2r_{s}$ and
$\alpha\equiv\pi/2$ definitely cannot be a solution to Eq. (IV). The phonon
sphere, specified in Section IV as made from circular rays at $r=2r_{s}$ in
radial gravity approximation, cannot actually accommodate rays. It is
interesting that, according to Nitta, Chiba and Sugiyama, phonon orbits become
unstable inside the phonon sphere Nitta .

##  VIII Wheeler equation

Figure 5: Real part of the solution to the Wheeler equation ($L=0$,
$k=12.57/r_{s}$) in Eq. (18) shown as a function of $r$. A section of the
solution, over a shaded range of 0.0249 in $r/r_{s}$ in the main frame, is
shown in the inset for clarity. A section of the solution in the inset is also
shaded for clarity. There can be an infinite number of such zoom-in
procedures, like a Russian doll with infinite layers. No definite boundary
value can be imposed on the event horizon for the Fourier expansion over
$[\,0,\,r_{s}\,]$.

For the Schwarzschild metric and weak electromagnetic waves, with negligible
effect on spacetime curvature, the Wheeler equation assumes the form

$\displaystyle\frac{d^{2}R}{dx^{2}}+\left[k^{2}-\frac{L(L+1)}{r^{2}}\,e^{\nu}\right]R=0$
(18)

where $k=\omega/c$, $\omega$ is the circular frequency of the waves, $L$
angular momentum, $e^{\nu}=1-r_{s}/r$ and $x=r+r_{s}\ln|r/r_{s}-1|$ the so-
called tortoise coordinate Fiziev . If we add $3(r_{s}/r^{3})\,e^{\nu}$ to the
terms in the square brackets in Eq. (18) then we find the Regge-Wheeler
equation for gravitational waves which, unlike the Wheeler equation, is
perturbational and approximate Regge .

Eq. (18) is reduced to the Riccati-Bessel equation in flat spacetime, where
$r_{s}\rightarrow 0$, $x\rightarrow r$ and $e^{\nu}\rightarrow 1$. If we let
$R(r)=rZ(r)$ then Eq. (18) is further reduced to the standard spherical Bessel
equation in $Z(r)$ and its first and second order derivatives in $r$
Abramowitz . In curved spacetime, however, it becomes very challenging to
study Eq. (18), partly due to the simultaneous presence of both $x$ and $r$.
Fiziev found Eq. (18) can be transformed into Heun’s equation, with an
explicit transform to remove $x$. He also found a number of exact solutions,
mostly in the form of power series Fiziev . Now the challenge lies in
understanding physical implications of these solutions. For example Fiziev was
not certain if a complete set can be found in a functional space to present a
natural basis of normal modes for an expansion of the solution to Eq. (18)
below the evwnt horizon Fiziev .

It is interesting that when $L=0$ for electromagnetic disturbances without
angular momentum, that is light-rays propagating in the radial direction, the
presence of $r$ in Eq. (18) is automatically removed, and the equation is
reduced to the Helmholtz equation, with a simple solution in the tortoise
coordinate:

$\displaystyle R(x)\propto\exp(i\omega t)\exp(-ikx)$ (19)

of which the real part is evaluated and shown in FIG. 5. It represents a
harmonic wave propagating from $r=0$ to $\infty$. Clearly we are justified to
expect the void below the event horizon being filled with electromagnetic
disturbances, that serves as the premise of our analysis.

In Eq. (19) we have $\omega t-kx=\mbox{const.}$ for the location of a wave
front, giving $\omega dt-kdx=\omega dt-kdr/(1-r_{s}/r)=0$, or
$dr/dt=(1-r_{s}/r)\,c$ as the speed of the wave. Letting $\alpha=0$ in Eqs.
(III) and (III) we also find
$dr/dt=d\ell/dt=(1-r_{s}/r)d\ell^{\prime}/dt^{\prime}=(1-r_{s}/r)\,c$.
Apparently solutions to the Wheeler equation can be identified as light-rays
in our black hole optics, at least in the radial direction, for the very
reason why solutions to the Maxwell equation were identified as light-rays
first time in history.

In Eq. (19) the wave propagates outwardly, on account of the term
$\exp(-ikx)$. The wave can also propagate inwardly, because we can replace
$\exp(-ikx)$ with $\exp(ikx)$. This endorses our understanding that gravity
cannot force a ray to make a U-turn and pull it back to the centre, if the ray
is launched outwardly towards the event horizon at a right angle. It must be
time dilation and length contraction that impedes and eventually stops the
escaping ray on its outward trajectory.

The effect of length contraction is clearly appreciable from FIG. 5. The real
part of $R(r)$ oscillates, pitch progressively higher as $r\rightarrow r_{s}$
indicating that the wave fronts of $R(r)$ are squeezed tighter the closer the
distance from the event horizon. Indeed, in the lower part of FIG. 1, the wave
fronts from the Huygens principle are squeezed tighter towards the event
horizon.

In FIG. 5 we have to shade sections of $r$ close to the event horizon to hide
high pitches for clarity, indicating $R(r)$ has no definite value at $r=r_{s}$
(that is $x=\infty$). Therefore it may not be realistic to expect for a well-
defined expansion of electromagnetic disturbances in the interior of the black
hole, in terms of normal modes (solutions to the Wheeler equation with a
definite boundary value at $r=r_{s}$ due to some physics) Fiziev .

##  IX conclusions

The theory for the gravitational and electromagnetic fields is represented in
an elegant set of compact equations, namely those of Einstein-Maxwell Bonnor ;
Lovelock ; Papapetrou ; Linet ; Bachelot ; Bachelot2 ; Gutsunaev ; Choquet ;
Kroon ; Blue ; Ovsiyuk ; Fernandes , or the Wheeler equation coupled with the
the Schwarzschild metric when spherical symmetry is assumed Wheeler . With an
explicit substitution Fiziev transformed the Wheeler equation into Heun’s
equation which is known to have exact solutions both above and below the event
horizon, in the form of spherical waves propagating both inwardly and
outwardly Fiziev . Therefore the void below the event horizon may not be empty
but filled with electromagnetic disturbances (light-rays) constantly trying to
cross the horizon from below and escape.

To pull back or to slow down to a standstill? That is the question when the
task is to prevent a ray from crossing a certain barrier. It turns out that,
from the optical point of view, light-rays may not be kept below the event
horizon of a black hole by deflection, which cannot force the rays to make a
U-turn and fall back when they are propagating towards the horizon to escape.
It must be length contraction and time dilation that lend deflection a helping
hand and slow the escaping rays to a standstill.

The optical point of view is in accordance with Einstein’s vision with respect
to light-rays in a gravitational field, in terms of the Huygens principle,
when he outlined his plan to develop the general theory of relativity Einstein
. It is immediately clear from his vision that deflection alone cannot stop
light-rays from escaping across the event horizon, because the wavelets in the
Huygens principle cannot propagate in the backward direction. His vision also
accommodates the possibility that light-rays may be slowed down to a
standstill in a strong gravitational field, although at that time the concept
of black hole has not yet been envisaged.

Oppenheimer and Snyder listed three reasons to explain why a collapsing star
may close itself off from any communication with a distant observer, apart
from via its gravitational field. Two of these, the Doppler effect and
gravitational red-shift, are devices from general relativity due to length
contraction and time dilation. The remaining reason, concerning the aperture
of the external Oppenheimer-Snyder cone in Section VI, is due to gravitational
deflection Oppenheimer . It certainly helps to stop light-rays from escaping,
but may be compromised by the tendency of the light-rays from below the event
horizon to encounter the horizon at right angles.

Our black hole optics interprets and verifies the observation of Fiziev that
the Wheeler equation has exact solutions both above and below the event
horizon, in the form of waves propagating in both the outward and inward
directions. Fiziev looks forward to identifying some physics, to impose a
boundary condition upon the event horizon, that would enable him to find a
natural basis of normal modes for a well-defined expansion of any small
perturbation of the metric in the interior of the black hole Fiziev . Our
black hole optics could become a useful alternative to help tackle this
problem, because it entirely is based on the Schwarzschild metric, with no
need for additional physics.

## References

  * (1) W. B. Bonnor, Static Magnetic fields in general relativity Proc. Phys. Soc. A 67, 225 (1954).
  * (2) D. Lovelock, A spherically symmetric solution of the Maxwell-Einstein equations, Commun. math. Phys. 5, 257 (1967).
  * (3) A. Papapetrou, Dipole electromagnetic radiation in a Schwarzschild space: the wavetail to order m/r, J. Phys. A 8, 313 (1975)
  * (4) B. Linet, Dipole field solution of Maxwell’s equations in the Schwarzschild metric, J. Phys. A 8, 328 (1975).
  * (5) A. Bachelot, Scattering by Black-Hole for Electromagnetic Fields, in P. C. Sabatier (eds) Inverse Methods in Action (Berlin, Springer 1990).
  * (6) A. Bachelot, Gravitational scattering of electromagnetic field by Schwarzschild black-hole, Ann. Inst. Henri Poincaré 54, 261 (1991).
  * (7) Ts. I. Gutsunaev and S. L. Él’sgol’ts, A new class of solutions of the magnetostatic Einstein-Maxwell equations, Zh. Eksp. Teor. Fiz. 104, 2257 (1993).
  * (8) Y. Choquet-Bruhat1, P. T. Chruściel and J. Loizelet, Global solutions of the Einstein–Maxwell equations in higher dimensions, Class. Quantum Grav. 23 7383 (2006).
  * (9) J. A. V. Kroon, The Maxwell field on the Schwarzschild space–time: behaviour near spatial infinity, Proc. R. Soc. A 463, 2609 (2007).
  * (10) P. Blue, Decay of the Maxwell field on the Schwarzschild manifold, J. Hyper. Diff. Eq. 5, 807 (2008).
  * (11) E. M. Ovsiyuk, On solutions of Maxwell equations in the space-time of Schwarzschild black hole, Non-linear pheno. complex system. 15, 84 (2012).
  * (12) P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu and N. Sanchis-Gual, Spontaneous scalarisation of charged black holes: coupling dependence and dynamical features, Class. Quantum Grav. 36, 134002 (2019), corrigendum in ibid 37, 049501 (2020).
  * (13) P. P. Fiziev, Exact solutions of Regge-Wheeler equation and quasi-normal modes of compact objects, Class. Quantum Grav. 23, 2447 (2006).
  * (14) J. A. Wheeler, Geons, Phys. Rev. 97, 511 (1955).
  * (15) A. Ronveaux (ed) Heun’s differential equations (Oxford, Oxford, 1995).
  * (16) T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957).
  * (17) A. Einstein, Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes, Ann Physik 35, 898 (1911), English translation in The principle of relativity (New York, Dover, 1952).
  * (18) C. Huygens, Treatise on light (London, MacMillan, 1912).
  * (19) J. R. Oppenheimer and H. Snyder, On continued gravitational contraction, Phys. Rev. 56, 455 (1939).
  * (20) M. Abramowitz and A. Stegun Handbook of mathematical functions (New York, Dover, 1972).
  * (21) D. Nitta, T. Chiba and N. Sugiyama, Shadows of colliding black holes, Phys. Rev. D. 84, 063008 (2011).