Discontinuities in the Evolution of Properties of Kerr Black Holes in the
Extremal Limit

###### Abstract

Ever since its discovery by Roy Kerr in 1963, the geometry around rotating,
electrostatically-neutral black holes, otherwise known as Kerr black holes,
has significantly contributed to theoretical developments in the fields of
general relativity, thermodynamics and beyond. Extremal Kerr black holes, in
which the spin parameter of the rotating black hole is at a maximum without
breaching certain postulates, has especially been of interest to the research
community due to its sensitivity to higher order corrections to Einstein’s
theory of relativity. In this paper, we review some unresolved open questions
regarding the discontinuity of certain properties of the Kerr black hole as it
approaches the extreme limit. These include the form of the innermost stable
circular orbit (ISCO), the nature of vanishing entropy, and the disappearance
of the trapped surface as a Kerr black hole approaches extremality. We
conclude that current theories are likely not suitable for studies of extremal
black holes, and we posit that further development in theory may likely
require a quantum approach to gravity. $G=c=1$ and Einstein summation
convention are assumed throughout this paper.

###### Contents

  1. 1 Introduction
  2. 2 ISCO of extremal vs. near-extremal Kerr black holes
    1. 2.1 Extremal case
    2. 2.2 Near-extremal case
    3. 2.3 Reconciling the discontinuity
  3. 3 Entropy of black holes
    1. 3.1 Origin of entropy in black holes
    2. 3.2 Absence of entropy in extremal case
    3. 3.3 Beyond semi-classical theories
  4. 4 The disappearance of the trapped surface in the extremal limit
    1. 4.1 Trapped surfaces
    2. 4.2 Trapped surfaces in Kerr spacetime
  5. 5 The problem
  6. 6 An optimistic answer
    1. 6.1 New theories?
    2. 6.2 Where to look
  7. 7 A pessimistic answer
  8. 8 Conclusion
  9. 9 Data availability
  10. 10 Conflicts of interest

##  1 Introduction

Black holes, formed as a consequence of supernovae which represent potential
endpoints in the life cycle of the most massive of stars, are regions of
spacetime separated from infinity by an event horizon [1]. A black hole’s
angular momentum is largely derived from the rotation of its collapsing source
star as it proceeds to a supernova. The black hole may also be
electrostatically charged, but in this instance it would be swiftly
neutralized by attracting matter of the opposite charge into the hole. This
rotating, uncharged black hole is known as a Kerr black hole.

The geometry around the Kerr black hole is described by the Kerr metric, which
in Boyer-Lindquist coordinates is represented as the following [2]

$\displaystyle\begin{split}ds^{2}&=-\left(1-\frac{2Mr}{\rho^{2}}\right)dt^{2}-\frac{4Marsin^{2}\theta}{\rho^{2}}d{\phi}dt+\frac{\rho^{2}}{\Delta}dr^{2}+\\\
&\rho^{2}d\theta^{2}+\frac{sin^{2}\theta}{\rho^{2}}\left[\left(r^{2}+a^{2}\right)^{2}-a^{2}{\Delta}sin^{2}\theta\right]d\phi^{2}\end{split}$
(1)

Where $\Delta(r)=r^{2}-2Mr+a^{2}$,
$\rho^{2}(r,\theta)=r^{2}+a^{2}cos^{2}\theta$, and $a=J/M$ is angular momentum
per unit mass.

By inspection, one can deduce that there are two singularities in the metric,
one of which occurs at $\Delta=0$ and the other occurs at $\rho=0$. Through
the analysis of the various scalar quantities one can make by contracting the
Riemann curvature tensor for the metric, it is possible to show that the
$\Delta=0$ singularity is merely a coordinate singularity and can be mitigated
simply by switching to a more appropriate coordinate system. The coordinate
singularities represent the event horizons of the Kerr black hole (analogous
to $r=2M$ in the Schwarzschild case) [2]. The $\rho=0$ singularity is an
intrinsic singularity (analogous to $r=0$ in Schwarzschild case). Thus, to
find the event horizons, we utilise the $\Delta=0$ to find the inner horizon
$r_{-}$and outer horizon $r_{+}$of the black hole to lie at

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ (2)

From this expression, it is apparent that if $M>a$, then there there are two
distinct event horizons. If $M<a$, then the event horizons vanish and we are
left with a naked singularity, something deemed non-physical by the weak
cosmic censorship conjecture [3, 4, 5]. Hence, it is impossible for a black
hole to have $a>M$. In 1974, Kip Thorne proved that the spin of Kerr black
holes would be limited to $a=0.998M$ due to the black hole preferentially
swallowing photons of negative angular momentum emitted by the accretion flow
[6, 7]. While recent papers have discussed the possibility of the existence of
black holes beyond this limit [7, 8, 9], it remains valid that in practice $a$
can never truly reach $M$ without breaching thermodynamic laws (see Section
7). In the theoretical case where $a=M$, the two event horizons merge into
one, otherwise known as an extremal Kerr black hole. The differences in the
geometry of spacetime around extremal and non-extremal Kerr black holes are
contrasted in the conformal diagrams in Fig. 1 below.

Figure 1: Conformal diagrams showing (i) the regions of spacetime in the
vicinity of non-extremal Kerr black holes, and; (ii) the regions of spacetime
in the vicinity of extremal Kerr black holes. Here, dotted lines represent
lines of constant $r$. Region I is asymptotically flat, Region II contains
trapped surfaces (see Section 4.1), and Region III contains the singularity.
The two horizons $r_{+}$ and $r_{-}$ can be seen to merge into one horizon at
$r=m$ as shown in (ii). Sourced from [10].

Near-extremal Kerr black holes are defined as rotating black holes which have
the property $(r_{+}-r_{-})/r_{1}<<1$, where $r_{1}$ is the $r$ coordinate of
the event horizon in the extremal case of a black hole with identical mass
[11].

Extremal black holes serve as a very useful theoretical tool for testing
higher-order corrections to Einstein’s theory of relativity, a theory
considered to be a low energy approximation to a more complete theory [12, 13,
14]. This is because it can be shown that generic stationary, axisymmetric
perturbations generate singularities on the horizon, and when one extends the
near-horizon geometry to a full asymptotically flat solution one necessarily
introduces these singular perturbations [12]. Hence, extremal black holes are
sensitive to higher-order corrections to Einstein’s theory, and in this paper
a discussion of potential failures of Einstein’s theory of general relativity
are highlighted in the form of discontinuities of certain Kerr black hole
properties in the near-extremal limit. These concerns may be effectively
addressed by theories of quantum gravity.

##  2 ISCO of extremal vs. near-extremal Kerr black holes

###  2.1 Extremal case

In this subsection, we analyze the ISCO of the extremal Kerr black hole and
establish its correspondence to null geodesics on the event horizon with
$d\tilde{\phi}/d\tilde{t}=1/2M$, following the work presented in [11].
However, the validity of this correspondence remains unclear in the case of
the near-extremal Kerr black hole (see Section 2.2). Because the Boyer-
Lindquist coordinate system is divergent as $\Delta$ approaches 0, we make use
of the Doran coordinate system yielding the Kerr metric [15]

$\displaystyle\begin{split}ds^{2}&=-d\tilde{t}^{2}+\left[\sqrt{\frac{2Mr}{\rho^{2}}}(d\tilde{t}-asin^{2}{\theta}d\tilde{\phi})+\sqrt{\frac{\rho^{2}}{r^{2}+a^{2}}}dr\right]^{2}\\\
&+\rho^{2}d\theta^{2}+\left(r^{2}+a^{2}\right)sin^{2}{\theta}d\tilde{\phi^{2}}\end{split}$
(3)

which are related with the Boyer-Lindquist coordinates by

$\displaystyle d\tilde{t}=dt+\frac{\beta}{1-\beta^{2}}dr$ (4) $\displaystyle
d\tilde{\phi}=d\phi+\frac{a\beta}{\left(1-\beta^{2}\right)\left(r^{2}+a^{2}\right)}dr$
(5)

Here $\rho$ follows the same definition as in the Boyer-Lindquist form and
$\beta$ is defined by

$\displaystyle\beta=\sqrt{\frac{2Mr}{r^{2}+a^{2}}}$ (6)

Restricting ourselves to orbits in the equatorial plane where $\theta=\pi/2$,
this results in the metric

$\displaystyle\begin{split}ds^{2}&=-\left(1-\frac{2M}{r}\right)d\tilde{t^{2}}-\frac{4M^{2}}{r}d\tilde{t}d\tilde{\phi}+2\sqrt{\frac{2Mr}{r^{2}+M^{2}}}\tilde{dt}dr-\\\
&2M\sqrt{\frac{2Mr}{r^{2}+M^{2}}}drd\tilde{\phi}+\frac{r^{2}}{r^{2}+M^{2}}dr^{2}+\\\
&\left(r^{2}+M^{2}+\frac{2M^{3}}{r}\right)d\tilde{\phi}^{2}\end{split}$ (7)

The metric is independent of the variables $\tilde{t}$ and $\tilde{\phi}$, and
hence $p_{\tilde{\phi}}$ and $p_{\tilde{t}}$ are conserved along geodesics
where $\vec{p}$ is the four-momentum of the test particle [16]. Defining the
conserved quantities $\mathcal{E}=-p_{\tilde{t}}$ and $L=p_{\tilde{\phi}}$
yields

$\displaystyle\mathcal{E}=\left(1-\frac{2M}{r}\right)p^{\tilde{t}}-\sqrt{\frac{2Mr}{r^{2}+M^{2}}}p^{r}+\frac{2M^{2}}{r}p^{\tilde{\phi}}$
(8) $\displaystyle
L=-\frac{2M^{2}}{r}p^{\tilde{t}}-M\sqrt{\frac{2Mr}{r^{2}+M^{2}}}p^{r}+\left(r^{2}+M^{2}+\frac{2M^{3}}{r}\right)p^{\tilde{\phi}}$
(9)

Analyzing circular orbits on the horizon where $r=M$ and $p^{r}=0$, it is
possible to show that $L=2M\mathcal{E}$ [11]. The scalar created when the
four-momentum is contracted with itself at $r=M$ yields

$\displaystyle\vec{p}\cdot\vec{p}=\mathcal{E}^{2}$ (10)

If $\mathcal{E}=0$, then it follows that $L=0$ and this represents null
geodesics on the horizon of the extremal black hole. Using $\mathcal{E}=0$, it
follows that the angular velocity $\Omega$ of the particle is given by

$\displaystyle\Omega=\frac{d\tilde{\phi}}{d\tilde{t}}=\frac{p^{\tilde{\phi}}}{p^{\tilde{t}}}=\frac{1}{2M}$
(11)

###  2.2 Near-extremal case

Summarising the analysis for non-extremal Kerr geometry in the same coordinate
system and using the results presented in [11], one obtains

$\displaystyle\mathcal{E}=\left(1-\frac{2M}{r}\right)p^{\tilde{t}}-\sqrt{\frac{2Mr}{r^{2}+a^{2}}}p^{r}+\frac{2aM}{r}p^{\tilde{\phi}}$
(12) $\displaystyle
L=-\frac{2aM}{r}p^{\tilde{t}}-a\sqrt{\frac{2Mr}{r^{2}+a^{2}}}p^{r}+\left(r^{2}+a^{2}+\frac{2Ma^{2}}{r}\right)p^{\tilde{\phi}}$
(13)

For circular orbits $p^{r}=0$. An expression for $p^{\tilde{t}}$ and
$p^{\tilde{\phi}}$ can be obtained by application of Cramer’s rule. One can
then prove that

$\displaystyle\vec{p}\cdot\vec{p}=-\mathcal{E}p^{\tilde{t}}+Lp^{\tilde{\phi}}$
(14)

For circular orbits at $r=r_{0}$, it is possible to recast Eq. 14 in the
following form

$\displaystyle\vec{p}\cdot\vec{p}=\frac{1}{\Delta}\left[(L^{2}-a^{2}\mathcal{E}^{2})-r_{0}^{2}\mathcal{E}^{2}-\frac{2M}{r_{0}}(L-a\mathcal{E})^{2}\right]$
(15)

For the extremal limit, we apply $a{\rightarrow}M$. Given we are in the
vicinity of the event horizon, we apply $\Delta=(r_{0}-M)^{2}\rightarrow 0$.
Taking these limits, we obtain [11]

$\displaystyle\vec{p}\cdot\vec{p}=-\frac{(L-2M\mathcal{E})^{2}}{(r_{0}-M)^{2}}$
(16)

A difficulty arises when we insert the exact extremal case i.e.
$L=2M\mathcal{E}$ and $r_{0}=M$, into Eq. 16, leading to ambiguous
interpretations for whether we obtain the true null geodesics from the exactly
extremal case given both numerator and denominator are concurrently zero.

###  2.3 Reconciling the discontinuity

As presented by [11], there is no agreement of the ISCO in the limiting case
of the near-extremal Kerr black hole with the exactly extremal case. The limit
may however be considered to be continuous, as suggested by [17] where it is
stated that orbits at $r=r_{0}$ possess the property (to leading order)

$\displaystyle\frac{\Omega-\Omega_{H}}{\Omega_{H}}=-\frac{3(r_{0}-r_{+})}{4r_{+}}$
(17)

Where $\Omega_{H}$ is the horizon angular velocity, defined by

$\displaystyle\Omega_{H}=\frac{a}{a^{2}+r_{+}^{2}}$ (18)

as shown in [1]. It can be seen that $\Omega_{H}$ is a continuous function.
For orbits at the outer horizon $r_{0}=r_{+}$ Eq. 17 vanishes so
$\Omega=\Omega_{H}$ is also continuous.

One possible approach in reconciling this discontinuity involves considering
the event horizon as a Killing horizon. In this approach, the event horizon is
defined as a hypersurface where the Killing vector is null.

In a stationary and non-static spacetime such as the region outside the outer
horizon of a Kerr black hole, the event horizon is also the Killing horizon
for the Killing vector $\chi^{\mu}$ defined in the Boyer-Lindquist coordinate
system by

$\displaystyle\chi^{\mu}=K^{\mu}+{\Omega}_{H}R^{\mu}$ (19)

as discussed in [1]. Here $K^{\mu}=(\partial_{t})^{\mu}$,
$R^{\mu}=(\partial_{\phi})^{\mu}$, with $\Omega_{H}$ a constant. It has been
shown that for the Kerr metric, $\Omega_{H}$ can be identified with the
horizon angular velocity. Transforming this expression into Doran coordinates,
we obtain

$\displaystyle\chi^{\mu^{\prime}}=\frac{{\partial}x^{\mu^{\prime}}}{{\partial}t}+{\Omega_{H}}\frac{{\partial}x^{\mu^{\prime}}}{{\partial}\phi}$
(20)

where primed and unprimed coordinates represent Doran coordinates and Boyer-
Lindquist coordinates, respectively. Using Eq. 4 and 5, the only modifications
required to make Eq. 19 valid in Doran coordinates is a change of $R^{\mu}$
from $({\partial_{\phi}})^{\mu}$ to $({\partial}_{\tilde{\phi}})^{\mu}$ and a
change of $K^{\mu}$ from $({\partial_{t}})^{\mu}$ to
$({\partial_{\tilde{t}}})^{\mu}$.

Null surfaces such as Killing horizons possess the property of being unable to
have two linearly independent null tangent vectors [1]. This means the four-
momentum of the particle, a null tangent vector to one of the geodesics that
make up the generator for the null hypersurface, must be a multiple of the
Killing vector $\vec{\chi}$. From this, we can conclude that

$\displaystyle\Omega=\frac{d\tilde{\phi}}{d{\tilde{t}}}=\frac{d\tilde{\phi}/d\lambda}{d\tilde{t}/d\lambda}=\frac{p^{\tilde{{\phi}}}}{p^{\tilde{t}}}=\frac{{\chi}^{\tilde{\phi}}}{{\chi}^{\tilde{t}}}=\Omega_{H}$
(21)

Where $\lambda$ is some affine parameter. For extremal Kerr black holes the
circular null geodesic on the horizon has $\Omega=1/2M$ (Section 2.1), which
is exactly $\Omega_{H}$ in the extremal case, making this equation valid for
the exactly extremal case. Eq. 21 is continuous, yielding the same result in
the limiting case of near-extremal black holes, thereby potentially mitigating
the issue of discontinuity.

##  3 Entropy of black holes

###  3.1 Origin of entropy in black holes

At first review, it may be considered counter-intuitive for black holes to
have non-zero entropy. According to statistical mechanics $S=kln\Omega$ where
$S$ is the entropy, $k$ is Boltzmann’s constant, and $\Omega$ is the number of
microstates associated with a particular macrostate [18]. According to the no-
hair theorem of black holes [19] recently proven to be true for isolated,
asymptotically flat, stationary, axisymmetric black holes (such as Kerr black
holes in $f(R)$ theories of gravity) [20], the characteristics of a black hole
are fully determined by its mass, charge, and angular momentum. Hence, one may
naively conclude that black holes have no entropy as there is only one
configuration of charge, mass, and angular momentum that could create a
particular black hole. It should however be noted that the no-hair theorem may
not be valid in alternate theories of gravity, as discussed in [21, 22, 23].

If there is a single time-independent field without event horizons, such as
the case of a hypothetical naked singularity, then there is only one
configuration of stress energy that could generate the field, and this would
correspond to zero entropy [24]. However, it is important to consider that the
black hole must have been formed by the collapse of matter, and the contents
of that matter are shielded from the outside observers by the non-stationary
region (non-stationary meaning all Killing vectors are spacelike) between the
inner and outer horizons. This region is present for all non-extremal black
holes, including Reissner-Nordstrom black holes and Kerr-Newman black holes
[1]. The contents of the material inside this region cannot be determined
outside the hole, and this generates the microstates that give the black hole
its entropy.

According to Bekenstein, the entropy of black holes originates from the
ignorance of the internal configurations of the black hole to outside
observers [24, 25]. We may therefore expect that the entropy of a black hole
may in fact be quite large; the black hole could be made from anything and
arranged in any shape, pattern, or size, as long as its net charge, net
angular momentum, and total mass parameters are valid. In the Bekenstein
approach, a black hole of one solar mass would have an entropy of
approximately 18 orders of magnitude larger than the thermodynamic entropy of
the Sun in our solar system [25].

###  3.2 Absence of entropy in extremal case

It has been proven that the entropy of a black hole is proportional to the
area of its event horizon, with the proportionality constant being $1/4\hbar$
in units where $G=c=1$ [26, 27, 28, 29]. This is true for black holes close to
the extremal limit, but many research papers have pointed out that this
equation breaks down for the exactly extremal black hole [24, 30, 31]. This is
to be expected, as in the extremal limit the two event horizons merge into
one, and the non-stationary region, which is what gives the black hole its
entropy in the first place, vanishes. Here, we outline a review of how
classical results break down, reviewing the work by Howard [29]. Suppose the
entropy of the black hole is given by some arbitrary function of the area of
an event horizon

$\displaystyle S_{BH}=f(A)$ (22)

If the area of the event horizon is perturbed by some small amount
${\delta}A$, then f(A) is changed by some amount ${\delta}f(A)$, given by

$\displaystyle{\delta}f(A)=\frac{df(A)}{dA}{\delta}A$ (23)

This change in event horizon area must have been caused by an alteration in
the internal configurations of the black hole, and this could occur if matter
falls into the black hole. Suppose a particle were to fall into the black
hole. After it crosses the outer horizon, there is no information about the
particle that can be obtained for outside observers. Hence, the particle could
either still exist or not exist with equal probability, and this gives two
possible microstates to the macrostate. Thus, the minimum entropy change is
[29]

$\displaystyle({\delta}f(A))_{min}=kln(\Omega)=kln2$ (24)

For Kerr black holes, the first law of thermodynamics reads [32]

$\displaystyle dM=TdS+\Omega_{H}dJ$ (25)

Which for small perturbations in black hole mass, entropy and angular momentum
can be recast in the following form

$\displaystyle{\delta}M-\Omega_{H}{\delta}J=T{\delta}S$ (26)

The black hole’s event horizon area will change accordingly by [1]

$\displaystyle{\delta}A=\frac{8{\pi}a({\delta}M-\Omega_{H}{\delta}J)}{{\Omega_{H}}\sqrt{M^{2}-a^{2}}}$
(27)

As extremal black holes have zero temperature [1], the expression in Eq. 26 is
equal to zero, and hence the numerator in Eq. 27 also equals zero. However,
the denominator in Eq. 27 vanishes too, and so the expression for ${\delta}A$
is undefined. Hence, no conclusions can be drawn for the value of $df(A)/dA$
indicating a failure of the classical approach.

###  3.3 Beyond semi-classical theories

It is pertinent to note that while the Bekenstein entropy equation can be
proven under certain assumptions [1], there are many alternate modern theories
regarding the entropy of black holes that extend beyond semi-classical
methods. Some theories of quantum gravity explain the entropy by attributing
it to the quantum field correlations between the exterior and interior of the
black hole [30]. Others relate the black hole entropy to the ordinary entropy
of its thermal atmosphere [33]. Attempts have also been made to explain the
entropy of black holes in the context of Sakharov’s theory of induced gravity,
in which the dynamical aspects of gravity arise from the collective
excitations of massive fields [33]. More recent approaches involve the
framework of quantum geometry [33]. The holographic principle attributes the
entropy to the encoding of the degrees of freedom in a gravitating system to
its spatial boundaries [34]. String theorists have also made attempts to
explain the entropy by using methods of microstate counting. It has been found
that a string theory approach gives a non-vanishing entropy for exactly
extremal black holes, and gives a value which agrees with the classical
prediction of the value of entropy calculated when taking the near-extremal
limit of non-extremal black holes [30, 33]. For the We consider extremal black
holes to have no entropy, as there are far more research papers which hold
this viewpoint [29, 31, 24, 26].

##  4 The disappearance of the trapped surface in the extremal limit

###  4.1 Trapped surfaces

A trapped surface is defined by a compact two-dimensional submanifold embedded
in original manifold such that the expansion for all future directed light
rays is negative everywhere on the submanifold [1].

Figure 2: Overview of the definition of trapped surfaces, adapted from [10].
Here a trapped surface is defined as a compact two-dimensional submanifold
embedded in original manifold. See text for a further description.

The expansion $\theta$ is defined by

$\displaystyle\theta=\nabla_{\mu}{q}^{\mu}$ (28)

where ${q}^{\mu}=dx^{\mu}/d\lambda$ is the null vector representing the light
ray. Light rays with this property are said to converge [1]. From Fig. 2,
consider a sphere $S$ which emits a flash of light. At some later time, the
light emitted from a point $p$ on the sphere $S$ would form a sphere $S_{3}$
which touches the wavefront of the ingoing light $S_{2}$ and outgoing light
$S_{1}$. If the expansion is less than zero, then the surface area of the
wavefront will be smaller than S, and if both the ingoing and outgoing
wavefronts satisfy this condition then matter inside, which cannot travel
faster than light, are trapped inside the surface whose area falls to zero
within a finite time [10].

###  4.2 Trapped surfaces in Kerr spacetime

For these calculations, we work with the metric induced by the original metric
on the spatial slices generated by the null geodesic as time progresses,
defined by [11]

$\displaystyle\gamma_{{\mu}{\nu}}=g_{{\mu}{\nu}}+l_{\mu}n_{\nu}+n_{\mu}l_{\nu}$
(29)

Where $l^{\mu}$ represent an ingoing future directed null vector and $n^{\mu}$
represents an outgoing one. Here $g_{{\mu}{\nu}}$ is the metric in Boyer-
Lindquist coordinates. Of course, we could scale the vectors by arbitrary
constant scalars and the vectors would still represent future directed null
vectors, and hence for the sake of simplicity we choose the two vectors such
that

$\displaystyle l^{\mu}n_{\mu}=-1$ (30)

Then the two null rays can be represented as follows [11]

$\displaystyle l^{\mu}=\frac{1}{\Delta}(r^{2}+a^{2},-\Delta,0,a)$ (31)
$\displaystyle n^{\mu}=\frac{1}{2r^{2}}(r^{2}+a^{2},\Delta,0,a)$ (32)

Using this, one can calculate the expansion for both these null vectors. They
are [11]

$\displaystyle\theta_{(l)}=-\frac{2}{r}$ (33)
$\displaystyle\theta_{(n)}=\frac{\Delta}{r^{3}}$ (34)

$\theta_{(l)}$ always converges, but this is to be expected as it is ingoing.
The ingoing null ray emanating into spacetime from the surface of a two-sphere
embedded in Minkowski spacetime converges too. Both of these are negative for
$r_{-}<r<r_{+}$, which indicates the existence of trapped surfaces. However,
if the black hole is extremal then $r_{-}=r_{+}$ and this region would vanish.
This means $\Delta$ is never negative and so there is no trapped surface in
the extremal case. This can also be seen from Fig. 1 as Region II in diagram
(ii) vanishes.

##  5 The problem

Most concepts in physics involve the concept of differentiability, something
lost if discontinuities are present. Consequently, having discontinuities in
certain physical properties such as entropy or the form of the ISCO could
potentially violate known physics. In classical statistical mechanics, entropy
is a function of state which can be derived from the partition function, a
continuous function [18]. In the classical theory of general relativity,
Einstein’s field equation link the energy-momentum of a system to it’s
spacetime curvature in a continuous manner, which would not permit a sudden
change in the form of the ISCO as $a{\rightarrow}M$. The disappearance of the
trapped surface further shows the continuity issues with viewing extremal Kerr
black holes as the limiting case of near-extremal Kerr black holes. These
discontinuities in properties spark consequences which violate known physics,
and hence present an open problem in the realm of theoretical physics.

##  6 An optimistic answer

###  6.1 New theories?

General relativity is one of the simplest possible metric theories of gravity,
and it has successfully withstood all experimental tests conducted so far. The
equation that link the metric tensor with the stress energy tensor is known as
Einstein’s field equation, and one way to derive it is through the principle
of least action. In n dimensions, the equation for the action S is given by
[1]

$\displaystyle S=\int{\mathcal{L}({\Phi^{i},{\nabla_{\mu}{\Phi}}^{i}}})d^{n}x$
(35)

where $\mathcal{L}$ is the Lagrange density, and ${\Phi}^{i}(x)$ a set of
fields which act as the dynamical variables. In general relativity, the
dynamical variable is the metric $g_{{\mu}{\nu}}$, and it is necessary to
construct a scalar from the metric that could serve as the Lagrange density
[1]. Since the laws of general relativity permits us to analyze local problems
in local Lorentz frames where the equations of special relativity can be used
[16], it follows that metric can be set to it’s canonical form and first
derivatives all set to zero at any point in the manifold. Hence, any
nontrivial scalar must have at minimum second derivatives of $g_{{\mu}{\nu}}$.
Naturally, the Riemann curvature tensor comes to mind. It can be shown that
any nontrivial tensor made from products of the metric, it’s derivatives, and
it’s second derivatives can be expressed in terms of the metric and the
Riemann curvature tensor [1]. The Ricci scalar R is a rank 0 tensor made from
products of the metric, it’s derivatives, and it’s second derivatives. It is
also the only independent scalar one could construct from the Riemann
curvature tensor [1]. Hence, the only scalar choice for the Lagrange density
which is no higher than second order in the derivatives of the metric tensor
is the Ricci scalar. Technically, multiples would also suffice, but they don’t
affect how we minimize the action. The Hilbert action $S_{H}$ is given by

$\displaystyle S_{H}=\int{\sqrt{-g}Rd^{n}x}$ (36)

Where $g$ represents the determinant of the metric tensor. It can be shown
that minimizing this function yields Einstein’s equation in vacuum [1].

But why is it acceptable to only consider terms no larger than second order?
What about terms like $R^{2}$, $R_{{\mu}{\nu}}R^{{\mu}{\nu}}$,
${\nabla}^{\mu}R{\nabla}_{\mu}R$? Why rule them out? While no experiments have
revealed these terms, theories regarding quantum gravity have suggested the
existence of these terms [13, 35, 36]. For example, one problem that has
plagued Einstein’s theory is that when one attempts to consistently quantize
general relativity, one discovers that the theory is not renormalizable. As it
turns out, this obstacle can be removed if one adds the right combination of
higher-order Lagrange densities to the original Lagrange density [1, 37].

The author holds the view that perhaps the non-physical discontinuities one
encounters when taking the extremal limit of a black hole could be due to our
current theory of relativity being incomplete, as it is just a classical
theory and not a quantum one. After all, the discontinuity problem of entropy
vanishes if one analyzes the problem from the point of view of microstates
using string theory, and as mentioned earlier in the paper these black holes
are incredibly sensitive to higher-order corrections to Einstein’s theory. So
it isn’t too far fetched to believe that corrections from string theory or
quantum physics can be the key to solving the other problems too. Also, while
discontinuities are indeed non-physical in classical theories, they occur all
the time in quantum physics (for example the possible eigenvalues of the
Hamiltonian operator in bound states). It is entirely possible that when we
look back at this problem in the future equipped with a more complete theory
of quantum gravity these problems will fade away.

###  6.2 Where to look

In particular, I believe that string theory is most likely to resolve the
discrepancies of the discontinuites that occur in the extremal limit. Many
theorems in classical physics, for example the equipartition theorem, require
that the thermal energy is much larger than the energy gap between quantized
energy levels so that quantum effects can be washed out [18]. This is clearly
not the case for extremal black holes as they have vanishing temperature, so
it follows that to model extremal black holes quantum theories must be used.
Hence, it comes as no surprise that Einstein’s classical theory of relativity
breaks down. What we are searching for then is a theory that can unify quantum
physics and gravitation, and this is precisely what string theory is.

String theory can fix many problems in general relativity, like for example
the discontinuity in black hole entropy in the extremal limit and the problems
that occur when one attempts to quantize Einstein’s theory [38]. There are
other reasons to believe that string theory is a good unified theory of
quantum physics and gravity. For example, there are no adjustable parameters
in string theory, as opposed to in the Standard Model of particle physics
where there are about twenty [38]. The absence of adjustable parameters mean
we know for a fact that researchers aren’t just adjusting the parameters to
fit other known theories, giving any confirmations to the theory more value.
There is an abundance of research papers on the topic of black holes that make
use of string theory with many different approaches to explain the physics
behind them (see [39] for a list of approaches), and the author believes that
there is a reason to this. The author would suggest any future researchers who
want to delve into the topic of the discontinuity of properties of Kerr black
holes in the extremal limit to look into string theory, as the author believes
that this theory is most likely to bear fruit.

##  7 A pessimistic answer

Another possibility is that it is not Einstein’s theory that is incomplete,
but merely just that an unrealistic system is analyzed. It has been proven
that black holes cannot reach $a=M$ as mentioned previously in the paper. To
further prove that extremal black holes cannot form, the author devised a
thought experiment of his own.

We propose a hypothetical situation where a black hole has just formed.
Initially it would not be extremal as extremal black holes have zero
temperature, which according to the third law of black hole thermodynamics
(The weaker version of the third law of thermodynamics, to which black holes
obey) cannot be obtained within a finite amount of steps [40, 28] and hence
cannot form as a result of matter collapse. To increase it’s angular momentum
per unit mass it would have to change it’s internal configurations. One way it
could do this is by absorbing matter with positive angular momentum to
increase the value of angular momentum per unit mass. Suppose at some instance
it is very very close to extremality. At this instant a tennis ball is thrown
at it which has just the right mass and angular momentum to turn it into an
extremal Kerr black hole. Initially, the black hole and tennis ball would both
have entropy, but after the tennis ball falls in the black hole becomes
extremal and the entropy vanishes. This is a clear violation of the
generalized second law, which states that the net entropy of matter and black
holes can never decrease in an isolated system [28, 26].

Another way it could change it’s internal configurations is by emitting
Hawking radiation. Suppose again we have a black hole that is very very close
to extremality, and by emitting Hawking radiation it reduces it’s mass just
enough such that $a/M=1$ and it becomes extremal. Before the radiation is
emitted, the black hole has entropy, but after the radiation is emitted the
entropy vanishes, again violating the generalized second law of
thermodynamics. Hence, we conclude that for black holes close to extremality
Hawking radiation must affect it’s physical parameters in a way such that it
evolves away from extremality, as also suggested by other studies [41]. As
neither the emission nor the absorption of particles can generate an extremal
black hole without breaching the generalized second law of thermodynamics, it
is clear that extremal Kerr black holes are non-physical.

Utilizing general relativity to study extremal Kerr black holes can be likened
to employing special relativity to examine scenarios where the object’s
velocity exceeds the speed of light. In such cases, peculiar conclusions may
arise, such as the Lorentz factor becoming imaginary, but that’s not because
the theory is flawed, it’s because an impossible system is being analyzed. So
it is also possible that the non-physical discontinuities coming up in the
theory of extremal Kerr black holes is merely because we are dealing with an
unrealistic system, and that further analysis of it may not be of much value.

##  8 Conclusion

This paper has shown that evidently the details of the theory of extremal
black holes is still lacking, as there are many non-physical discontinuities
in the evolution of the physical properties of the black hole as it approaches
extremality. First, It is revealed that while the ISCO of the extremal black
hole can be shown to be represented by null geodesics on the horizon of the
black hole, it remains unclear whether this holds true in the extremal case.
We then went on to show that while non-extremal black holes have entropy that
obey the Bekenstein-Hawking formula, the entropy for extremal black holes
vanish although it’s event horizon area does not, leading to a disagreement
with the formula and another discontinuity. Finally, we showed that the
trapped surface disappears when a black hole reaches extremality.

As of March 2024, there are no concrete explanations for these
discontinuities. Of course, it could just be because a non-physical system is
analyzed, but the author is more tempted to believe that extremal black holes
could hold the key to new, potentially revolutionizing physics. Gravity is the
weakest of the four fundamental forces, and hence little is known about it’s
effects on small scales where usually other forces such as electromagnetic
forces dominate. In the extreme conditions of black holes, gravity is the
force that dominates over the other forces, and this could lead to failure of
some of our classic models and would require a quantum theory to explain.
Progress is currently being made by these non-classical theories to explain
the black hole’s entropy [30, 42, 43, 44, 45], and it may just be a matter of
time before experts in the field develop a more complete theory where all the
discontinuity problems vanish, and we will then look back at these problems in
the same way we view the ultraviolet catastrophe today. Not only will this
help us better understand black holes, but will also be helpful in analysis of
other areas where gravity is strong such as the interior of neutron stars and
the properties of the universe during the very first moments of the big bang.
In that sense, extremal black holes could act as a testing site for these new
theories, as they are very sensitive to higher order corrections of spacetime
curvature. Indeed, the possibilities are boundless, and the further research
can be pursued by future researchers.

##  9 Data availability

No new data was used in this paper.

##  10 Conflicts of interest

The authors declare no conflicts of interest.

## References

  * [1] S. M. Carroll, Spacetime and geometry: an introduction to general relativity.  Harlow: Pearson Education, pearson new international edition ed., 2014.
  * [2] S. A. Teukolsky, “The Kerr Metric,” Classical and Quantum Gravity, vol. 32, p. 124006, June 2015.  arXiv:1410.2130 [astro-ph, physics:gr-qc, physics:hep-th].
  * [3] M. Zhao, M. Tang, and Z. Xu, “Testing The Weak Cosmic Censorship Conjecture in Short Haired Black Holes,” Feb. 2024.  arXiv:2402.16373 [gr-qc].
  * [4] S. Shaymatov, N. Dadhich, and B. Ahmedov, “The higher dimensional Myers-Perry black hole with single rotation always obeys the Cosmic Censorship Conjecture,” The European Physical Journal C, vol. 79, p. 585, July 2019.  arXiv:1809.10457 [astro-ph, physics:gr-qc, physics:hep-th].
  * [5] K. Landsman, “Penrose’s 1965 singularity theorem: From geodesic incompleteness to cosmic censorship,” General Relativity and Gravitation, vol. 54, p. 115, Oct. 2022.  arXiv:2205.01680 [gr-qc, physics:math-ph, physics:physics].
  * [6] H. Kunde, “On Kip Thorne’s limit for rotation of black holes,” 2021.
  * [7] A. J. Benson and A. Babul, “Maximum Spin of Black Holes Driving Jets,” Monthly Notices of the Royal Astronomical Society, vol. 397, pp. 1302–1313, Aug. 2009.  arXiv:0905.2378 [astro-ph].
  * [8] S. Aretakis, G. Khanna, and S. Sabharwal, “An observational signature for extremal black holes,” Jan. 2024.  arXiv:2307.03963 [gr-qc, physics:hep-th, physics:math-ph].
  * [9] A. Arbey, J. Auffinger, and J. Silk, “Evolution of primordial black hole spin due to Hawking radiation,” Monthly Notices of the Royal Astronomical Society, vol. 494, pp. 1257–1262, May 2020.  arXiv:1906.04196 [astro-ph, physics:gr-qc].
  * [10] S. Hawking, G. F. R. Ellis, and A. Ashtekar, The large scale structure of space-time.  Cambridge monographs on mathematical physics, Cambridge, United Kingdom: Cambridge University Press, 50th anniversary edition ed., 2023.
  * [11] P. Pradhan and P. Majumdar, “Extremal Limits and Kerr Spacetime,” The European Physical Journal C, vol. 73, p. 2470, June 2013.  arXiv:1108.2333 [gr-qc].
  * [12] G. T. Horowitz, M. Kolanowski, G. N. Remmen, and J. E. Santos, “Extremal Kerr black holes as amplifiers of new physics,” Physical Review Letters, vol. 131, p. 091402, Aug. 2023.  arXiv:2303.07358 [astro-ph, physics:gr-qc, physics:hep-ph, physics:hep-th].
  * [13] S. Silveravalle, “The gravitational field of isolated objects in quadratic gravity,” Il Nuovo Cimento C, vol. 45, pp. 1–4, June 2022.  arXiv:2202.00999 [gr-qc, physics:hep-th].
  * [14] G. Gabadadze, “A New Gravitational Action For The Trace Anomaly,” Physics Letters B, vol. 843, p. 138031, Aug. 2023.  arXiv:2301.13265 [gr-qc, physics:hep-ph, physics:hep-th].
  * [15] T. Jacobson, “Where is the extremal Kerr ISCO?,” Classical and Quantum Gravity, vol. 28, p. 187001, Sept. 2011.  arXiv:1107.5081 [gr-qc, physics:hep-th].
  * [16] B. F. Schutz, A First Course in General Relativity.  Cambridge: University Press, second edition, 13th printing ed., 2018.
  * [17] S. E. Gralla, A. P. Porfyriadis, and N. Warburton, “Particle on the innermost stable circular orbit of a rapidly spinning black hole,” Physical Review D, vol. 92, p. 064029, Sept. 2015.
  * [18] S. Blundell and K. M. Blundell, Concepts in thermal physics.  Oxford: Oxford university press, 2nd ed ed., 2010.
  * [19] R. Ruffini and J. A. Wheeler, “Introducing the black hole,” Physics Today, vol. 24, pp. 30–41, Jan. 1971.
  * [20] J. Sultana and D. Kazanas, “A no-hair theorem for spherically symmetric black holes in $R^2$ gravity,” General Relativity and Gravitation, vol. 50, p. 137, Nov. 2018.  arXiv:1810.02915 [gr-qc].
  * [21] Y. Brihaye, C. Herdeiro, E. Radu, and D. H. Tchrakian, “Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension,” Journal of High Energy Physics, vol. 2017, p. 37, Nov. 2017.  arXiv:1710.03833 [gr-qc, physics:hep-th].
  * [22] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, “Dilatonic Black Holes in Higher-Curvature String Gravity II: Linear Stability,” Physical Review D, vol. 57, pp. 6255–6264, May 1998.  arXiv:hep-th/9703192.
  * [23] J. D. Bekenstein, “Black hole hair: twenty–five years after,” May 1996.  arXiv:gr-qc/9605059.
  * [24] A. Edery and B. Constantineau, “Extremal black holes, gravitational entropy and nonstationary metric fields,” Classical and Quantum Gravity, vol. 28, p. 045003, Feb. 2011.  arXiv:1010.5844 [gr-qc, physics:hep-th].
  * [25] J. D. Bekenstein, “Black Holes and Entropy,” Physical Review D, vol. 7, pp. 2333–2346, Apr. 1973.
  * [26] C. O. Lousto, “The Fourth Law of Black Hole Thermodynamics,” Nuclear Physics B, vol. 410, pp. 155–172, Dec. 1993.  arXiv:gr-qc/9306014.
  * [27] A. Strominger and C. Vafa, “Microscopic Origin of the Bekenstein-Hawking Entropy,” Physics Letters B, vol. 379, pp. 99–104, June 1996.  arXiv:hep-th/9601029.
  * [28] S. Carlip, “Black Hole Thermodynamics,” International Journal of Modern Physics D, vol. 23, p. 1430023, Oct. 2014.  arXiv:1410.1486 [gr-qc].
  * [29] E. M. Howard, “Geometric aspects of Extremal Kerr black hole entropy,” Journal of Modern Physics, vol. 04, no. 03, pp. 357–363, 2013.  arXiv:1511.00594 [gr-qc].
  * [30] S. M. Carroll, M. C. Johnson, and L. Randall, “Extremal limits and black hole entropy,” Journal of High Energy Physics, vol. 2009, pp. 109–109, Nov. 2009.  arXiv:0901.0931 [hep-th].
  * [31] F. Belgiorno and M. Martellini, “Black Holes and the Third Law of Thermodynamics,” International Journal of Modern Physics D, vol. 13, pp. 739–770, Apr. 2004.  arXiv:gr-qc/0210026.
  * [32] M. Johnstone, M. M. Sheikh-Jabbari, J. Simon, and H. Yavartanoo, “Extremal Black Holes and First Law of Thermodynamics,” Physical Review D, vol. 88, p. 101503, Nov. 2013.  arXiv:1305.3157 [hep-th].
  * [33] R. M. Wald, “The Thermodynamics of Black Holes,” Living Reviews in Relativity, vol. 4, p. 6, Dec. 2001.  arXiv:gr-qc/9912119.
  * [34] A. C. Wall, “A Survey of Black Hole Thermodynamics,” July 2018.  arXiv:1804.10610 [gr-qc, physics:hep-th].
  * [35] S. Capozziello and G. Lambiase, “Higher-Order Corrections to the Effective Gravitational Action from Noether Symmetry Approach,” General Relativity and Gravitation, vol. 32, pp. 295–311, Feb. 2000.  arXiv:gr-qc/9912084.
  * [36] E. Velasco-Aja and J. Anero, “Quantum corrections to Einstein’s equations,” Sept. 2022.  arXiv:2209.06938 [gr-qc].
  * [37] J. Bezerra-Sobrinho and L. G. Medeiros, “Modified Starobinsky inflation by the $R\ln\left( \square\right) R$ term,” Journal of Cosmology and Astroparticle Physics, vol. 2023, p. 039, Jan. 2023.  arXiv:2202.13308 [astro-ph, physics:gr-qc].
  * [38] B. Zwiebach, A first course in string theory.  Cambridge ; New York: Cambridge University Press, 2nd ed ed., 2009.
  * [39] R. C. Myers, “Black Holes and String Theory,” July 2001.  arXiv:gr-qc/0107034.
  * [40] M. Socolovsky, “Black Holes and the Third Law of Thermodynamics Revisited,” Journal of High Energy Physics, Gravitation and Cosmology, vol. 09, no. 02, pp. 499–505, 2023.  arXiv:2304.07890 [gr-qc].
  * [41] S. Di Gennaro and Y. C. Ong, “How Not to Extract Information From Black Holes: Cosmic Censorship as a Guiding Principle,” Physics Letters B, vol. 829, p. 137112, June 2022.  arXiv:2103.05516 [gr-qc, physics:hep-th].
  * [42] S. De Haro, J. van Dongen, M. Visser, and J. Butterfield, “Conceptual Analysis of Black Hole Entropy in String Theory,” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, vol. 69, pp. 82–111, Feb. 2020.  arXiv:1904.03232 [hep-th, physics:physics].
  * [43] S. R. Das, “Black Hole Entropy and String Theory,” Feb. 1996.  arXiv:hep-th/9602172 version: 1.
  * [44] N. Dorey, R. Mouland, and B. Zhao, “Black Hole Entropy from Quantum Mechanics,” Journal of High Energy Physics, vol. 2023, p. 166, June 2023.  arXiv:2207.12477 [hep-th].
  * [45] Y. Nomura and S. J. Weinberg, “Black Holes, Entropies, and Semiclassical Spacetime in Quantum Gravity,” Journal of High Energy Physics, vol. 2014, p. 185, Oct. 2014.  arXiv:1406.1505 [gr-qc, physics:hep-th].


Unitary rotation of
pixellated polychromatic images
Alejandro R. Urzúa and Kurt Bernardo Wolf

Instituto de Ciencias Físicas

Universidad Nacional Autónoma de México

Av. Universidad s/n, Cuernavaca, Morelos 62251, México

###### Abstract

Unitary rotations of polychromatic images on finite two-dimensional pixellated
screens provide invertibility, group composition, and thus conservation of
information. Rotations have been applied on monochromatic image data sets,
where we now examine closer the Gibbs-like oscillations that appear due to
discrete ‘discontinuities’ of the input images under unitary transformations.
Extended to three-color images we examine here the display of color at the
pixels where, due to the oscillations, some pixel color values may fall
outside their required common numerical range $[0,1]$, between absence and
saturation of the red, green, and blue formant color images.

##  1 Introduction

In geometric and in wave optics, transformations that respect the Hamiltonian
structure of the models are called canonical; they are termed linear when the
phase space coordinates are linearly mapped. Images displayed on screens can
undergo linear maps representing free propagation, rotation, gyration,
squeezing, fractional Fourier and other transformations for one- or two-
dimensional images on line or plane screens.

The discrete version of these systems and transformations, are based on the
finite oscillator model [1, 2, 3], which considers finite pixellated line or
plane screens that bear the set of generally complex data values that form the
image. The requirement that the transformations be unitary to conserve Hilbert
space properties of the image data sets (orthogonality, reality), requires
them to be elements of a group, of which screen rotations form a one-parameter
subgroup, to guarantee invertibility and proper concatenation for two or more
such transformations. In particular, when the pixel values are real, they
should remain so under rotation.

For this article to be essentially self-contained, in Sect. 2 we remind the
reader of the basic finite oscillator model for $D=1$ and $D=2$-dimensional
finite pixellated images along Cartesian coordinates [4]. We emphasize that
this rotation algorithm provides the only unitary rotation coefficients for
the image pixels on an $N_{x}\times N_{y}$ generally rectangular screen [5].
For simplicity we address here only square $N\times N$ images and screens.

In this paper we extend unitary rotations to polychromatic pixellated arrays
that carry the three component color values (for red, green and blue) as
generally used in the applied literature. In the polychromatic case, the
ranges of the pixel values are restricted to the common interval $[0,1]$,
between absence and saturation of each color. The main problem that appears
now is that of Gibbs-like oscillations in one or more color component images
of pixel values that may fall outside that basic $[0,1]$ range. This must be
addressed with an appropriate display tactic, which is laid out in Sect. 3.
Finally, in the concluding Sect. 4, we comment further on linear unitary
transformations of pixellated data that are required to conserve information
and Hamiltonian structure.

##  2 Finite one- and two-dimensional images

The finite oscillator model [6] is based on the Lie algebra of spin su($2$),
with generators $\\{J_{1},J_{2},J_{3}\\}$ of commutation relations
$[J_{i},J_{j}]=\varepsilon_{i,j,k}J_{k}$ ($i,j,k$ cyclic), whose
representation multiplets, of dimension $N=2j+1$ (with $j$ a non-negative
integer or half-integer), are interpreted to contain its $N$ equidistant
eigenstates. The eigenvalues of the generator $Q:=J_{1}$ characterize the $N$
pixel positions on a screen, $P:=J_{2}$ is momentum, while those of $K:=J_{3}$
number the $N$ ‘energy’ states. The overlap between the position and mode
eigenbases yields the finite oscillator wavefunctions. This finite model
mimics the usual ‘continuous’ harmonic oscillator based on the Heisenberg-Weyl
algebra (of commutator $[\overline{Q},\overline{P}]={\rm i}\hbar{\it 1}$,
where the spectrum of position $\overline{Q}$ and of momentum $\overline{P}$
are the full real line), whose energy eigenstates of
$\overline{H}:={\textstyle\frac{1}{2}}(\overline{P}^{2}+\overline{Q}^{2})$ are
equally-spaced with only a lower bound.

###  2.1 The one-dimensional finite oscillator and screen

For $D=1$-dimensional screens of $N$ pixels in a line, the finite quantization
of the harmonic oscillator replaces the commutator
$[\overline{Q},\overline{P}]={\rm i}{\it 1}$ of the continuous model with
$[Q,P]={\rm i}K$ in their $N=2j+1$-dimensional representation [1]. Finite
quantization thus assigns $N\times N$ su($2$) matrix representations $\bf Q$
and $\bf P$ to position and momentum $q$ and $p$, as follows:

$\displaystyle{\hskip-15.0pt}\hbox{position}:\quad q\mapsto Q:=J_{1},\quad{\bf
Q}=\|Q_{q,q^{\prime}}\|,$ (1) $\displaystyle
Q_{q,q^{\prime}}=q\,\delta_{q,q^{\prime}},\qquad
q,q^{\prime}\in\\{-j,-j{+}1,\ldots\,,j\\},\quad N=:2j+1,$
$\displaystyle{\hskip-15.0pt}\hbox{momentum}:\quad p\mapsto P:=J_{2},\quad{\bf
P}=\|P_{q,q^{\prime}}\|,$ (2) $\displaystyle P_{q,q^{\prime}}=-{\rm
i}{\textstyle\frac{1}{2}}\sqrt{(j{-}q)(j{+}q{+}1)}\,\delta_{q+1,q^{\prime}}+{\rm
i}{\textstyle\frac{1}{2}}\sqrt{(j{+}q)(j{-}q{+}1)}\,\delta_{q-1,q^{\prime}},$
$\displaystyle{\hskip-15.0pt}\hbox{and their commutator}:\quad
K:=J_{3},\quad{\bf K}=\|K_{q,q^{\prime}}\|,$ (3) $\displaystyle
K_{q,q^{\prime}}={\textstyle\frac{1}{2}}\sqrt{(j{-}q)(j{+}q{+}1)}\,\delta_{q+1,q^{\prime}}+{\textstyle\frac{1}{2}}\sqrt{(j{+}q)(j{-}q{+}1)}\,\delta_{q-1,q^{\prime}}.$

These three operators are thus represented by self-adjoint $N\times N$
matrices in the vector space with inner product
$\sum_{q=-j}^{j}f_{q}g_{q}^{*}$, and generate the Lie algebra su($2$) with
commutation relations

$[J_{i},J_{j}]={\rm
i}\,\varepsilon_{i,j,k}J_{k},\quad\hbox{i.e.},\quad[K,Q]=-{\rm i}P,\
[K,P]={\rm i}Q,\ [Q,P]={\rm i}K,$ (4)

where the first two will stand in place of the two Hamiltonian commutators in
the continuous model, $[\overline{H},\overline{Q}]=-{\rm i}\overline{P}$ and
$[\overline{H},\overline{P}]={\rm i}\overline{Q}$. The eigenvalue spectra of
the three su($2$) generators are
$\Sigma(Q)=\Sigma(P)=\Sigma(K)=\\{-j,-j{+}1,\ldots\,,j\\}=:|_{-j}^{j}$, and we
remark that the position operator $Q=J_{1}$, is here diagonal. Finally, the
eigenvalues of the mode operator $K+j{\it 1}$ will count its eigenstates by
the mode number $n:=\kappa+j$ as

$\ \Sigma(K+j{\it 1}):=\kappa+j=n\in\\{0,\,1,\,\ldots,\,2j\\}.$ (5)

Using Dirac notation, let the orthogonal and complete eigenbases of the
position and mode operators, $J_{1}$ and $J_{3}$, be

$Q|j,q\rangle_{\scriptscriptstyle 1}=q|j,q\rangle_{\scriptscriptstyle 1},\
q|_{-j}^{j},\qquad K|j,n\rangle_{\scriptscriptstyle
3}=\kappa|j,n\rangle_{\scriptscriptstyle
3}=(n{-}j)|j,n\rangle_{\scriptscriptstyle 3},\ n|_{0}^{2j}.$ (6)

One-dimensional finite signals or discrete images ${\bf
F}=\\{F_{q}\\}_{q=-j}^{j}$ are then given by $N=(2j{+}1)$-dimensional vectors
of components

$F_{-j}={}_{\scriptscriptstyle 1}\langle j,-j|{\bf F}\rangle,\quad
F_{-j+1}={}_{\scriptscriptstyle 1}\langle j,-j{+}1|{\bf
F}\rangle,\quad\cdots\quad F_{j}={}_{\scriptscriptstyle 1}\langle j,j|{\bf
F}\rangle.$ (7)

The overlap between the two bases in (6) leads to a three-term difference
equation that yields the Wigner ‘little-$d$’ functions [7] for the angle
${\textstyle\frac{1}{2}}\pi$ between the two axes, which are the finite
oscillator wavefunctions [1]:

$\begin{array}[]{rcl}\Psi_{n}(q)&:=&{}_{\scriptscriptstyle 1}\langle
j,q|j,n\rangle_{\scriptscriptstyle
3}=d^{j}_{n-j,q}({\textstyle\frac{1}{2}}\pi)=\Psi_{q+j}(n{-}j)\\\\[7.0pt]
&=&\displaystyle\frac{(-1)^{n}}{2^{j}}\sqrt{\bigg{(}{2j\atop
n}\bigg{)}\bigg{(}{2j\atop
q{+}j}\bigg{)}}\,K_{n}(q{+}j;{\textstyle\frac{1}{2}};2j),\end{array}$ (8)

that are given in terms of the square root of a binomial coefficient in $q$, a
discrete version of a Gaussian, and symmetric Kravchuk polynomials of degree
$n$,
$K_{n}(q{+}j;{\textstyle\frac{1}{2}};2j)=K_{q+j}(n;{\textstyle\frac{1}{2}};2j)={}_{2}F_{\\!1}(-n,-2j{-}q;-2j;2)$,
that are discrete analogues of Hermite polynomials. The set of functions
$\\{\Psi_{n}(q)\\}_{n=0}^{2j}$, and also the set
$\\{\Psi_{n}(q)\\}_{q=-j}^{j}$, form orthonormal bases for the $N$-dimensional
space of signals (i.e., one-dimensional pixellated images on linear screens),
so $\sum_{q=-j}^{j}\Psi_{n}(q)\Psi_{n^{\prime}}(q)=\delta_{n,n^{\prime}}$ and
$\sum_{n=0}^{2j}\Psi_{n}(q)\Psi_{n}(q^{\prime})=\delta_{q,q^{\prime}}$.

Finally, we write the general form of the Wigner ‘little-$d$’ functions [7,
Sect. 3.6],

$\begin{array}[]{rcl}d^{j}_{m^{\prime},m}(\beta)&:=&d^{j}_{m,m^{\prime}}(-\beta)=\sqrt{(j+m^{\prime})(j-m^{\prime})(j+m)(j-m)}\\\\[5.0pt]
&\times&\displaystyle\sum_{k}\frac{(-1)^{m^{\prime}-m+k}(\cos{\textstyle\frac{1}{2}}\beta)^{2j+m-m^{\prime}-2k}(\sin{\textstyle\frac{1}{2}}\beta)^{m^{\prime}-m+2k}}{k!\,(j+m-k)!\,(m^{\prime}-m+k)!\,(j-m^{\prime}-k)!},\end{array}$
(9)

where the summation extends over the integer range
$\hbox{max}(0,m{-}m^{\prime})\leq k\leq\hbox{min}(j{-}m)$ for $m>m^{\prime}$,
while for $m<m^{\prime}$ the reflection formula in (8) applies.

###  2.2 Finite two-dimensional images

In $D=2$ dimensions, the finite oscillator Lie algebra is $\hbox{\sf
su($2$)}_{x}\oplus\;\hbox{\sf su($2$)}_{y}=\hbox{\sf so($4$)}$, i.e., the
four-dimensional orthogonal Lie algebra, which contains in particular the
generator of an SO($2$) group of rotations in the $x$-$y$ plane. (Lie algebras
are denoted by lower-case letters, their generated Lie groups by upper-case
letters.) On a square screen, the so($4$) algebra represents finite images
$\bf F$ by $N\times N$ matrices of entries $\\{F_{q_{x},q_{y}}\\}$, where
$-j\leq q_{x},q_{y}\leq j$ are integer-spaced. Generic rectangular pixellated
screens $N_{x}\times N_{y}$ have been considered in [5] with general
representations of so($4$), but square ones are simpler and suffice for this
study.

Finite 2D images can be characterized also by a second set of $N^{2}$ pixel
values that follow polar coordinates responding to the subalgebra chain
$\hbox{\sf so($4$)}\supset\hbox{\sf so($3$)}\supset\hbox{\sf so($2$)}$ [2].
The rotation of pixellated images in a square screen is then accomplished by
transforming from the Cartesian to the polar basis, rotating through
multiplicative phases in the polar basis, and transforming back to the
Cartesian basis.

In two dimensions marked by orthogonal axes $x,\,y$, the algebra $\hbox{\sf
su($2$)}_{x}\oplus\;\hbox{\sf su($2$)}_{y}$ provides two sets of generators
(1)–(3) that mutually commute and provide, as in (6), their common eigenbases
with $j_{x}=j=j_{y}$ (where we omit writing $j$ henceforth),

$\begin{array}[]{ll}Q_{x}|q_{x},q_{y}\rangle_{\scriptscriptstyle
1}=q_{x}|q_{x},q_{y}\rangle_{\scriptscriptstyle 1},\
q_{x}|_{-j}^{j},&K_{x}|n_{x},n_{y}\rangle_{\scriptscriptstyle
3}=(n_{x}{-}j)|n_{x},n_{y}\rangle_{\scriptscriptstyle 3},\ n_{x}|_{0}^{2j},\\\
Q_{y}|q_{x},q_{y}\rangle_{\scriptscriptstyle
1}=q_{y}|q_{x},q_{y}\rangle_{\scriptscriptstyle 1},\
q_{y}|_{-j}^{j},&K_{y}|n_{x},n_{y}\rangle_{\scriptscriptstyle
3}=(n_{y}{-}j)|n_{x},n_{y}\rangle_{\scriptscriptstyle 3},\
n_{y}|_{0}^{2j},\end{array}$ (10)

defining the $N^{2}$ states $\\{|q_{x},q_{y}\rangle_{\scriptscriptstyle 1}\\}$
of the position basis and the $N^{2}$ states
$\\{|n_{x},n_{y}\rangle_{\scriptscriptstyle 3}\\}$ of the mode basis. The
overlap between these two orthogonal eigenbases provides the 2D finite
Cartesian oscillator wavefunctions,

$\Psi_{n_{x},n_{y}}(q_{x},q_{y})={}_{\scriptscriptstyle 1}\langle
q_{x},q_{y}|n_{x},n_{y}\rangle_{\scriptscriptstyle
3}=\Psi_{n_{x}}(q_{x})\,\Psi_{n_{y}}(q_{y}).$ (11)

As before, the set of functions $\\{\Psi_{n_{x},n_{y}}(q_{x},q_{y})\\}$ forms
orthonormal bases for the $N^{2}$-dimensional space of pixellated two-
dimensional images with respect to the indices $q_{x},q_{y}$ and with respect
to $n_{x},n_{y}$.

Two-dimensional images ${\bf F}=\\{F_{q_{x},q_{y}}\\}_{q_{x},q_{y}=-j}^{j}$
are thus represented by the values of $N\times N$ matrices built with the
adjoint of the position basis $|q_{x},q_{y}\rangle_{\scriptscriptstyle 1}$ in
(10), as was done in (7) for one dimension,

${\bf F}=\left(\begin{array}[]{ccc}{}_{\scriptscriptstyle
1}\langle-j,-j|&\cdots&{}_{\scriptscriptstyle 1}\langle-j,j|\\\
\vdots&\ddots&\vdots\\\ \phantom{-}{}_{\scriptscriptstyle 1}\langle
j,-j|&\cdots&\phantom{-}{}_{\scriptscriptstyle 1}\langle
j,j|\end{array}\right)\\!|{\bf
F}\rangle=\left(\begin{array}[]{ccc}F_{-j,-j}&\cdots&F_{-j,j}\\\
\vdots&\ddots&\vdots\\\ F_{j,-j}&\cdots&F_{j,j}\end{array}\right),$ (12)

Next, according to the commutation relations (4), $K_{x}$ and $K_{y}$ generate
independent rotations in the $(Q_{x},P_{x})$ and $(Q_{y},P_{y})$ phase planes.
Their sum and difference have the eigenvalue sets

$\displaystyle K:=K_{x}+K_{y},$ $\displaystyle\quad
n:=n_{x}+n_{y}\in\\{0,1,\ldots,4j\\},$ (13) $\displaystyle M:=K_{x}-K_{y},$
$\displaystyle\quad\\!m:=n_{x}-n_{y}\in\\{-2j,-2j{+}1,\ldots,2j\\}.$ (14)

There follows the definition of a distinct position-momentum basis [2] that is
classified by the so($4$) generators

$Q^{\circ}_{x}:=Q_{x}+Q_{y},\quad Q^{\circ}_{y}:=P_{x}-P_{y},\qquad
P^{\circ}_{x}:=P_{x}+P_{y},\quad P^{\circ}_{y}:=-Q_{x}+Q_{y}.$ (15)

With $K$ and $M$, these operators satisfy the commutation relations

$\begin{array}[]{l}\Big{[}K,\,\Big{\\{}{Q_{x}^{\circ}\atop
P_{x}^{\circ}}\Big{]}={\rm i}\Big{\\{}{\phantom{-}P_{x}^{\circ}\atop-
Q_{x}^{\circ},}\quad\Big{[}K,\Big{\\{}{Q_{y}^{\circ}\atop
P_{y}^{\circ}}\Big{]}={\rm i}\Big{\\{}{\phantom{-}P_{y}^{\circ}\atop-
Q_{y}^{\circ},}\\\\[5.0pt] \Big{[}M,\Big{\\{}{Q_{x}^{\circ}\atop
Q_{y}^{\circ}}\Big{]}={\rm i}\Big{\\{}{\phantom{-}Q_{y}^{\circ}\atop-
Q_{x}^{\circ},}\quad\Big{[}M,\Big{\\{}{P_{x}^{\circ}\atop
P_{y}^{\circ}}\Big{]}={\rm i}\Big{\\{}{\phantom{-}P_{y}^{\circ}\atop-
P_{x}^{\circ},}\end{array}\qquad[K,M]=0,$ (16)

that interpret $K$ as the generator of isotropic fractional Fourier-Kravchuk
transforms [2, 3], and $M$ as the generator of rotations of the phase space
operators in (15). The commuting operators $K,\,M$ define the orthonormal
polar eigenbasis $\\{|n,m\rangle_{\scriptscriptstyle P}\\}$,

$K|n,m\rangle_{\scriptscriptstyle P}=(n-2j)|n,m\rangle_{\scriptscriptstyle
P},\ n|_{0}^{4j},\quad M|n,m\rangle_{\scriptscriptstyle
P}=m|n,m\rangle_{\scriptscriptstyle P},\ m|_{-2j}^{2j}.$ (17)

The overlap between the position basis and the polar basis determines the 2D
finite polar oscillator wavefunctions [2],

$\begin{array}[]{rcl}&&\Lambda^{j}_{n,m}(q_{x},q_{y}):={}_{\scriptscriptstyle
1}\langle q_{x},q_{y}|n,m\rangle_{\scriptscriptstyle P}\\\\[5.0pt]
&&{\qquad}=\displaystyle(-1)^{(|m|-m)/2}\\!\\!\\!\sum_{n_{x}+n_{y}=n}\\!\\!\\!(-{\rm
i})^{n_{y}}d^{n/2}_{\frac{1}{2}(n_{x}-n_{y}),\frac{1}{2}m}({\textstyle\frac{1}{2}}\pi)\,\Psi_{n_{x},n_{y}}(q_{x},q_{y}).\end{array}$
(18)

Here also, the set of functions $\\{\Lambda^{j}_{n,m}(q_{x},q_{y})\\}$ provide
orthonormal bases for the $N^{2}$-space of images, with respect to the indices
$q_{x},q_{y}$ and with respect to $n,m$.

By construction, the functions (18) transform under rotations through
multiplication by phases,

${\cal R}(\theta)\Lambda^{j}_{n,m}(q_{x},q_{y}):=e^{-{\rm i}\theta
M}\Lambda^{j}_{n,m}(q_{x},q_{y})=e^{-{\rm
i}m\theta}\Lambda^{j}_{n,m}(q_{x},q_{y}),$ (19)

The rotation of finite images ${\cal R}(\theta):|{\bf F}\rangle\mapsto|{\bf
F}^{(\theta)}\rangle$ from the set $\\{F_{q_{x},q_{y}}\\}$ to
$\\{F_{\\!q_{x},q_{y}}^{(\theta)}\\}$ can thus proceed as

$\displaystyle F_{\\!q_{x},q_{y}}^{(\theta)}$ $\displaystyle=$
$\displaystyle{}_{\scriptscriptstyle 1}\langle q_{x},q_{y}|\,{\cal
R}(\theta)\,|{\bf F}\rangle$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum_{n,m;q_{x}^{\prime},q_{y}^{\prime}}\\!\\!\\!\\!{}_{\scriptscriptstyle
1}\langle
q_{x},q_{y}|n,m\rangle_{\scriptscriptstyle\\!P}\,{}_{\scriptscriptstyle
P\\!}\langle n,m|{\cal
R}(\theta)|n,m\rangle_{\scriptscriptstyle\\!P}{}_{\scriptscriptstyle
P\\!}\langle n,m|q_{x}^{\prime},q_{y}^{\prime}\rangle_{\scriptscriptstyle
1}\,{}_{\scriptscriptstyle 1}\langle q_{x}^{\prime},q_{y}^{\prime}|{\bf
F}\rangle$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum_{n,m;q_{x}^{\prime},q_{y}^{\prime}}\\!\\!\\!\\!{}_{\scriptscriptstyle
1}\langle q_{x},q_{y}|n,m\rangle_{\scriptscriptstyle\\!P}\,e^{-{\rm
i}m\theta}{}_{\scriptscriptstyle P\\!}\langle
n,m|q_{x}^{\prime},q_{y}^{\prime}\rangle_{\scriptscriptstyle
1}{}_{\scriptscriptstyle 1}\langle q_{x}^{\prime},q_{y}^{\prime}|{\bf
F}\rangle{\qquad}$ $\displaystyle=$
$\displaystyle\sum_{q_{x}^{\prime},q_{y}^{\prime}}R(q_{x},q_{y};q_{x}^{\prime},q_{y}^{\prime};\theta)\,F_{q^{\prime}_{x},q^{\prime}_{y}}$
(21)

with $n|_{0}^{2j}$, $m|_{-2j}^{2j}$ and
$q^{\prime}_{x},q^{\prime}_{y}|_{-j}^{j}$, where the kernel of rotation is
given by

$R(q_{x},q_{y};q_{x}^{\prime},q_{y}^{\prime};\theta):=\sum_{n,m}\Lambda^{j}_{n,m}(q_{x},q_{y})\,e^{-{\rm
i}m\theta}\,\Lambda^{j}_{n,m}(q_{x}^{\prime},q_{y}^{\prime})^{*}.$ (22)

An alternative expression for this kernel can be written as [2]

$\begin{array}[]{l}\displaystyle
R(q_{x},q_{y};q_{x}^{\prime},q_{y}^{\prime};\theta):=\sum_{\mu,\mu^{\prime}}\Psi_{n_{x},n_{y}}(q_{x},q_{y})\,d^{n/2}_{\mu,\mu^{\prime}}(2\theta)\,\Psi_{n_{x}^{\prime},n_{y}^{\prime}}(q_{x}^{\prime},q_{y}^{\prime})^{*},\\\
\mu={\textstyle\frac{1}{2}}(n_{x}-n_{y}),\
\mu^{\prime}={\textstyle\frac{1}{2}}(n_{x}^{\prime}-n_{y}^{\prime});\quad
n_{x}+n_{y}=n=n_{x}^{\prime}+n_{y}^{\prime}.\end{array}$ (23)

When the input image is a matrix of real numbers, after rotation it will
remain a real matrix, because the rotation kernel (22) is real. Due to the
orthonormality of the bases, from (22) or (23), we see that
$R(q_{x},q_{y};q_{x^{\prime}},q_{y^{\prime}};0)=\delta_{q_{x},q_{x^{\prime}}}\delta_{q_{y},q_{y^{\prime}}}$
is the unit transformation. Similarly, the product of two rotations satisfies
group composition
$\sum_{q_{x^{\prime}},q_{y^{\prime}}}R(q_{x},q_{y};q_{x^{\prime}},q_{y^{\prime}};\theta_{1})\\*R(q_{x^{\prime}},q_{y^{\prime}};q_{x^{\prime\prime}},q_{y^{\prime\prime}};\theta_{2})=R(q_{x},q_{y};q_{x^{\prime\prime}},q_{y^{\prime\prime}};\theta_{1}+\theta_{2})$,
and the inverse is
$R(q_{x},q_{y};q_{x^{\prime}},q_{y^{\prime}};-\theta)=R(q_{x^{\prime}},q_{y^{\prime}};q_{x},q_{y};\theta)^{*}$.
This shows that the matrices in the set (22) are unitary, ${\bf
R}(-\theta)={\bf R}(\theta)^{\dagger}$, and is a representation of the group
SO($2$) of plane rotations; it is reducible in the polar basis but irreducible
in the Cartesian basis. As an example of this rotation algorithm we provide
Fig. 1 that we proceed to examine and comment.

###  2.3 The ‘Gibbs-like’ oscillation phenomenon

As is evident in Fig. 1, a ubiquitous consequence of the unitarity of the
rotation algorithm is that rotated images display oscillations due to
‘discrete discontinuities’, i.e., large differences between values of
neighboring pixels. This oscillation pattern reminds us of that appearing in
another context: the summation of truncated Fourier series, known as the Gibbs
phenomenon. Because their origin is different (i.e., it is not a type of Gibbs
oscillation), we shall call these oscillations Gibbs-like; here we address
this phenomenon for the first time.

Figure 1: Left: A $50\times 50$ monochromatic image (letter “R”) of pixel
values 1, over a background of pixel values 0. Center: Rotation of this image
by $\frac{1}{4}\pi$ ($45^{\circ}$) with clipping of the over- and under-shoots
beyond the obligate range $[0,1]$. Right: Same rotation with normalization of
the pixel values to fit this range. Both output images exhibit ‘Gibbs-like’
oscillations due to the sharp boundaries of the input image. Whereas clipping
leads to a more contrasted image, it also entails loss of information in pixel
values; normalization retains information subject to translation and scale,
although it appears less contrasted.

Linear normalization is the main tactic to keep the grey-tone scale within the
specified range $[0,1]$, between absence and saturation of the pixel values.
We thus distinguish between the data-image whose pixel values can have any
range, and the screen-image which is output for display, where pixel values
are all within $[0,1]$. This tactic maps the data- to screen-values, ${\bf
F}\mapsto\overline{\bf F}$, using the extreme values,

$F_{q_{x},q_{y}}\mapsto\overline{F}_{q_{x},q_{y}}=\frac{F_{q_{x},q_{y}}-s_{N}}{S_{N}-s_{N}},\quad\begin{array}[]{l}s_{N}\,:=\min_{q_{x},q_{y}}\,\\{F_{q_{x},q_{y}}\\}<0,\\\
S_{N}:=\max_{q_{x},q_{y}}\,\\{F_{q_{x},q_{y}}\\}>1.\end{array}$ (24)

The screen-image will thus have a minimum of complete blacks and whites, with
no information loss.

Figure 2: Unitary rotations of a centered Kronecker delta (25) for $N=11$, 31,
51, with their normalized rotated images. Left: the original discrete
Kronecker function
$\Delta^{{{\scriptscriptstyle(}}N)}_{{\scriptscriptstyle(}x{\scriptscriptstyle)}}(q_{x},q_{y})$
of 0 and 1. Center and Right: their images rotated by $\theta=\frac{1}{8}\pi$
and $\frac{1}{4}\pi$. Below each, the (non-normalized) values of anti-diagonal
pixels, with their largest over- and undershoots. For $N=11$ and 31, the over-
or under-shoots are marked by $+$ and $-$.

Figure 3: Unitary rotations of the unit step function (26) for $N=10$, 30, 50,
with normalized rotated images, organized as in Fig. 2. Left: the step
function
$\Theta^{{{\scriptscriptstyle(}}N)}_{{\scriptscriptstyle(}x{\scriptscriptstyle)}}(q_{x},q_{y})$.
Center and Right: rotatated images for $\theta=\frac{1}{8}\pi$ and
$\frac{1}{4}\pi$.

Whereas in the familiar trigonometric Gibbs phenomenon, a unit step function
$\Theta(q):={\textstyle\frac{1}{2}}(1+\hbox{sign}\,q)$ entails a maximal
overshoot of ${\sim}8.95\%$ at a positions ever closer to the discontinuity as
the number of series terms grows (see [9, Fig. 4.11 at p. 167]), to calibrate
the Gibbs-like phenomenon for $N\times N$ images under unitary rotation, let
us consider a Kronecker delta at the center of the screen with an odd number
$N=2j+1$ of pixel rows and columns,

$\Delta^{{{\scriptscriptstyle(}}N)}_{{\scriptscriptstyle(}x{\scriptscriptstyle)}}(q_{x},q_{y}):=\\{0\hbox{
for }q_{y}<0,\ 1\hbox{ at }q_{y}=0,\ 0\hbox{ for }q_{y}>0\\},$ (25)

and a discrete step function along the $x$-axis for even $N=2j+1$ rows and
columns

$\Theta^{{{\scriptscriptstyle(}}N)}_{{\scriptscriptstyle(}x{\scriptscriptstyle)}}(q_{x},q_{y}):=\\{0\hbox{
for }q_{y}\leq-{\textstyle\frac{1}{2}},\ 1\hbox{ for
}q_{y}\geq{\textstyle\frac{1}{2}}\\},$ (26)

shown for rotations of $\frac{1}{8}\pi$ and $\frac{1}{4}\pi$ in Figs. 2 and 3
respectively. The resulting oscillations are recognizably parallel to the
lines of discontinuity. To evince this oscillation behavior we plot below the
pixel values along the main anti-diagonal of each screen

A first observation is that, unlike the continuous case, the over- and under-
oscillations are not symmetrical. This is because while in the Gibbs case
$e^{0{\rm i}x}=1$ belongs to the Fourier series terms and allows the addition
of constants, in the discrete case $\Psi_{0,0}(q_{x},q_{y})$ is not a constant
that could be freely added to all pixel values. Most importantly, we note that
the over- and under-shoots diminish with increasing $N$: as the pixellation
becomes finer, the finite oscillator states approximate those of continuous
systems —as expected. Yet, although the delta and step functions lead to
rotated images that are relatively simple to write analytically, they are not
very useful to elucidate certain systematic features of their rotated images
on the pixellated screen, such as the the behavior on and near the screen
vertices and edges, as can be seen in those figures.

##  3 Three-color pixellated images

With a qualitative understanding of oscillations due to discrete Kronecker
deltas and step functions, we now extend the algorithm from rotations of
monochromatic finite $N\times N$ pixellated images presented in the previous
section, to polychromatic images within the standard three-color scheme of RGB
(Red-Green-Blue) technology. Each channel, R, G or B, is a monochromatic
image; polychromatic images (indicated by sans serif font) are the direct sum
of the three components where the pixel values for each color are all in the
range $[0,1]$,

${\sf F}={\bf F}^{\scriptscriptstyle\rm R}\oplus{\bf F}^{\scriptscriptstyle\rm
G}\oplus{\bf F}^{\scriptscriptstyle\rm B},\quad{}_{\scriptscriptstyle
1\\!}\langle q_{x},q_{y}|{{\sf
F}}\rangle=\left[\begin{array}[]{l}\scriptstyle{}_{\scriptscriptstyle
1\\!}\langle q_{x},q_{y}|{\bf F^{\scriptscriptstyle\rm R}}\rangle\\\\[-3.0pt]
\scriptstyle{}_{\scriptscriptstyle 1\\!}\langle q_{x},q_{y}|{\bf
F^{\scriptscriptstyle\rm G}}\rangle\\\\[-3.0pt]
\scriptstyle{}_{\scriptscriptstyle 1\\!}\langle q_{x},q_{y}|{\bf
F^{\scriptscriptstyle\rm B}}\rangle\end{array}\right]\\!\\!,\quad
0\leq{}_{\scriptscriptstyle 1\\!}\langle q_{x},q_{y}|{\bf
F^{\scriptscriptstyle\rm X}}\rangle\leq 1,$ (27)

where X stands for R, G or B, with values 0 for absence and 1 for saturation
in each color.

Under rotation we let each component undergo the same transformation
(2.2)–(22) detailed in the previous section,

${{\sf F}}^{(\theta)}:={\cal R}(\theta){{\sf F}}={\cal R}(\theta){\bf
F}^{\scriptscriptstyle\rm R}\oplus{\cal R}(\theta){\bf
F}^{\scriptscriptstyle\rm G}\oplus{\cal R}(\theta){\bf
F}^{\scriptscriptstyle\rm B}.$ (28)

The RGB color scheme is an industrial standard [14] that entails an additive
colorimetric scale, that is, each of the elements of each channel sums to
define an individual color [11, 12]. Physically, each of the colors in the RGB
scheme is associated with the way the human eye perceives and processes light
with the photoreceptor cells in the retina [13]. The chromatic response is
intrinsic to the aggregation of the three channels and independent of the
luminescence, which means that the white point of the scale is the algebraic
sum of the three channel values, and independent of the illumination [21].
This color scheme is strictly positive within a range of bits or bytes
displayed in digital or analogue devices [15, 16]. Although pixel values
outside the standard range have been also used as representing complementary
colors [18, 17], we forego this alternative interpretation because it would
introduce dubious changes in the color rendering of black or white
backgrounds.

Here we address the joint display of the three data-image matrices to
acceptably transform color images. The normalization of pixel values (24) now
requires the extreme values among the three colors to define, for
$Y\in\\{R,G,B\\}$,

$F_{q_{x},q_{y}}^{{\scriptscriptstyle\rm
Y}}\mapsto\overline{F}_{q_{x},q_{y}}^{{\scriptscriptstyle\rm
Y}}=\frac{F_{q_{x},q_{y}}^{{\scriptscriptstyle\rm
Y}}-s_{N}}{S_{N}-s_{N}},\quad\begin{array}[]{l}s_{N}\,:=\min_{X;\,q_{x},q_{y}}\,\\{F_{q_{x},q_{y}}^{{\scriptscriptstyle\rm
X}}\\}<0,\\\
S_{N}:=\max_{X;\,q_{x},q_{y}}\,\\{F_{q_{x},q_{y}}^{{\scriptscriptstyle\rm
X}}\\}>1.\end{array}$ (29)

Two examples of this unitary rotation algorithm for three-color images are
presented in Figs. 4, for images with high and low pixel discontinuity
contrasts.

Figure 4: Left: Two RGB $52\times 52$ pixellated images (above a pseudo-Mario,
below, the well-known Lena). Center and Right: Their unitary rotation by
$\theta=\frac{1}{8}\pi$ and $\frac{1}{4}\pi$ with the normalization (29); the
data image of the former can be further rotated by $\frac{1}{8}\pi$ to obtain
the data-image of the latter, because the unitary transformations satisfy
composition.

We should remark that the computation of all figure rotations in this report
has been performed with an algorithm that enhances the speed of the previously
used rotation algorithms [2, 5] that needs the computation of the special
functions in the kernel (22). Previous reports of unitary transformations did
not focus on the computational efficiency because one was more interested in
the mathematical features of the transformations. Yet for example in [5] a
grayscale rectangular image of $61\times 37=2257$ pixels was calculated in
$\sim$78 hours, here each of the three layers of Fig. 4 has $52\times 52=2704$
pixels that were computed in $\sim$250 seconds, while the full RGB image took
$\sim$12 minutes to complete. This improvement is due to two factors: first,
the optimization of the routine to calculate the elements in (22) was done
through a three-term recurrence relation (instead of library functions), and
that a high-level scripting language such as Julia [20] was used, allowing the
implementation of a syntactically natural transformation algorithm that
simultaneously profits from the multi-threading in modern CPU’s. For more
technical details, such as the binomial overflow as $N$ grows, the algorithm
we used in all the previous figures is provided by an Open Source code shared
on a public repository [8], to fulfill the principle of reproducibility for
the results here presented.

##  4 Conclusions

We remarked above that the unitary rotation algorithm for pixellated images
based on the finite oscillator model is unique, and can be inverted or
concatenated. Yet we should underline that the unitary rotation algorithm
requires that the value at every pixel in the rotated image depend on the
values of all pixels of the input image. We are thus dealing with an
$\mathcal{O}(N^{4})$ dependence in the number of operations needed to compute
a rotation. The obvious advantage of information conservation comes thus at
the price of a lengthier computation time. For commercial applications, where
$N\approx 10^{3}$ is common, the more common image rotation algorithms based
on interpolation in smaller regions of an image, have none of the former
advantages, nor the last disadvantage.

The results presented for three basic colors of course also apply to any
number of quantities that may be attached to the pixel values in pre-ordained
ranges. The unitary rotations can also be made to act on the latter quantities
covariantly, such as would be the case for a discrete and finite vector field,
whose two components will rotate together with the data-image. Also, three-
and higher-dimensional pixellated (‘voxellated’) images, subject to rotations
[22] can be subject to considerations similar to those analyzed here.

Lastly, we should place the group of rotations of pixellated screens within
the wider context of the finite oscillator model, which uses basically the
rotation Lie subalgebra chain $\hbox{\sf so($3$)}\supset\hbox{\sf so($2$)}$ to
provide the pixel coordinates and a phase space interpretation for the system,
as opposed to their usual numbering through the $N$-point Fourier transform.
In the latter, the position space forms a discrete torus whose two discrete
$(x,y)$ coordinates curl into circles, but where rotations between them cannot
be applied without breaking the torus. In the finite oscillator model, the
manifold harboring the coordinates is a sphere, which can be freely rotated
and also gyrated in phase space, among other linear canonical transformations.

## Acknowledgments

We thank the support of the Universidad Nacional Autónoma de México through
the dgapa-papiit project AG100120 Óptica Matemática. A.R. Urzúa acknowledges
the support of the National Council for Science and Technology (conacyt)
through the Becas Postdoctorales program (2021).

## References

  * [1] N.M. Atakishiyev, L.E. Vicent and K.B. Wolf, Continuous vs. discrete fractional Fourier transforms, J. Comp. Appl. Math. 107, 73–95 (1999).
  * [2] N.M. Atakishiyev, G.S. Pogosyan, L.E. Vicent, and K.B. Wolf, Finite two-dimensional oscillator II. The radial model, J. Phys. A 34, 9399–9415 (2001).
  * [3] K.B. Wolf and L.E. Vicent, The Fourier U(2) group and separation of discrete variables, Sigma 7, 053, 18p. (2011).
  * [4] N.M. Atakishiyev, G.S. Pogosyan, L.E. Vicent and K.B. Wolf, Finite two-dimensional oscillator. I: The Cartesian model, J. Phys. A 34, 9381–9398 (2001).
  * [5] A.R. Urzúa and K.B. Wolf, Unitary rotation and gyration of pixellated images on rectangular screens, J. Opt. Soc. Am. A 33, 642–647 (2016).
  * [6] N.M. Atakishiyev, G.S. Pogosyan, and K.B. Wolf, Finite models of the oscillator, Phys. Part. Nuclei 36, 521–555 (2005).
  * [7] L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics. Theory and Application. Encyclopedia of Mathematics and its Applications, Vol. 8 (Addison-Wesley Publ. Co., Reading, Mass.,1981).
  * [8] A.R. Urzúa (2022), Polychromatic Unitary Rotations, version 0.1.0 [Computer software], in https://github.com/rurz/UnitRots.
  * [9] K.B. Wolf, Integral Transforms in Science and Engineering (Plenum Press, New York, 1979).
  * [10] C. van Trigt, Open problems in color constancy: discussion, J. Opt. Soc. Am. A 31, 338–347 (2014)
  * [11] R.G. Kuehni, Color space and its divisions, Color Research & Application 26, 209–222 (2001).
  * [12] I. Lissner and P. Urban, Toward a unified color space for perception-based image processing, IEEE Transactions on Image Processing 21, 1153–1168 (2012).
  * [13] G. Wyszecki and W.S. Stiles, Color Science Vol. 8 (Wiley, New York, 1982).
  * [14] S. Süsstrunk, R. Buckley, and S. Swen, Standard rgb color spaces, Color and Imaging Conference 1999, 127–134 (1999).
  * [15] H-R. Kang,Color Technology for Electronic Imaging Devices, (SPIE Optical Engineering Press, Bellingham, Wash., 1997).
  * [16] Z. Shen. Three-Color Tunable Organic Light-Emitting Devices, Science 276, 2009–2011 (1997).
  * [17] R.W. Pridmore, Complementary colors: The structure of wavelength discrimination, uniform hue, spectral sensitivity, saturation, chromatic adaptation, and chromatic induction, Color Research & Application 34, 233–252 (2009).
  * [18] R.W. Pridmore, Complementary colors: A literature review, Color Research & Application 46, 482–488 (2020).
  * [19] A.R. Urzúa, “Rotaciones y giraciones en pantallas cartesianas rectangulares” (M.Sc. thesis, UNAM, 2016), https://repositorio.unam.mx/contenidos/95764.
  * [20] J. Bezanson, A. Edelman, S. Karpinski, and V.B. Shah, Julia: A Fresh Approach to Numerical Computing. In: SIAM Review 59, 65–98 (2017). https://doi.org/10.1137/141000671.
  * [21] H. Grassmann, Zur Theorie der Farbenmischung, Ann. Phys. Chem. 165, 69 (1853).
  * [22] G. Krötzsch, K. Uriostegui and K.B. Wolf, Unitary rotations in two-, three-, and $D$-dimensional Cartesian data arrays, J. Opt. Soc. Am. A 31, 1531–1535 (2014).