# Charged Throats in the Hořava–Lifshitz Theory

Alvaro Restuccia<EMAIL_ADDRESS>Departamento de Física, Facultad
de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta,
Chile.  Francisco Tello-Ortiz<EMAIL_ADDRESS>Departamento de Física,
Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170,
Antofagasta, Chile.

###### Abstract

A spherically symmetric solution of the field equations of the Hořava–Lifshitz
gravity–gauge vector interaction theory is obtained and analyzed. It describes
a charged throat. The solution exists provided a restriction on the relation
between the mass and charge is satisfied. The restriction reduces to the
Reissner–Nordström one in the limit in which the coupling constants tend to
the relativistic values of General Relativity. We introduce the correct charts
to describe the solution across the entire manifold, including the throat
connecting an asymptotic Minkowski space–time with a singular 3+1 dimensional
manifold. The solution external to the throat on the asymptotically flat side
tends to the Reissner–Nordström space–time at the limit when the coupling
parameter, associated with the term in the low energy Hamiltonian that
manifestly breaks the relativistic symmetry, tends to zero. Also, when the
electric charge is taken to be zero the solution becomes the spherically
symmetric and static solution of the Hořava–Lifshitz gravity.

##  I Introduction

The Hořava–Lifshitz (HL hereinafter) theory stands nowadays as a strong
candidate to describe a quantum description of the gravitational interaction
r1 ; r4 . The proposal breaks the relativistic symmetry by introducing an
anisotropic scale of temporal and spatial coordinates. In addition, it
introduces terms with higher order spatial derivatives in the potential,
compatible with the geometric structure of the formulation, which make the
theory, in the UV regime, renormalizable by power counting. Although the
original proposal given in r1  was incomplete, since it suffered from
instabilities at all energy scales and strong coupling r4 ; r5 ; r6 , a
subsequent analysis r7  overcame these problems by including a new terms in
the Hamiltonian compatible with the symmetries of the original proposal. The
anisotropic behavior becomes manifest by the introduction of the so–called
critical dynamical exponent $z$. Furthermore, the exponent $z$ describes
different scale energies, for example the case $z=1$ corresponds to the IR
point whilst $z=3$ the UV one 111Obviously, the value of the exponent $z$ at
the UV point depends on the spatial dimension $d$.. The HL theory can be seen
as general framework containing two classes of theories: i) the projectable
ones, namely when the lapse function is restricted to be a purely time
function $N=N(t)$ and ii) the non–projectable ones where the lapse function is
more general $N=N(t,\vec{x})$. Both theories are not equivalent since they do
not propagate the same spectrum, besides its constraints structure is also
different. Along the years the feasibility of the HL theory has been widely
studied, for example in r8 ; r9 ; r15  the Hamiltonian formulation was
analyzed, and from the phenomenological point of view in r14  the radiation of
gravitational waves of self–gravitating binary systems in the low–energy limit
was investigated. On the cosmological background within the framework of the
projectable version, interesting solutions have been obtained r23 ; r24 ; r25
; r26  and also in the standard model domain r27  and four–fermion Gross–Neveu
like models r28  novels results were presented.

In a more particular context than the quantum and cosmological aspects that
this proposal offers, it is especially interesting as well as intriguing to
investigate the possible existence of exotic structures such as wormholes (WH
from now on). Since the pioneering work by Morris and Thorne m1 ; m2 , the
study of WH in the GR framework has laid the foundations for both the
geometric shape that they must adopt, and the type of matter that must support
them. WHs are visualized as connecting two relatively distant regions or
universes located on an asymptotically flat space–times. This type of WH is
known in the literature as traversable or passable WH m1 ; m2 ; m3 . Of
course, this is conditioned to a strong restriction, which implies the
violation of the energy conditions m3 , mainly the null (NEC): $\rho+p_{r}\geq
0$ and $\rho+p_{t}\geq 0$, and weak (WEC): $\rho\geq 0$, $\rho+p_{r}\geq 0$
and $\rho+p_{t}\geq 0$ conditions, being $\rho$ the energy–density, $p_{r}$
and $p_{t}$ the radial and lateral pressures, respectively. This type of
matter is called as _exotic matter_ m1 ; m2 ; m3 . In the references m4 ; m5
it was shown that in the background of GR it is not possible to obtain a
passable WH satisfying at the same time the energy conditions. In this
respect, during last years dozens of articles were devoted in obtaining WH
solutions within the arena of modified gravity theories satisfying the energy
conditions. For example, in the background of $f(R)$ gravity cosmological WH
solutions and WH space–times satisfying different equations of state n1 ; n2 ;
n3 ; n4 ; n5 ; n8 ; n6 ; n7  (and references therein) were studied. However,
as occurs in GR, in the $f(R)$ gravity arena it is not possible to get a
traversable WH space–time satisfying the energy conditions, unless the theory
propagates ghost fields n9 ; n10 . On the other hand, in the $f(R,T)$ gravity
n11 ; n12 ; n13 ; n14 ; n15 , Brans–Dicke gravity n16 ; n17 , Einstein–Cartan
theory n18  and extra dimensions n19 ; n20  scenarios, it is possible to meet
at the same time both the traversability and energy conditions. This is so,
because the extra terms or fields introduced in these kind of theories can be
seen as corrections to the Einstein field equations.

In considering the HL theory, these objects have been widely explored by
taking into account both the projectable and non–projectable versions. For
instance, in the projectable case, in wh1  traversable WHs connecting two
asymptotically de Sitter/anti–de Sitter regions were studied, in wh2
asymptotically flat WHs threading by phantom energy were obtained. More
recently, the so–called Euclidean WHs have been analyzed in the Euclidean path
integral approach to quantum gravity wh3 , also within the branch of 2d causal
dynamical triangulations wh4  this kind of object were explored. On the other
hand, in the non–projectable branch it has been shown that these
configurations are not allowed, instead of it one obtains a throat r31 ; r32 .
The main difference between a WH and a throat, relies on the fact that the
former is connecting necessarily two asymptotically flat regions. In the
models found in r31 ; r32  the structure is asymptotically flat only in one
side, whilst is divergent at the other one. Interestingly, the throat solution
does not need to be supported by any kind of exotic matter distribution, that
is, this solution was obtained by solving only the fields equations without
any assumption on the energy–momentum tensor r31  (vacuum solution).

Taking advantage of these remarks, in the present paper we study charged
throats within the non–projectable HL background. The motivation behind this
analysis, is supported on recent articles r29 ; r30  working out the
anisotropic gravity–vector gauge coupling in 3+1 dimensions. This formulation
is obtained, starting from the the 4+1 dimensional non–projectable HL theory
after a dimensional reduction _a la_ Kaluza–Klein. The resulting 3+1
dimensional theory exhibits the same features as the original proposal r1
ı.e., the anisotropic gravity-vector gauge coupling respects the
$\text{Diff}_{\mathcal{F}}(\mathcal{M})$ symmetries r29 ; r30  and is
power–counting renormalizable arxiv . In this case, in contrast with the
solution provided in r31 , here we have a spherical throat supported by and
static spatial electric field. This contribution is coming from the gauge
sector of the reduced theory. The analysis will we performed at the low energy
regime. In this case the higher order derivatives terms in the potential of
the HL Hamiltonian are dismissed and we are left with the electromagnetic
coupling to HL gravity in the standard way. Despite the higher non–linear
behavior of the resulting equation, we have thoroughly demonstrated the
existence of a global minimum describing the existence of a throat connecting
to regions, one of them asymptotically flat while the other presenting an
essential singularity. Besides, the asymptotic behavior has been studied
finding out that the solution tends to the well–known Reissner–Nordström
solution in GR. To support the feasibility of the present study we have
incorporated a graphical analysis, where the assigned values to the coupling
constants $k$, $\alpha$ and $\beta$ satisfy the known restrictions arising
from experimental data. In fact, $\alpha$ and $\beta$ must be very near to 0
and 1 respectively feno5 ; feno6 . Consequently, the parameter $k$ we
introduce satisfies $k^{2}=1-\alpha/2\beta>0$, $\beta>0$.

The article is organized as follows: In Sec. II are presented the 3+1
dimensional reduced theory and its equations of motion. In Sec. III the main
equations to be solved for the spherical charged throat are provided. Next, in
Sec. IV the resulting differential equation is solved, showing the existence
of the throat. Sec. V shows the asymptotic behavior of the obtained solution
and in Sec. VI the full metric charts of the charged throat are presented.
Finally, Sec. VII concludes the work.

##  II The Anisotropic Gravity–Vector Gauge Interaction in the Hořava-Lifshitz
Framework

The Hamiltonian formulation of the pure electromagnetic–gravitational
interaction in the HL scenario was obtained and studied in r29 ; r30 . In that
work the 3+1 Hamiltonian was obtained from a Kaluza–Klein reduction from a 4+1
dimensional HL gravity. At this point, we can take the dilaton scalar field be
1 in the Hamiltonian as one does in GR when following a Kaluza–Klein reduction
to obtain the Einstein–Maxwell Hamiltonian. Otherwise one can leave the
dilaton field unconstrained and obtained a formulation not equivalent to the
previous one. In this case it is not possible to take the dilaton to be one in
the field equations without restricting the other fields of the model. Both
models are consistent and compatible with the geometric symmetries of the HL
proposal. When the dilaton is taken to be 1, that is at its ground state, the
Hamiltonian density describing this interaction is expressed as follows

$\begin{split}\mathcal{H}=\frac{N}{\sqrt{\gamma}}\bigg{[}p^{ij}p_{ij}+\frac{p^{i}p_{i}}{2}+\frac{\lambda}{\left(1-4\lambda\right)}\left(p^{ij}\gamma_{ij}\right)^{2}-\gamma\beta\left(R-\frac{1}{4}F_{ij}F^{ij}\right)-\gamma\alpha
a_{i}a^{i}\bigg{]}&\\\
-\Lambda\partial_{i}p^{i}-\Lambda_{j}\left(2\nabla_{i}p^{ij}+p^{i}\gamma^{jk}F_{ik}\right)-\sigma
P_{N}+\tilde{V}\left(\gamma_{ij},A_{k},N\right),\end{split}$ (1)

where $\lambda\neq 1/4$. In the above expression
$\tilde{V}\left(\gamma_{ij},A_{k},N\right)$ represents the higher order
spatial derivatives terms in the potential of the theory and, $R$ is the
curvature associated with the $3$-dimensional metric $\gamma_{ij}$.
Furthermore, $\Lambda$, $\Lambda_{j}$ and $\sigma$ Lagrange multipliers and
the two form $F_{ij}dx_{i}\wedge dx_{j}$ is defined by

$F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}.$ (2)

From now on we will dismiss the higher order derivative terms contained in the
potential $\tilde{V}$. Physically, it corresponds to consider the low energy
Hamiltonian. So, Following r29 , the first class constraints associated to (1)
are given by

$\displaystyle\partial_{i}p^{i}$ $\displaystyle=$ $\displaystyle 0$ (3)
$\displaystyle 2\nabla_{i}p^{ij}+p^{i}\gamma^{jk}F_{ik}$ $\displaystyle=$
$\displaystyle 0,$ (4)

and the second class one is

$\begin{split}H_{N}\equiv\frac{1}{\sqrt{\gamma}}\bigg{[}p^{ij}p_{ij}+\frac{p^{i}p_{i}}{2}+\frac{\lambda}{\left(1-4\lambda\right)}\left(p^{ij}\gamma_{ij}\right)^{2}-\beta\gamma
R+\frac{\beta}{4}\gamma F_{ij}F^{ij}\bigg{]}+\alpha\sqrt{\gamma}a_{i}a^{i}\\\
+2\alpha\sqrt{\gamma}\nabla_{i}a^{i}=0.\end{split}$ (5)

Next, the Hamiltonian density (1) provides the following evolution equations

$\displaystyle\dot{\gamma}_{ij}$ $\displaystyle=$
$\displaystyle\frac{2N}{\sqrt{\gamma}}\left[p_{ij}+\frac{\gamma_{ij}\lambda}{\left(1-4\lambda\right)}p^{lm}\gamma_{lm}\right]+\nabla_{i}\Lambda_{j}+\nabla_{j}\Lambda_{i},$
(6) $\displaystyle\dot{A}_{i}$ $\displaystyle=$
$\displaystyle\frac{N}{\sqrt{\gamma}}p_{i}+\partial_{i}\Lambda-\Lambda_{j}\gamma^{jk}F_{ik},$
(7)
$\begin{split}\dot{p}^{ij}=\frac{N}{2}\frac{\gamma^{ij}}{\sqrt{\gamma}}\left[p^{lk}p_{lk}+\frac{p^{l}p_{l}}{2}+\frac{\lambda}{\left(1-4\lambda\right)}\left(p^{lm}\gamma_{lm}\right)^{2}\right]-\frac{N}{\sqrt{\gamma}}\bigg{[}2p^{ik}p^{j}_{\
k}+\frac{p^{i}p^{j}}{2}&\\\
+\frac{2\lambda}{\left(1-4\lambda\right)}\left(p^{lm}\gamma_{lm}\right)p^{ij}\bigg{]}+N\sqrt{\gamma}\beta\left[\frac{R}{2}\gamma^{ij}-R^{ij}\right]+\beta\sqrt{\gamma}\bigg{[}\nabla^{(i}\nabla^{j)}N&\\\
-\gamma^{ij}\nabla_{k}\nabla^{k}N\bigg{]}+\frac{\beta}{2}N\sqrt{\gamma}\Bigg{[}F^{in}F^{\
j}_{n}-\frac{\gamma^{ij}}{4}F_{mn}F^{mn}\Bigg{]}+\alpha
N\sqrt{\gamma}\bigg{[}\frac{\gamma^{ij}}{2}a_{k}a^{k}&\\\
-a^{i}a^{j}\bigg{]}-\nabla_{k}\bigg{[}2p^{k(i}\Lambda^{j)}-p^{ij}\Lambda^{k}\bigg{]}-\Lambda^{(i}\gamma^{j)m}p^{l}F_{lm},\end{split}$
(8)
$\dot{p}^{i}=\beta\partial_{j}\left(N\sqrt{\gamma}F^{ji}\right)+\partial_{k}\left(\Lambda^{k}p^{i}-\Lambda^{i}p^{k}\right).$
(9)

##  III The Spherically Symmetric and Static Equations

As we are interested in the study of spherically symmetric, static and charged
configurations we consider the following line element describing the
3–dimensional manifold

$ds^{2}_{(3)}=\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (10)

together with a lapse function $N=N(r)$ and a shift $\Lambda_{i}=0$, where
$d\Omega^{2}\equiv\ d\theta^{2}+\text{sin}^{2}\theta d\varphi^{2}$. Besides,
the staticity assumption implies that the magnetic field is null ı.e,
$F_{ij}=0$, $\dot{A}_{i}=0$ and $\dot{\gamma}_{ij}=0$ then from Eq. (6) one
obtains $p^{ij}=0$ for $\lambda$ different from 1 and $1/4$. The latter means
that the terms containing the dimensionless constant $\lambda$ are canceled on
the field equations. At this point it is worth mentioning that the static
electric field is encoded in $p^{i}$. The field equations (7)–(8) reduce then
to

$\frac{N}{\sqrt{\gamma}}p_{i}+\partial_{i}\Lambda=0,$ (11)
$\begin{split}\frac{1}{4}\frac{\gamma^{ij}}{\gamma}p^{k}p_{k}-\frac{1}{2\gamma}p^{i}p^{j}+\beta\left(\frac{\gamma^{ij}}{2}R-R^{ij}\right)+\frac{\beta}{N}\left(\nabla^{(i}\nabla^{j)}N-\gamma^{ij}\nabla_{k}\nabla^{k}N\right)&\\\
+\alpha\left(\frac{\gamma^{ij}}{2}a_{k}a^{k}-a^{i}a^{j}\right)=0.\end{split}$
(12)

Now, taking the trace of the above expression (12) one gets

$\frac{1}{4\gamma}p^{k}p_{k}+\frac{1}{2}\beta
R-2\frac{\beta}{N}\nabla_{i}\nabla^{i}N+\frac{\alpha}{2}a_{k}a^{k}=0.$ (13)

From Eq. (5) one arrives to

$\frac{1}{2}\frac{1}{\gamma}p^{k}p_{k}-\beta R-\alpha
a^{i}a_{i}+2\alpha\frac{\nabla_{i}\nabla^{i}N}{N}=0.$ (14)

Combining Eqs. (13) and (14) one gets

$\frac{1}{2}p^{k}p_{k}+\left(\alpha-2\beta\right)\frac{\nabla_{i}\nabla^{i}N}{N}=0.$
(15)

Due to the spherical symmetry the only non vanishing component in (11) is the
radial one. So, we have

$\frac{p_{r}}{\sqrt{\gamma}}=-\frac{\partial_{r}\Lambda(r)}{N}.$ (16)

Raising the indices with the $3$-dimensional metric tensor $\gamma^{ij}$ from
(16) one gets

$\frac{p^{r}}{\sqrt{\gamma}}=-\frac{f(r)}{N}\partial_{r}\Lambda(r).$ (17)

Now from (3) and (11) we get

$\frac{r^{2}}{N}\sqrt{f}\partial_{r}\Lambda=A,$ (18)

being $A$ a new integration constant. We then have for the static electric
field

$\sqrt{\frac{p_{r}p^{r}}{\gamma}}=\frac{|A|}{r^{2}}.$ (19)

Next, putting together Eqs. (15), (16), (17) and (18) we arrive at the
following equation

$\frac{1}{4}\partial_{r}\Lambda^{2}+\frac{1}{2}\left(\alpha-2\beta\right)\partial_{r}N^{2}=\frac{CN}{r^{2}\sqrt{f}},$
(20)

where $C$ is an integration constant. By replacing (18) on the right hand side
of (20) we obtain

$\frac{\Lambda^{2}}{4}+\frac{1}{2}\left(\alpha-2\beta\right)N^{2}=\frac{C}{A}\Lambda+B,$
(21)

where $B$ is an integration constant. Eq. (21) can be reorganized as follows

$\left(\frac{\alpha}{2}-\beta\right)N^{2}+\left(\frac{\Lambda}{2}-\frac{C}{A}\right)^{2}=\frac{C^{2}}{A^{2}}+B\equiv
E.$ (22)

Equation (12) has only two independent components, the $r-r$ and the
$\theta-\theta$ components. The complete set of field equations then reduces
to the $\theta-\theta$ component of Eq. (12), its trace (13), Eq. (15) or
equivalently Eqs. (22), (17) and (18).

For the $3$-dimensional metric under consideration we have the following
components of the Ricci tensor

$R^{rr}=-\frac{f}{r}\partial_{r}f,\quad
R^{\theta\theta}=-\frac{r\partial_{r}f+2f-2}{2r^{4}}\quad\mbox{and}\quad
R^{\varphi\varphi}=csc^{2}(\theta)R^{\theta\theta},$ (23)

and the Ricci’s scalar is given by

$R=-\frac{2}{r^{2}}\left(f-1+r\partial_{r}f\right).$ (24)

So, the $\theta-\theta$ component of Eq. (12), after the replacement of (13)
on it, is

$\frac{\beta}{2r^{4}}\left(r\partial_{r}f+2f-2\right)+\frac{\beta}{Nr^{3}}f\partial_{r}N+\frac{\beta}{r^{2}N}\nabla_{i}\nabla^{i}N=0,$
(25)

while Eq. (13), after the replacement of Eqs. (16), (17) and (18) on it, is

$-\frac{\beta}{r^{2}}\left(r\partial_{r}f+f-1\right)+\frac{\alpha}{2}f\left(\frac{\partial_{r}N}{N}\right)^{2}-2\frac{\beta}{N}\nabla_{i}\nabla^{i}N=-\frac{A^{2}}{4r^{4}}.$
(26)

Finally, from Eqs. (25) and (26) we get

$\left(\sqrt{f}+r\sqrt{f}\frac{\partial_{r}N}{N}\right)^{2}-\left(r\sqrt{1-\frac{\alpha}{2\beta}}\sqrt{f}\frac{\partial_{r}N}{N}\right)^{2}=1-\frac{A^{2}}{4\beta
r^{2}}.$ (27)

Consequently, the independent field equations reduce to (17), (18), (22), (25)
or (26) and (27).

##  IV The Charged Throat Solution

We consider a non–trivial electrostatic field, hence $A\neq 0$. We will now
introduce a new variable $\chi$. This new variable will be the correct one to
describe the metric on the throat. Eq. (27) can be solved in terms of the new
variable $\chi$. From now on we shall employ prime to denotes derivatives with
respect to the radial coordinate $r$,

$kr\sqrt{f}\frac{N^{\prime}}{N}=\pm\sqrt{1-\frac{A^{2}}{4\beta
r^{2}}}\text{Sinh}(\chi),$ (28)
$\sqrt{f}\left(1+r\frac{N^{\prime}}{N}\right)=\pm\sqrt{1-\frac{A^{2}}{4\beta
r^{2}}}\text{Cosh}(\chi).$ (29)

It then follows

$\sqrt{f}=\pm\sqrt{1-\frac{A^{2}}{4\beta
r^{2}}}\left(\text{Cosh}(\chi)-\frac{1}{k}\text{Sinh}(\chi)\right),$ (30)

in (28)–(30) we take + for $\text{Cosh}(\chi)\geq\frac{1}{k}\text{Sinh}(\chi)$
and - for $\text{Cosh}(\chi)<\frac{1}{k}\text{Sinh}(\chi)$, and

$\frac{N^{\prime}}{N}=\frac{\text{Sinh}(\chi)}{r\left(k\text{Cosh}(\chi)-\text{Sinh}(\chi)\right)}.$
(31)

From (19) and (22) we have

$\Lambda=2\frac{C}{A}+2\sqrt{E-\left(\frac{\alpha}{2}-\beta\right)N^{2}},$
(32) $\frac{\Lambda^{\prime}}{N}=2\beta
k^{2}\frac{N^{\prime}}{\sqrt{E-\left(\frac{\alpha}{2}-\beta\right)N^{2}}}=\frac{A}{r^{2}\sqrt{f}}.$
(33)

Now using (28) we obtain

$N^{2}=\frac{EA^{2}}{\left(2k\beta\right)^{2}\left(r^{2}\text{Sinh}^{2}(\chi)-\frac{A^{2}}{4\beta}\text{Cosh}^{2}(\chi)\right)},$
(34)

we will show that its right hand member is positive. From (31) and (34) we
finally obtain a first order differential equation determining $r$ as a
function of $\chi$. Explicitly it reads

$\begin{split}\frac{dr}{d\chi}=-\left(\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}-\frac{1}{k}\right)r\left(1-\frac{A^{2}}{4\beta
r^{2}}\right)\left(1-\frac{A^{2}}{4k\beta
r^{2}}\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right)^{-1}.\end{split}$
(35)

In the particular case in which $A=0$ this differential equation was solved
explicitly in r31 . Moreover, the expressions for $\sqrt{f}$ in (30) and
$N^{2}$ in (34) reduces to the ones in that reference. In order to analyze the
first order differential equation (35) and following the theorem of existence
and uniqueness of the solution to ordinary differential equations we give the
initial data. At $\hat{\chi}$ we give $\hat{r}$ defined as follows:
$\hat{\chi}$ is the solution of the equation

$\frac{\text{Cosh}(\hat{\chi})}{\text{Sinh}(\hat{\chi})}=\frac{1}{k},$ (36)

and we take $\hat{r}$ any real number satisfying

$\hat{r}>\frac{|A|}{2k\sqrt{\beta}}L$ (37)

We have already assumed $k^{2}=1-\frac{\alpha}{2\beta}>0$, $\beta>0$ in
agreement with the experimental data. We assume now, $\alpha>0$, hence we have
$k<1$. There exists then the solution $\hat{\chi}$ to Eq. (36). $L$ is defined
as
$L=(1/k)[(1+k)^{1+k}/(1-k)^{1-k}]^{1/2k}=e^{(\hat{\chi}/k)}/\text{Sinh}(\hat{\chi})$,
it is $L>1$ and when $k$ goes to 1, then $L$ tends to 2. We notice that if
$\alpha=0$ there does not exist $\hat{\chi}$, in this case the field equations
on the previous section, after a redefinition in order to eliminate $\beta$,
are the corresponding ones in GR. We then have at $\hat{\chi}$

$\frac{dr}{d\chi}\bigg{|}_{\hat{\chi}}=0,$ (38)
$\frac{d^{2}r}{d\chi^{2}}\bigg{|}_{\hat{\chi}}=\frac{\hat{r}}{\left(\text{Sinh}\hat{\chi}\right)^{2}}\left(1-\frac{A^{2}}{4\beta\hat{r}^{2}}\right)\left(1-\frac{A^{2}}{4k^{2}\beta\hat{r}^{2}}\right)^{-1}>0,$
(39)

hence we have a local minimum at $\hat{\chi}$. We will show now that it is a
global minimum of $r=r(\chi)$.

Let us denote $h(\chi,r(\chi))$ the right hand member of Eq. (39). Since at
$\hat{\chi}$, $r(\hat{\chi})=\hat{r}$ satisfies the bound (37), then by
continuity of the factors in $h(\chi,r(\chi))$ there exists a neighborhood $U$
of $\hat{\chi}$ such that

$1-\frac{A^{2}}{4\beta r^{2}}>0,\quad 1-\frac{A^{2}}{4k^{2}\beta{r}^{2}}>0,$
(40)

for all $\chi\in U$. Let us denote $U_{+}\equiv\\{\chi\in
U:\chi>\hat{\chi}\\}$ and $U_{-}\equiv\\{\chi\in U:\chi<\hat{\chi}\\}$.

Then for all $\chi\in U_{+}$

$\frac{\text{Cosh}({\chi})}{\text{Sinh}({\chi})}<\frac{1}{k},$ (41)

hence $h(\chi,r(\chi))>0$ and $dr/d\chi>0$. Consequently, $r(\chi)$ is a
monotonically increasing function of $\chi$. Then

$\frac{dr}{d\chi}>\left(\frac{1}{k}-\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right)\left(1-\frac{A^{2}}{4\beta
r^{2}}\right)r>0,$ (42)

where the right hand member is a monotonically increasing function of $\chi$,
for all $\chi\in U_{+}$. But now we can extend the neighborhood $U_{+}$, for
all $\chi>\hat{\chi}$ and (40), (41) and (42) remain valid for the extension,
the function $r(\chi)$ is then a monotonically increasing function for all
$\chi>\hat{\chi}$, with an infimum value at $\hat{r}$. We can also bound
$dr/d\chi$ from above for all $\chi>\hat{\chi}$. In fact

$0<\frac{dr}{d\chi}<\left(\frac{1}{k}-\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right)r\left(1-\frac{A^{2}}{4\beta
r^{2}}\right)\left(1-\frac{A^{2}}{4k^{2}\beta r^{2}}\right)^{-1}.$ (43)

Hence $dr/d\chi$ is bounded from above and from below by strictly positive
differentiable functions of $\chi$. The solution $r(\chi)$ extends then as a
differentiable function for all $\chi>\hat{\chi}$ and
$r(\chi)\rightarrow+\infty$ when $\chi\rightarrow+\infty$. Indeed,
$\frac{dr}{d\chi}\sim\left(\frac{1}{k}-1\right)r$ when
$\chi\rightarrow+\infty$.

We may now analyze the solution for $\chi<\hat{\chi}$. We consider $\chi\in
U_{-}$ and introduce

$\omega(\chi)=\chi-\frac{1}{k}\text{Ln}\left[\sqrt{1-\frac{A^{2}}{4\beta
r^{2}}}\right],$ (44)

then

$\frac{d\omega}{d\chi}=1-\frac{A^{2}}{4rk\beta\left(r^{2}-\frac{A^{2}}{4\beta}\right)}\frac{dr}{d\chi}.$
(45)

So, we obtain

$\frac{dr}{d\chi}=\frac{d\omega}{d\chi}\left(\frac{1}{k}-\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right)r\left(1-\frac{A^{2}}{4\beta
r^{2}}\right)\left(1-\frac{A^{2}}{4k^{2}\beta r^{2}}\right)^{-1}.$ (46)

We notice that

$\frac{1}{k}-\frac{\text{Cosh}(\hat{\chi})}{\text{Sinh}(\hat{\chi})}<0,$ (47)

for all $\chi\in U_{-}$ and it is a monotonically increasing function of
$\chi$. From the theorem of existence and uniqueness of the solution to the
differential equation (35) we conclude that there exists $r(\chi)$ satisfying
the initial data $r(\hat{\chi})=\hat{r}$ at least in a neighborhood $U_{-}$ of
$\hat{\chi}$, and it is differentiable. We have for $\chi\in U_{-}$:
$\frac{dr}{d\chi}<0$. Hence $\frac{d\omega}{d\chi}>1$. We then obtain the
following bound for $\frac{dr}{d\chi}$

$-\frac{dr}{d\chi}>-\left(\frac{1}{k}-\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right)r>0,$
(48)

for all $\chi\in U_{-}$. We can integrate both side of (48) from $\chi$ to
$\hat{\chi}$, obtaining

$r>\hat{r}\frac{e^{\chi/k}}{\text{Sinh}(\chi)}\frac{\text{Sinh}(\hat{\chi})}{e^{\hat{\chi}/k}}=\frac{\hat{r}}{L}\frac{e^{\chi/k}}{\text{Sinh}(\chi)},$
(49)

and also

$1>1-\frac{A^{2}}{4k\beta
r^{2}}\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}>1-\frac{A^{2}}{4k\beta}\frac{L^{2}}{\hat{r}^{2}}>0,$
(50)

where we have used (49) and (37). Additionally from equations (35) and (50) we
get

$-\frac{1}{r}\frac{dr}{d\chi}<\left(\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}-\frac{1}{k}\right)\left(1-\frac{A^{2}L^{2}}{4\beta
k\hat{r}^{2}}\right)^{-1},$ (51)

which implies

$r<\frac{\hat{r}}{L}\frac{e^{\chi/k}}{\text{Sinh}(\chi)}\left(1-\frac{A^{2}L^{2}}{4\beta
k\hat{r}^{2}}\right)^{-1}.$ (52)

The differential function $r(\chi)$ is a monotonically decreasing function ,
bounded, from below by (49) and from above by (52), the function can then be
extended for all $\chi<\hat{\chi}$ with the same properties and bounds. Its
infimum values is obtained at $\hat{r}=r(\hat{\chi})$ and when
$\chi\rightarrow 0^{+}$ then $r(\chi)\rightarrow+\infty$. We conclude then
that $r(\chi)$ is a differentiable function with a global minimum at
$\hat{\chi}$, it is monotonically decreasing for $0<\chi<\hat{\chi}$ and
monotonically increasing for $\hat{\chi}<\chi$. Also, $r\rightarrow+\infty$
when $\chi\rightarrow 0^{+}$ and $r\rightarrow+\infty$ when
$\chi\rightarrow+\infty$. The previous analysis is depicted in Fig. 1, where
the left panel shows the existence of the global minimum
$\hat{r}=r(\hat{\chi})$ describing the throat (blue curve). To compare the
resulting solution with the GR limit, we have plotted the solution of (35) it
for the values $\\{\beta;\alpha;k\\}=\\{1;0;1\\}$ represented by the red
curve. As it is appreciated, for the GR limit there isn’t a throat. Besides,
the middle and right panels are displaying the behavior of the $\chi$
coordinate in terms of the radial $r$ one. As was mentioned , it is clear the
one to one correspondence between both variables away from the throat.
Moreover, the radial coordinate $r$ is divergent when $\chi\rightarrow 0^{+}$
or $\chi\rightarrow+\infty$, the $\chi$ is the correct one to describe the
charged throat.

Figure 1: Left panel: The throat located at the point $(\hat{\chi},\hat{r})$
(blue curve) and the GR limit (red curve), against $\chi$. Middle panel: The
$\chi$ variable as a function of the radial $r$ one, for points
$\chi<\hat{\chi}$. Right panel: The $\chi$ variable versus the radial $r$ one
for points $\chi>\hat{\chi}$.

##  V Asymptotic Behaviour of the Solutions

In this section we analyze the asymptotic behavior of the solution.
Interestingly, this analysis can be done either, starting from the original
equations without the introduction of the variable $\chi$ or from the previous
analysis in terms of $\chi$. So, from (15) and (19) we obtain

$\frac{\nabla_{i}\nabla^{i}N}{N}=\frac{A^{2}}{4k^{2}r^{4}\beta}.$ (53)

We replace it on (25) to get

$\frac{r}{2}f^{\prime}+f-1+rf\frac{N^{\prime}}{N}=-\frac{A^{2}}{4k^{2}r^{2}\beta}.$
(54)

From (26) we have

$\frac{1}{r^{2}}\left(rf^{\prime}+f-1\right)-\frac{\alpha}{2\beta}f\left(\frac{N^{\prime}}{N}\right)^{2}=\frac{A^{2}}{2\beta
r^{4}}\left(\frac{1}{2}-\frac{1}{k^{2}}\right).$ (55)

We notice from (30), that for $\chi<\hat{\chi}$, $\sqrt{f}$ is bounded when
$r\rightarrow+\infty$. We can then expand $f$, asymptotically when
$r\rightarrow+\infty$, as

$f=f_{0}+\frac{a}{r}+\frac{b}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{3}}\right).$
(56)

After replacing this expression for $f$ in the previous equations we obtain,
assuming $a\neq 0$,

$f_{0}=1,$ (57)

and

$b=-\frac{\alpha}{2\beta}\frac{a^{2}}{4}+\frac{A^{2}}{2\beta}\left(\frac{1}{k^{2}}-\frac{1}{2}\right).$
(58)

In the particular case $\alpha=0$, $\beta=1$ we obtain $b=\frac{A^{2}}{4}$,
$f$ coincides in this case with the Reissner-Nordström solution in GR. From
the above equations we also obtain the lapse $N$ as follows

$N^{2}=N^{2}_{0}\left[1+\frac{a}{r}+\frac{A^{2}}{4k^{2}r^{2}\beta}+\mathcal{O}\left(\frac{1}{r^{3}}\right)\right].$
(59)

where we shall take $N_{0}=1$. As before, if $\alpha=0$ and $\beta=1$ we then
obtain the same asymptotic behaviour of the metric as in the
Reissner–Nordström space-time in GR. In order to express the parameter $a$ in
terms of the initial data $\\{\hat{r},\hat{\chi}\\}$, we consider the
asymptotic expansion of $f$ from the expansion (30). Therefore, we take

$\chi=\frac{m}{r}+\frac{n}{r^{2}}+\mathcal{O}(\frac{1}{r^{3}}).$ (60)

From the differential equation (35) we obtain

$n=\frac{m^{2}}{k}-\frac{A^{2}}{4\beta k},$ (61)

and from the expansion of (30) we get

$a=-\frac{2}{k}m,$ (62)

that is

$a=-\frac{2}{k}\lim_{\chi\rightarrow 0^{+}}r(\chi)\text{Sinh}(\chi).$ (63)

The above limit exits and it is bounded by the following expressions

$\frac{\hat{r}}{L}\leq\lim_{\chi\rightarrow
0^{+}}r(\chi)\text{Sinh}(\chi)\leq\frac{\hat{r}}{L}\left(1-\frac{A^{2}L^{2}}{4\beta
k\hat{r}^{2}}\right)^{-1}.$ (64)

##  VI The Metric of the Charged Throat

We use two coordinate charts to describe the 4–dimensional metric describing
the charged throat solution of the electromagnetic–gravitational HL field
equations. In one chart the radial coordinate $r$ is used. In that chart
defined as $r>r(\hat{\chi}-\epsilon)$ that is, $r$ evaluated at
$\hat{\chi}-\epsilon$ or equivalently $0<\chi<\hat{\chi}-\epsilon$, with
$\epsilon>0$, the metric is given by

$ds^{2}=-N^{2}(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (65)

where $f(r)$ is given by

$f(r)=\left(1-\frac{A^{2}}{4r^{2}\beta}\right)\left(\text{Cosh}\chi(r)-\frac{1}{k}\text{Sinh}\chi(r)\right)^{2},$
(66)

and $N^{2}$ by (34). When $0<\chi<\hat{\chi}$ we have shown that the solution
to the field equations satisfies $r>\hat{r}$, where $\hat{r}$ is the value of
$r$ at the throat. It is the infimum of $r$ on that interval. Hence

$f(r)>\left(1-\frac{A^{2}}{4\hat{r}^{2}\beta}\right)\left(\text{Cosh}(\hat{\chi}-\epsilon)-\frac{1}{k}\text{Sinh}(\hat{\chi}-\epsilon)\right)^{2}>0,\quad
0<\chi<\hat{\chi}-\epsilon$ (67)

and $f(r)\rightarrow 1$ when $r\rightarrow+\infty$. The function $\chi(r)$ is
the solution to

$-\frac{d\chi}{dr}=\frac{1}{r}\left(\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}-\frac{1}{k}\right)^{-1}\left(1-\frac{A^{2}}{4\beta
r^{2}}\right)^{-1}\left(1-\frac{A^{2}}{4k\beta
r^{2}}\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right),$ (68)

we have shown that the solution $\chi(r)$ or equivalently $r(\chi)$ exist,
they are differentiable functions and there is a one to one correspondence
between $\chi$ and $r$, moreover $\frac{dr}{d\chi}<0$ for $0<\chi<\hat{\chi}$.
All the factors on the right hand side are positive on that range of $\chi$.
We now show that $N^{2}$ is also well defined for $0<\chi<\hat{\chi}$. In
fact, the denominator of the expression of $N^{2}$, Eq. (34), is bounded from
below by

$\left(r\text{Sinh}(\chi)\right)^{2}-\frac{A^{2}}{4\beta}\left(\text{Cosh}(\chi)\right)^{2}\geq\left(\frac{\hat{r}^{2}}{L^{2}}-\frac{A^{2}}{4\beta}\right)\left(\text{Cosh}(\chi)\right)^{2}\geq\frac{\hat{r}^{2}}{L^{2}}-\frac{A^{2}}{4\beta}>0.$
(69)

Moreover, the $\lim r(\chi)\text{Sinh}(\chi)$ when $\chi\rightarrow 0^{+}$
exits and it is bounded (64). We then take the integration constant $E$ such
that

$\lim_{\chi\rightarrow 0^{+}}N^{2}=1.$ (70)

The second chart in order to describe the throat solution is defined in terms
of $\chi$ for $\hat{\chi}-\epsilon<\chi$. The metric is given by

$ds^{2}=-N^{2}(r(\chi))dt^{2}+\left(\frac{dr}{d\chi}\right)^{2}\frac{d\chi^{2}}{f(r(\chi))}+r^{2}(\chi)d\Omega^{2},$
(71)

where $r(\chi)$ is the solution of the differential equation (35). It is a
differentiable function with a minimum at $\hat{r}=r(\hat{\chi})$.. The
expression

$\left(\frac{dr}{d\chi}\right)^{2}\frac{1}{f(r(\chi))}=\frac{1}{\left(\text{Sinh}(\chi)\right)^{2}}\left(1-\frac{A^{2}}{4\beta
r^{2}}\right)\left(1-\frac{A^{2}}{4\beta
kr^{2}}\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\right)^{-2},$ (72)

is non–singular and bounded away from zero on the throat
$\hat{\chi}-\epsilon<\chi<\hat{\chi}+\epsilon$. Indeed,

$1-\frac{A^{2}}{4\beta r^{2}}\geq
1-\frac{A^{2}}{4\beta\hat{r}^{2}}>0\quad\forall\quad\chi,$ (73)
$1-\frac{A^{2}}{4\beta kr^{2}}\frac{\text{Cosh}(\chi)}{\text{Sinh}(\chi)}\geq
1-\frac{A^{2}}{4\beta k^{2}\hat{r}^{2}}>0\quad\forall\quad\chi\geq\hat{\chi},$
(74)

and the bound (50) for all $\chi<\hat{\chi}$. Furthermore, $N^{2}$ is well
defined on the throat, since

$\left(r\text{Sinh}(\chi)\right)^{2}-\frac{A^{2}}{4\beta}\left(\text{Cosh}(\chi)\right)^{2}\geq\left(\hat{r}^{2}-\frac{A^{2}}{4\beta
k^{2}}\right)\left(\text{Sinh}(\chi)\right)^{2}>0\quad\forall\quad\chi>\hat{\chi},$
(75)

and it is also bounded (69) for all $\chi<\hat{\chi}$. Henceforth, we have a
regular metric on the throat. On the other hand, away from the throat, the
expression (72) has a singularity when $\chi\rightarrow+\infty$, corresponding
to $r\rightarrow+\infty$. It is an essential singularity of the metric (71).
This behavior is shown on Fig. 3, where it is clear that for points
$\chi>\hat{\chi}$ the function $N$ tends to zero, while the function $f$ tends
to infinite.

Figure 2:  Left panel: The lapse function $N$ against the radial coordinate
$r$, for points $\chi<\hat{\chi}$ (dashed black line). As can be seen, this
function shows an asymptotically flat behavior $N\rightarrow 1$ when
$r\rightarrow+\infty$. Middle panel: The lapse function $N$ versus the radial
coordinate $r$, for points $\chi>\hat{\chi}$ (dashed black line). As it is
appreciated, $N$ tends to $0$ making the metric (71) singular. Right panel:
The function $f$ against the radial coordinate $r$, for points
$\chi<\hat{\chi}$ (dashed black line). As can be seen, this function shows an
asymptotically flat behavior $f\rightarrow 1$ when $r\rightarrow+\infty$.

On the other hand, the functions $N$ and $f$ of the metric (71) in terms of
the $\chi$ variable are depicted in Fig. 3, left and right panel respectively.

At this point it is illustrative to compare the charged throat solution with
well–known Morris–Thorne (MT) WH space–time m1 ; m2 ; m3 . Of course, the
comparison can be done only from the geometrical point of view.

Before going into the comparison, it is instructive to briefly go over the
characteristics of the MT metric. So, in terms of the proper radial distance
$l$, the most general spherically, symmetric, static and non–rotating WH
space–time connecting two asymptotically flat regions is being describing by
m1 ; m2 ; m3

$ds^{2}_{\text{MT}}=-e^{-\phi(l)}dt^{2}+dl^{2}+r^{2}(l)\left[d\theta^{2}+\text{sin}^{2}\theta
d\varphi^{2}\right],$ (76)

where the proper length $l$ and functions $\phi(l)$ and $r(l)$ should satisfy
some requirements:

  * •

The coordinate $l$ covers the entire range $(-\infty,+\infty)$.

  * •

The assumed absence of event horizons implies that $\phi(l)$ must be
everywhere finite.

  * •

In order for the spatial geometry to tend to an appropriate asymptotically
flat limit one must impose,

$\lim_{l\rightarrow\pm\infty}\frac{r(l)}{|l|}=1\quad\mbox{and}\quad\lim_{l\rightarrow\pm\infty}\phi(l)=\phi_{\pm},$
(77)

where $\phi_{\pm}$ must be finite.

  * •

The radius of the WH throat is defined by

$\hat{r}=\text{min}\\{r(l)\\},$ (78)

where without of generality one can take the WH throat to occurs at $l=0$.

The explicit relation between the radial coordinate $r$ and the proper radial
distance $l$ is given by

$r(l)=\sqrt{\varrho^{2}+l^{2}},$ (79)

being $\varrho$ a constant. Then it is no hard to note that when
$l\rightarrow-\infty$ then $r\rightarrow+\infty$ and when
$l\rightarrow+\infty$ then $r\rightarrow+\infty$. In the present solution, the
equivalence situation corresponds to take the $\chi$ coordinate as the proper
radial distance $l$ in the MT space–time. However there is a small difference,
the range that $\chi$ covers is $[0,+\infty)$. Although this is not an
impediment to compare both models. As was pointed out earlier, when
$\chi\rightarrow 0^{+}$ the radial coordinate goes to $r\rightarrow+\infty$
and when $\chi\rightarrow+\infty$ the $r\rightarrow+\infty$, just as MT
solution. On the other hand, the WH throat of the MT solution is located when
$l\rightarrow 0$ yielding from Eq. (79) to

$r(0)=|\varrho|,$ (80)

$|\varrho|$ is identified as $r(0)=\hat{r}$ the WH throat size. Now, for the
charged HL throat this occurs when $\chi\rightarrow\hat{\chi}$, hence
$r(\hat{\chi})=\hat{r}$. As can be seen from the line element (76) the
solution is completely regular when it is described in term of the proper
radial distance. Of course, replacing the value $l=0$ in (76) and fixing both
the time and the equatorial angle ı.e, $\theta=\pi/2$ the metric (76) reduce
to

$ds^{2}_{\text{MT}}=\hat{r}^{2}d\varphi^{2}.$ (81)

The above expression is describing the WH throat region, a circle with radius
$\hat{r}$. Respect to the line element (71) when the $\chi$ is used, the
metric is completely regular. In fact, from Fig. 4 it is clear that
$g_{\chi\chi}$ metric component is completely regular and non vanishing at
$\hat{\chi}$. On the other hand, if the MT solution is expressed in canonical
or Schwarzschild–like coordinates, the line element (76) becomes

$ds^{2}_{\text{MT}}=-e^{\Phi(r)_{\pm}}dt^{2}+\frac{dr^{2}}{1-\frac{b(r)_{\pm}}{r}}+r^{2}\left[d\theta^{2}+\text{sin}^{2}\theta
d\varphi^{2}\right],$ (82)

where the function $\phi(r)$ is identified as the red–shift function and
$b(r)$ is the shape function. In this coordinates, the WH throat occurs at
$b(\hat{r})=\hat{r}$ m1 ; m2 ; m3 . Therefore, it is clear from (82) that
$g_{rr}$ metric component is indeterminate making the metric degenerates. For
our solution the same situation happens when the metric is expressed in terms
of the radial coordinate $r$ ı.e., $\chi=\chi(r)$, where the line element (71)
degenerates. This fact is corroborated in right panel of Fig. 3 where $f$ is
vanishing at the throat $\hat{\chi}$. Hence the throat area is best described
in terms of $l$ for the MT model and in terms of $\chi$ in the present case.
In other words, $l$ and $\chi$ are the “good coordinates” when it comes to
describing the throat in the mentioned models, respectively. Finally, in
comparing the $g_{tt}$ components of the MT and (71) solution, it is evident
that the lapse function $N$ is playing the role of the ref–shift function
$\phi$. Independently if one is on the chart described by $r$ or $\chi$, the
lapse function $N$ is always regular and non vanishing at the throat, thus
there is not event horizon as required (see Figs. 2–3).

Figure 3:  Left panel: The lapse function $N$ trend versus the $\chi$
coordinate. As can be seen, this object is completely regular at the throat
$\chi=\hat{\chi}$. Right panel: The behavior of the $f$ function against the
$\chi$ variable. It is observed that at the throat, $f(\hat{\chi})=0$ making
the metric (71) singular.  Figure 4:  The trend of the $g_{\chi\chi}$ metric
component of the line element (71) versus $\chi$.

##  VII Concluding remarks

We obtain a static spherically symmetric solution of the gauge vector–gravity
HL equations r29 ; r30 , in the low energy regime where only the $z=1$
potential terms are considered. The solution depends on three parameters, the
low energy couplings $\alpha$ and $\beta$ of the HL gravity and the electric
charge. The metric describes a charged throat connecting a space–time which in
the limit when $\alpha$ tends to 0 and $\beta$ to 1 becomes the
Reissner–Nordström solution of Einstein–Maxwell theory and an asymptotic
singular manifold. The singularity is an essential one. To obtain the solution
we introduce a new coordinate, which allows to describe it along the throat.
We solve the field equations up to the nonlinear differential equation (35).
The solution is proven to exist starting from an initial data, satisfying the
restriction (37) at the throat and then showing its extension to an
asymptotically flat space–time on one side and to an asymptotically singular
manifold on the other side. We also describe the asymptotic behavior on the
regular side and relate it to the initial data. We may also identify the
coefficient $a$ of the asymptotic expansion with $M$ the mass of the system
and the integration constant $A$ with $Q$ the electric charge. From (37) and
(64) we conclude that the solution exists when $M$ satisfies the restriction
$M>|A/2k^{2}\beta^{1/2}|$. We notice that in the relativistic limit the
restriction reduces to $M>|A/2|$ the condition in the Reissner–Nordström
solution of GR to have two horizons, the event horizon and the internal Cauchy
horizon. We have then obtained the anisotropic generalization, in the HL
framework, of the Reissner–Nordström solution of GR. Hence, although there are
not horizons in the HL solution the physical relevant relation between the
mass and the charge is also obtained as in the relativistic case. Although we
have shown the existence of the solution and characterized it by using bounds
and estimates, we do not have an explicit formula for the metric. We thus
explicitly obtain the solution by following a numerical approach. The
numerical solution of (35) is depicted by the left panel of Fig. 1. The blue
curve shows the existence of the throat located at the point
$(\hat{\chi},\hat{r})$, whilst the red curve displays the GR limit. On the
other hand the middle and right panels in Fig. 1 exhibit the behavior of
$\chi$ is terms of $r$. The one to one correspondence is corroborated and it
is clear from these panels, that the $\chi$ variable is the good one to
describe the throat region, since $r(\chi)$ has a global minimum at the
throat. As was mentioned in the relativistic limit, the asymptotic behavior of
this solution yields the well–known Reissner–Nordstöm solution. The asymptotic
solution is given by Eqs. (56) and (59), where in the relativistic limit ı.e,
$\beta\rightarrow 1$ and $\alpha\rightarrow 0$, the parameters $a$ and $b$ can
be interpreted as $-2M$ and $Q^{2}$ respectively, where $M$ is Schwarzschild
mass and $Q$ the electric charge.

In considering the full metric given by Eq. (71), its graphical representation
in Fig. 2 depicts the behavior for $N$ and $f$ as functions of the radial
coordinate $r$. Interestingly, both functions for points $\chi<\hat{\chi}$
tend to one, hence the metric (71) is asymptotically flat, describing a
Minkowski space–time. Those are the reasons why we called the solution a
throat and not a WH, because following the definition given in m1 ; m2 ; m3  a
WH structure is asymptotically flat for points below and above the throat. In
this case we have an essential singularity on one side and an asymptotically
flat region on the other side. On the other hand, the line element expressed
in terms of $\chi$ allows to compare with the usual WH behavior at the throat.
In fact, the Fig. 3 depicts the trend of $N$ and $f$ as functions of $\chi$,
as can be observed $N$ at $\hat{\chi}$ is not zero while $f$ is. The
geometrical behavior is exactly the same as the MT solution m1 ; m2 ; m3
where a traversable WH solution must satisfy $\phi(\hat{r})\neq 0$ and
$b(\hat{r})=\hat{r}$. Of course, it is evident from (82) that evaluating the
shape function $b(r)$ at the WH throat, the line element becomes singular.

Finally, it is remarkable to point out that this kind of solution are
characteristic of the non–projectable HL theory, without introducing any kind
of exotic matter distribution r31  as it is required in the framework of GR m1
; m2 ; m3 . The throat is supported by the static electric field, introduced
by the 3+1 dimensional reduced theory r29 ; r30  after performing the
Kaluza–Klein program, acting as matter content. Concerning this point, the
energy conditions, Bianchi’s identities for the matter sector and other
treatment performed in relativistic theories when considering this type of
solutions, are still not known in the framework of the HL theory.

## Acknowledgements

F. Tello-Ortiz thanks the financial support by the CONICYT PFCHA/DOCTORADO-
NACIONAL/2019-21190856 and project ANT-1956 at the Universidad de Antofagasta,
Chile.

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# Parity and Partition of the Rational Numbers

Peter Lynch and Michael Mackey
School of Mathematics & Statistics, University College
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We define an extension of parity from the integers to the rational numbers.
Three parity classes are found — even, odd and ‘none’. Using the 2-adic
valuation, we partition the rationals into subgroups with a rich algebraic
structure.

The natural density provides a means of distinguishing the sizes of countably
infinite sets. The Calkin-Wilf tree has a remarkably simple parity pattern,
with the sequence ‘odd/none/even’ repeating indefinitely. This pattern means
that the three parity classes have equal natural density in the rationals. A
similar result holds for the Stern-Brocot tree.

The natural numbers $\mathbb{N}$ split nicely into two subsets, the odd and
even numbers

$\mathbb{N}_{\mathrm{O}}=\\{1,3,5,7,\dots\\}\,,\qquad\mathbb{N}_{\mathrm{E}}=\\{2,4,6,8,\dots\\}\,.$

Stopping at some number $2N$, the odd and even numbers are equinumerous.
Stopping at $2N+1$, the odds are slightly ahead, but as $N$ gets larger, the
ratio of odd to even numbers tends to $1$. So, we can say informally that
there are the same number of odd and even integers. This will be made precise
below by defining densities for the sets $\mathbb{N}_{\mathrm{O}}$ and
$\mathbb{N}_{\mathrm{E}}$. Similar arguments apply to the integers
$\mathbb{Z}$, which split into two subsets

$\displaystyle\mathbb{Z}_{\mathrm{O}}$ $\displaystyle=$
$\displaystyle\\{\dots\ -3,-1,+1,+3,+5,\dots\\}$
$\displaystyle\mathbb{Z}_{\mathrm{E}}$ $\displaystyle=$
$\displaystyle\\{\dots\ -4,-2,\ \ 0,+2,+4,\dots\\}\,.$

The integers form an abelian group $(\mathbb{Z},+)$ under addition. The even
numbers form an additive subgroup of $(\mathbb{Z},+)$, with index
$[\mathbb{Z}:\mathbb{Z}_{\mathrm{E}}]=2$ and two cosets
$\mathbb{Z}_{\mathrm{E}}$ and
$\mathbb{Z}_{\mathrm{E}}+1=\mathbb{Z}_{\mathrm{O}}$. This definition provides
a bijection between the two cosets, which have the same cardinality.

##  1 Parity.

The distinction between odd and even numbers is called _parity_. The even/odd
concept is defined only for the integers. The distinction does not apply to
fractions or irrational numbers, but one may wonder if there is a natural way
to extend the concept of parity to larger sets of numbers.

What characteristics might one require of such an extension? The definition
would have to agree with the traditional definition for the integers, so $5$
would continue to be odd and $10$ even. In addition, the usual ‘rules of
parity’ might be required:

  1. 1.

The sum of two even numbers is even; the product is even.

  2. 2.

The sum of two odd numbers is even; the product is odd.

  3. 3.

The sum of an even and an odd number is odd; the product is even.

  4. 4.

An odd number plus $1$ is even; an even number plus $1$ is odd.

Table 1 shows the effects of addition and multiplication on the ring of
integers.

Table 1: Addition table (left) and multiplication table (right) for
$\mathbb{Z}$.

$\boldsymbol{+}$ | $\mathbf{even}$ | $\mathbf{odd}$
---|---|---
$\mathbf{even}$ | $even$ | $odd$
$\mathbf{odd}$ | $odd$ | $even$

$\boldsymbol{\times}$ | $\mathbf{even}$ | $\mathbf{odd}$
---|---|---
$\mathbf{even}$ | $even$ | $even$
$\mathbf{odd}$ | $even$ | $odd$

If the concept of parity is extended to larger sets of numbers, some of the
properties indicated above may have to be sacrificed. For rational numbers, we
might define a number $q=m/n$ to be even if the numerator $m$ is even and odd
if $m$ is odd. But then $\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$, meaning that
two odd rationals would add to yield another odd one.

We will distinguish between ‘odd’ and ‘uneven’ rationals: assuming $m$ and $n$
to be relatively prime integers, $(m,n)=1$, we will adopt the following
definition:

$\mbox{For any rational number\ }q=\frac{m}{n}\,,\quad\begin{cases}q\ \mbox{is
\emph{even} if $m$ is even},\\\ q\ \mbox{is \emph{uneven} if $m$ is
odd}\,.\end{cases}$ (1)

## A three-way split.

There is a simple way of separating the rational numbers into three subsets:

$\mbox{For any rational\ }q=\frac{m}{n}\,,\quad\begin{cases}q\ \mbox{has
parity \emph{even} if $m$ is even and $n$ is odd,}\\\ q\ \mbox{has parity
\emph{odd} if $m$ is odd and $n$ is odd,}\\\ q\ \mbox{has parity \emph{none}
if $m$ is odd and $n$ is even.}\end{cases}$ (2)

The term _none_ is an initialism for ‘neither odd nor even’. Corresponding to
this three-way partition, we define three subsets of the rationals:

$\displaystyle\mbox{Even:\ \ }\mathbb{Q}_{\mathrm{E}}$ $\displaystyle=$
$\displaystyle\\{q\in\mathbb{Q}:q=\textstyle{\frac{2m}{2n+1}\ \mbox{for some}\
m,n\in\mathbb{Z}}\\}$ $\displaystyle\mbox{Odd:\ \ }\mathbb{Q}_{\mathrm{O}}$
$\displaystyle=$
$\displaystyle\\{q\in\mathbb{Q}:q=\textstyle{\frac{2m+1}{2n+1}\ \mbox{for
some}\ m,n\in\mathbb{Z}}\\}$ $\displaystyle\mbox{None:\ \
}\mathbb{Q}_{\mathrm{N}}$ $\displaystyle=$
$\displaystyle\\{q\in\mathbb{Q}:q=\textstyle{\frac{2m+1}{2n}\ \mbox{for some}\
m,n\in\mathbb{Z}}\\}\,.$

These three sets are mutually disjoint and
$\mathbb{Q}=\mathbb{Q}_{\mathrm{E}}\uplus\mathbb{Q}_{\mathrm{O}}\uplus\mathbb{Q}_{\mathrm{N}}$.
It is immediately obvious that
$\mathbb{Z}_{\mathrm{E}}\subset\mathbb{Q}_{\mathrm{E}}$ and
$\mathbb{Z}_{\mathrm{O}}\subset\mathbb{Q}_{\mathrm{O}}$, confirming that the
definition of parity for the rationals is an extension of the usual meaning
for the integers. We see that the even and odd rationals respect the four
‘rules of parity’ listed above.

###  1.1 “Twice as many uneven as even fractions”.

The rational numbers are countable: they can be put into one-to-one
correspondence with the natural numbers. We can list all rationals in $(0,1)$
in a sequence where, for each $n$ in turn, all (new) numbers $m/n$ with $m<n$
are listed in order. For $n_{\mathrm{max}}=8$ we have

$\displaystyle\bigl{\\{}\textstyle{\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{5}{6},\frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7},\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8}}\bigr{\\}}\,.$
(3)

Rearrangement in increasing order of magnitude gives the Farey sequence
$F_{8}$.

A Mathematica program was written to count the proportion of rationals in each
parity class in the interval $(0,1)$, with denominators less than or equal to
$n$, for a range of cut-off values $n\leq n_{\mathrm{max}}$. The ratios are
plotted in Fig. 1. As $n$ increases, the ratios of numbers with parity even,
odd and none all tend to the limit $\frac{1}{3}$. Colloquially, there are an
equal number of rationals with parity even, odd and none, and “twice as many
uneven as even rationals”.

Figure 1: Parity ratio $r$ for rationals $m/n$ of parity even (solid line,
blue online), odd (dashed line, red online) and none (dotted line, black
online) for $n\leq n_{\mathrm{max}}=20$.

##  2 The Density of subsets of $\mathbb{N}$.

In pure set-theoretic terms, the set of even positive numbers is “the same
size” as the set of all natural numbers; both are infinite countable sets.
However, cardinality is a blunt instrument: with the usual ordering, every
second natural number is even and, intuitively, we feel that there are half as
many even numbers as natural numbers. The concept of _density_ provides a
means of expressing the relative sizes of sets that is more discriminating
than cardinality.

Density — also called natural or asymptotic density — is defined for many
interesting subsets of $\mathbb{N}$, although not for all subsets. Assume a
subset $A$ of $\mathbb{N}$ is enumerated as $\\{a_{1},a_{2},\dots\\}$. We
define the density of $A$ in $\mathbb{N}$ as the limit, if it exists,

$\rho_{\mathbb{N}}(A)=\lim_{n\to\infty}\ \frac{|\\{a_{k}:a_{k}\leq
n\\}|}{n}\,.$ (4)

Thus, if the fraction of elements of $A$ among the first $n$ natural numbers
converges to a limit $\rho_{\mathbb{N}}(A)$ as $n$ tends to infinity, then $A$
has density $\rho_{\mathbb{N}}(A)$ [7]. More generally, if
$A=\\{a_{1},a_{2},a_{3},\dots\\}$ is a subset of a countable set $X$
enumerated as $\\{x_{1},x_{2},x_{3},\dots\\}$, we define the density of $A$ in
$X$ — if it exists — as

$\rho_{X}(A)=\lim_{n\to\infty}\
\frac{|A\cap\\{x_{1},x_{2},\dots,x_{n}\\}|}{n}\,.$ (5)

For $X=\mathbb{N}$, we usually write $\rho_{\mathbb{N}}(A)$ as $\rho(A)$. For
$A=\mathbb{N}_{\mathrm{E}}$ or $A=\mathbb{N}_{\mathrm{O}}$, we have
$\rho(A)=\textstyle{\frac{1}{2}}$, as might be expected. This is consistent
with our intuitive notion that $50\%$ of the natural numbers are even and
$50\%$ are odd.

Let us now rearrange the natural numbers into a set $F$ such that there are
_twice as many even as odd numbers in $F$_. We reorder $\mathbb{N}$ so that
each odd number is followed by two even ones:

$F=\\{1,2,4,3,6,8,5,10,12,\ \dots\ ,2n-1,4n-2,4n,\dots\\}\,.$

It is easy to see that $\rho_{F}(\mathbb{N}_{\mathrm{E}})=\frac{2}{3}$ and
$\rho_{F}(\mathbb{N}_{\mathrm{O}})=\frac{1}{3}$. Proceeding further, we can
construct a set $H$ in which the $n$-th odd number is followed by $n$ even
numbers. We find that $\rho_{H}(\mathbb{N}_{\mathrm{E}})=1$, so that “almost
all the elements of $H$ are even”.

These examples make it clear that density depends strongly on the ordering of
the reference set. Our intuition is guided by the usual (natural) ordering of
the natural numbers and the alternation between odd and even numbers leads us
to the conclusion that, somehow, they are equal in number, each comprising
“half” of the set of natural numbers. Density relative to $\mathbb{N}$ is
consistent with this intuition.

With the ordering $\\{0,+1,-1,+2,-2,\ldots\\}$ of the integers, the densities
defined by (5) are

$\rho_{\mathbb{Z}}(\mathbb{Z}_{\mathrm{E}})=\textstyle{\frac{1}{2}}\,,\qquad\rho_{\mathbb{Z}}(\mathbb{Z}_{\mathrm{O}})=\textstyle{\frac{1}{2}}\,.$

We will prove that, for the rational numbers with the Calkin-Wilf and Stern-
Brocot orderings defined below,

$\rho_{\mathbb{Q}}(\mathbb{Q}_{\mathrm{E}})=\rho_{\mathbb{Q}}(\mathbb{Q}_{\mathrm{O}})=\rho_{\mathbb{Q}}(\mathbb{Q}_{\mathrm{N}})=\tfrac{1}{3}\,.$
(6)

##  3 Partitioning the rationals.

Table 2: Addition table (left) and multiplication table (right) for
$\mathbb{Q}$. The entry ‘ _any_ ’ indicates that the result may be in any of
the three parity classes.

$\boldsymbol{+}$ | $\mathbf{even}$ | $\mathbf{odd}$ | $\mathbf{none}$
---|---|---|---
$\mathbf{even}$ | $even$ | $odd$ | $none$
$\mathbf{odd}$ | $odd$ | $even$ | $none$
$\mathbf{none}$ | $none$ | $none$ | $\underline{any}$

$\boldsymbol{\times}$ | $\mathbf{even}$ | $\mathbf{odd}$ | $\mathbf{none}$
---|---|---|---
$\mathbf{even}$ | $even$ | $even$ | $\underline{any}$
$\mathbf{odd}$ | $even$ | $odd$ | $none$
$\mathbf{none}$ | $\underline{any}$ | $none$ | $none$

In Table 2 we show the results of adding and multiplying numbers from the
three parity classes. The most important thing to notice is that, if we
confine attention to only the even and odd rationals, the tables are identical
to the addition and multiplication tables for $\mathbb{Z}$ (Table 1). The
entry _“any”_ in the tables indicates a number that is a ratio of two even
numbers and that may, after reduction, be in any of the three parity classes.
Examining the left panel of Table 2, we see that $(\mathbb{Q}_{\mathrm{E}},+)$
is an additive (normal) subgroup of $(\mathbb{Q},+)$. In Table 2 (right panel)
we show the results of multiplying numbers from the three parity classes.
Restricting attention to the even and odd rationals only — omitting those with
no parity — we define

$\mathbb{Q}_{\mathrm{P}}:=\mathbb{Q}_{\mathrm{E}}\uplus\mathbb{Q}_{\mathrm{O}}\,.$
(7)

This is the set of all rationals whose denominators are odd numbers in
$\mathbb{Z}$. It is closed under addition and multiplication and forms a
commutative subring of the field $\mathbb{Q}$. Moreover, since there are no
divisors of zero, $\mathbb{Q}_{\mathrm{P}}$ is an integral domain [4].
Although $\mathbb{Q}_{\mathrm{P}}$ is not an ideal of $\mathbb{Q}$ (fields do
not have proper ideals), it is a (normal) subgroup of $(\mathbb{Q},+)$. So, we
may enquire about its index $[\mathbb{Q}:\mathbb{Q}_{\mathrm{P}}]$ and its
quotient group $\mathbb{Q}/\mathbb{Q}_{\mathrm{P}}$.

Somewhat out of context, we mention the easily-proved observation that all
three parity classes, $\mathbb{Q}_{\mathrm{E}}$, $\mathbb{Q}_{\mathrm{O}}$ and
$\mathbb{Q}_{\mathrm{N}}$, are (topologically) dense in the rationals.

### 2-Adic valuation and the “degree of evenness”.

_All multiples of 2 are even, but_ _some are more even than others._

The $p$-adic valuation — or $p$-adic order [6] — of an integer $n$ is the
function

$\nu_{p}(n)=\begin{cases}\max\\{k\in\mathbb{N}:p^{k}\mid n\\}&\mbox{for\
}n\neq 0\\\ \infty&\mbox{for\ }n=0\end{cases}$

This is extended to the rational numbers $m/n$:

$\nu_{p}\left(\frac{m}{n}\right)=\nu_{p}(m)-\nu_{p}(n)\,.$

It is easily proved that, for any rationals $q_{1}$ and $q_{2}$,

$\nu_{p}(q_{1}+q_{2})\geq\min\\{\nu_{p}(q_{1}),\nu_{p}(q_{2})\\}\,,$ (8)

with equality holding if $\nu_{p}(q_{1})\neq\nu_{p}(q_{2})$.

We shall be concerned exclusively with the case $p=2$. We note that
$\mathbb{Q}_{\mathrm{P}}=\\{q\in\mathbb{Q}:\nu_{2}(q)\geq 0\\}$ and
$\mathbb{Q}_{\mathrm{E}}=\\{q\in\mathbb{Q}:\nu_{2}(q)>0\\}$. The “degree of
evenness” of a number can be expressed in terms of the 2-adic valuation. For
an integer $n$, the 2-adic valuation is the largest natural number $k$ such
that $2^{k}$ divides $n$. It is normally written $\nu_{2}(n)$ or
$\mathop{\mathrm{missing}}{ord}_{2}(n)$. For even integers, $\nu_{2}(n)>0$;
for odd integers, $\nu_{2}(n)=0$. By convention, $\nu_{2}(0)=\infty$ (since
zero is divisible by every power of $2$).

If we write a rational number $q$ in the form $2^{k}(2m+1)/(2n+1)$ with
$k\in\mathbb{Z}$, then $\nu_{2}(q)=k$. Odd rationals have order $0$ and
rationals with no parity have negative 2-adic order. In particular, half
integers have 2-adic order equal to $-1$. In summary,

For $q$ rational with parity even, $\displaystyle\nu_{2}(q)>0\,,$ For $q$
rational with parity odd, $\displaystyle\nu_{2}(q)=0\,,$ For $q$ rational with
parity none, $\displaystyle\nu_{2}(q)<0\,.$

The 2-adic order clearly identifies the parity classes of the rationals, and
it provides a means of partitioning them into finer-grain parity classes. The
resulting partition reveals a wealth of algebraic structure. For all
$k\in\mathbb{Z}$, we define the set of all rational numbers with 2-adic
valuation $k$:

$\mathit{Q}_{k}=\\{q\in\mathbb{Q}:\nu_{2}(q)=k\\}\qquad\mbox{and}\qquad\mathit{Q}_{\infty}=\\{\
0\ \\}\,.$

The union of all the $\mathit{Q}$-sets comprises the entire set of rationals

$\mathbb{Q}=\\{\ 0\ \\}\uplus\biguplus_{k=-\infty}^{\infty}\mathit{Q}_{k}\,.$

We illustrate the subsets $\mathit{Q}_{k}$ in Fig. 2. The vertical axis is the
2-adic valuation $\nu_{2}$. Each subset $\mathit{Q}_{k}$ is represented by a
horizontal dotted line. We remark that all odd rationals are in
$\mathit{Q}_{0}$ and all even rationals are in $\bigcup_{k>0}\mathit{Q}_{k}$.
For all $K\in\mathbb{Z}$, we define

$\mathbb{Q}_{\mathrm{K}}\equiv\\{\ 0\ \\}\uplus\biguplus_{k\geq
K}\mathit{Q}_{k}$ (9)

and observe that $\mathbb{Q}_{\mathrm{K}}$ is a subgroup of $\mathbb{Q}$. We
write this as $\mathbb{Q}_{\mathrm{K}}\ \text{\large$\trianglelefteq$}\ \
\mathbb{Q}$. Note, in particular, that
$\mathbb{Q}_{0}=\mathbb{Q}_{\mathrm{P}}$ and
$\mathbb{Q}_{1}=\mathbb{Q}_{\mathrm{E}}$. There is an infinite chain of
subgroups, starting with $\mathbb{Q}_{\infty}=\\{\ 0\ \\}$ and extending
through all the $\mathbb{Q}_{K}$ groups to the full group of rationals:

$\mathbb{Q}_{\infty}=\\{\ 0\ \\}\ \text{\large$\trianglelefteq$}\ \ \cdots\
\text{\large$\trianglelefteq$}\ \ \mathbb{Q}_{2}\
\text{\large$\trianglelefteq$}\ \ \mathbb{Q}_{1}\
\text{\large$\trianglelefteq$}\ \ \mathbb{Q}_{0}\
\text{\large$\trianglelefteq$}\ \ \mathbb{Q}_{-1}\
\text{\large$\trianglelefteq$}\ \ \mathbb{Q}_{-2}\
\text{\large$\trianglelefteq$}\ \ \cdots\ \text{\large$\trianglelefteq$}\ \
\mathbb{Q}\,.$

### Dyadic rational numbers.

A dyadic rational is a number that can be expressed as a fraction whose
denominator is a power of two. The usual definition of the dyadic rational
numbers [1, pg. 122] is

$\mathbb{D}=\left\\{\frac{z}{2^{m}}:z\in\mathbb{Z},m\in\mathbb{Z}\right\\}\,.$

Note that the integers are included in the set of dyadic rationals. A
convenient alternative definition is

$\mathbb{D}=\\{2^{k}(2\ell-1):k\in\mathbb{Z},\ell\in\mathbb{Z}\\}\uplus\\{\ 0\
\\}\,,$

since all the numbers of the form $2^{k}(2\ell-1)$ are in $\mathit{Q}_{k}$.
Moreover, the expression of each number in this form is unique. We also define
the sets

$\mathit{D}_{k}=\\{2^{k}(2\ell-1):\ell\in\mathbb{Z}\\}\qquad\mbox{and}\qquad\mathit{D}_{\infty}=\\{\
0\ \\}\,,$

and note that
$\mathbb{D}=\biguplus_{k}\mathit{D}_{k}\,\uplus\,\mathit{D}_{\infty}$. We see
that $\mathit{D}_{0}=\mathbb{Z}_{\mathrm{O}}$ and
$\biguplus_{k>0}\mathit{D}_{k}=\mathbb{Z}_{\mathrm{E}}$.

The dyadics correspond to all real numbers with finite binary expansions, and
also to the set of surreal numbers born on finite days [3]. The dyadic
rational numbers form a ring between the ring of integers and the field of
rational numbers:

$\mathbb{Z}\ \text{\large$\trianglelefteq$}\ \mathbb{D}\
\text{\large$\trianglelefteq$}\ \mathbb{Q}\,.$

The sets $\mathit{D}_{k}$ are indicated in Fig. 2 by the marked points in
$\mathit{Q}_{k}$. The vertical axis is the 2-adic valuation $\nu_{2}$. For
each $k$, $\mathit{D}_{k}\subset\mathit{Q}_{k}$. The (black) dots at level
$k=0$ are the odd integers. The (blue) dots at positive $k$-levels are the
even integers. The (red) dots at each negative level $k<0$ are the dyadic
fractions, with odd numerator and denominator $2^{k}$. Zero sits, like an
angel, on top of the tree.

By analogy with the definition (9) of the $\mathbb{Q}_{K}$-sets, we construct
a countable infinity of subgroups of $\mathbb{D}$:

$\mathbb{D}_{K}:=\\{\ 0\ \\}\uplus\biguplus_{k\geq K}\mathit{D}_{k}\,.$

Particular cases of the $\mathbb{D}$-sets include

$\mathbb{D}_{-\infty}=\mathbb{D}\,,\qquad\mathbb{D}_{-1}=\textstyle{\frac{1}{2}}\mathbb{Z}\,,\qquad\mathbb{D}_{0}=\mathbb{Z}\,,\qquad\mathbb{D}_{1}=\mathbb{Z}_{\mathrm{E}}\,,\qquad\mathbb{D}_{\infty}=\\{\
0\ \\}\,.$

There is an infinite chain of subgroups starting with $\mathbb{D}_{\infty}$
and extending through all the $\mathbb{D}_{K}$ groups to the full group of
dyadic rationals:

$\mathbb{D}_{\infty}=\\{\ 0\ \\}\ \text{\large$\trianglelefteq$}\ \ \cdots\
\text{\large$\trianglelefteq$}\ \ \mathbb{D}_{2}\
\text{\large$\trianglelefteq$}\ \ \mathbb{D}_{1}\
\text{\large$\trianglelefteq$}\ \ \mathbb{D}_{0}\
\text{\large$\trianglelefteq$}\ \ \mathbb{D}_{-1}\
\text{\large$\trianglelefteq$}\ \ \mathbb{D}_{-2}\
\text{\large$\trianglelefteq$}\ \ \cdots\ \text{\large$\trianglelefteq$}\ \
\mathbb{D}\,.$

Figure 2: Partition of the rational numbers. The vertical axis is the 2-adic
valuation $\nu_{2}$. Each (dense) subset $\mathit{Q}_{k}$ is represented by a
horizontal dotted line. The sets $\mathit{D}_{k}$ are indicated by the marked
points in $\mathit{Q}_{k}$. The totality of these comprises the dyadic
rationals $\mathbb{D}$.

Readers familiar with the theory of $p$-adic numbers may wish to show that
$\mathbb{Q}_{\mathrm{P}}$ is the ring of rational-valued 2-adic integers,
$\mathbb{Q}\cap\mathbb{Z}_{2}$, and the dyadic rational numbers may be
expressed as

$\mathbb{D}=\mathbb{Q}\cap\bigcap_{p\ \mathrm{odd}}\mathbb{Z}_{p}\,.$

##  4 Cosets of $\mathbb{Q}_{\mathrm{P}}$ in $\mathbb{Q}$.

In the following section we show that any two rationals $q_{1}$ and $q_{2}$
with distinct, negative 2-adic orders are representatives of distinct cosets:
$q_{1}+\mathbb{Q}_{\mathrm{P}}\neq q_{2}+\mathbb{Q}_{\mathrm{P}}$. Thus, if
$q_{1}+\mathbb{Q}_{\mathrm{P}}=q_{2}+\mathbb{Q}_{\mathrm{P}}$ then
$\nu_{2}(q_{1})=\nu_{2}(q_{2})$. Consequently, there is at least one coset for
each $k<0$ and therefore an infinite number of cosets. However, it is clear
that $\nu_{2}(q_{1})=\nu_{2}(q_{2})$ does not imply equality of cosets;
consider, for example, $q_{1}=\frac{1}{4}$ and $q_{2}=\frac{3}{4}$, since
$(q_{2}-q_{1})=\textstyle{\frac{1}{2}}\not\in\mathbb{Q}_{\mathrm{P}}$.

We now investigate the cosets $q+\mathbb{Q}_{\mathrm{P}}$ in
$\mathbb{Q}/\mathbb{Q}_{\mathrm{P}}$. First, we note that if
$q\in\mathit{Q}_{-k}$ then $q+\mathbb{Q}_{\mathrm{P}}\subset\mathit{Q}_{-k}$,
but there is no guarantee that $q+\mathbb{Q}_{\mathrm{P}}$ is equal to
$\mathit{Q}_{-k}$. Suppose $q_{1}$ and $q_{2}$ are in $\mathit{Q}_{-k}$ for
some $k>0$. If they represent the same coset then
$q_{1}-q_{2}\in\mathbb{Q}_{\mathrm{P}}$. However, it is easily seen that
$\nu_{2}(q_{1}-q_{2})$ may assume any value greater than $-k$:

$q_{1}-q_{2}=2^{\ell-k}\frac{o_{1}}{o_{2}}\quad\mbox{for}\quad\ell>0\,.$

Thus, $q_{1}-q_{2}$ may be in any of the following sets:

$\mathit{Q}_{-k+1}\,,\mathit{Q}_{-k+2}\,,\mathit{Q}_{-k+3}\,,\dots\,,\mathit{Q}_{-1}\,,\mathbb{Q}_{\mathrm{P}}\,.$

Clearly, $q_{1}-q_{2}\in\mathbb{Q}_{\mathrm{P}}$ if and only if $\ell\geq k$,
whence

$q_{1}+\mathbb{Q}_{\mathrm{P}}=q_{2}+\mathbb{Q}_{\mathrm{P}}\qquad\mbox{{iff}}\qquad\ell\geq
k\,.$

For each $k>0$, we define a set of values

$q_{k}^{\ell}=2^{-k}(2\ell-1)\in\mathit{Q}_{-k}\quad\mbox{for}\quad\ell=1,2,3,\dots,2^{k-1}\,.$
(10)

We note that these are the first $2^{k-1}$ positive values in
$\mathit{D}_{-k}$. We show in the following section that these are
representatives of $2^{k-1}$ cosets, which are all distinct and which provide
a disjoint partition of $\mathit{Q}_{-k}$. This analysis provides explicit
expressions for each of the infinite set of cosets of
$\mathbb{Q}_{\mathrm{P}}$ in $\mathbb{Q}$:

$q_{k}^{\ell}+\mathbb{Q}_{\mathrm{P}}\,,\qquad\ell=1,2,3,\dots 2^{k-1}\,,\quad
k=1,2,\dots\,.$ (11)

### Scale invariance of the structure.

Figure 3: Partition of the rational numbers. The vertical axis is
$\mu_{2}=2^{\nu_{2}}$. Each subset $\mathit{Q}_{k}$ is represented by a
horizontal dotted line. The sets $\mathit{D}_{k}$ are indicated by the marked
points in $\mathit{Q}_{k}$. The dyadic numbers in $\mathbb{D}$ all appear on a
pencil of lines emanating from the origin.

We notice that the diagram in Fig. 2 has a scaling invariance: if the
horizontal axis is stretched by a factor of 2 and the diagram translated one
unit in the vertical, the dyadic rationals occupy the same set of points. We
have chosen to analyse the quotient group
$\mathbb{Q}/\mathbb{Q}_{\mathrm{P}}$. However, a similar analysis could be
done for any subgroup $\mathbb{Q}_{\mathrm{K}}$, with directly analogous
results.

In Fig. 3, we re-plot Fig. 2 with vertical axis $\mu_{2}=2^{\nu_{2}}$. The
dyadic numbers in $\mathbb{D}$ now appear on a pencil of lines with slopes
$1/(2\ell-1)$, emanating from the origin.

### Density of $\mathit{Q}_{k}$: a heuristic discussion.

The set $\mathit{Q}_{-1}=\textstyle{\frac{1}{2}}+\mathbb{Q}_{\mathrm{P}}$ is a
coset of $\mathbb{Q}_{\mathrm{P}}$. It can be visualized as a copy of
$\mathbb{Q}_{\mathrm{P}}$ shifted by a distance $\textstyle{\frac{1}{2}}$. We
argue heuristically that $\mathit{Q}_{-1}$ is “as dense as
$\mathbb{Q}_{\mathrm{P}}$”.

More generally, for any $k$, there is a natural correspondence between
elements of $\mathit{Q}_{k}$ and elements of $\mathit{Q}_{k-1}$:

$\frac{1}{2^{k}}\left(\frac{2m+1}{2n+1}\right)\in\mathit{Q}_{k}\longleftrightarrow\frac{1}{2^{k-1}}\left(\frac{2m+1}{2n+1}\right)\in\mathit{Q}_{k-1}\,.$

Thus, $\mathit{Q}_{k-1}$ may be visualized as a compressed version of
$\mathit{Q}_{k}$. Since $\mathit{Q}_{k-1}$ is “twice as dense as
$\mathit{Q}_{k}$”, we may argue that we should have twice as many cosets in
$\mathit{Q}_{k-1}$ as there are in $\mathit{Q}_{k}$. This is consistent with
what is proved rigorously below.

##  5 Formal proof of the coset structure for $\mathbb{Q}_{\mathrm{P}}$.

In this section, we give rigorous proofs of some of the results considered
heuristically in the discussion above. Lemmas 1 and 2 give conditions for
cosets to be equal. Proposition 1 gives explicit representatives
$q_{k}^{\ell}$ for each of the distinct cosets. In the following, we
abbreviate the $2$-adic valuation $\nu_{2}$ to $\nu$.

######  Lemma 1.

Suppose $q_{1},q_{2}\in\mathbb{Q}_{\mathrm{N}}$ and
$q_{1}+\mathbb{Q}_{\mathrm{P}}=q_{2}+\mathbb{Q}_{\mathrm{P}}$. Then
$\nu(q_{1})=\nu(q_{2})$.

###### Proof.

For the cosets to be equal, we must have
$q_{1}-q_{2}\in\mathbb{Q}_{\mathrm{P}}$. Suppose that $\nu(q_{1})=-k$ and
$\nu(q_{2})=-\ell$ with $k\neq\ell\in\mathbb{N}$. Without loss of generality,
we may assume that $\ell=k+d$ with $d>0$. Then, using (8),

$\nu(q_{1}-q_{2})=\min(\nu(q_{1}),\nu(-q_{2}))=\min(\nu(q_{1}),\nu(q_{2}))=\min(-\ell,-k)=-\ell\,.$

Thus, $q_{1}-q_{2}\in\mathit{Q}_{-\ell}$, so that
$q_{1}+\mathbb{Q}_{\mathrm{P}}\neq q_{2}+\mathbb{Q}_{\mathrm{P}}$.
Consequently, a necessary condition for equality of the cosets is that $q_{1}$
and $q_{2}$ have the same 2-adic valuation, $\nu(q_{1})=\nu(q_{2})$. ∎

The next lemma strengthens this to a necessary and sufficient condition.

######  Lemma 2.

Let $q_{1}$ and $q_{2}$ be in $\mathbb{Q}_{\mathrm{N}}$. Then
$q_{1}+\mathbb{Q}_{\mathrm{P}}=q_{2}+\mathbb{Q}_{\mathrm{P}}$ if, and only if,
$\nu(q_{1})=\nu(q_{2})=-k$ for some $k\in\mathbb{N}$ and
$\nu(a_{1}b_{2}-a_{2}b_{1}))\geq k$ where $q_{1}={a_{1}}/{(2^{k}b_{1})}$ and
$q_{2}={a_{2}}/{(2^{k}b_{2})}$.

###### Proof.

Let $q_{1}$ and $q_{2}$ be in $\mathbb{Q}_{\mathrm{N}}$. Suppose that
$q_{1}+\mathbb{Q}_{\mathrm{P}}=q_{2}+\mathbb{Q}_{\mathrm{P}}$. Lemma 1 tells
us that $\nu(q_{1})=\nu(q_{2})=-k$ for some $k\in\mathbb{N}$. We may thus
write

$q_{1}=\frac{a_{1}}{2^{k}b_{1}}\quad\mathrm{and}\quad
q_{2}=\frac{a_{2}}{2^{k}b_{2}}$ (12)

where $a_{i},b_{i}$ are odd, and so

$q_{1}-q_{2}=\frac{a_{1}b_{2}-a_{2}b_{1}}{2^{k}b_{1}b_{2}}.$ (13)

By hypothesis, this is an element of $\mathbb{Q}_{\mathrm{P}}$ which implies
$2^{k}|(a_{1}b_{2}-a_{2}b_{1})$, as asserted.

For the converse, if $\nu(q_{1})=\nu(q_{2})=-k$, then (12), and hence (13),
holds. The condition $\nu(a_{1}b_{2}-a_{2}b_{1})\geq k$ then implies
$\nu(q_{1}-q_{2})\geq 0$ and so $q_{1}-q_{2}\in\mathbb{Q}_{\mathrm{P}}$. ∎

######  Proposition 1.

For each $k\in\mathbb{N}$, let
$q_{k}^{\ell}=({2\ell-1})/{2^{k}}\in\mathit{Q}_{-k}$ for
$\ell=1,\ldots,2^{k-1}$. These numbers generate $2^{k-1}$ distinct
$\mathbb{Q}_{\mathrm{P}}$-cosets, which comprise all the cosets of
$\mathbb{Q}_{\mathrm{P}}$ by elements of $\mathit{Q}_{-k}$.

###### Proof.

It is clear that $q_{k}^{\ell}={(2\ell-1)}/{2^{k}}$ lies in $\mathit{Q}_{-k}$.
If two of these numbers, $q_{k}^{\ell}$ and $q_{k}^{\ell^{\prime}}$ say,
generate the same coset, then
$q_{k}^{\ell}-q_{k}^{\ell^{\prime}}\in\mathbb{Q}_{\mathrm{P}}$ and so, by
Lemma 2, $2^{k}$ divides
$2{\ell^{\prime}}-1-(2\ell-1)=2({\ell^{\prime}}-\ell)$ and hence $2^{k-1}$
divides $\ell-\ell^{\prime}$. As $1\leq\ell,\ell^{\prime}\leq 2^{k-1}$, the
only way this can occur is if $\ell=\ell^{\prime}$. This shows that all
$2^{k-1}$ cosets generated by $q_{k}^{\ell}$ are distinct.

Next, we show that these exhaust all possible cosets by elements of
$\mathit{Q}_{-k}$. For this, given $q\in\mathit{Q}_{-k}$ we need to show that
$q+\mathbb{Q}_{\mathrm{P}}=q_{k}^{\ell}+\mathbb{Q}_{\mathrm{P}}$ for some
$\ell\in\\{1,2,\ldots 2^{k-1}\\}$. As $q=n/(2^{k}m)$ for $m$ and $n$ odd, by
Lemma 2, this amounts to showing $\nu(1.n-(2\ell-1)m)\geq k$, or that $2^{k}$
divides $n+m-2\ell m$. We may let $n+m=2s$ ($s\in\mathbb{Z}$) and examine
instead whether $2^{k-1}$ divides $s-\ell m$ for some
$\ell\in\\{1,\ldots,2^{k-1}\\}$.

Notice that

$\displaystyle s-\ell m\equiv s-\ell^{\prime}m(\\!\\!\\!\\!\\!\mod 2^{k-1})$
$\displaystyle\iff 2^{k-1}\mbox{\ divides\ }m(\ell-\ell^{\prime})$
$\displaystyle\overset{\mbox{$m$\ odd}}{\iff}2^{k-1}\mbox{\ divides\
}(\ell-\ell^{\prime})\iff\ell=\ell^{\prime}$

as again $1\leq\ell,\ell^{\prime}\leq 2^{k-1}$. In particular, by the
pigeonhole principle, all $2^{k-1}$ possible $2^{k-1}$-remainders, including
0, are contributed by $s-\ell m$, $\ell=1,\ldots,2^{k-1}$. So $2^{k-1}$
divides $s-\ell m$ for some $\ell$ and, for this $\ell$ we have $q\in
q_{k}^{\ell}+\mathbb{Q}_{\mathrm{P}}$, as stated. ∎

##  6 Densities of the parity classes.

Figure 4: The initial rows of the Calkin-Wilf tree (left) and Stern-Brocot
tree (right).

There are many exhaustive sequences of rationals other than (3), one
attractive option being the Calkin-Wilf tree [2]. The Calkin-Wilf tree is
complete: it includes all positive rational numbers and each such number
occurs precisely once. The tree starts with the root value ${1}/{1}$, and
everything springs from this root (see Fig. 4, left panel). Each rational in
the tree has two “children”: for the entry $m/n$, the children are $m/(m+n)$
and $(m+n)/n$. The “left child” $m/(m+n)$ is always smaller that $1$ while the
“right child” $(m+n)/n$ is always greater that $1$ (mnemonic: the _children_
are “top over sum” and “sum over bottom”).

The pattern of parity from one row of the Calkin-Wilf diagram to the next is
simple. Denoting odd parity, even parity and no parity by $o$, $e$ and $n$
respectively, the parity transfer rules are as follows:

$\displaystyle\phantom{o}\phantom{\swarrow}{\ e\
}\phantom{\searrow}\phantom{o}\qquad\quad\phantom{o}\phantom{\swarrow}{\ o\
}\phantom{\searrow}\phantom{o}\qquad\quad\phantom{o}\phantom{\swarrow}{\ n\
}\phantom{\searrow}\phantom{o}\qquad\quad$
$\displaystyle\phantom{o}{\swarrow}\phantom{\ o\
}{\searrow}\phantom{o}\qquad\quad\phantom{o}{\swarrow}\phantom{\ o\
}{\searrow}\phantom{o}\qquad\quad\phantom{o}{\swarrow}\phantom{\ o\
}{\searrow}\phantom{o}\qquad\quad$ (14)
$\displaystyle{e}\phantom{\swarrow}\phantom{\ o\
}\phantom{\searrow}{o}\qquad\quad{n}\phantom{\swarrow}\phantom{\ o\
}\phantom{\searrow}{e}\qquad\quad{o}\phantom{\swarrow}\phantom{\ o\
}\phantom{\searrow}{n}\qquad\quad$

We will now show that the elements of the Calkin-Wilf tree are remarkably
regular, with the pattern $(o,n,e)$ repeating interminably. Thus, the parity
of any specific term in the tree can easily be deduced. We also prove that,
with the ordering of $\mathbb{Q}$ corresponding to the Calkin-Wilf tree, the
three parity classes all have the same density.

######  Theorem 1.

Let $\mathbb{Q}^{+}$ be ordered with the Calkin-Wilf tree. Then
$\mathbb{Q}^{+}$ may be partitioned into three parity classes,
$\mathbb{Q}_{\mathrm{E}}^{+}$, $\mathbb{Q}_{\mathrm{O}}^{+}$ and
$\mathbb{Q}_{\mathrm{N}}^{+}$, each having asymptotic density $\frac{1}{3}$.

###### Proof.

The parity classes for the first few rows of the Calkin-Wilf tree are shown in
Fig. 5. Odd rows follow a pattern $(one)^{k}o$ for some $k\geq 0$ and even
rows follow a pattern $ne(one)^{k}$. This is clearly true for the first few
rows. Using the transfer rules (14), and arguing inductively, it is clear that
a row with pattern $(one)^{k}o$ is followed by a row with pattern
$ne(one)^{2k}$. This, in turn, is followed by a row with pattern
$(one)^{4k+1}o$. The full sequence begins

$o\ \quad\ ne\ \quad\ (one)o\ \quad\ ne(one)^{2}\ \quad\ (one)^{5}o\ \quad\
ne(one)^{10}\ \quad\ \cdots$

We conclude that, if the entire tree is written row by row as a sequence, the
parity follows an unvarying pattern, with _odd_ followed by _none_ followed by
_even_. The parity of an element at any position $N$ is immediately deduced
from $m=N(\mbox{mod}\ 3)$. The pattern also implies that the three parity
classes have equal densities. ∎

Figure 5: Parity classes of terms in the Calkin-Wilf tree.

The Calkin-Wilf tree enumerates the positive rationals $\mathbb{Q}^{+}$. This
enumeration, which we write $\\{q_{n}:n\in\mathbb{N}\\}$, can be extended in a
natural way to the full set of rationals: we enumerate $\mathbb{Q}$ by
$\\{0,q_{1},-q_{1},q_{2},-q_{2},\dots\\}$. With this ordering, the rational
numbers split into three parts, each of asymptotic density $\frac{1}{3}$.

To summarise, the parity classes of elements of the Calkin-Wilf tree follow a
simple pattern if arranged in a single sequence: the pattern $\\{o,n,e\\}$
repeats indefinitely (see Fig. 5). As a result, the densities of the parity
classes in $\mathbb{Q}$ are all equal for this ordering:

$\rho_{\mathbb{Q}}(\mathbb{Q}_{\mathrm{E}})=\rho_{\mathbb{Q}}(\mathbb{Q}_{\mathrm{O}})=\rho_{\mathbb{Q}}(\mathbb{Q}_{\mathrm{N}})=\tfrac{1}{3}\,.$
(15)

The Stern-Brocot tree [5, pg. 116] is another ordering of $\mathbb{Q}$ very
similar to the Calkin-Wilf tree. The numbers at each level are formed from the
mediants of adjacent pairs of numbers above (Fig. 4, right panel). The mediant
of two (reduced) rationals, $m_{1}/n_{1}$ and $m_{2}/n_{2}$ is defined as
$M(m_{1}/n_{1},m_{2}/n_{2}):=(m_{1}+m_{2})/(n_{1}+n_{2})$. We note that the
parity of the mediants of two numbers of different parity is the third parity:

$M(e,o)=n\,,\qquad M(o,n)=e\,,\qquad M(n,e)=o\,.$ (16)

We now show that, with the ordering of the Stern-Brocot tree, (15) holds true.

######  Theorem 2.

For the order of $\mathbb{Q}$ induced by the Stern-Brocot process, the
asymptotic density of each parity class, $\mathbb{Q}_{\mathrm{E}}$,
$\mathbb{Q}_{\mathrm{O}}$ and $\mathbb{Q}_{\mathrm{N}}$, is $\frac{1}{3}$.

###### Proof.

The Stern-Brocot tree is generated starting from level $0$ with the boundary
elements $\left[\frac{0}{1}\right]$ and $\left[\frac{1}{0}\right]$,
representing $0$ and $\infty$ and with parities $[e]$ and $[n]$. To get each
subsequent level we add, between each pair of adjacent numbers, the mediant of
that pair, retaining all numbers already generated. The results, for the first
few levels, are shown in Fig. 4 (right panel). The parity pattern for the
first few levels is

$[e]o[n]\quad\ [e]noe[n]\quad\ [e]oneoneo[n]\quad\ [e]noenoenoenoenoe[n]\,.$

Using the transfer rules (16), an odd row with parity $(eon)^{K}$ is followed
by even row with parity $e(noe)^{2K-1}n$. This in turn is followed by an odd
row with parity $(eon)^{4K-1}$. By an inductive argument, it follows that the
parity pattern for an odd row $k$ is $(eon)^{K}$, where $K=(2^{k}+1)/3$ and,
for an even row $k$, is $e(noe)^{K}n$ where $K=(2^{\ell}-1)/3$. This implies
that the asymptotic densities are equal for all three parity classes.

The Stern-Brocot tree enumerates the positive rationals $\mathbb{Q}^{+}$. This
enumeration, is easily extended to the full set of rationals, as was done
above for the Calkin-Wilf tree. Then the rational numbers split into three
parts, each of asymptotic density $\frac{1}{3}$. ∎

The determination of the densities of parity classes for the ordering
corresponding to the Farey sequences is left as a challenge for readers.

##  7 Conclusion.

We have extended parity from the integers to the rational numbers. Three
parity classes — even, odd and ‘none’ — were found. The even and odd rationals
$\mathbb{Q}_{\mathrm{E}}$ and $\mathbb{Q}_{\mathrm{O}}$ follow the usual rules
of parity. The union of these forms an additive subgroup
$\mathbb{Q}_{\mathrm{P}}$ of $\mathbb{Q}$.

Using the 2-adic valuation, we partitioned $\mathbb{Q}$ into subsets and found
a chain of subgroups, each having a quotient group of cosets. We constructed a
complete set of representatives for the cosets of $\mathbb{Q}_{\mathrm{P}}$.

The Calkin-Wilf tree was found to have a remarkably simple parity pattern,
with the sequence ‘odd/none/even’ repeating indefinitely. Using the natural
density, which provides a means of distinguishing the sizes of countably
infinite sets, we showed that, with the Calkin-Wilf ordering, the three parity
classes are equally dense in the rationals. The same conclusion holds for the
Stern-Brocot tree.

Finally, we remark that, while this study used only 2-adic numbers, there is
potential for broad extensions and generalizations using more general $p$-adic
numbers.

### Acknowledgments.

We are grateful to Tom Laffey, Emeritus Professor, School of Mathematics &
Statistics, University College Dublin, for reading a draft of this paper and
to Tony O’Farrell, Emeritus Professor at Maynooth University for guidance on
$p$-adic numbers.

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