# Solutions of a charged scalar field in five-dimensional helicoid solution
with electromagnetic field

Tolga Birkandan
Department of Physics E-mail<EMAIL_ADDRESS>Istanbul Technical
University  34469 Istanbul  Turkey

###### Abstract

We study a charged and massive scalar field in the background of the Nutku-
Ghezelbash-Kumar metric which is obtained by the addition of a time coordinate
to the Nutku helicoid metric in a non-trivial way. The angular part of the
Klein-Gordon equation can be written as a double confluent Heun equation. The
radial equation cannot be solved in terms of a known function in its general
form. However, in some special cases, the radial equation can also be written
explicitly as a double confluent Heun equation. We study the full radial
equation numerically and observe that the electromagnetic field parameter
defines an effective cut-off on the range of the radial coordinate. Finally,
we obtain a quasi-exact solution with an approximation.

##  1 Introduction

Gravitational instantons are Euclidean exact solutions of Einstein’s field
equations in vacuum which admit hyper-Kähler structure and their properties
can be studied by a Euclidean version of the Newman-Penrose formalism [1, 2,
3, 4, 5].

Nutku’s instanton metric corresponds to the helicoid minimal surface which is
a realization of Weierstrass’ general local solution for minimal surfaces. It
has a quadratic Killing tensor and a curvature singularity at the origin. The
metric corresponds to the catenoid minimal surface with a reparametrization
[6, 7, 8].

Klein-Gordon and Dirac equations have been studied for the Nutku helicoid
metric in its original form, and they have been solved in terms of Mathieu
functions [7, 9, 10]. A time coordinate can be added to the metric in a
trivial way as $ds^{2}=-dt^{2}+ds_{\text{Nutku}}^{2}$ and the wave equations
can be solved in terms of the double confluent Heun (DCH) functions or the
Mathieu functions as in the original case [10, 11, 12].

In this paper, we study a charged and massive scalar field in the background
of the Nutku-Ghezelbash-Kumar (NGK) metric which is based on the Nutku
helicoid metric with a time coordinate added in a non-trivial manner in the
Einstein-Maxwell theory[13]. The angular part of the Klein-Gordon equation
studied in this background yields a DCH equation. In some limits, the radial
part can be written as a DCH equation, and the numerical solution of the
radial equation also signals a DCH-like behavior.

Heun-type equations are known to appear in many applications in gravity,
especially as solutions of the wave equations [14, 15, 16, 17, 18, 19, 20].
Our present work and the other works in the literature which study the wave
equations for the metrics based on the Nutku instanton yield DCH functions as
solutions. In all cases, these solutions can be transformed into a Mathieu
equation as in the wave equations written for the original Nutku metric.

The paper is organized in the following way: In the second section, we present
the five-dimensional NGK metric, in the third section we study the charged and
massive scalar field by separating the Klein-Gordon equation into its angular
and radial parts where we obtain exact and quasi-exact solutions for some
limits. In this section, we also study the full radial equation numerically to
observe an effective radial cut-off. We summarize our results in the
conclusions section, and we present the transformation of the DCH equation to
the Mathieu equation for a special case that we observe in the studies
associated with the Nutku instanton in an appendix.

##  2 The 5D helicoid metric with EM field

The Nutku helicoid metric with a non-trivial addition of the time coordinate
(the NGK metric) is given as

$\displaystyle ds^{2}=$
$\displaystyle-\frac{dt^{2}}{(H(r))^{2}}+\frac{H(r)}{\sqrt{1+\frac{a^{2}}{r^{2}}}}\bigg{[}dr^{2}+(a^{2}+r^{2})d\theta^{2}$
$\displaystyle+\bigg{(}1+\frac{a^{2}sin^{2}\theta}{r^{2}}\bigg{)}dy^{2}+\bigg{(}1+\frac{a^{2}cos^{2}\theta}{r^{2}}\bigg{)}dz^{2}-\frac{a^{2}sin(2\theta)dydz}{r^{2}}\bigg{]},$
(1)

where $r\in(0,\infty)$, $\theta\in[0,2\pi]$, and the Killing coordinates $y$
and $z$ are periodic on a 2-torus [6, 8, 13]. Here, the metric function

$H(r)=1+ac_{2}sinh^{-1}\bigg{(}\frac{r}{a}\bigg{)},$ (2)

is related with the electromagnetic (EM) potential. The only non-zero
component of the EM potential is

$A_{t}=\sqrt{\frac{3}{2}}\frac{1}{H(r)},$ (3)

in five-dimensions as given in [13]. In order to prevent a signature change in
the metric, $a\geq 0$ and $c_{2}\geq 0$ will be required. $a\rightarrow 0$ is
the Minkowski limit for the metric.

##  3 The charged scalar field

The Klein-Gordon equation for a massive and charged scalar field $\Phi$ can be
given as

$[(\nabla^{\nu}-iqA^{\nu})(\nabla_{\nu}-iqA_{\nu})-\mu^{2}]\Phi=0,$ (4)

where $\mu$ is the mass and $q$ is the charge of the scalar field. We can
employ the Ansatz

$\Phi(t,r,\theta,y,z)=e^{-i\omega t}e^{im_{1}y}e^{im_{2}z}\varphi(r,\theta),$
(5)

that yields

$\displaystyle\bigg{\\{}$
$\displaystyle(a^{2}+r^{2})\frac{\partial^{2}}{\partial
r^{2}}+\frac{\partial^{2}}{\partial\theta^{2}}+r\frac{\partial}{\partial
r}+r\sqrt{a^{2}+r^{2}}H(r)\bigg{(}\sqrt{6}qH(r)\omega+(H(r))^{2}\omega^{2}+\frac{3}{2}q^{2}-\mu^{2}\bigg{)}-(m_{1}^{2}+m_{2}^{2})r^{2}$
$\displaystyle-a^{2}(m_{1}^{2}-m_{2}^{2})cos^{2}(\theta)-2a^{2}m_{1}m_{2}sin(\theta)cos(\theta)-a^{2}m_{2}^{2}\bigg{\\}}\varphi(r,\theta)=0.$
(6)

The radial and angular parts can be separated using
$\varphi(r,\theta)=R(r)S(\theta)$, namely,

$\displaystyle\frac{d^{2}R(r)}{dr^{2}}+\frac{r}{a^{2}+r^{2}}\frac{dR(r)}{dr}+\frac{r\sqrt{a^{2}+r^{2}}H(r)\bigg{[}\sqrt{6}qH(r)\omega+(H(r))^{2}\omega^{2}+\frac{3}{2}q^{2}-\mu^{2}\bigg{]}-(m_{1}^{2}+m_{2}^{2})r^{2}-\lambda}{a^{2}+r^{2}}R(r)=0,$
(7)
$\displaystyle\frac{d^{2}S(\theta)}{d\theta^{2}}-\bigg{[}a^{2}(m_{1}cos\theta+m_{2}sin\theta)^{2}-\lambda\bigg{]}S(\theta)=0,$
(8)

where $\lambda$ is the separation constant.

###  3.1 The angular equation

The angular part of the Klein-Gordon equation can be understood better if we
perform the change of variable $v=-tanh(i\theta)$. Then we obtain

$\displaystyle\frac{d^{2}S(v)}{dv^{2}}+\frac{2v}{v^{2}-1}\frac{dS(v)}{dv}+\frac{(a^{2}m_{2}^{2}-\lambda)v^{2}-2ia^{2}m_{1}m_{2}v-a^{2}m_{1}^{2}+\lambda}{(v-1)^{3}(v+1)^{3}}S(v)=0.$
(9)

This equation has two irregular singular points of rank-1 located at
$v=\\{-1,1\\}$ which corresponds to a DCH equation [14, 15].

Let us define the general form of the DCH equation that has been used in
studying the case where the time component was introduced in a trivial manner
[10], and this will be useful for us while comparing our solutions in the
$\mu=q=c_{2}=0$ limit. The general form of the DCH equation reads

$\frac{d^{2}H_{D}}{dx^{2}}+{\frac{2\,{x}^{5}-\alpha\,{x}^{4}-4\,{x}^{3}+2\,x+\alpha}{\left(x-1\right)^{3}\left(x+1\right)^{3}}}\frac{dH_{D}}{dx}+{\frac{\beta{x}^{2}+\left(\gamma+2\,\alpha\right)x+\delta}{\left(x-1\right)^{3}\left(x+1\right)^{3}}}H_{D}=0,$
(10)

and the irregular, rank-1 singular points are located at $x=\\{-1,1\\}$. The
radius of convergence of the local solution
$H_{D}(\alpha,\beta,\gamma,\delta,x)$ around the ordinary point $x=0$ is
$|x|<1$. For a detailed analysis of the equation, one can see [14, 15, 21].

We can compare the equations (9) and (10) to find the parameters in the DCH
equation, namely,

$\displaystyle\alpha$ $\displaystyle=0,$ (11) $\displaystyle\beta$
$\displaystyle=a^{2}m_{2}^{2}-\lambda,$ (12) $\displaystyle\gamma$
$\displaystyle=-2ia^{2}m_{1}m_{2},$ (13) $\displaystyle\delta$
$\displaystyle=\lambda-a^{2}m_{1}^{2},$ (14)

needed to show that the angular part of the Klein-Gordon equation can be
solved in terms of DCH functions with these parameters.

We should note that the angular equation is related with the quadratic Killing
tensor of the original Nutku metric [10, 8]. It was solved in terms of Mathieu
functions in [10] and here, we present it as a DCH equation.

In [10], it was shown that one can transform the DCH equation to a Mathieu
type equation via some transformations in the Nutku helicoid case. In our
appendix, we will follow the same procedure for the general form of the DCH
equation (10) with $\alpha=0$ to show that it can be transformed into the
Mathieu equation. This transformation is also valid for our angular equation
(9) and the future DCH-type results which will have $\alpha=0$.

###  3.2 The radial equation

If we perform the change of variable $r=asinh(x)$ in Eq. (7), we can write the
EM metric parameter $H(r)$ as

$H(x)=1+c_{2}ax.$ (15)

and we get

$\frac{d^{2}R(x)}{dx^{2}}+\mathcal{F}(x)R(x)=0,$ (16)

where

$\displaystyle\mathcal{F}(x)=$
$\displaystyle\frac{a^{2}}{2}(m_{1}^{2}+m_{2}^{2})-\lambda-\frac{a^{2}}{2}(m_{1}^{2}+m_{2}^{2})cosh(2x)+\frac{a^{2}}{2}\bigg{(}\sqrt{6}q\omega(H(x))^{2}+\omega^{2}(H(x))^{2}+\frac{3}{2}q^{2}-\mu^{2}\bigg{)}sinh(2x).$
(17)

Another change of coordinate $x=tanh^{-1}(u)$, brings the radial equation (16)
in a form that the singularities can be analyzed better. The new form of the
radial equation can be given as

$\displaystyle\frac{d^{2}R(u)}{du^{2}}+\frac{2u}{u^{2}-1}\frac{dR(u)}{du}+\frac{F_{1}(u)+F_{2}(u,c_{2})}{(u-1)^{3}(u+1)^{3}}R(u)=0,$
(18)

where

$\displaystyle
F_{1}(u)=[(m_{1}^{2}+m_{2}^{2})a^{2}-\lambda]u^{2}-a^{2}\bigg{(}\frac{3}{2}q^{2}-\sqrt{6}q\omega+\omega^{2}-\mu^{2}\bigg{)}u+\lambda,$
(19) $\displaystyle
F_{2}(u,c_{2})=\bigg{[}-a^{2}\omega^{2}(tanh^{-1}(u))^{2}c_{2}^{2}-a\omega(\sqrt{6}q+3\omega)tanh^{-1}(u)c_{2}-(2\sqrt{6}q\omega-\mu^{2}+\frac{3}{2}q^{2}+3\omega^{2})\bigg{]}a^{3}c_{2}utanh^{-1}(u).$
(20)

The equation (18) has two irregular singular points of rank-1 located at
$u=\\{-1,1\\}$, which is the signature of a DCH equation. The range of the
radial coordinate is transformed as $u\in(0,1)$.

If we set $c_{2}=0,q=0,\mu=0$, then we get

$\displaystyle\frac{d^{2}R(u)}{du^{2}}+\frac{2u}{u^{2}-1}\frac{dR(u)}{du}+\frac{[(m_{1}^{2}+m_{2}^{2})a^{2}-\lambda]u^{2}-a^{2}\omega^{2}u+\lambda}{(u-1)^{3}(u+1)^{3}}R(u)=0,$
(21)

which corresponds to the following set of the DCH parameters in Eq. (10),

$\displaystyle\alpha$ $\displaystyle=0,$ (22) $\displaystyle\beta$
$\displaystyle=(m_{1}^{2}+m_{2}^{2})a^{2}-\lambda,$ (23) $\displaystyle\gamma$
$\displaystyle=-a^{2}\omega^{2},$ (24) $\displaystyle\delta$
$\displaystyle=\lambda.$ (25)

This result agrees with the result of the case that the time coordinate has
been added in a trivial manner in [10].

The case with $c_{2}=0,q\neq 0,\mu\neq 0$ also yields the DCH equation with
parameters

$\displaystyle\alpha$ $\displaystyle=0,$ (26) $\displaystyle\beta$
$\displaystyle=(m_{1}^{2}+m_{2}^{2})a^{2}-\lambda,$ (27) $\displaystyle\gamma$
$\displaystyle=-a^{2}\bigg{(}\frac{3}{2}q^{2}+\sqrt{6}q\omega+\omega^{2}-\mu^{2}\bigg{)},$
(28) $\displaystyle\delta$ $\displaystyle=\lambda.$ (29)

We know that the $tanh^{-1}(u)$ terms have no effect on the singularity
structure of Eq. (18), and it still has the singularity structure of the DCH
equation (10). However, if $c_{2}\neq 0$, i.e. in the presence of an EM
potential, the form of the radial equation with $F_{2}(u,c_{2})$ does not meet
with the DCH equation. Moreover, if we assume that Eq. (18) is a DCH equation,
then its parameter $\alpha$ should be zero, therefore it should be possible to
transform it to the Mathieu equation using the method we studied in the
Appendix. However, when we compare Eq. (35) with the radial equation given in
(16), we observe that the coefficient of $sinh(2x)$ in Eq. (16) has a radial
dependence, unlike Eq. (35) which prevents us from using this method to obtain
a Mathieu-type solution, or a DCH solution with $\alpha=0$ correspondingly.

We can study Eq. (18) numerically to see the behavior of the radial solutions.
We employed the SageMath software system in our numerical analysis [22] as
explained in [23]. As an alternative, one can use the integral series approach
given in [24, 25].

Figure 1: Numerical solutions of the radial equation (18) with {$q=0,\mu=0$}
for different $c_{2}$ values. The numerical values of the other parameters are
{$\lambda=1,a=3,m_{1}=0.3,m_{2}=0.5,\omega=10$}. Figure 2: Numerical solutions
of the radial equation (18) with {$q=6,\mu=15$} for different $c_{2}$ values.
The numerical values of the other parameters are
{$\lambda=1,a=3,m_{1}=0.3,m_{2}=0.5,\omega=10$}.

(a) Varying mass ($q=6$).

(b) Varying charge ($\mu=15$).

Figure 3: Numerical solutions of the radial equation (18) for several mass and
charge values. The numerical values of the other parameters are
{$\lambda=1,a=3,m_{1}=0.3,m_{2}=0.5,\omega=10,c_{2}=0.5$}.

As one can see from the Figures (1, 2, 3), changes in the mass and charge
values affect the behavior of the solution slightly, and the EM field (via the
parameter $c_{2}$) does not change the general character of the solution
except the decaying behavior and the “effective” radial range where the radial
solution has a visible oscillatory nature. We can claim that $c_{2}$ puts an
“effective” cut-off on the physical range as the solutions approach to zero
after a radial distance from the origin.

In Figures (1) and (2), one can observe that the behavior of the solutions
with a non-zero $c_{2}$ is very similar to the ones with the solutions of the
DCH type (with zero $c_{2}$). Thus we can claim that the solutions with non-
zero $c_{2}$ values might correspond to a form of a “deformed” DCH equation.
Then, we might attempt to find a quasi-exact solution using some
approximations.

For $|u|<1$, we have $tanh^{-1}(u)=\sum_{k=0}^{\infty}\frac{u^{2k+1}}{2k+1}$.
Let us make the approximation $tanh^{-1}(u)\sim u$ in the vicinity of $u=0$
and keep only the quadratic term in $u$ in eq. (20) in order to preserve the
DCH form. Then we can write this equation as a DCH equation with the
parameters

$\displaystyle\alpha$ $\displaystyle=0,$ (30) $\displaystyle\beta$
$\displaystyle=-a^{3}\bigg{(}\frac{3}{2}q^{2}+2\sqrt{6}q\omega+3\omega^{2}-\mu^{2}\bigg{)}c_{2}+(m_{1}^{2}+m_{2}^{2})a^{2}-\lambda,$
(31) $\displaystyle\gamma$
$\displaystyle=-a^{2}\bigg{(}\frac{3}{2}q^{2}+\sqrt{6}q\omega+\omega^{2}-\mu^{2}\bigg{)},$
(32) $\displaystyle\delta$ $\displaystyle=\lambda.$ (33)

It should be emphasized that the range cut-off cannot be read from this
solution, however, it can serve as a “quasi-exact” solution for small $u$
values as can be seen from Figure (4). It can also be seen that our quasi-
exact DCH solution requires small $c_{2}$ values for a better fit.

In this limit, the effect of the EM field can be eliminated when the
coefficient of $c_{2}$ in eq. (31), namely
$\frac{3}{2}q^{2}+2\sqrt{6}q\omega+3\omega^{2}-\mu^{2}$ is zero.

(a) $c_{2}=0.1$

(b) $c_{2}=0.5$

(c) $c_{2}=1$

(d) $c_{2}=5$

Figure 4: Full radial equation (18) (solid curve) vs. small $u$ approximation
determined by (30-33) (dotted curve). The numerical values of the other
parameters are {$\lambda=1,a=3,m_{1}=0.3,m_{2}=0.5,\omega=10,q=6,\mu=15$}.

##  4 Conclusion

We studied a charged and massive scalar field in the background of the Nutku-
Ghezelbash-Kumar metric which was obtained by the addition of a time
coordinate to the Nutku helicoid metric in a non-trivial way, using the
source-free Einstein-Maxwell equations.

The charged and massive scalar field could be separated into its radial and
angular parts. The angular part was free of the physical parameters charge and
mass, and it could be written as a double confluent Heun equation. However,
the full radial part could not be solved in terms of a known function in its
general form.

When the charge, mass, and the parameter associated with the electromagnetic
field were set to zero, we recovered the double confluent Heun solution in the
literature written for the Nutku metric with a trivially added time
coordinate. The charged and massive scalar field without the electromagnetic
field parameter also yielded the double confluent Heun equation.

The full radial equation could be analyzed numerically to see the effect of
the physical parameters on the solution. The changes in the charge and mass
values affected the solution slightly but more importantly, the change in the
electromagnetic field parameter set a cut-off on the effective range of the
radial coordinate.

We managed to find a “quasi-exact” solution for the radial equation when we
approximated the arctangent function for small radial values. Neglecting the
terms higher than quadratic in radial coordinate, we could write the equation
in terms of the double confluent Heun parameters. We also required a small
electromagnetic field parameter in order to prevent the effect of the higher-
order terms. In this limit, we also found an expression that eliminates the
electromagnetic coupling for the special values of the physical parameters.

## Acknowledgement

The author would like to thank Prof. Mahmut Hortaçsu for stimulating
discussions.

## Appendix: Transformation of the double confluent Heun equation to the
Mathieu equation for $\alpha=0$

In [10], the transformation of the DCH equation to the Mathieu equation was
studied using the parameters related with the spacetime and the fields. Here,
we will give a more general approach to the method by using the general form
of the DCH equation with $\alpha=0$ which is associated with all cases in the
literature where the Nutku helicoid is involved. It should be noted that the
variable names $y$, $z$, and $u$ are not related with the metric coordinates
used in the main text.

We will start with the general form of the DCH equation given in Eq. (10) and
set $\alpha=0$ as our results and some other result from the literature have
this property [10, 11, 12]. Then the equation becomes

$\frac{d^{2}H_{D}}{dx^{2}}+{\frac{2\,{x}^{5}-4\,{x}^{3}+2\,x}{\left(x-1\right)^{3}\left(x+1\right)^{3}}}\frac{dH_{D}}{dx}+{\frac{\beta{x}^{2}+\gamma
x+\delta}{\left(x-1\right)^{3}\left(x+1\right)^{3}}}H_{D}=0.$ (34)

Following the procedure given in [10], we perform a change of variable
$x=tanhy$ to get

$\frac{d^{2}H_{D}}{dy^{2}}+\bigg{(}\frac{\beta-\delta}{2}-\frac{\gamma}{2}sinh(2y)-\frac{\beta+\delta}{2}cosh(2y)\bigg{)}H_{D}=0.$
(35)

and we set

$\displaystyle A$ $\displaystyle=-\frac{\gamma}{2},$ (36) $\displaystyle B$
$\displaystyle=\frac{\beta-\delta}{2},$ (37) $\displaystyle C$
$\displaystyle=-\frac{\beta+\delta}{2}.$ (38)

Then we use the transformations $z=e^{-2y}$, $u=\sqrt{\frac{C-A}{C+A}}z$,
$w=\frac{1}{2}\big{(}u+\frac{1}{u}\big{)}$ consecutively to obtain

$(1-w^{2})\frac{d^{2}H_{D}}{dw^{2}}-w\frac{dH_{D}}{dw}-\bigg{(}\frac{\beta-\delta}{8}-\frac{\sqrt{(\beta+\delta)^{2}-\gamma^{2}}}{8}w\bigg{)}H_{D}=0,$
(39)

which corresponds to an algebraic form of the Mathieu equation. If we perform
another transformation $p=cos^{-1}\bigg{(}\sqrt{\frac{w+1}{2}}\bigg{)}$, we
obtain

$\frac{d^{2}H_{D}}{dp^{2}}+\bigg{(}\frac{\delta-\beta}{2}+\frac{\sqrt{(\beta+\delta)^{2}-\gamma^{2}}}{2}cos(2p)\bigg{)}H_{D}=0,$
(40)

which is the standard form of the Mathieu equation [26].

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