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\begin{document}
\begin{titlepage}
\vspace*{5mm}
\begin{center}{\Large \bf Electrostatic self--energy in $QED_2$ on curved background}
\end{center}
\begin{center}
\vskip 1cm {\bf H. Mohseni Sadjadi \footnote
{e-mail:heinms\_2000@yahoo.com}}

\vskip 1cm {\it Physics Department, College of Basic Sciences \\
 Shahed University, PO Box 15875-5794} \\
{\it Tehran, Iran }
\end{center}
\vskip 2cm

\begin{abstract}
By considering the vacuum polarization, we study the effects of
geometry on electrostatic self--energy of a test charge near the
black hole horizon and also in regions with strong and weak
curvature in static two dimensional curved backgrounds. We discuss
the relation of ultraviolet behavior of the gauge field propagator
and charge confinement.
\end{abstract}
\begin{center}
{\bf PACS numbers:} 11.10.Kk, 11.15.Kc, 12.20.Ds \\
{\bf Keywords:} electrostatic self--energy, static space--time, meson \\
\end{center}
\end{titlepage}
\newpage
\section{Introduction}
One of the interesting areas in physics, is the study of behavior
of classical and quantum fields in curved background and
investigating how their properties are affected by the curvature
of the space--time. It also provides a lot of insights into
important problems such as black hole entropy, Hawking radiation,
quantum theory of gravity and so on.

It is well known that the change in the geometry of the space
associated with the gravitational field, deforms electromagnetic
field, inducing a self force on a point charge at rest in a static
curved space--time\cite{black}.

Also the presence of boundary condition like the boundary
condition at conical singularity produced for example by a cosmic
string or a point mass, alters the electromagnetic field of a
point charge which after subtracting the infinite part, leads to a
finite self force \cite{cone}.

A renewed investigation has been appeared in this subject in order
to study the upper bound on the entropy of charged object by
requiring the validity of thermodynamics of black--holes. This
problem is studied in \cite{thermo}, for classical black-hole
backgrounds, in the absence of dynamical fermions, i.e.
disregarding vacuum polarization.

Another subject of studies in gauge field theory is the screening
and confinement of charges. The static potential between external
charges, which can be obtained from the Wilson loop expectation
value, carries important information of infrared behavior of gauge
fields which is suggested to be responsible for confinement,
binding the quarks and anti--quarks into $q\bar q$ pairs (infrared
slavery). Because of computational hurdles in four dimensions one
can consider these problems in lower dimensional models, as a
laboratory to study physical effects which can may be carried out
to the real world.

In this letter we study the influence of curvature on the self
energy of static charges, by considering the effect of vacuum
polarization in two dimensional static space--times. We show that
in order to explain the confining behavior of $QED_2$ on a curved
background, considering ultraviolet behavior of gauge fields and
self energy of external charges is necessary, in other words our
method of studying confinement involves the behavior of two point
function in the ultraviolet regime instead of the infrared. This
is related to the fact that in usual calculations in four
dimensional flat space--time the self energy of a test charge is
infinite and is subtracted from the potential energy, but on
curved space--times the finite part of self--energy is not always
zero and is a function of charge position
\cite{black},\cite{cone}. This force must be considered (beside
the mutual interaction of charges) in studying mesonic structure.
We show that, for $QED_2$, the self force can prohibit a single
charge to be in some region of the space unless it is coupled to
another opposite test charge, forming mesonic $q{\bar q}$
structure. We also obtain electrostatic self--force using the heat
kernel method up to the adiabatic order four.
\section{ Geometrical effects on charge confinement and mesonic structure}
A general static two dimensional surface can be described by the
metric \be \l{1} ds^2={\sqrt g}(x)(dt^2-dx^2),  \ee where ${\sqrt
g}(x)$ is the conformal factor.\footnote{ Euclidean version of
this space--time is $ds^2={\sqrt g}(x)(dt^2+dx^2)$, whose one of
the geodesics is the straight line parallel to the $x$ axis.} On
this space--time, $QED_2$ consisting of charged matter field
interacting with an abelian gauge field in two dimensions is
described by Lagrangian \be \l{2} L={\sqrt g}(x)
(\psi^{\dagger}\gamma^{\mu}(\nabla_\mu -ieA_\mu)\psi ) +{1\over
2{\sqrt g}(x)}F^2), \ee where $\gamma^\mu $ are the curved space
counterparts of Dirac gamma matrices. $\nabla_{\mu}$ is the
covariant derivative, including the spin connection, acting on
fermionic fields. $e$ is the charge of dynamical fermions. The
dual field strength $F$, is described through $F=\epsilon ^{\mu
\nu}
\partial_{\mu}A_{\nu}$,
where $ \epsilon^{\mu \nu}=\epsilon_{\mu \nu}$ and $
\epsilon^{01}=-\epsilon_{10}=1$.
 By integrating out matter fields one can obtain one loop
 effective action for the gauge field \cite{schwcurved}
 \be \l{3}
 L_{eff.}={1\o 2{\sqrt g}(x)}F^2 +{\mu^2\o 2}F{1\o {\partial^2}}F,
\ee where $\mu ={e\o {\sqrt \pi}}$. In static case and using the
Coulomb gauge $A_1=0$ this Lagrangian reduces to \be \l{4}
 L_{eff.}=
{1\o 2{\sqrt g}(x)}({dA_0\over dx})^2 +{\mu^2\o 2}A_0^2. \ee Hence
in the presence of vacuum polarization,  the gauge field has
gained a mass. As a consequence, one may expect the replacement of
the Coulomb force by a finite range force. We introduce two static
opposite charges located at $x=a$ and $x=b$, described by the
covariantly conserved current \be \l{5} J^0(x) ={e'\o{\sqrt
g}(x)}(\delta (x-b) -\delta (x-a)), \ \ \ \ J^1=0. \ee  The gauge
field's equation of motion is \be \l{6} {d\o dx}{1\o {\sqrt
g}(x)}{dA_0\o dx}- \mu^2 A_0= e'(\delta (x-b)-\delta (x-a)). \ee
The Green function of the elliptic operator ${d\o dx} {1\o {\sqrt
g}(x)}{d\o dx}-\mu^2$ satisfies \be \l{7}  ({d\o dx}{1\o {\sqrt
g}(x)}{d\o dx}- \mu^2)G(x,x')=\delta (x,x'). \ee  In terms of
$G(x,x')$ the energy of external charges is obtained \be \l{8}
E=\int T^0_0dx=-\int L_{eff.}dx=-{{e'^2}\o
2}[G(a,a)+G(b,b)-2G(a,b)]. \ee This is the energy measured by an
observer whose velocity $u^\mu =(g^{-1/4}(x),0)$ is parallel to
the direction of the time--like killing vector of the space--time
and ${-e'^2\o 2}G(x,x)$ is the self--energy of a static point
charge located at $x$.

In order to see the effect of the curvature \be \l{9} R(x)={1\over
{\sqrt g}(x) }{d\over dx}{1\over {\sqrt g}(x)}{d\over dx}{\sqrt
g}(x), \ee on the energy we write the equation (\ref{7}) as \be
\l{10} {1\over \sqrt g(x)}{d\over dx}{1\over \sqrt g(x)}{d\over
dx}\sqrt g(x) {\tilde G(x,x')}-\mu^2 {\tilde G(x,x')}={\delta
(x,x')\over {\sqrt g(x)}}, \ee where $G(x,x')=\sqrt g(x){\tilde
G(x,x')}$. This equation can be rewritten as \be
\l{11}(R(x)-\mu^2){\tilde G(x,x')}+{1\over g(x)}({d{\sqrt
g}(x)\over dx})({d{\tilde G(x,x')}\over dx})+{1\over {\sqrt
g}(x)}{d^2{\tilde G(x,x')}\over dx^2}={\delta(x,x')\over
{\sqrt{g}}(x)}, \ee which is equivalent to \be
\l{12}(R(x)-\mu^2)G(x,x')+{1\over {\sqrt g}(x)}({d{\sqrt
g}(x)\over dx})({d\over dx}{1\over {\sqrt g}(x)}
G(x,x'))+{d^2\over dx^2} ({G(x,x')\over {\sqrt
g}(x)})=\delta(x,x'). \ee Hence in the strong curvature limit
$|R(x)|\gg \mu^2$, the Green function is approximately unaffected
by dynamical fermions. In other words if we assume the same
boundary condition for the gauge field in the presence and absence
of dynamical fermions, the vacuum polarization doesn't change the
confining or screening phase of the system. To elucidate this
subject and to emphasize how the ultraviolet behavior of $QED_2$
on curved space--time is concerned in confinement of test charges
let us give an example. Consider the following space--time in
conformal coordinates \be \l{13} ds^2={{dt^2-dx^2}\over x^m}
;\;\;\;\;x> 0 \ee where $m=2-1/(k+1)$; $k\neq 0,-{1/2}$, this is
one of the classical solutions of two dimensional scale invariant
gravity \cite{scaleinv} with a curvature singularity. Here we
assume that this is only a classical background for $QED_2$. The
homogenous solutions $G_h(x)$ of the equation (\ref{7}) satisfy
\be \l{14}{d\over dx}x^m{d\over dx}G_h(x)-\mu^2 G_h(x)=0.\ee

For $m\neq 2$, by defining $z\equiv x^{1-{m\over 2}}$ and
$G_h(x)\equiv x^{{1-m}\over 2}u$, we obtain \be
\l{15}z^2{d^2u\over dz^2}+z{du\over dz}-({(m-1)^2\over
(m-2)^2}+{4\mu^2 z^2\over(m-2)^2})u=0. \ee Hence
\begin{eqnarray}\l{16}
G _h(x)=  x^{{1-m}\over 2}\left\{ \begin{array}{ll}
 I_{|{{m-1}\over {m-2}}|}(\sqrt{4\mu^2\over
(2-m)^2}x^{1-{m\over 2}})& \\
K_{|{{m-1}\over {m-2}}|}(\sqrt{4\mu^2\over
(2-m)^2}x^{1-{m\over 2}}). & \\
\end{array} \right.
\end{eqnarray}
Therefore the Green function is
\begin{eqnarray}\l{17}
 G(x,x')=-{2 \over
|m-2|}(x_{<}x_{>})^{1-m\over 2}\left\{
\begin{array}{ll}
I_{{{m-1}\over {m-2}}}({2\mu\over {m-2}}x_{>}^{1-{m\over 2}})
K_{{{m-1}\over {m-2}}}({2\mu\over {m-2}}x_{<}^{1-{m\over 2}})&
\textrm{if $m>2$}\\
I_{|{{m-1}\over {m-2}}|}({2\mu\over {2-m}}x_{<}^{1-{m\over 2}})
K_{|{{m-1}\over {m-2}}|}({2\mu\over {2-m}}x_{>}^{1-{m\over 2}})&
\textrm{if $m<2$},
\end{array} \right.
\end{eqnarray} where $x_{>(<)}$ is the bigger (smaller) of $x,x'$ and $I$, $K$
are modified Bessel functions.
This Green function Satisfies Dirichlet boundary condition at
$x=0$ and at $x=\infty$. At the coincidence limit the Green
function is
 \be \l{18}
G(x,x)=-{2 \over |m-2|}x^{1-m}I_{|{{m-1}\over
{m-2}}|}\big({2\mu\over {|m-2|}}x^{1-{m\over 2}}\big )
K_{|{{m-1}\over {m-2}}|}({2\mu\over |m-2|}x^{1-{m\over 2}}).
 \ee
 In terms of the scalar curvature $R(x)={m\over x^{2-m}}$, this
 relation becomes
 \be \l{19}
 G(x,x)=-{2 \over |m-2|}\Big({m\over R(x)}\Big)^{{1-m}\over {2-m}}I_{|{{m-1}\over
{m-2}}|}\Big({2\mu\over {|m-2|}}({R(x)\over m})^{-{1\over 2}}\Big)
K_{|{{m-1}\over {m-2}}|}\Big({2\mu\over {|m-2|}}({R(x)\over
m})^{-{1\over 2}}\Big).
 \ee
 In the strong curvature limit, or in regions where ($|R(x)|\gg \mu^2$), by considering
 the asymptotic behavior of Bessel functions
 this relation becomes
 \be \l{20}
 \lim_{R\to \infty}G(x,x)=-|{k+1\over k}|({R(x)\over m})^k,
 \ee
 which is $\mu$ independent as anticipated. $k$ is defined after the equation (\ref{13}).
 In regions where the curvature is weak $|R(x)|\ll \mu^2$, we
 have
 \be \l{21}
 G(x,x)=-{1\over 2\mu}({R(x)\over m})^{m\over 2(2-m)}.
\ee
 Now we show the relation of electrostatic self--energy and
confinement. Assume that $m>2$; the energy needed to locate a
single charge $e'$ in the region $R(x)\ll \mu^2$ or $x\simeq 0$
(in the coordinate (\ref{13})), is \be \l{22}
E_{self.}(x)={{e'}^2\over 4\mu}({R(x)\over m})^{m\over 2(2-m)},\ee
which tends to infinity. In other words there is a great repulsive
force on an external charge near $x=0$ prohibiting to have single
charges in this region. The same procedure occurs in the region
$x\simeq 0$, for $m<2$ (when $|R(x)|\to \infty$). So in these
regions following the equation (\ref{8}) only charges forming
mesonic structure may survive. These two opposite charges must be
near together in order to obtain a finite energy for the system.
Note that our criterion for confinement is not based on the
behavior of the energy in the infrared (where the geodesic
distance of external charges tends to infinity).

A similar phenomenon (but not due to gravitational effects) occurs
in dual bag model. In that case quarks and anti--quarks must form
mesonic structure in order to avoid divergences in static
potential, i.e only color singlets have finite energy because the
divergence term appears as a multiplier of total external charges
\cite{dual}. In our case besides this constraint, test charges
must be close together.

The repulsive forces on single charges besides the curvature of
the space--time is related to the boundary conditions imposed on
the gauge fields in defining the vacuum of the system. These
repulsive forces may also be arisen on a surface with constant
curvature (adS or dS space--times, obtained for example by taking
$m=2$ in the previous example)\cite{sad}.

Maxwell field theory (disregarding vacuum polarization) in 2+1
dimensional conical space--times, despite the null curvature of
the manifold exhibits also a repulsive force on charges.

For Maxwell theory on a four dimensional Schwarzschild black-hole
a test charge near the horizon is repelled by an image charge
inside the horizon. In these case one must subtract the infinite
parts to obtain a renormalized Green function or the finite part
of self--energy. On a two dimensional static space--time, in
contrast to the Maxwell theory in three and four dimensions, as we
will discuss later,  electrostatic self--energy in $QED$ is a well
defined function.

In  Schwarzschild coordinate, we consider a non--extremal two
dimensional static black hole described by the metric \be
\l{23}ds^2=f(r)dr^2-{1\over f(r)}dr^2. \ee At the horizon $r=h$,
$f(h)=0$. In this coordinate the equation (\ref{7}) becomes \be
\l{24}\Big(f(r)({d^2\over
dr^2})-\mu^2\Big)G(r,r')=f(r)\delta(r,r').\ee Near the (bifurcate)
horizon, i.e. $r \simeq h$, $r>h$, we have $f(r)=\kappa (r-h)$,
where $2\kappa$ denotes the surface--gravity. Assuming the gauge
field tends to zero at infinity and is well behaved at the
horizon, the two point function of the gauge field becomes \be
\l{25} G(r,r')=2(r_{>}-h)^{1\over 2}(r_{<}-h)^{1\over
2}K_1(2\mu\sqrt{{r_>-h\over \kappa}})I_1(2\mu\sqrt{{r_<-h\over
\kappa}}),\ee where $r_{<(>)}$ is the smaller (bigger) of $r$ and
$r'$. The self energy near the horizon is then \be \l{26}
E_{self.}(x)=e'^2(r-h)K_1\Big(2\mu \sqrt {{r-h}\over
\kappa}\Big)I_1\Big(2\mu \sqrt {{r-h}\over \kappa}\Big).\ee In
contrast to the four dimensional case, there is an attractive
force on the test charge near the horizon, which by considering
the asymptotic behavior of Bessel functions is independent of
vacuum polarization: this attractive force may be related to an
image charge inside the horizon, when these charges are near
together, that is near the horizon, the effect of vacuum
polarization may be disregarded (self--energy is independent of
$\mu$). At the horizon the self--energy is zero, and in contrast
to the previous example ultraviolet behavior of the Green function
doesn't lead to charge confinement.
\section{Heat kernel expansion of electrostatic self--energy}

In this part we study the short distance behavior of Green
function $G(x,x')$, using the heat kernel of positive elliptic
operator $O:=-{d\o dx} {1\o \sqrt g}{d\o dx}+\mu^2$.  We write the
heat kernel in the form \be \l{27} h(\tau
;x,x')=\sum_{n=0}^{\infty}{\tau^{(n-{1\over 2})}\over
\sqrt{4\pi}}\exp({-\sigma \over 2\tau}-\mu^2 \tau)a_n(x,x'),\ee
which satisfies \be \l{28} Oh(\tau;x,x')+{\partial
h(\tau;x,x')\over
\partial \tau}=0,\ee where $\sigma={1\over 2}\mid
\int_{x'}^{x}g^{1\over 4}(y)dy\mid^2$, is one half of the square
of geodesic distance between $(t,x)$ and $(t,x')$ and $\tau$ is
the proper--time parameter. $G(x,x')$ is given by \be
\l{29}G(x,x')=-\int_{0}^{\infty}h(\tau;x,x')d\tau,\ee provided \be
\l{30} h(0;x,x')=\delta( x,x').\ee Therefore \be
\l{31}G(x,x')=-{1\over \sqrt \pi} \sum_{n=0}^\infty ({\sigma \over
2\mu^2})^{({n\over 2}+{1\over 4})}K_{n+{1\over 2}}(\mu
\sqrt{2\sigma})a_{n}(x,x') \ee and for $\sigma=0$,\be \l{32}
G(x,x)=-{1\over 2\sqrt \pi} \sum_{n=0}^{\infty}(\mu^2)^{-n-{1\over
2}}\Gamma(n+{1\over 2})a_n(x,x),\ee which is regular.

Under a scale transformation parameterized by the positive number
$\lambda$,
\begin{eqnarray}\l{33}
&&{\sqrt g}(x)\rightarrow \lambda {\sqrt g}(x)
\nonumber \\
&& R(x)\rightarrow {1\over \lambda}R(x),
\end{eqnarray}\
the Green function becomes
 $G_{\lambda}(\mu^2)=\lambda G(\mu^2
\lambda)$ ${}$or${}$ $ {1\over \lambda} G_{\lambda}({ \mu^2\over
\lambda})= G(\mu^2)$. We have written the $\mu$ dependence of $G$
explicitly. Using (\ref{32} ), we obtain \be \l{34}
a_{n}(\lambda)={a_n(\lambda=1)\over{\lambda^{n-{1\over 2}}}}. \ee
Hence As a polynomial, $a_n$ consists only of $mth$ power of $g$
(including also its derivatives), where $m={1\over 4}-{n\over 2}$.
In $a_n$, the order of derivatives is $2n$. For example as we will
see, in $a_1(x,x)$, only the terms $g^{-{9\over 4}}(x)g'^2(x)$ and
$g^{-{5\over 4}}(x)g''(x)$ are present.

In order to obtain heat--kernel coefficients we use the relations
\be \l{35}[\sigma]=[\sigma']=0,{}[\sigma^{(2)}]={\sqrt g}(x),{}
[\sigma^{(3)}]={3g'\over 4{\sqrt g}}(x). \ee We have shown $\sigma
(x,x)$ by $[\sigma]$ and $\prime$ denotes the first derivative and
$(n)$ the nth
 derivative with respect to $x$.

By solving the equation (\ref{28}) for the Seeley coefficients, we
obtain a recursion relation
\begin{eqnarray} \l{36}&-&{1\over 2}g^{-{3\over 2}}(x)g'(x)a'_{n}(x,x')+g^{-{1\over
2}}(x)a^{(2)}_n(x,x')-(n+1)a_{n+1}(x,x'
)+ {} \nonumber \\
& & {} {1\over 8}g'(x)\sigma'(x,x')g^{-{3\over
2}}(x)a_{n+1}(x,x')-g^{-{1\over 2}}\sigma'(x,x')a'_{n+1}(x,x')=0.
\end{eqnarray}
For $n<0$, $a_{n<0}=0$. In order to satisfy (\ref{30}) we must
have $a_{0}(x,x')=g^{1\over 8}(x)g^{1\over 8}(x')$. Taking the
diagonal value of (\ref{36}) yields \be \l{37}-{1\over
2}g^{-{3\over 2}}(x)g'(x)[a'_{n}]+g^{-{1\over
2}}(x)[a^{(2)}_n]-(n+1)[a_{n+1}]=0.\ee

For $n=0$ \be \l{38}[a_1]=-{1\over 2}g^{-{3\over
2}}(x)g'(x)[a'_0]+g^{-{1\over2}}[a_0^{(2)}], \ee hence \be \l{39}
[a_1]=-{11\over 64}g^{-{9\over 4}}(x){g'}^2(x)+{1\over
8}g^{-{5\over 4}}(x)g^{(2)}(x).\ee For $[a_2]$ we require the
diagonal part of $a_1$  derivatives: $[a_1']$ and $[a_1^{(2)}]$.
Differentiating (\ref{36}) with respect to $x$ gives
\begin{eqnarray} \l{40}& &\Big(-{1\over 2}g^{-{3\over
2}}(x)g^{(2)}(x)+{3\over 4}g^{-{5\over
2}}{g'}^2(x)\Big)[a'_n]-g^{-{3\over
2}}(x)g'(x)[a_n^{(2)}]+g^{-{1\over 2}}[a_n^{(3)}]
-\nonumber \\
& & (n+2)[a'_{n+1}]+{1\over 8}g^{-1}(x)g'(x)[a_{n+1}]=0.
\end{eqnarray}
For $n=0$ \be \l{41} [a'_1]=-{1\over 4}g^{-{9\over
4}}(x)g'(x)g^{(2)}(x)+{99\over 512}g^{-{13\over 4}}{g'}^3+{1\over
16}g^{-{5\over 4}}(x)g^{(3)}(x).\ee Another differentiation of
(\ref{36}) with respect to $x$ in the limit $x\rightarrow x'$
leads to the following equation
\begin{eqnarray}\l{42}
&& \Big(-{1\over 2}g^{-{3\over 2}}(x)g^{(3)}(x)+{9\over
4}g^{-{5\over 2}}(x)g'(x)g^{(2)}(x)-{15\over 8}g^{-{7\over
2}}(x){g'}^3(x)\Big)[a'_n]+\nonumber \\
&&\Big(-{3\over 2}g^{(2)}(x)g^{-{3\over 2}}(x)+{9\over
4}{g'}^2(x)g^{-{5\over 2}}(x)\Big)[a_n^{(2)}]+g^{-{1\over 2}}[a^{(4)}_n]
-{3\over 2}g'(x)g^{-{3\over 2}}(x)[a_n^{(3)}]+\nonumber \\
&&\Big({1\over 4}g^{(2)}(x)g^{-1}(x)-{9\over
32}{g'}^2(x)g^{-2}(x)\Big)[a_{n+1}]-{1\over
2}g'(x)g^{-1}(x)[a'_{n+1}]-\nonumber \\
&&(n+3)[a^{(2)}_{n+1}]=0.
\end{eqnarray}
Therefore we find \begin{eqnarray}\l{43}&& [a_1^{(2)}]={-{23\over
96}}g^{-{9\over 4}}(x)g'(x)g^{(3)}(x)+{1\over 24}g^{-{5\over
4}}(x)g^{(4)}(x)-{31\over 192}g^{-{9\over
4}}(x){g^{(2)}}^2(x)-\nonumber \\
& & {1947\over 4096}g^{-{17\over 4}}(x){g'}^4(x)+{213\over
256}g^{-{13\over 4}}(x){g'}^2(x)g^{(2)}(x).\end{eqnarray}
Equations (\ref{37}), (\ref{41} ), (\ref{43}) yield
\begin{eqnarray}\l{44}
[a_2]=&&{245\over 512}g^{-{15\over
4}}(x){g'}^2(x)g^{(2)}(x)-{2343\over 8192}g^{-{19\over
4}}(x){g'}^4(x)-{26\over 192}g^{-{11\over
4}}(x)g'(x)g^{(3)}(x)-\nonumber \\
&&{31\over 384}g^{-{11\over 4}}(x){g^{(2)}}^2(x) +{1\over
48}g^{-{7\over 4}}(x)g^{(4)}(x).
\end{eqnarray}
One can continue this method to obtain other $[a_n]$.
Heat kernel
coefficient can be expressed in terms of the scalar curvature
$R={1\over 2}g^{-{3\over 2}}(x)g^{(2)}(x)-{1\over 2}g^{-{5\over
2}}(x)g'^2(x)$ and $\kappa(x)={1\over 2}g'(x)g^{-{1}}(x)$ (at the
horizon of a black--hole, $2\kappa$ is the surface --gravity).
\begin{eqnarray}\l{45} &&[a_1]={1\over 4}g^{1\over 4}(x)R(x)-{3\over
16}g^{-{1\over
4}}(x)\kappa^2(x)\nonumber \\
&& [a_2]={1\over 24}g^{-{1\over 4}}(x)R^{(2)}(x)-{1\over
8}g^{-{1\over
4}}(x)\kappa (x)R'(x)-{1\over 32}g^{1\over 4}(x)R^2(x)+\nonumber \\
&&{23\over 192}g^{-{1\over 4}}(x)\kappa^2(x)R(x)-{23\over
512}g^{-{3\over 4}}(x)\kappa^4(x).
\end{eqnarray}

Hence the self--energy of the charge $e'$ up to the fourth
adiabatic order is
\begin{eqnarray}\l{46}
E_{self.}(x)&=&{e'^2\over 4}g^{-{1\over 4}}(x) \Big
[\mu^{-1}g^{1\over 2}(x)+{1\over 8}\mu^{-3}\Big(g^{1\over
2}(x)R(x)-{3\over 4}\kappa^2(x)\Big)+\nonumber \\
&& {3\over 32}\mu^{-5}\Big({1\over 3}R^{(2)}(x)-
\kappa (x)R'(x)-{1\over 4}g^{1\over 2}(x)R^2(x)+\nonumber \\
&&{23\over 24}\kappa^2(x)R(x)-{23\over
64}g^{-1}(x)\kappa^4(x)\Big)\Big].
\end{eqnarray}

Note heat kernel expansion like the WKB method can not be applied
at the horizon $h$,  i.e. where $g(h)=0$. To obtain an expression
for the self energy near the horizon one can expand the metric and
follows the steps after the equation (\ref{23}). For example
Considering asymptotic behavior of Bessel function for large
arguments in equation (\ref{26}) gives the first term in equation
(\ref{46}). This is similar to the relation of Airy functions and
the WKB solutions in solving the Schrodinger equation for a slowly
varying potential. The first term of (\ref{46}) is also the same
as (\ref{22}) obtained in weak curvature limit. Note that in the
flat case (\ref{46}) and therefore the self energy of test charge
in our previous examples tends to the well known result
$E_{self.}={e'^2\over 4\mu}$. \vskip 1cm
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\end{document}
