\documentstyle[aps]{revtex} 
\begin{document} 
\section*{{Reply to comment by Zaslavskii on extremal black hole action}}
\begin{abstract} 
It is shown that Zaslavskii's misunderstanding of our published proof of the
irrelevance of all extremal black hole configurations (whether with equal
charge and mass or not) rests on his refusal to see the essential difference
between the correct inequality governing extremal and non-extremal actions
and his incorrect version.
\end{abstract}

\bigskip

Recently in a comment \cite{z} on our already published reply
\cite{GMrep} Zaslavskii expressed the opinion that our relation
$I_n(m,q-\epsilon)<I_e(m,q)$ \cite{GMrep} between non-extremal and
extremal on-shell Reissner - Nordstr\"{o}m actions
is essentially the same as the
relation $I_n(m,q)<I_e(m,q)$ \cite{zcom}.  But the latter relation, which
is his misinterpretation of a statement in our Letter \cite{GM}, leads him
astray, when he relies on the {\it equality} of the charges on the two sides
at the top of p. 2 (\cite{z}:v.1). On p. 4, while commenting on the
former (correct) inequality, he forgets this and claims that the fact
{\it "that charges on both sides ... may (be) slightly different from
each other, is not crucial"}! This contradicts his use of the inequality.

Our point was that for each extremal configuration $(m,q)$, there is a
non-extremal configuration $(m,q-\epsilon)$ of lower action. This follows
trivially from the first inequality, but cannot be understood from the
second inequality (which does not exist anywhere in \cite{GMrep,GM}
and is Zaslavskii's invention) because in the case $m=q$ one would 
need a non-extremal configuration with $m=q$, but such a thing does not 
exist, whereas non-extremal configurations with $q=m-\epsilon$ of course do. 
For extremal configurations only with $m\ne q$, the second relation can be 
used. In all cases, the corresponding non-extremal configurations have 
lower action, so that extremal configurations cannot be physically relevant 
in the approximation considered.

His other comments (on, {\it e.g.}, the evaluation of path integrals under
the approximations considered in \cite{GM}) show further misconceptions
which, however, are not related to the main point involved in \cite{GM}.

\begin{references}
\bibitem{z} O. B. Zaslavskii, hep-th 9804090 (1998)
\bibitem{GMrep} A. Ghosh and P. Mitra, Phys. Rev. Letters, {\bf 80}, 3413 (1998)
\bibitem{zcom} O. B. Zaslavskii, Phys. Rev. Letters, {\bf 80}, 3412 (1998)
\bibitem{GM} A. Ghosh and P. Mitra, Phys. Rev. Letters, {\bf 78}, 1858 (1997)
\end{references}
\bigskip
\noindent{A. Ghosh}\\
{CERN, CH-1211 Geneva 23, Switzerland}\\
and P. Mitra\\
{Saha Institute of Nuclear Physics,
Block AF, Bidhannagar,
Calcutta 700 064, India}
\end{document}
% Here is the text of the published reply
\documentstyle[aps]{revtex}
\begin{document}
\section*{{Ghosh and Mitra Reply to Comment on "Understanding the area proposal ..."}}

The  Comment \cite{zasl} criticizes its own Eq. (1). 
This equation was neither written   nor  used in \cite{GM}.

\bigskip

The technical observation made in \cite{zasl} is that
configurations with $\alpha^2\sim (r-r_+)^{-1}$ near the horizon ($r\sim
r_+$)  and with extremal topology, {\it i.e.,} ${b'\over\alpha}=0$ at the
horizon, have {\it curvatures} diverging at the horizon like
$(r-r_+)^{-1}$, because $R$ has a term proportional to $ (\alpha
b)^{-1}(r^2b'/\alpha)'$ \cite{york}. 
However, it is trivial to see that the integral of a
curvature behaving in that manner is well defined near the horizon, where
$\sqrt g\propto\alpha b\to 0$.  This is consistent with
the equation for the extremal action given in \cite{GM}
on the basis of the
calculation indicated in \cite{GMcom}.

Further, it is to be emphasized that these configurations were not explicitly
used by us, nor do they need to be used implicitly, as explained below,
and are completely irrelevant for the conclusion drawn by us.  The final,
physical configuration in our approach \cite{GM} comes from the
non-extremal sector and has finite curvature as well as finite action and
leads to finite thermodynamic quantities.

The main misunderstanding of \cite{zasl} is expressed in Eq. (1), which is
not a correct representation of what we said in \cite{GM}. The issue is the
determination of the configuration(s) of minimum action from the set of
configurations of non-extremal topology and of extremal topology.
We first argued that the configuration with minimum action must be one
of non-extremal topology. To prove this, it is sufficient to show that
for each extremal configuration with parameters $m_1, q_1$, there exists
a non-extremal configuration with parameters $m_2, q_2$, such that
$$I_n(m_2,q_2)<I_e(m_1,q_1).$$ In view of the discontinuity in the forms
of the action between the two topologies, it is clear that this inequality
is satisfied for $m_2=m_1,~q_2=q_1$. But it is also true that there
exist small, but nonzero values of $\epsilon$ such that the above
inequality of actions is satisfied with $m_2=m_1, ~q_2=q_1-\epsilon$. {\it
This point is overlooked in \cite{zasl}.} Since for each set of $q_1,m_1$,
{\it whether equal or not}, there exist non-extremal configurations
with lower action, the minimum cannot occur within the full set of
configurations with extremal topology. Once it is established that only
non-extremal configurations need to be considered in the search for the
configuration with minimum action, simple calculations lead to the
area formula.

%The above argument demonstrates the (semiclassical) irrelevance of {\it
%all} extremal configurations.  If some of these
%configurations were to be eliminated from the beginning, that would make the
%problem logically {\it simpler}, for then there would be {\it fewer}
%configurations of extremal topology which would have to be shown to be
%irrelevant, {\it i.e.,} for which one would have to find non-extremal
%configurations with lower action! Thus the inclusion
%or exclusion of the configurations highlighted in
%\cite{zasl} does not alter either our proof or our conclusion.

A further criticism of \cite{GM} is made in \cite{zasl} apparently in the
belief that our work was based on the {\it assumption} of zero entropy.
Again, this assumption was not used there. The expression for the
extremal action can be explicitly calculated along the lines of \cite{york}
with appropriate boundary conditions as indicated in \cite{GMcom}.  Thus
each of the objections raised in \cite{zasl} is unfounded.

%
\begin{references}
\bibitem{zasl} O. B. Zaslavskii, preceding Comment
\bibitem{GM} A. Ghosh and P. Mitra, Phys. Rev. Letters, {\bf 78}, 1858 (1997)
\bibitem{york} H. W. Braden, J. D. Brown, B. F. Whiting and J. W. York,
Phys. Rev. {\bf D42}, 3376 (1990)
\bibitem{GMcom} A. Ghosh and P. Mitra, Phys. Rev. Letters, {\bf 77}, 4848 (1996)
\end{references}
\bigskip
\noindent{A. Ghosh\footnote{e-mail Amit.Ghosh@cern.ch}\\
{CERN, CH-1211 Geneva 23, Switzerland}\\
and P. Mitra\footnote{e-mail mitra@tnp.saha.ernet.in}}\\
{Saha Institute of Nuclear Physics,
Block AF, Bidhannagar,
Calcutta 700 064, India}
\end{document}

