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\title{Instability in self-gravitating systems of particles with nearly
critical charges}
\author{Kiyoshi Shiraishi\\
Faculty of Science, Yamaguchi University,\\
Yoshida, Yamaguchi-shi, Yamaguchi 753, Japan}
\date{}
\begin{document}
\maketitle
%\begin{abstract}

%\end{abstract}


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\section{Introduction}
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The physics of self-gravitating particle system has been investigated
not only by numerical simulations but also by analytic studies. We can
consider the case that there are other long-range forces acting between
particles; for instance, the electromagnetic forces, and/or the force
mediated by a massless scalar field. In general, elementary particles or
states which appear in recent models of unified theory have various
interactions other than gravitation. Thus, the study of the system of the
particles with many kinds of ``charges'' may reveal some aspects of the
very early stage, or beginning of our universe. 

Further, recent studies on
string theories and so-called M theory have shown that the elementary
excitations correspond to BPS or nearly BPS black holes. Among the BPS
black holes, static forces are cancelled out. There is, however,
velocity-dependent forces among them. When the many particle system of
(nearly) BPS states is studied, it must be important to take the
velocity-dependent interactions into account.

In this talk, we consider the extension of the method of gaussian
integration used by de Vega et al.~\cite{1} to the case in the presence
of velocity-dependent interactions.

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\section{Review of gaussian integration technique}
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One can write the Hamiltonian of $N$-particle system with the Newtonian
potential as 
\be
H_N=\sum_{\ell=1}^N\frac{p_{\ell}^2}{2m}-Gm^2\sum_{\ell0$ and $3Gm^2-\sigma^2 < 0$, the static
configuration is impossible at sufficiently high temperature because
$\mu^2$ becomes negative at high temperature. 

If we regard the particles
as charged dilatonic black holes~\cite{5}, this situation can occurs when
$a^2 > 3$ (where $a$ is the dilatonic coupling constant). This fact
corresponds to the statement that black holes look like ``elementary
particles'' if $a^2 > 3$~\cite{6}.

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\section{Conclusion}
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In this talk, I have shown almost nothing to do with the property of
the system governed by the long range forces, but the possible method to
approch the study in terms of the effective field theory. For nearly BPS
particles, the inclusion of the velocity-dependent
interactions is essential. The field theoretic study of this exotic
matter system is under
investigation, which is very parallel to~\cite{1}.

We note that since we deal with the grand canonical formalism, we can
take the coalescence of particles (i.e. the change and transfer in the
particle number) into consideration.
This consideration is regarded as a model of the system of interacting 
black holes.

As we treat isothermal system, we wish to take the temperature as the
Hawking temperature in the presence of a black hole. To incorporate such a
background effect, we must extend the formulation to the case with fully
curved spacetime and inclusion of radiation (and other possible fields).
The work is in preparation.

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\section*{Note added}
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We have developed semi-general-relativistic formalism for the gravitating
particles since after the workshop. We consider the Hamiltonian of
Einstein-Hilbert gravity coupled to particles. Then we integrate out the
momenta of particle using gaussian approximation. If the transfer energy
between particles is negligible, the partition function can be
represented by the path integral of the variables.

Here, for simplicity, the shift vector is assumed to vanish. Then the
grand canonical partition function in $(1 + d)$-dimensional spacetime is
written as:
\be
Z_G=\int [DN][D\gamma_{ij}]\exp\left\{\int\left[
\frac{\beta}{16\pi G}N\,\, {}^{(d)}\!\!R+z\left(\frac{m}{2\pi\beta
N}\right)^{d/2}e^{-\beta Nm}\right]
\sqrt{\gamma}d^dx\right\} , 
\ee
where $N$ is the lapse function, $\gamma_{ij}$ is the $d$-dimensional
metric and ${}^{(d)}\!\!R$ is the scalar curvature
constructed from  $\gamma_{ij}$.

The ``classical'' equation for $N$ is
\be
\nabla^2N=4\pi
G\left(2\frac{d-2}{d-1}Nm+\frac{d^2}{d-1}\frac{1}{\beta}
\right)z\left(\frac{m}{2\pi\beta
N}\right)^{d/2}e^{-\beta Nm}\, . 
\ee
We emphasize that the $O(v^2)$
velocity-dependent interaction is {\it automatically} incorporated. 

The gravitational potential is involved in $N$. The coefficient of the
$1/\beta$ seemingly differs from the previous approach. But this is not
the real defect because there is $N^{d/2}$ in the right hand side and the
gravitational potential is also included in the laplacian in the left hand
side through $\gamma_{ij}$. The correct flat-space limit yields the
agreement with the previous result.%
\footnote{To see this, substitute the metric (for $d\ne 2$) as
$\gamma_{ij}\rightarrow N^{-\frac{2}{d-2}}\gamma_{ij}$ in eq.~(20).}
(The interpretation is possible that the half of the
averaged energy of a particle is shared with the gravitational potential,
which cannot be neglected in this case.)

Taking other forces into account is straightforward.

One more comment is in order: even in the $(1+2)$-dimensions, isothermal
spheres can exist! (but global topology of space seems to be curious.)

Further development and field theoretic study in this line will be
reported elsewhere.

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\begin{thebibliography}{99}

\bibitem{1} H.~J.~de~Vega, N.~Sanchez and F.~Combes, Phys. Rev. {\bf D54}
(1996), 6008; Nature {\bf 383}  (5 Sept. 1996), 56. 

\bibitem{2} S.~Chandrasekhar, {\it An Introduction to
the Study of Stellar Structure} (Univ. of  Chicago, 1939).

\bibitem{3} L.~D.~Landau
and E.~M.~Lifshitz, {\it The Classical Theory of Fields} (Pergamon,
Oxford, 1962).

\bibitem{4} K.~Shiraishi and T.~Maki, Phys. Rev. {\bf D53} (1996), 3070.

\bibitem{5} G.~W.~Gibbons and K.~Maeda, Nucl. Phys. {\bf B298} (1988),
741.

D.~Garfinkle, G.~T.~Horowitz and A.~Strominger, Phys. Rev. {\bf D43}
(1991), 3140.

\bibitem{6} C.~F.~E.~Holzhey and F.~Wilczek, Nucl. Phys. {\bf B380}
(1992), 447.


\end{thebibliography}

\end{document}
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