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Non-relativistic field theory of ``extreme black holes''
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Poster for YKIS'99 BLACK HOLES AND GRAVITATIONAL WAVES
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\title{Non-relativistic field theory of ``extreme black holes''}
\author{Yoshitaka~Degura and Kiyoshi~Shiraishi\\ Yamaguchi University, Japan}
\date{2PM Tuesday, June 29 -- 1PM Wednesday, June~30}
\begin{document}
\maketitle
\begin{abstract}
We study a low-energy effective action for
a scalar field coupled with electromagnetic,
gravitational, and dilatonic fields.
The charges of the scalar field are assumed
to satisfy a critical relation, which is the same as
that of ``extreme black holes''.
Using this action, we investigate statistical
dynamics for the gas of ``extreme black holes''.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Effective Lagrangian}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Classcal action for a particle with mass $m$ and charge $e$,
which is coupled to gravitational, electromagnetic, and dilaton
fields:
\be
I=-\int\left[m e^{a\phi}+e A_{\mu}\frac{dx^{\mu}}{ds}\right]ds,
\ee
where $\phi$ is the dilaton field.
This action leads to an equation:
\be
g^{\mu\nu}\left(P_{\mu}+e A_{\mu}\right)
\left(P_{\nu}+e A_{\nu}\right)+m^2 e^{2a\phi}=0,
\ee
where
\be
P_{\mu}\equiv m e^{a\phi}g_{\mu\nu}\frac{dx^{\nu}}{ds}-e A_{\mu}.
\ee
One can find that a ``Wave equation''
\be
g^{\mu\nu}\left(P_{\mu}+e A_{\mu}\right)
\left(P_{\nu}+e A_{\nu}\right)\varphi+m^2 e^{2a\phi}\varphi=0
\ee
can be obtained from:
\be
S_m=\int d^4x\sqrt{-g}\left[-\varphi^{*}e^{-a\phi}
g^{\mu\nu}\left(P_{\mu}+e A_{\mu}\right)
\left(P_{\nu}+e A_{\nu}\right)\varphi-m^2 e^{a\phi}\varphi^{*}
\varphi\right].
\ee
\bigskip
The total action is:
\be
S=\int d^4x\frac{\sqrt{-g}}{16\pi}\left[R-2
\left(\nabla\phi\right)^2-e^{-2a\phi}F^2\right]+S_m,
\ee
which yields the following field equations:
\be
\nabla^2\phi+\frac{a}{2}e^{-2a\phi}F^2+4\pi a\left[
e^{-a\phi}\varphi^{*}\left(P+eA\right)^2\varphi-
e^{a\phi}m^2\varphi^{*}\varphi\right]=0
\label{eq:D}
\ee
\bea
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R&=&
2\left[\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}
\left(\nabla\phi\right)^2\right]
+e^{-2a\phi}\left[2F^2_{\mu\nu}-
\frac{1}{2}g_{\mu\nu}F^2\right] \nn
&+&16\pi\left\{\frac{}{}e^{-a\phi}
\Re\left[\frac{}{}\varphi^{*}\left(P_{\mu}+e A_{\mu}\right)
\left(P_{\nu}+e A_{\nu}\right)\varphi\right.\right. \nn
&-&\left.\left.\frac{1}{2}g_{\mu\nu}
\varphi^{*}\left(P+eA\right)^2\varphi\right]-\frac{1}{2}g_{\mu\nu}
e^{a\phi}m^2\varphi^{*}\varphi\right\}
\label{eq:E}
\eea
\be
\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}e^{-2a\phi}F^{\mu\nu}\right)=
8\pi e e^{-a\phi}\varphi^{*}g^{\nu\lambda}
\left(P_{\lambda}+e A_{\lambda}\right)\varphi
\label{eq:F}
\ee
\bigskip
We take ansatze:
\be
ds^2=-U^{-2}\left(dt+B_{i}dx^i\right)^2+U^2d\vx^2
\ee
\be
U=V(\vx)^{\frac{1}{1+a^2}}
\ee
\be
e^{-2a\phi}=V^{\frac{2a^2}{1+a^2}}
\ee
\be
A_0=\frac{1}{\sqrt{1+a^2}}\left(1-\frac{1}{V}\right)
\ee
We have known, for a static configuration ($A_i=B_i=0$),
a vacuum solution satisfies
\be
\partial^2V=0,
\ee
which describes an arbitrary number of ``extreme black holes'',
provided that
\be
\frac{e}{m}=\sqrt{1+a^2}.
\ee
We take this relation hereafter.
\bigskip
Low-energy limit is considered:
\be
-P_0-m=E-m<
–ß‚é