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\begin{document}

\preprint{}

\title{A possible catastrophic energy release}

\author{She-Sheng Xue}

\email{xue@icra.it}

\affiliation{ICRA, INFN and
Physics Department, University of Rome ``La Sapienza", 00185 Rome, Italy}

\date{August, 2000}

%\date{\today}


\begin{abstract}
On the analogy of the Casimir effect, we present an effect of quantum-field fluctuations, 
attributed to gravitational field coupling to the zero-point energy
of virtual particles in the vacuum. In the process of black hole's formation, such an effect could cause
tremendous energy release, possibly describing a scenario of energetic sources for 
observed gamma ray bursts. The conclusion is based on the analysis of the zero-point energy 
of quantum field theories in curved spacetime and classical dynamics
of gravitational collapse. These analyses show that the zero-point energy is gravitationally 
red-shifted, which causes the vacuum becomes unstable and quantum-mechanically 
decays, leading to energy release. The experimental test of such an effect is proposed.  
  

\end{abstract}

\pacs{04.62.+v, 04.70.Bw, 04.70.Dy}

\maketitle


\vskip0.2cm
\noindent{\it Introduction.}\hskip0.3cm 
The mystery of energetic sources generating gamma ray bursts,  
a prompt emission of extremely huge energy in order of a second, has stimulated many 
studies in connection with electromagnetic 
properties of black holes\cite{dr,prx,putten,koide}. We present an alternative 
scenario on the basis of gravitational field interacting with the zero-point energy of virtual 
particles in the vacuum. Analyzing the energy-spectrum of virtual particles in 
the Schwarzschild geometry around a gravitationally 
collapsing matter, we show that the vacuum gains tremendous energy from gravitational field
in the process of black hole's formation. The reason that this energy has to be released is 
discussed, analogously with the Casimir effect\cite{casimir}.
 
The vacuum has a very rich physical content in the context of relativistic 
quantum field theories. It consists of extremely large number of virtual 
particles and anti-particles. The quantum-field fluctuations of the vacuum are 
creations and annihilations of these virtual particles and anti-particles. As a consequence, 
the vacuum energy (zero-point energy) does not vanish and is given by,
\begin{equation}
{\cal E}_o\!=\! {1\over2}\int {d^3 xd^3 k\over(2\pi)^3}\epsilon(|{\bf k}|),\hskip0.2cm
\epsilon(|{\bf k}|)\!=\!\sqrt{|{\bf k}|^2+m^2},
\label{eve}
\end{equation}
where $\epsilon(|{\bf k}|)$ is the energy-spectrum of virtual particles and the 
summation is over all possible states of quantum-field fluctuations in phase space. 

In the description of renormalizable and perturbative quantum field theories, 
the vacuum energy (\ref{eve}) is dropped and set to be zero by the normal 
ordering of creation and annihilation 
operators, for the reason that the absolute value of the physical energy can be 
determined only up 
to a constant. Quantum-field fluctuations of the vacuum
is treated by the renormalization of theories.

However, as shown by the Casimir effect that was experimentally 
evidenced\cite{exp}, the vacuum-energy (\ref{eve}) of virtual photons
is not just a trivial constant.
The Casimir effect can be physically understood as the following: the continuous 
energy-spectrum (\ref{eve}) of virtual photons is modified by boundary 
conditions to be discrete one, the ``new'' vacuum energy 
corresponding to the modified 
energy-spectrum is smaller than the ``old'' vacuum energy 
corresponding to the continuous energy-spectrum (\ref{eve}). The vacuum
gains the Casimir energy ${\cal E}_c=-{\pi^2\over720 a^3}$, where $a$ is 
a distance separating two plates.
The vacuum becomes energetically unstable and has to quantum-mechanically 
fluctuate (decay) from the ``old'' vacuum state to the 
``new'' vacuum state. This leads to releasing the Casimir energy, as a result, 
an attractive force between two plates is observed as the Casimir effect.  

Analogously, as recently shown in ref.\cite{xuem}, an external magnetic field modifies 
continuous and non-degenerate negative-energy spectrum of virtual fermions to a discrete and 
degenerate spectrum (Laudau levels), leading to vacuum decay and possibly releasing photons. 
Though such an effect is difficult to be measured in a ground Laboratory, it could
account for astrophysics events, for instance, the anomalous X-ray pulsar\cite{xray}.   

In this article, we attempt to study a similar effect in case of an external 
gravitational field, which modifies the energy-spectrum (\ref{eve}). Such an effect 
possibly provides a scenario of energetic source accounting for gamma 
ray bursts. In addition, a possible experimental test of such an effect in a 
ground laboratory is proposed.
   
\vskip0.2cm
\noindent{\it General formulation.}\hskip0.3cm
In order to clearly illustrate physics content,
we consider a complex scalar field $\phi$ and 
the simplest coordinate-invariant action is given by ($\hbar=c=G=1$, the 
Planck scale $\Lambda_p=1$)
\begin{equation}
S = {1\over2}\int
d^4x\sqrt{-g}\Big[g^{\mu\nu}\phi_{,\mu}\phi^*_{,\nu}+(m^2+\xi {\cal R})
\phi\phi^*\Big],
\label{action}
\end{equation}
where $m$ is particle mass and ${\cal R}$ the Riemann scalar. The effective action 
$S_{\rm eff}$ is defined as
\begin{equation}
S_{\rm eff}=-i\ln \langle 0|0\rangle,\hskip0.2cm\langle 0|0\rangle=
\int [{\cal D}\phi{\cal D}\phi^*]\exp(iS),
\label{action1}
\end{equation}
where $\langle 0|0\rangle$ is the transition amplitude from vacuum to vacuum in curved 
spacetime with geometry $g_{\mu\nu}(x)$. The averaged energy-momentum tensor 
$\langle T_{\mu\nu}\rangle$ of the vacuum is given by:
\begin{equation}
\langle T_{\mu\nu}(x)\rangle = 
{2\over\sqrt{-g}}{\delta S_{\rm eff}\over \delta
g^{\mu\nu}(x)}.
\label{e1t}
\end{equation}

The quantum scalar field $\phi$ can be in principle expressed in terms of
a complete and orthogonal basis of quantum-field states $u_k(x)$:
\begin{equation}
\phi(x)=\sum_k\Big(a_k u_k(x)+
a^\dagger_ku^*_k(x)\Big),\hskip0.1cm \left [a_k,a^\dagger_{k'}\right ]=\delta_{k,k'}
\label{de}
\end{equation}
where $a^\dagger_k$ and $a_k$ are creation and annihilation operators
of the $k$-th quantum-field state. The quantum-field states $\{u_k(x)\}$ obey the 
following equation,
\begin{equation}
(\Delta_x + m^2+\xi {\cal R})u_k(x)=0,
\label{eq}
\end{equation}
where $\Delta_x$ is the Laplacian operator in curved spacetime.  
Based on this basis $\{u_k(x)\}$, we can compute the transition amplitude from vacuum
to vacuum as
\begin{eqnarray}
\langle 0|0\rangle\! &\!=\!&\!{\det}^{- 1}\left({\cal M}\right),\label{fb}\\
{\cal M}_{k,k'}\! &\!=\!&\!\int d^4x\sqrt{-g}u_k(x)(\Delta_x + m^2+\xi {\cal R})u^{*}_{k'}(x).
\nonumber
\end{eqnarray}
Diagonizing the hermitian matrix ${\cal M}$, we obtain the effective action (\ref{action1}):
\begin{equation}
iS_{\rm eff}=\int\sqrt{-g} {d^4xd^4k\over (2\pi)^4}
\ln(\lambda^2_k),
\label{zr}
\end{equation}
where $\lambda^2_k$ denotes the $k$-th eigen-value of the
matrix ${\cal M}$. The vacuum-energy density 
$\langle T_{\circ\circ}\rangle$ (\ref{e1t}) is given by,
\begin{equation}
\langle T_{\circ\circ}\rangle \simeq g_{\circ\circ}(x)\int{d^4k\over (2\pi)^4}
\ln (\lambda^2_k),
\label{e1'}
\end{equation}
where we approximately neglect the functional variation $\delta 
g_{\mu\nu}(x)$ of eigen-values $\lambda^2_k$ in the 
logarithmic function. The vacuum energy ${\cal E}$ is given by\cite{book} 
\begin{equation}
{\cal E}=\sqrt{h}d^3x\langle  T^\circ_\circ\rangle,
\label{energy0}
\end{equation}
where $h_{ij}$ is the matrix in the 3-dimensional space.

\vskip0.2cm
\noindent{\it Energy-spectrum of the vacuum.}\hskip0.3cm
As a preliminary study, we assume the geometry of spacetime outside of the collapsing 
mass $M$
($r>2M$) is stationary and spherical, e.g., the Schwarzschild geometry,
\begin{equation}
ds^2\!=\!-\!g(r)dt^2\!+\!g^{\!-\!1}(r)dr^2\!+\!r^2d\Omega,\hskip0.1cm
g(r)\!\equiv\! (1\!-\!{2M\over r})
\label{sg}
\end{equation}
where $\Omega$ is the spherical solid angle and $r,\theta,\phi,t$
are the Schwarzschild coordinates. The Riemann scalar ${\cal R}=0$. 

In the asymptotically flat space, $2M/r\rightarrow 0 $ and $g(r)\rightarrow 1$,
$u_k(x)\sim h_l((k_r^\circ r)Y_{lm}(\theta, \phi)e^{i\omega_\circ t}$, where 
$Y_{lm}(\theta, \phi)$
and $h_l((k_r^\circ r)$ are standard spherical harmonic and Hankel functions. 
$\omega_\circ$ is the
energy-spectrum and the radial momentum $k_r^\circ$ is the eigenvalue of the 
operator given by
\begin{equation}
\hat k_r^\circ={1\over ir}{\partial\over\partial r}r,\hskip0.3cm (\hat k_r^\circ)^2
=-\left({\partial^2\over\partial r^2}+{2\over r}{\partial\over\partial r}\right).
\label{pr}
\end{equation}
On the basis of Hankel functions $h_l(k_rr)$, the energy-spectrum of the 
vacuum is given by
\begin{equation}
\omega^2_\circ=\left((k_r^\circ)^2+{l(l+1)\over r^2}+m^2\right).
\label{eigen0}
\end{equation}

While, for $2M/r \not=0$,
$u_k(x)\sim R_{l\omega}(r)Y_{lm}(\theta,\phi)e^{i\omega t}$ and
$\omega$ is the energy-spectrum. The radial function $R_{l\omega}(r)$ obeys
the Regge and Wheeler equation,  
\begin{equation}
\left[\omega^2-\hat k_r^2+i{2M\over r^2}\hat k_r-V_l(r)\right]R_{l\omega}(r)=0,
\label{eq2}
\end{equation}
where $\hat k_r=g(r)\hat k_r^\circ$, $\hat k_r^2=g^2(r)(\hat k_r^\circ)^2$ and
\begin{equation}
V_l(r)=g(r)\left[{l(l+1)\over r^2}+{2M\over r^3}+m^2\right].
\label{p}
\end{equation}
$R_{l\omega}(r)$ and $h_l(k_r^\circ r)$
obey Eq.(\ref{eq2}) respectively for $2M/r \not=0$ and $2M/ r\rightarrow 0$.
Both $R_{l\omega}(r)$ and $h_l(k_r^\circ r)$ satisfy the same boundary conditions at
$r\rightarrow\infty$. Since $h_l(k_r^\circ r)$ is a complete set of orthogonal functions,  
we can express $R_{l\omega}(r)$ in terms
of $h_l(k_r^\circ r)$  
\begin{equation}
R_{l \omega}(r)\!=\!\int {dk_r^\circ\over2\pi} \big[b_{l \omega}(k_r^\circ) 
h_l(k_r^\circ r)
\!+\!{\rm h.c.}\big].
\label{rj}
\end{equation}
For simplicity, 
we discuss only s-wave case ($l=0$). Substituting Eq.(\ref{rj}) into Eq.(\ref{eq2}), for
$b_{\circ \omega}(k_r^\circ) \not=0$ we have,
\begin{equation}
\omega^2=k_r^2-{2Mi\over r^2}k_r+V(r),
\label{sm2'}
\end{equation}
where $k_r= g(r) k_r^\circ$, $ k_r^2= g^2(r) (k_r^\circ)^2$ and $V(r)$ is 
Eq.(\ref{p}) without the angular term $l(l+1)/r^2$.
The eigen-value $\lambda^2_k$ in Eq.(\ref{zr}) is then given by,
\begin{equation}
\lambda^2_k=\omega^2-\left(k_r^2-{2Mi\over r^2}k_r+V(r)\right).
\label{eigen}
\end{equation}
As discussed in the previous paper 
\cite{xuec}, the imaginary part in Eq.(\ref{sm2'}) gives rise to particle
creations of the Hawking type and will be thus disregarded in the following discussions.

We find that the energy-spectrum (\ref{sm2'}) of the vacuum is modified by gravitational field, 
in contrast with the energy-spectrum (\ref{eigen0}) of the vacuum in the absence 
of gravitational field. This difference is originated from 
gravitational field interacting with energy-momentum ($\omega,k_r$) of virtual particles
in the vacuum.

Virtual particles in the vacuum are not in mass-shell. The variations of their 
energy $(\bar\omega)$ and 
momentum $(\bar k_r)$ are described by the Heisenberg uncertainty relationships:
\begin{equation}
\Delta \bar t\Delta\bar\omega\simeq 1,\hskip0.3cm  \Delta \bar r\Delta\bar k_r\simeq 1,
\label{uncert1}
\end{equation}
in a local inertial coordinate system $(\bar r,\bar t)$. 
The equivalence 
principle tells us that the Heisenberg uncertainty relationships are unaffected by the 
presence of a gravitational field:
\begin{equation} 
\Delta t\Delta \omega\simeq 1,\hskip0.3cm  \Delta r\Delta k_r\simeq 1.
\label{uncert2}
\end{equation}  
Eqs.(\ref{uncert1},\ref{uncert2}) and the gravitational time dilation between 
local inertial and arbitrary 
coordinate systems $\Delta \bar t = g^{1\over2}(r)\Delta t$ lead us to obtain:
\begin{equation}
\Delta \omega= g^{1\over2}(r)\Delta \omega_\circ,\hskip0.3cm \omega=g^{1\over2}(r)\omega_\circ.
\label{shift}
\end{equation}
This indicates that energy-levels $\omega$ and its width 
$\Delta \omega$ of virtual particles is gravitationally red-shifted from corresponding energy-levels $\omega_\circ$ 
and its width $\Delta \omega_\circ$ of virtual particles in the absence of
gravitational field.

\vskip0.2cm
\noindent{\it Vacuum-energy density.}\hskip0.3cm
The vacuum-energy density Eq.(\ref{e1'}) can be expressed as
\begin{equation}
\langle T_{\circ\circ}\rangle={1\over 4\pi r^2}\int{d\omega dk_r\over (2\pi)^2}
\int_0^\infty {ds\over s}e^{is(\lambda^2_k+i\bar\epsilon)},
\label{zr1}
\end{equation}
where logarithmic function in Eq.(\ref{e1'}) is represented by an $s$-integration and
infrared convergence at $s\rightarrow 0$ is insured by $i\bar\epsilon$ prescription 
($\bar\epsilon\rightarrow 0$). Performing the Wick rotation $\omega\rightarrow i\omega$ and 
integrating $\omega$ over $[0,\infty]$, we obtain,
\begin{equation}
\langle T_{\circ\circ}\rangle = {1\over 16\pi r^2\sqrt{i\pi}}\int{dk_r\over 
(2\pi)}\int_0^\infty
{ds\over s^{3\over2}}
e^{-is(\alpha + i\bar\epsilon)},
\label{seff}
\end{equation}
where $\alpha=(k_r^2+V(r))$.

In order to compute the integration over ``$s$'' in Eq.(\ref{seff}), we introduce a complex
variable $z=-{1\over2}+\delta$ ($|\delta| \rightarrow 0 $) and use the following integral
representation of the $\Gamma(z)$-function by an analytical continuation 
for ${\rm Im}(\alpha) <0$:
\begin{equation}
\int_0^\infty e^{-i\alpha s}s^{z-1}ds=(i\alpha)^{-z}\Gamma(z).
\label{int}
\end{equation}
In the neighborhood of singularity, where $|\delta| \rightarrow 0$ and 
$z\rightarrow -{1\over2}$, we have 
\begin{equation}
\Gamma(z)=-2\sqrt{\pi},\hskip0.2cm
\alpha^{-z}=\sqrt{k_r^2+V(r)}
\label{ana}
\end{equation}
As a result, we cast the vacuum-energy density Eq.(\ref{seff}) to be:
\begin{equation}
\langle T_{\circ\circ}\rangle ={1\over2}{1\over 4\pi r^2}\int{dk_r\over (2\pi)}\sqrt{k_r^2+V(r)},
\label{real2}
\end{equation}
as expected. Comparing the radial momentum $k^\circ_r$ in the range 
$[-\Lambda_p,\Lambda_p]$, we neglect 
$V(r)\ll\Lambda_p$ and approximately obtain the leading contribution,
\begin{equation}
\langle  T_{\circ\circ}\rangle \simeq g^2(r){1\over (4\pi)}{\Lambda_p^2\over 4\pi r^2}.
\label{tl}
\end{equation}
The radial momentum should be quantized as $k^\circ_r\!\sim\! {n\over r}$\cite{xuec} due to 
gravitational potential well $V(r)$. However, we do not consider this, since the quanta 
$\delta k^\circ_r\sim {1\over r}$ is very small and spectrum is almost continuous.

\vskip0.2cm
\noindent{\it Energy gain.}\hskip0.3cm
Using Eqs.(\ref{energy0}) and (\ref{tl}), we define the surface vacuum-energy ${\cal E}_s$ 
at radius $r$,
\begin{equation}
{\cal E}_s=4\pi r^2drg^{-{3\over2}}(r)\langle  T_{\circ\circ}\rangle=g^{1\over2}(r){\cal E}^\circ_s,
\label{senergy}
\end{equation}
for $dr=\Lambda_p^{-1}$ and ${\cal E}^\circ_s\simeq {\Lambda_p\over4\pi}$, which
is the corresponding surface vacuum-energy in the absence of gravitational field
$(M=0)$. Eq.(\ref{senergy}) shows that the surface vacuum-energy in the presence of gravitational field is red-shifted, compared with its counterpart in the absence of
gravitational field. This coincides with gravitational red-shift of energy-spectrum
Eq.(\ref{shift}) of the vacuum. For $r\gg 2M$, we approximately have
\begin{equation}
{\cal E}_s\simeq {\cal E}^\circ_s + \Delta {\cal E}_s,\hskip0.5cm 
\Delta {\cal E}_s=-{M{\cal E}^\circ_s\over r},
\label{de0}
\end{equation}
which indicates that the vacuum gains the energy 
$\Delta {\cal E}_s$ from gravitational field. 
 
This phenomenon of the vacuum gaining the gravitational energy could be experimentally
verified by measuring the Casimir energy ${\cal E}_c=-{\pi^2\over720 a^3}$ 
at different altitude $r_2$ and $r_1$ above the Earth. Analogous to Eq.(\ref{senergy}), 
the Casimir energy ${\cal E}_c$ should be modified by the gravitation field 
of the Earth in the following way:
\begin{equation}
|{\cal E}_c(r_2)|\!=\!\left({g(r_2)\over g(r_1)}\right)^{1\over2}|{\cal E}_c(r_1)|, 
\hskip0.2cm 
g(r)\!=\!1\!-\!{2M_\oplus\over r},
\label{fermi}
\end{equation}
where $M_\oplus$ is the mass of the Earth. This implies that 
$|{\cal E}_c(r_2)|>|{\cal E}_c(r_1)|$ for $r_2>r_1$. 
Given $r_1=r_\oplus$, 
${M_\oplus\over r_\oplus}\simeq 7.1\cdot 10^{-10}$ and 
$\Delta r=r_2-r_1=10^6$cm, we obtain
\begin{equation} 
|{\cal E}_c(r_2)|\simeq (1+O(10^{-12}))
|{\cal E}_c(r_\oplus)|.
\label{exp}
\end{equation}
Test of this very small energy-gain, modifying the Casimir energy, 
seems to be very difficult for current experiments.

\vskip0.2cm
\noindent{\it Energy release.}\hskip0.3cm
Such an energy gain (\ref{de0}) could be enormous, in a gravitational collapse 
approaching to the formation of black hole's horizon.
We make a simple model that a massive star of mass $M$ and radius $R$ undergoes 
a spherical collapse and its mass-density $\rho=M/{4\pi\over3}R^3$ is uniform. 
Considering that in the collapsing process, the surface of the star moves 
inward from $R+dR$ to $R$, we have the surface vacuum-energy ${\cal E}_s$ 
in the surface-area $4\pi R^2$ changes from
\begin{equation}
{\cal E}'_s\!=\!(1\!-\!{2M'\over R})^{1\over2}{\cal E}_s^\circ
\hskip0.5cm {\rm to}\hskip0.5cm
{\cal E}_s\!=\!(1\!-\!{2M\over R})^{1\over2}{\cal E}_s^\circ, 
\label{deltae2}
\end{equation}
where $M'=M-\delta M$ and 
\begin{equation}
\delta M=M(1-{R^3\over (R+d R)^3}),
\label{deltam}
\end{equation}
which is the mass distributed in the spatial shell $R\rightarrow R+dR$ and falling into the 
spherical zone $r<R$. ${\cal E}'_s>{\cal E}_s$, for $M'<M$.
The vacuum-energy gain from gravitational field is 
\begin{equation}
d {\cal E}={\cal E}_s - {\cal E}'_s<0. 
\label{deltae3}
\end{equation}

We approximately adopt a simplified model for the gravitational collapse, in which 
the dynamical equation describing the process of spherically collapsing 
is given by\cite{scoll},
\begin{equation}
dt=-{2Mh(R)\over g(R)\sqrt{h^2(R)-g(R)}}dR,\hskip0.2cm h(R)=1-{2M\over4R}.
\label{coll}
\end{equation}
By using Eqs.(\ref{deltae3}) and (\ref{coll}), we compute the rate of energy gain 
$d {\cal E}/d t$ in the spatial shell $4\pi R^2dR$ that the surface of the collapsing 
star sweeps in the time interval $dt$. Given the initial condition that at the moment 
$t_\circ=0$ of starting the collapsing process, the radial size of the collapsing star $R_\circ=100(2M)$ and star's mass $M=10M_\odot$, we compute the rate of vacuum-energy 
gain $d {\cal E}/d t$, plotted in Fig.(\ref{rate}) as a function of $R$ in the unit of $2M$. 
The result shows that the rate $d {\cal E}/d t$ rapidly increases to $10^{57}$erg/sec, 
as the surface $R(t)$ of the collapsing star 
moves, almost in the speed of light, inward to the horizon. Whereas, in the vicinity of the 
horizon, the collapsing process becomes slow and the rate $d {\cal E}/d t$ decreases.

Due to this vacuum-energy gain $d{\cal E}$ (\ref{deltae3}), 
vacuum states become energetically unstable, have to spontaneously undergo a quantum
transition to lower energy states by quantum-field fluctuations. 
This is exactly analogous to the phenomenon of the Casimir effect described in the 
introduction section. As a consequence, the vacuum-energy $d {\cal E}$ (\ref{deltae3}) 
gained from gravitational field must be released and deposited in the region 
from $r=2M$ extending to $r=R_\circ$.

Which quantum transition makes this vacuum-energy $d {\cal E}$ be released. 
One of possibilities is spontaneous photon emissions, analogous to the
spontaneous photon emission taking place in the atomic physics. Such a spontaneous 
photon emission is induced by the four-photon interacting vertex in the QED. 
The rate of the quantum transition must be very fast, $\sim\alpha^4\tilde\omega$ for high-energy
$\tilde\omega\sim m_e$. These will be discussed in a 
future work. As shown in Fig.(\ref{rate}), 
the spatial density of energy release can be very large, as the collapsing process 
approaching to the formation of black hole's horizon $R=2M$, so that energy of photons spontaneously emitted can be larger than the threshold of producing electron and 
positron pairs. These pairs, on the other hand, annihilate into
two photons. As a consequence, a dense and energetic plasma of photons, electron and 
position pairs, called ``{\it dyadosphere}'' \cite{prx}, could be formed. 
Using equation (\ref{coll}) and the rate 
of vacuum-energy release (Fig.(\ref{rate})), we can obtain that
total amount energy:
\begin{equation}
E_{\rm total}=\int_{2M}^{R_\circ}d {\cal E} \lesssim {3\over 2\pi} M\simeq 8.6 
10^{54}{\rm erg},
\label{rdem}
\end{equation}
is released in a very short time, about $0.1$ second for the collapsing process 
from $R=R_\circ$ to $R=2M$. These qualitatively agree to the characteristic of 
energetic sources for gamma ray bursts.    
    
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\begin{figure}
\includegraphics[width=\hsize]{power}
%\vspace{5cm}
\caption{The rate of energy release $d {\cal E}/d t$ (erg/sec) as a function of the radius $R$
in unit of $2M$.} 
\label{rate}
\end{figure}

\newpage
%\begin{figure}
%\vspace{5cm}
%\caption{The spatial density of energy release (erg/cm$^3$) as a function of radius $r$ (cm).}
%\label{density}
%\end{figure}
%\bibitem{grb} T.~Piran, Phys.~Rep.{\bf 314}, 575 (1999).
\end{document}
