RELATIVISTIC FLUIDS AND THE PHYSICS OF GRAVITATIONAL COLLAPSE
0909.3474.tex
\documentstyle[12pt]{article}
\begin{document}
\Large
\title{RELATIVISTIC FLUIDS AND THE PHYSICS OF GRAVITATIONAL COLLAPSE}
\date{}
\maketitle
\section{Questions}
\begin{itemize}
\item Why is the study of gravitational collapse important for General Relativity?
\item Why is the gravitational collapse a highly dissipative process. 
\item Is there any life between quasi--equilibrium and non--equilibrium?
\item What happens if we relax  the Pascal principle? (and why?)
\item How does the electric charge and dissipation affects the evolution of massive stars?
\end{itemize}
\newpage
\section{Why is the gravitational collapse a highly dissipative process?}

\begin{equation}
E\approx-\frac{G M^2}{R}\approx -\frac{6.6 \times 10^{-8}g^{-1}cm^3 s^{-2}10^{66}g^2}{10^6 cm}
\end{equation}

$M=M_{\odot}\approx 2. 10^{33} g$;\quad \ $R\approx 10 Km$

\begin{equation}
E\approx -10^{53} erg.
\end{equation}

\begin{equation}
E_{in}\approx kT
\end{equation}

$k\approx 1.3 \times 10^{-16} erg. K^{-1}$

\begin{equation}
T\approx 10^{69} K
\end{equation}
\begin{equation}
L\approx \sigma T^4 R^2
\end{equation}

$\sigma \approx 5.6 \times 10^{-5}erg.cm^{-2}s^{-1}K^{-4}$
\begin{equation}
L\approx 10^{283}erg.s^{-1}
\end{equation}


\begin{equation}
t\approx (10)^{-230}s
\end{equation}
\newpage

\section{ Non--comoving (Bondi 1964).}


\begin{equation}
ds^2=e^{\nu} dt^2 - e^{\lambda} dr^2 -
r^2 \left( d\theta^2 + sin^2\theta d\phi^2 \right),
\label{metric}
\end{equation}

\begin{equation}
G^\nu_\mu=8\pi T^\nu_\mu,
\label{Efeq}
\end{equation}

\[ \left(\begin{array}{cccc} 
\rho + \epsilon    &  - q -\epsilon  &   0     &   0    \\
- q - \epsilon &  P_r + \epsilon    &   0     &   0    \\
0       &   0       & P_\bot  &   0    \\
0       &   0       &   0     &   P_\bot
\end{array} \right). \]

\begin{equation}
\frac{\rho + P_r \omega^2 }{1 - \omega^2} +\frac{2\omega
q}{1-\omega^2}+\frac{\epsilon(1+\omega)}{1-\omega}=-\frac{1}{8 \pi}\Biggl\{-\frac{1}{r^2}+e^{-\lambda}
\left(\frac{1}{r^2}-\frac{\lambda'}{r} \right)\Biggr\},
\label{fieq00}
\end{equation}

\begin{equation}
\frac{ P_r + \rho \omega^2}{1 - \omega^2} +
\frac{2\omega
q}{1-\omega^2}+\frac{\epsilon(1+\omega)}{1-\omega}=-\frac{1}{8 \pi}\Biggl\{\frac{1}{r^2} - e^{-\lambda}
\left(\frac{1}{r^2}+\frac{\nu'}{r}\right)\Biggr\},
\label{fieq11}
\end{equation}

\begin{eqnarray}
P_\bot = -\frac{1}{8 \pi}\Biggl\{\frac{e^{-\nu}}{4}\left(2\ddot\lambda+
\dot\lambda(\dot\lambda-\dot\nu)\right) \nonumber \\
 - \frac{e^{-\lambda}}{4}
\left(2\nu''+\nu'^2 -
\lambda'\nu' + 2\frac{\nu' - \lambda'}{r}\right)\Biggr\},
\label{fieq2233}
\end{eqnarray}

\begin{equation}
\frac{(\rho + P_r) \omega e^{\frac{\lambda +\nu}{2}}}{1 - \omega^2} +\frac{q e^{\frac{\lambda+\nu}{2}}}{1-\omega ^2}(1+\omega
^2)+\frac{e^{\frac{\lambda +\nu}{2}}
\epsilon(1+\omega)}{1-\omega}=-\frac{\dot\lambda}{8 \pi r}.
\label{fieq01}
\end{equation}



\begin{equation}
T^\mu_{\nu;\mu}=0.
\label{dTmn}
\end{equation}


\begin{equation}
\left(-8\pi T^1_1\right)'=\frac{16\pi}{r} \left(T^1_1-T^2_2\right)
+ 4\pi \nu' \left(T^1_1-T^0_0\right) +
\frac{e^{-\nu}}{r} \left(\ddot\lambda + \frac{\dot\lambda^2}{2}
- \frac{\dot\lambda \dot\nu}{2}\right),
\label{T1pn}
\end{equation}


\begin{equation}
P'_r=-\frac{m + 4 \pi P_r r^3}{r \left(r - 2m\right)}\left(\rho+P_r\right)+
\frac{2\left(P_\bot-P_r\right)}{r},
\label{Prp}
\end{equation}

\begin{equation}
m(r,t)=\frac{r}{2}(1-e^{-\lambda})= 4\pi \int_0^r{T_0^0 r^2 dr}
\label{mT00n}
\end{equation}


\subsection{Equilibrium $\omega=0$, and quasi--equilibrium $\omega^2 \approx 0$}


\begin{equation}
\ddot\lambda + \frac{\dot\lambda^2}{2} -
\frac{\dot\nu \dot\lambda}{2} = 8 \pi r e^{\nu}
\left[(P_r +\epsilon)'+ \left(\rho + P_r + 2 \epsilon\right) \frac{\nu'}{2} -
2 \frac{P_\bot - P_r - \epsilon}{r}\right].
\label{lsps}
\end{equation}

\begin{equation}
\ddot\lambda \approx \dot\nu \dot\lambda \approx
\dot\lambda^2 \approx 0,
\label{lp0}
\end{equation}

$$
\ddot\nu \approx 0.
%\label{nu2p}
$$


\begin{equation}
\dot\omega \approx O(\ddot\lambda, \dot\lambda \omega, \dot\nu \omega).
\label{Omo}
\end{equation}


\begin{equation}
O(\omega^2) = {\dot\lambda}^2 = {\dot\nu}^2 =
\dot\lambda \dot\nu = \ddot\lambda = \ddot\nu = 0,
\label{om2}
\end{equation}


\subsection{Effective variables and the post--quasi--static approximation: The life between quasi--equilibrium and non--equilibrium}


\begin{equation}
\tilde\rho=T^0_0= \frac{\rho + P_r \omega^2 }{1 - \omega^2} +
\frac{2 q\omega}{(1 - \omega^2)} + \epsilon\frac{(1+\omega)}{1-\omega},
\label{rhoeffec}
\end{equation}

\begin{equation}
\tilde P=-T^1_1=\frac{ P_r + \rho \omega^2}{1 - \omega^2} +
\frac{2 q \omega}{(1 - \omega^2)}+\epsilon\frac{(1+\omega)}{1-\omega}.
\label{peffec}
\end{equation}


\begin{equation}
m =\frac{r}{2}(1-e^{-\lambda})= 4 \pi \int^{r}_{0}{r^2 \tilde\rho dr}, \\
\label{m1}
\end{equation}

\begin{equation}
\nu = \nu_\Sigma + \int^{r}_{r_\Sigma}\frac{2 (4 \pi r^3  \tilde
P+m)}{r(r-2m)} dr.
\label{nu}
\end{equation}


\subsection{The algorithm for modelling spheres out of equilibrium}


\begin{enumerate}
\item  Take an interior (seed) solution to Einstein equations, representing a fluid
distribution of matter in equilibrium, with a given

$$\rho_{st}=\rho(r)\,\qquad\, P_{r\, st}= P_{r}(r)$$

\item  Assume that the $r$--dependence of $\tilde P$ and $\tilde\rho$ is the
same as that of $P_{r\, st}$ and $\rho_{st}$, respectively.

\item  Using equations (\ref{nu}) and (\ref{m1}), with the $r$ dependence of
$\tilde P$ and $\tilde\rho$, one gets $m$ and $\nu$ up to some functions of
$t$, which will be specified below.

\item  For these functions of $t$ one has three ordinary differential equations
(hereafter referred to as surface equations), namely:
\begin{enumerate}
\item  $\omega=\dot r_\Sigma\,e^{(\lambda-\nu)/2}$ evaluated  on $r=r_{\Sigma}$.

\item  $T^\mu_{r;\mu}=0$ evaluated on $r=r_{\Sigma}$.

\item The equation relating the total mass loss rate with the energy flux
through the boundary surface.
\end{enumerate}
\item Depending on the kind of matter under consideration, the system of
surface equations described above may be closed with the additional
information provided by the transport equation
and/or the equation of state for the anisotropic pressure and/or
additional information about some of the physical variables evaluated on
the boundary surface (e.g. the
luminosity).

\item Once the system of surface equations is closed, it may be integrated for
any particular initial data.

\item  Feeding back the result of integration in the expressions for $m$ and
$\nu$, these two functions are completely determined.

\item  With the input from the point 7 above, and using field equations,
together with the equations of state and/or transport equation, all
physical variables may be found for any piece of
matter distribution.
\end{enumerate}
\newpage

\section{Relaxing the Pascal principle? (and why?)}
\begin{equation}
P'_r=-\frac{m + 4 \pi P_r r^3}{r \left(r - 2m\right)}\left(\rho+P_r\right)+
\frac{2\left(P_\bot-P_r\right)}{r},
\label{Prp}
\end{equation}

$\frac{P_r+P_\bot}{2}$
\begin{itemize}
\item What might be the origin of local anisotropy?
\end{itemize}
\begin{enumerate}
\item ``Exotic'' phase transition (e. g. pion condensate).
\item\ Magnetic fields
\item  Type II superconductor
\item Type P superfluid
\item Boson stars
\item Viscosity
\item Anisotropic velocity distributions
\item Two fluid systems
\end{enumerate}
\begin{itemize}
\item How does the properties of the locally anisotropic system differs from the locally isotropic one?
\end{itemize}
\begin{enumerate}
\item Cracking induced by perturbations of local isotropy of pressure.$\frac{\left(P_\bot-P_r\right)}{P_r}<<1$
\item Changes in the total mass allowed for a compact object
\end{enumerate}

\section{Comoving (Misner and Sharp 1964).}

\begin{equation}
ds^2_-=-A^2dt^2+B^2dr^2+(Cr)^2(d\theta^2+\sin^2\theta d\phi^2),
\label{1}
\end{equation}

\begin{eqnarray}
T_{\alpha\beta}^-&=&(\mu +
P_{\perp})V_{\alpha}V_{\beta}+P_{\perp}g_{\alpha\beta}+(P_r-P_{\perp})\chi_{
\alpha}\chi_{\beta}\nonumber \\
&+&q_{\alpha}V_{\beta}+V_{\alpha}q_{\beta}
+\epsilon l_{\alpha}l_{\beta}-2\eta\sigma_{\alpha\beta}, \label{3}
\end{eqnarray}

\begin{eqnarray}
&&V^{\alpha}V_{\alpha}=-1, \;\; V^{\alpha}q_{\alpha}=0, \;\;
\chi^{\alpha}\chi_{\alpha}=1, \;\; \nonumber \\
&&\chi^{\alpha} V_{\alpha}=0,\quad
\;\; l^{\alpha} V_{\alpha}=-1, \;\; l^{\alpha}l_{\alpha}=0, \label{4}
\end{eqnarray}

\begin{equation}
\sigma_{\alpha\beta}=V_{(\alpha
;\beta)}+a_{(\alpha}V_{\beta)}-\frac{1}{3}\Theta(g_{\alpha\beta}+V_{\alpha}V
_{\beta}),
\label{4a}
\end{equation}

\begin{equation}
a_{\alpha}=V_{\alpha ;\beta}V^{\beta}, \;\;
\Theta={V^{\alpha}}_{;\alpha}. \label{4b}
\end{equation}

\begin{eqnarray}
V^{\alpha}=A^{-1}\delta_0^{\alpha}, \;\;
&&q^{\alpha}=qB^{-1}\delta^{\alpha}_1, \;\;\nonumber \\
l^{\alpha}=A^{-1}\delta^{\alpha}_0+B^{-1}\delta^{\alpha}_1, \;\;
&&\chi^{\alpha}=B^{-1}\delta^{\alpha}_1, \label{5}
\end{eqnarray}

\begin{equation}
\sigma_{11}=\frac{2}{3}B^2\sigma, \;\;
\sigma_{22}=\frac{\sigma_{33}}{\sin^2\theta}=-\frac{1}{3}(Cr)^2\sigma,
 \label{5a}
\end{equation}

\begin{equation}
\sigma=\frac{1}{A}\left(\frac{\dot{B}}{B}-\frac{\dot{C}}{C}\right),\label{5b1}
\end{equation}

\begin{equation}
\sigma_{\alpha\beta}\sigma^{\alpha\beta}=\frac{2}{3}\sigma^2.
\label{5b}
\end{equation}

\begin{equation}
a_1=\frac{A^{\prime}}{A}, \;\;
\Theta=\frac{1}{A}\left(\frac{\dot{B}}{B}+2\frac{\dot{C}}{C}\right),
\label{5c}
\end{equation}


\subsection{The electromagnetic energy tensor and the Maxwell equations}
\begin{equation}
E_{\alpha\beta}^-=\frac{1}{4\pi}\left({F_{\alpha}}^{\gamma}F_{\beta\gamma}
-\frac{1}{4}F^{\gamma\delta}F_{\gamma\delta}g_{\alpha\beta}\right),
\label{6}
\end{equation}

\begin{eqnarray}
F_{\alpha\beta}=\phi_{\beta,\alpha}-\phi_{\alpha,\beta}, \label{7}
\\
{F^{\alpha\beta}}_{;\beta}=4\pi J^{\alpha}, \label{8}
\end{eqnarray}
\begin{equation}
\phi_{\alpha}=\Phi\delta_{\alpha}^0, \;\; J^{\alpha}=\varsigma
V^{\alpha}, \label{9}
\end{equation}

\begin{equation}
s(r)=4\pi\int^r_0\varsigma B(Cr)^2dr, \label{13}
\end{equation}

\begin{eqnarray}
\Phi^{\prime\prime}-\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}-2\frac{C
^{\prime}}{C}-\frac{2} {r}\right)\Phi^{\prime}=4\pi\varsigma
AB^2, \label{10} \\
\dot \Phi^{\prime}-\left(\frac{\dot{A}
}{A}+\frac{\dot{B}}{B}-2\frac{\dot{C}}{C}\right)\Phi^{\prime}=0.
\label{11}
\end{eqnarray}
\begin{equation}
\Phi^{\prime}=\frac{sAB}{(Cr)^2}. \label{12}
\end{equation}

\subsection{The Einstein equations}

\begin{equation}
G_{\alpha\beta}^-=8\pi(T_{\alpha\beta}^-+E_{\alpha\beta}^-).
\label{2}
\end{equation}

\begin{eqnarray}
&&8\pi(T_{00}^-+E_{00}^-)=8\pi(\mu+\epsilon)A^2+\frac{(sA)^2}{(Cr)^4}
\nonumber \\
&&=\left(2\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)\frac{\dot{C}}{C}
 +\left(\frac{A}{B}\right)^2\left\{-2\frac{C^{\prime\prime}}{C}
 +\left(2\frac{B^{\prime}}{B}
 -\frac{C^{\prime}}{C}\right)\frac{C^{\prime}}{C} \right.\nonumber\\
&&\left.+\frac{2}{r}\left(\frac{B^{\prime}}{B}-3\frac{C^{\prime}}{C}\right)
  -\left[1-\left(\frac{B}{C}\right)^2\right]\frac{1}{r^2}\right\},
  \label{14} 
  \end{eqnarray}
  \begin{eqnarray}
&&8\pi(T_{01}^-+E_{01}^-)=-8\pi(q+\epsilon)AB\nonumber\\
&&=-2\left(\frac{\dot{C}^{\prime}}{C}-\frac{\dot{B}}{B}\frac{C^{\prime}}{C}
 -\frac{\dot{C}}{C}\frac{A^{\prime}}{A}\right)
 +\frac{2}{r}\left(\frac{\dot{B}}{B}-\frac{\dot{C}}{C}\right),
 \label{15} 
 \end{eqnarray}
 \begin{eqnarray}
&&8\pi(T_{11}^-+E_{11}^-)=
 8\pi\left(P_r+\epsilon-\frac{4}{3}\eta\sigma\right)B^2
 -\frac{(sB)^2}{(Cr)^4}\nonumber \\
&&=-\left(\frac{B}{A}\right)^2\left[2\frac{\ddot{C}}{C}
 +\left(\frac{\dot{C}}{C}\right)^2
 -2\frac{\dot{A}}{A}\frac{\dot{C}}{C}\right] \label{16b}\nonumber \\
&&+\left(\frac{C^{\prime}}{C}\right)^2
 +2\frac{A^{\prime}}{A}\frac{C^{\prime}}{C}
 +\frac{2}{r}\left(\frac{A^{\prime}}{A}
 +\frac{C^{\prime}}{C}\right)
 +\left[1-\left(\frac{B}{C}\right)^2\right]\frac{1}{r^2},\nonumber \\
  \end{eqnarray}
 \begin{eqnarray}
&&8\pi(T_{22}^-+E_{22}^-)=\frac{8\pi}{\sin^2\theta}(T_{33}^-+E_{33}^-)
 \nonumber \\
&&=8\pi\left(P_{\perp}+\frac{2}{3}\eta\sigma\right)(Cr)^2
 +\left(\frac{s}{Cr}\right)^2 \nonumber \\
&&=-\left(\frac{Cr}{A}\right)^2\left[\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}
 -\frac{\dot{A}}{A}\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)
 +\frac{\dot{B}}{B}\frac{\dot{C}}{C}\right]\nonumber\\
&&+\left(\frac{Cr}{B}\right)^2\left[\frac{A^{\prime\prime}}{A}
 +\frac{C^{\prime\prime}}{C}-\frac{A^{\prime}}{A}\left(\frac{B^{\prime}}{B}
 -\frac{C^{\prime}}{C}\right)-\frac{B^{\prime}}{B}\frac{C^{\prime}}{C}\right.
 \nonumber \\
&& \left.+\frac{1}{r}\left(\frac{A^{\prime}}{A}-\frac{B^{\prime}}{B}
 +2\frac{C^{\prime}}{C}\right)\right].\label{17}
\end{eqnarray}

\begin{equation}
m=\frac{(Cr)^3}{2}{R_{23}}^{23}+\frac{s^2}{2Cr}
=\frac{Cr}{2}\left\{\left(\frac{r\dot{C}}{A}\right)^2
-\left[\frac{(Cr)^{\prime}}{B}\right]^2+1\right\}+\frac{s^2}{2Cr},
 \label{18}
\end{equation}

\subsection{The exterior spacetime and junction conditions}

\begin{equation}
ds^2=-\left[1-\frac{2M(v)}{r}+\frac{Q^2}{r^2}\right]dv^2-2drdv+r^2(d\theta^2
+r^2\sin\theta^2
d\phi^2) \label{1int}
\end{equation}

\begin{equation}
P_r-\frac{4\eta \sigma}{3}\stackrel{\Sigma}{=}q, \quad
m(t,r)\stackrel{\Sigma}{=}M(v), \quad s \stackrel{\Sigma}{=} Q\label{junction},
\end{equation}

\section{Dynamical equations}

The non trivial components of the Bianchi identities,
$(T^{-\alpha\beta}+E^{-\alpha\beta})_{;\beta}=0$, 
\begin{eqnarray}
&&(T^{-\alpha\beta}+E^{-\alpha\beta})_{;\beta}V_{\alpha}
 =-\frac{1}{A}(\dot{\mu}+\dot{\epsilon})
   -\left(\mu+P_r+2\epsilon-\frac{4}{3}\eta\sigma\right)
    \frac{\dot B}{AB}\nonumber\\
&&-2\left(\mu+P_{\perp}+\epsilon+\frac{2}{3}\eta\sigma\right)
  \frac{\dot C}{AC}\label{II18} \nonumber\\
&&-\frac{1}{B}(q+\epsilon)^{\prime}-2(q+\epsilon)\frac{(ACr)^{\prime}}{ABCr}
 \phantom{-\frac{1}{A}(\dot{\mu}+\dot{\epsilon})}=0,
  \end{eqnarray}
 \begin{eqnarray}
&&(T^{-\alpha\beta}+E^{-\alpha\beta})_{;\beta}\chi_{\alpha}
=\frac{1}{A}(\dot{q}+\dot{\epsilon})
+\frac{1}{B}\left(P_r+\epsilon-\frac{4}{3}\eta\sigma\right)^{\prime}
 \nonumber\\
&&+2(q+\epsilon)\frac{\dot B}{AB}+2(q +\epsilon)\frac{\dot C}{AC} \nonumber\\
&&
 +\left(\mu+P_r+2\epsilon-\frac{4}{3}\eta\sigma\right)\frac{A^{\prime}}{AB}
\label{III18}\nonumber\\
&&+2(P_r-P_{\perp}+\epsilon-2\eta\sigma)\frac{(Cr)^{\prime}}{BCr}
 -\frac{ss^{\prime}}{4\pi B(Cr)^4}=0,  
\end{eqnarray}



\begin{equation}
D_T=\frac{1}{A}\frac{\partial}{\partial t}, \label{16}
\end{equation}
\begin{equation}
D_R=\frac{1}{R^{\prime}}\frac{\partial}{\partial r}, \label{23a}
\end{equation}

\begin{equation}
R=Cr, \label{23aa}
\end{equation}

\begin{equation}
U=rD_TC=D_TR<0 \;\; \mbox{(in the case of collapse)}. \label{19}
\end{equation}

\begin{equation}
E \equiv \frac{(Cr)^{\prime}}{B}=\left[1+U^2-\frac{2m(t,r)}{Cr}
+\left(\frac{s}{Cr}\right)^2\right]^{1/2}.
\label{20}
\end{equation}


\begin{eqnarray}
D_Tm=-4\pi\left[\left
(P_r+\epsilon-\frac{4}{3}\eta\sigma\right)U+(q+\epsilon)E\right]R^2,
\label{22}
\end{eqnarray}

\begin{eqnarray}
D_Rm=4\pi\left[\mu+\epsilon+(q+\epsilon)\frac{U}{E}\right]R^2+\frac{s}{R}D_Rs.
\label{27}
\end{eqnarray}

\begin{equation}
m=\int^{R}_{0}4\pi R^2 \left[\mu +
\epsilon+(q+\epsilon)\frac{U}{E}\right]dR+\frac{s^2}{2R}+\frac{1}{2}\int^{R
}_{0}\frac{s^2}{R^2}dR
\label{27int}
\end{equation}
(assuming a regular centre to the distribution, so $m(0)=0$).

\begin{equation}
D_TU=-\frac{m}{R^2}-4\pi\left(P_r+\epsilon-\frac{4}{3}\eta\sigma\right)R
+\frac{s^2}{R^3}+\frac{EA^{\prime}}{AB}, \label{28}
\end{equation}

\newpage
\begin{eqnarray}
\left(\mu+P_r+2\epsilon-\frac{4}{3}\eta\sigma\right) D_TU = \nonumber\\
-\left(\mu+P_r+2\epsilon-\frac{4}{3}\eta\sigma\right)
\left[m
+4\pi\left(P_r+\epsilon-\frac{4}{3}\eta\sigma\right)R^3
-\frac{s^2}{R}\right]\frac{1}{R^2}\nonumber \\
-E^2\left [D_R\left(P_r+\epsilon-\frac{4}{3}\eta\sigma\right)\right.
+2(P_r-P_{\perp}+\epsilon-2\eta\sigma)\frac{1}{R}\nonumber \\
\left.-\frac{s}{4\pi R^4}D_Rs\right]%\nonumber \\
-E\left[D_Tq+D_T\epsilon+4(q+\epsilon)\frac{U}{R}
 +2(q+\epsilon)\sigma\right],\nonumber \\
\label{eqd}
\end{eqnarray}
\begin{equation}
Force= Mass \; (density) \times Acceleration
\label{Newton}
\end{equation}

\begin{eqnarray}
\int^{R}_{0}4\pi R^2 \left[\mu +
\epsilon+(q+\epsilon)\frac{U}{E}\right]dR+4\pi(P_r+\epsilon-\frac{4}{3}\eta\sigma)R^3&-&\nonumber \\-\frac{s^2}{2R}+\frac{1}{2}\int^{R
}_{0}\frac{s^2}{R^2}dR
\label{27intnue}
\end{eqnarray}




\begin{equation}
\int^{R}_{0}\frac{s^2}{R^2}dR>\frac{s^2}{R}  \;\; \mbox{(increases active grav. mass term)}\label{conI}
\end{equation}
\begin{equation}
\int^{R}_{0}\frac{s^2}{R^2}dR<\frac{s^2}{R}  \;\; \mbox{(decreases active grav. mass term)}\label{conII}
\end{equation}
\begin{equation}
\int^{R}_{0}\frac{s^2}{R^2}dR=\frac{s^2}{R}  \;\; \mbox{(no regenerative effect of charge)}\label{conIII}
\end{equation}
\begin{equation}
s\sim R  \label{conIV}
\end{equation}

\section{The transport equation}
\subsection{ Maxwell-Fourier law and causality}

\begin{eqnarray}
\vec q(\vec x,t) &=& - \kappa \vec \nabla T(\vec x,t)  \\
 \nonumber \\ \nonumber 
\end{eqnarray}

\begin{equation}
{\rm d}e = \gamma {\rm d}T \;\qquad  {\rm and} \;\qquad  
\frac{{\rm d}e}{{\rm d}t} = -
\vec{\nabla} \cdot \vec q
\end{equation}

\begin{equation}
\overline{\underline{\left|\quad\frac{\partial T}{\partial t} = \frac{\kappa}{\gamma}
\nabla^2T\quad\right|}}
\end{equation}
\\

\begin{equation}
T \sim \frac{1}{\sqrt t} \exp{\left[- \frac{(x-x_0)^2}{t}\right]}
\end{equation}
\\
\\

$t=0$ : $T= \delta (x-x_0)$

$t=\tilde t>0$ : $T \not = 0$
\\
\\


\begin{equation}
\vec q(\vec x, t + \tau) = - \kappa \vec \nabla T(\vec x, t)
\end{equation}

\begin{equation}
\overline{\underline{\left|\quad \tau \frac{\partial \vec q}{\partial t} + \vec q =
- \kappa \vec \nabla T\quad\right|}} \qquad {\rm Cattaneo}
\end{equation}
\\

\begin{equation}
\overline{\underline{\left|\quad \frac{\kappa}{\tau \gamma} \nabla^2 T =
\frac{\partial^2 T}{\partial t^2 } + \frac{1}{\tau} \; \frac{\partial
T}{\partial t}\quad\right|}}
\end{equation}
\\

\begin{equation}
c = \sqrt{\frac{\kappa}{\tau \gamma}}
\end{equation}

\begin{equation}
\vec q(\vec x,t) = - \frac{\kappa}{\tau} \int_{-\infty}^{t}{\exp{\left[-
\frac{(t-t')}{\tau}\right]} \cdot \vec \nabla T(\vec x,t') dt'}
\end{equation}

\begin{equation}
\vec q(\vec x,t) = - \int^{t}_{-\infty}{Q(t-t') \vec \nabla T(\vec x,t') dt'}
\end{equation}

\begin{equation}
\begin{array}{cccc}
{\rm if}  & Q = K \delta(t-t')  &  \Longrightarrow   &  \vec q = - \kappa \vec \nabla
T
\quad {\rm (Fourier)}   
\\ 
{\rm if}  &  Q =  {\rm constant} &   \Longrightarrow   &   \frac{\partial^2 T}{\partial
t^2} =
\frac{\kappa}{\gamma}
\nabla^2 T   
\end{array} 
\end{equation}
\\

\begin{equation}
\left\{\begin{array}{cccc}
\tau  & \approx  &  10^{-11} {\rm sec.}   &  {\rm (phonon-elect.)}   \\ 
\tau  &  \approx &  10^{-13} {\rm sec.}   &  {\rm (phonon-phonon)}   \\
\tau  &  \approx &  10^{-3} {\rm sec.}   &  HeII \simeq 1.2^{\circ}K  \\
\tau  &  \approx &  10^{-1}-10^{-4} {\rm sec.} & {\rm neutron-star-matter}
\end{array}\right\} 
\end{equation}

\begin{equation}
\tau
h^{\alpha\beta}V^{\gamma}q_{\beta;\gamma}+q^{\alpha}=-\kappa h^{\alpha\beta}
(T_{,\beta}+Ta_{\beta}) -\frac 12\kappa T^2\left( \frac{\tau
V^\beta }{\kappa T^2}\right) _{;\beta }q^\alpha ,  \label{21t}
\end{equation}

\begin{eqnarray}
D_Tq&=&-\frac{\kappa T^2q}{2\tau}D_T\left(\frac{\tau}{\kappa T^2}\right)
     -q\left(\frac{3}{2}\frac{U}{R}
     +\frac{1}{2}\sigma+\frac{1}{\tau}\right)\nonumber \\
    && -\frac{\kappa E}{\tau}D_RT 
     -\frac{\kappa T}{\tau E}D_TU \nonumber\\
&&-\frac{\kappa T}{\tau E}
  \left[m+4\pi\left(P_r+\epsilon-\frac{4}{3}\eta \sigma\right)R^3
 -\frac{s^2}{R}\right]\frac{1}{R^2}.\nonumber\\
\label{V3t}
\end{eqnarray}
\newpage
\begin{eqnarray}
&&\left(\mu+P_r+2\epsilon-\frac{4}{3}\sigma\eta\right)(1-\alpha)D_TU
 \nonumber \\
&=&(1-\alpha)F_{grav}+F_{hyd}+\frac{\kappa E^2}{\tau}D_RT \nonumber \\
&+&E\left[\frac{\kappa T^2q}{2\tau}D_T\left(\frac{\tau}{\kappa
   T^2}\right)-D_T\epsilon\right]
\nonumber \\
&-&Eq\left(\frac{5}{2}\frac{U}{R}+\frac{3}{2}\sigma-\frac{1}{\tau}\right)
 -2E\epsilon\left(2\frac{U}{R}+\sigma\right),
\label{V4}
\end{eqnarray}

\begin{eqnarray}
F_{grav}&&=-\left(\mu+P_r+2\epsilon -\frac{4}{3}\eta
\sigma\right)\nonumber\\
&\times&
\left[m+4\pi\left(P_r+\epsilon-\frac{4}{3}\eta \sigma
\right)R^3-\frac{s^2}{R}\right]\frac{1}{R^2},
\label{grav}\\
F_{hyd}&=& -E^2 \left[D_R \left(P_r+\epsilon -\frac{4}{3}\eta
\sigma\right) \right. \nonumber\\
&+&\left.2(P_r-P_{\perp}+\epsilon-2\eta\sigma)\frac{1}{R}-\frac{s}{4\pi
R^4}D_Rs\right], \nonumber\\\label{hyd}
\end{eqnarray}

\begin{equation}
\alpha=\frac{\kappa
T}{\tau}\left(\mu+P_r+2\epsilon-\frac{4}{3}\sigma\eta\right)^{-1}.
\label{alpha}
\end{equation}

\begin{equation}
\alpha \approx \frac{[\kappa][T]}{[\tau]([\mu]+[P_r]+2[\epsilon]-\frac{4}{3}[\eta \sigma])}
\times 10^{-42}
\label{un}
\end{equation}
 $[\kappa]$, $[T]$, $[\tau]$, $[\mu]$  en $erg. \, s^{-1} \, cm^{-1} \, K^{-1}$, $K$, $s$ 
$g.\, cm^{-3}$.

\begin{enumerate}
\item Pre-supernovae 

 (Mart\'\i nez 96) ($[\kappa] \approx
10^{37}$;
$[T] \approx 10^{13}$; $[\tau] \approx 10^{-4}$; $[\mu] \approx
10^{12}$ )  $\alpha \approx 1$. 

\item Inflation
 $\mu + P = 0$ $\Longleftrightarrow$  $\alpha = 1$ 

\end{enumerate}



\newpage
\begin{itemize}
\item Once the transport equation has been taken into account, then the
inertial energy density  and the ``passive gravitational mass density'',
i.e the factor multiplying $D_TU$ and
the first factor at the right of (\ref{eqd}) respectively (which of course
are the same, as expected from the equivalence principle), appear
diminished by the factor $1-\alpha$, a result
already obtained, but here generalized by the inclusion of
the viscosity and radiative phenomena.
\item As far as the right hand side of (\ref{V4}) is negative, the
system
keeps collapsing. However, let us assume that the  collapsing sphere
evolves in such a way that, for some region of the sphere, the value of
$\alpha$ increases  and  approaches the
critical value of
$1$. Then, as this process goes on, the ensuing decreasing of the
gravitational force term would eventually lead to a change of the sign
of the right hand side of (\ref{V4}). Since that would happen for small
values of the
effective inertial mass density, that would imply a strong bouncing of that
part of the  sphere, even for a small absolute value of the
right hand side of (\ref{V4}).

\item Observe that the charge does not enter into the definition of $\alpha$.
However it does affect the ``active gravitational mass'' (the factor within
the square bracket in
(\ref{grav})).

\item The repulsive Coulomb term (the last term in (\ref{hyd})) depends on
$D_{R}s$ and always opposes gravitation. Its effect is reinforced if
$D_{R}s$ is large enough to
violate (\ref{conI}), in which  case the charge will decrease the ``active
gravitational mass'' term in (\ref{grav}).
\end{itemize}
\newpage

\section{Conclusions}
\begin{itemize}
\item  Dissipative phenomena may play a relevant role in the dynamics of collapse. In particular, relaxational effects may drastically change the outcome of gravitational collapse.
\item The dynamical regime may be approached by means of succesive approximations.
\item Local anisotropy of pressure has to be taken into  consideration in the study of the structure and  evolution  of  massive stars.
\item Not only electric charge but also its distribution may be relevant in stellar structure and evolution.
\item Local (non--gravitational) effects may be crucial to determine the outcome of gravitational collapse.
\item Learn to deal with comoving and non--comoving frames!
\end{itemize}
\end{document}
