2.6.2 Ecuaciones de la RMHD en forma conservativa

1. Conservación local del número bariónico:
   $$\begin{eqnarray*} \nabla_{\mu}J^{\mu}=0 \end{eqnarray*}$$
   
   
   
   
   
   

2. Conservación local de la energía-momento:
   $$\begin{eqnarray*} \nabla_{\mu}T^{\mu\nu}=0 \end{eqnarray*}$$
   
   
   
   
   
   

3. Ecuaciones de Maxwell:
   $$\begin{eqnarray*} \nabla_{\mu}H^{\mu\nu}=0 \end{eqnarray*}$$
   
   
   
   
   
   

$J^{\mu}=\rho u^\mu$
$T^{\mu\nu}=\rho h^*u^\mu u^\nu +p^*g^{\mu \nu}-\mu b^\mu b^\nu$
$H^{\mu\nu}=u^\mu b^\nu -u^\nu b^\mu$
$u^\mu =W(1,v^x,v^y,v^z)$
$b^\mu =(b^0,b^x,b^y,b^z)$
$h^*=1+\varepsilon +p/\rho+\mu \mid b\mid ^2/\rho$
$p^*=p+\mu \mid b\mid ^2/2$
$u^\mu b_\mu =0$
$b_\mu b^\mu =\mid b\mid ^2\geq 0$
$\sigma ^{\mu\nu}=\sigma_0g^{\mu\nu} \mbox{\quad con\quad } \sigma_0\rightarrow\infty$
$EOS: p=(\gamma -1)\rho\epsilon$

$$\begin{eqnarray*} \vec{u} & = & (\rho, v^j, \varepsilon, b^j) \\ \vec{U} & = & ... ... b^jb^x, \\ & & \rho h^*W^2v^x-\rho Wv^x, W(v^jb^x-v^xb^j)) \\ \end{eqnarray*}$$