Electromagnetic fields in curved spacetimes.

Accurate description of astrophysical phenomena near black hole should take into account general relativistic effects. In particular, an important issue that has not received enough attention in the past due to its great complexity is the evolution of electromagnetic fields in curved spacetimes.

The conservation equation for the particle number in a single component perfect fluid can be written in covariant form:

\begin{displaymath}0 = \nabla_{\mu}T^{\mu\nu}, \\\end{displaymath} (1)
while the conservation of the energy-momentum tensor is:
\begin{displaymath}0 = \nabla_{\mu}J^{\mu},\end{displaymath} (2)

$J^{\mu}=\rho u^{\mu}$,

$T^{\mu\nu}=\rho h u^{\mu} u^{\nu} +p g^{\mu\nu}$,

$h=1+\varepsilon+p/\rho$ is the relativistic enthalpy, $\rho$ is the density, $\varepsilon$ is the internal energy, $p$ is the pressure; an equation of state (EOS) is needed and there are two EOS that are often used: the polytropic EOS $p=(\Gamma-1)\rho\varepsilon$, and the relativistic EOS: $p=(\Gamma-1)\rho(1+\varepsilon)$, where $\Gamma$ is the polytropic coefficient of the fluid.

Following the 3+1 formalism and the Valencia formalism, the metric is written as:

\begin{displaymath}n_{\mu} = (-\alpha,\vec{0}),\end{displaymath} (3)
\begin{displaymath}n^{\mu} = \frac{1}{\alpha}\left(1,-\beta^i\right),\end{displaymath} (4)
and we introduce ``conserved variables'' in terms of the ``primitive variables'' of the fluid: ${\bf w } = (\rho,v_i,\varepsilon)$:
$\displaystyle D$ $\textstyle =$ $\displaystyle \rho W,$ (5)
$\displaystyle S_j$ $\textstyle =$ $\displaystyle \rho h W^2 v_j,$ (6)
$\displaystyle E$ $\textstyle =$ $\displaystyle \rho h W^2-p,$ (7)

where $v^i = \gamma^{ij}v_j$ are the contravariant components of the Eulerian observers,

\begin{displaymath}v^i = \frac{u^i}{\alpha u^0} + \frac{\beta^i}{\alpha}.\end{displaymath} (8)
and $W$ is the Lorentz relativistic factor, $W=\alpha u^0 = \sqrt{1-v^2}$$v^2=\gamma_{ij} v^i v^j$. The eqs. (0.1-0.2) can now be rewritten as a conservative first order system:
\begin{displaymath}\frac{1}{\sqrt{-g}}\left( \frac{\partial\sqrt{\gamma}{\bf U}......}{\bf F}^i({\bf w})}{\partial x^i} \right) = {\bf S}({\bf w}),\end{displaymath} (9)
where the state vector is ($T$ denotes the transpose):
\begin{displaymath}\mathbf{U(w^i)} = [D,S_j,\tau]^T,\end{displaymath} (10)
with $\tau=E-D$; the flux vector is:
\begin{displaymath}{\bf F}^i({\bf w}) = \left[D\left( v^i-\frac{\beta^i}{\alp...... \left( v^i-\frac{\beta^i}{\alpha} \right) + p v^i\right]^T,\end{displaymath} (11)
and the corresponding sources are:
\begin{displaymath}{\bf S(w)} = \left[0,T^{\mu\nu}\left( \frac{\partial g_{...... x^{\mu}}-T^{\mu\nu} \Gamma^{0}_{\nu\mu} \right)\right]^T.\end{displaymath} (12)
Casting this system in seudo-linear form:
\begin{displaymath}\frac{1}{\sqrt{-g}}\left( \frac{\partial\sqrt{\gamma}{\bf U}......\bf U}}\frac{\partial \bf U}{\partial x^i}\right) = {\bf S},\end{displaymath} (13)
we can use its characteristic structure to apply High Resolution Shock Capturing (HRSC) schemes, i.e., we need the eigenvectors and eigenvalues of the system:
\begin{displaymath}{\cal A}^i = \frac{\partial {\bf F}^i}{\partial \bf U},\end{displaymath} (14)
which turn out to be:
$\displaystyle \lambda_0$ $\textstyle =$ $\displaystyle \alpha v^x-\beta^x \, \, \, \, \, \, \, \, \,{\rm (triple) },$ (15)
$\displaystyle \lambda_{\pm}$ $\textstyle =$ $\displaystyle \frac{\alpha}{1-v^2 c_s^2}\left[v^x(1-c_s^2)\pm c_s\sqrt{(1-v^2)[\gamma^{xx}(1-v^2 c_s^2)-v^x v^x(1-c_s^2)]}\right] - \beta^x,$ (16)

where $c_s$ is the local speed of sound, $hc_s^2=\chi+\kappa p/rho^2$, with $\chi=\partial p/\partial\rho$ and $\kappa=\partial p/\partial\varepsilon$. A complete set of eigenvectors is:

$\displaystyle {\bf r}_{0,1}$ $\textstyle =$ $\displaystyle \left[\frac{{\cal K}}{hW},v_x,v_y,v_z,1-\frac{{\cal K}}{hW}\right]^T,$ (17)
$\displaystyle {\bf r}_{0,2}$ $\textstyle =$ $\displaystyle \left[Wv_y,h(\gamma_{xy}+2W^2v_xv_y),h(\gamma_{yy}+2W^2v_yv_y),h(\gamma_{zy}+2W^2v_zv_y),Wv_y(2hW-1)\right]^T,$ (18)
$\displaystyle {\bf r}_{0,3}$ $\textstyle =$ $\displaystyle \left[Wv_z,h(\gamma_{xz}+2W^2v_xv_z),h(\gamma_{yz}+2W^2v_yv_z),h(\gamma_{zz}+2W^2v_zv_z),Wv_z(2hW-1)\right]^T,$ (19)
$\displaystyle {\bf r}_{\pm}$ $\textstyle =$ $\displaystyle \left[1,hW\left(v^x-\frac{v^x-\Lambda^x_{\pm}}{\gamma^{xx}-v^x\L......hW\frac{\gamma^{xx}-v^xv^x}{\gamma^{xx}-v^x\Lambda^x_{\pm}}\right)-1\right]^T,$ (20)


$\displaystyle {\cal K}$ $\textstyle =$ $\displaystyle \frac{\tilde\kappa}{\tilde\kappa-c_s^2},$ (21)
$\displaystyle \Lambda^x_{\pm}$ $\textstyle =$ $\displaystyle \tilde\lambda_{\pm}+\tilde{\beta^x},$ (22)

and $\tilde{\kappa}=\kappa/\rho$$\tilde{\lambda}=\lambda/\alpha$$\tilde{\beta}^x=\beta^x/\alpha$.

The eqs.(0.9) can be discretized as (assuming one dimension for simplicity):

\begin{displaymath}\frac{1}{\sqrt{-g}}\frac{\partial \sqrt{\gamma}{\bf U}_i}{\p......c{(f^*)_{i+1/2}-(f^*)_{i-1/2}}{\Delta x} \right]+ {\bf S}_i,\end{displaymath} (23)
where $(f^*)_{i\pm 1/2}$ is the ``numerical flux'' computed at the interfaces$i\pm 1/2$ of the spatial cell $i$, and is given by:
\begin{displaymath}(f^*)_{i+1/2} = \frac{1}{2}\left[ {\bf F}^R + {\bf F}^L -\......mbda_m\right\vert\Delta\tilde\omega_m\tilde {\bf r}_m \right],\end{displaymath} (24)
where $R$$L$ are the right and left interfaces of one cell, and $\{\tilde\lambda_m,\tilde {\bf r}_m\}_{m=1\ldots 5}$ are the eigenvalues and eigenvectors at the cell interface using the average of the primitive variables. Finally, $\{\Delta\tilde\omega_n\}_{n=1\ldots 5}$ are the jumps in the characteristic variables, given by:
\begin{displaymath}{\bf U}_R-{\bf U}_L = \sum_{m}{\Delta\tilde\omega_m\tilde {\bf r}_m}\end{displaymath} (25)
Adding the Maxwell equations is straightforward:
$\displaystyle \nabla^{(3)}_{i}E^{i}$ $\textstyle =$ $\displaystyle 4\pi\rho_e,$ (26)
$\displaystyle \nabla^{(3)}_{i}B^{i}$ $\textstyle =$ $\displaystyle 0,$ (27)
$\displaystyle \partial_{t} E^{i}$ $\textstyle =$ $\displaystyle \epsilon^{ijk}\nabla^{(3)}_{j}(\alpha B_k)-4\pi\alpha J^{i}+ \alpha K E^i + {\cal L}_{\beta}E^i,$ (28)
$\displaystyle \partial_{t} B^{i}$ $\textstyle =$ $\displaystyle -\epsilon^{ijk}\nabla^{(3)}_{j}(\alpha E_k)+ \alpha K B^i + {\cal L}_{\beta}B^i,$ (29)

plus the conservation of electric charge:

\begin{displaymath}\partial_{t}\rho_e = - \nabla^{(3)}_{i}(\alpha J^i)+ \alpha K \rho_e + \beta^i \nabla_i \rho_e,\end{displaymath} (30)
where $\nabla^{(3)}$ is the covariant derivative with respect to the spatial metric $\gamma_{ij}$$K={\rm tr}K_{ij}$ is the trace of the extrinsic curvature, $E^i$ and $B^i$ are the electric and magnetic fields as measured by an Eulerian observer, and$\rho_e$ is the electric charge density.

The conservation of the new energy-momentum tensor gives:

$\displaystyle 0$ $\textstyle =$ $\displaystyle \frac{1}{\sqrt{-g}}\partial_{t}(\sqrt{\gamma}\tau)+\frac{1}{\sqrt......c{1}{2}(E_{m}E^{m}-B_{m}B^{m}) \right)v^i + \epsilon^{ijk}E_{j}B_{k} \right] -$ (31)
    $\displaystyle \alpha \left[ \partial_{\nu}(\ln\alpha)T^{t\nu}+\Gamma^{t}_{\mu\nu}T^{\mu\nu} \right],$ (32)
$\displaystyle 0$ $\textstyle =$ $\displaystyle \frac{1}{\sqrt{-g}}\partial_{t}(\sqrt{\gamma}S_j) + \frac{1}{\sqr......\delta^{i}_{j}+\frac{1}{2}\delta^{i}_{j}(E_{m}E^{m}-B_{m}B^{m})-\right.\right.$ (33)
    $\displaystyle \left.\left.(E_{j}E^{i}-B_{j}B^{i})-v^{i}\epsilon_{jmn}E^{m}B^{n}\right)\right]-\Gamma^{\lambda}_{j\nu}g_{\lambda\alpha}T^{\alpha\nu},$ (34)

where we have redefined the energy density:

\begin{displaymath}\tau = W^2 \rho h-p-\rho W+\frac{1}{2}\left(E_{m}E^{m}-B_{m}B^{m}\right),\end{displaymath} (35)
and the momentum density:
\begin{displaymath}S_j = W^2 \rho h v_j + \epsilon_{jmn}E^{m}B^{n},\end{displaymath} (36)
From here we can attemp to find the eigenvalues and a set of eigenvectors and apply again HRSC to evolve electromagnetic fields on a curved background. This is the current task being carried out now.

Hugo Villegas