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:

(1) 
while the conservation of the energymomentum tensor is:

(2) 
where:
,
,
is the relativistic enthalpy,
is the density,
is the internal energy,
is the pressure; an equation of state (EOS) is needed and there are two
EOS that are often used: the polytropic EOS ,
and the relativistic EOS: ,
where
is the polytropic coefficient of the fluid.
Following the 3+1 formalism and the Valencia formalism, the metric is
written as:

(3) 

(4) 
and we introduce ``conserved variables'' in terms of the ``primitive variables''
of the fluid: :
where
are the contravariant components of the Eulerian observers,

(8) 
and
is the Lorentz relativistic factor, , .
The eqs. (0.10.2)
can now be rewritten as a conservative first order system:

(9) 
where the state vector is (
denotes the transpose):

(10) 
with ;
the flux vector is:

(11) 
and the corresponding sources are:

(12) 
Casting this system in seudolinear form:

(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:

(14) 
which turn out to be:
where
is the local speed of sound, ,
with
and .
A complete set of eigenvectors is:
with
and , , .
The eqs.(0.9) can be discretized
as (assuming one dimension for simplicity):

(23) 
where
is the ``numerical flux'' computed at the interfaces
of the spatial cell ,
and is given by:

(24) 
where
y
are the right and left interfaces of one cell, and
are the eigenvalues and eigenvectors at the cell interface using the average
of the primitive variables. Finally,
are the jumps in the characteristic variables, given by:

(25) 
Adding the Maxwell equations is straightforward:
plus the conservation of electric charge:

(30) 
where
is the covariant derivative with respect to the spatial metric ,
is the trace of the extrinsic curvature,
and
are the electric and magnetic fields as measured by an Eulerian observer,
and
is the electric charge density.
The conservation of the new energymomentum tensor gives:
where we have redefined the energy density:

(35) 
and the momentum density:

(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
20020122