B GR Hydrodynamical and MHD equations
In what follows, we will adopt the stress energy tensor of an ideal relativistic fluid,
where
,
, and
are the rest mass density, pressure, and fluid 4-velocity, respectively, and
is the relativistic specific enthalpy, with
the specific internal energy of the fluid.
The equations of ideal GR hydrodynamics [186] may be derived from the local GR conservation laws of
mass and energy-momentum:
where
denotes the covariant derivative with respect to the 4-metric, and
is the mass
current.
The 3-velocity
can be calculated in the form
where
is the Lorentz factor. The contravariant 4-velocity is then given by:
and the covariant 4-velocity is:
To cast the equations of GR hydrodynamics as a first-order hyperbolic flux-conservative system for the
conserved variables
,
, and
, defined in terms of the primitive variables
, we define
where
is the determinant of
. The evolution system then becomes
with
Here,
and
are the 4-Christoffel symbols.
Magnetic fields may be included in the formalism, in the ideal MHD limit under which we assume
infinite conductivity, by adding three new evolution equations and modifying those above to include
magnetic stress-energy contributions of the form
where the magnetic field seen by a comoving observer,
is defined in terms of the dual Faraday tensor
by the condition
where
represents twice the magnetic pressure. With magnetic terms included, we may rewrite
the stress-energy tensor in a familiar form by introducing magnetically modified pressure and enthalpy
contributions:
and redefine the conserved momentum and energy variables
and
accordingly:
Defining the (primitive) magnetic field 3-vector as
and the conserved variable
, which are related to the comoving magnetic field 4-vector
through the relations
we may rewrite the conservative evolution scheme in the form
where the magnetic field evolution equation is just the relativistic version of the induction equation. An
external mechanism to enforce the divergence-free nature of the magnetic field,
must also be
applied.