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.

Living Rev. Relativity 15, (2012), 8
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