In this section, we will review the ideal MHD. For more detials, one can see Ref. [25-28, 30].
In relativistic hydrodynamics, the electromagnetic strength tensor $ F^{\mu\nu} $ will be decomposed by the fluid velocity $ u^{\mu} $,
which gives,
Generally, we can parametrize the velocity as, $ u^{\mu} = \gamma(1,{ {v}}), $ with $ \gamma $ being $ 1/\sqrt{1-|{ {v}}|^{2}} $. We can prove that,
In the comoving frame of a fluid cell, i.e. local rest frame, $ u^{\mu} = (1,{\bf{0}}) $, the $ E^\mu $ and $ B^\mu $ will reduce to
The ideal MHD includes the energy-momentum and charge conservation equations coupled to the Maxwell's equations.
The energy-momentum conservation is given by,
where $ T^{\mu\nu} $ is the energy-momentum tensor. In an ideal fluid, this tensor can be decomposed as two parts,
The first term is the part for the matter
where $ \epsilon,P $ are energy density and pressure. $ \Pi $ and $ \pi^{\mu\nu} $ denote the bulk pressure and shear viscous tensor, respectively. In the first order relativistic hydrodynamics, $ \Pi $ and $ \pi^{\mu\nu} $ can be parameterized as,
where $ \zeta $ and $ \eta $ stand for the bulk and shear viscosities, respectively. The projector is defined as,
It is easy to prove that
where we have used that $ u^\mu u_\nu = 1 $.
Both $ \Pi $ and $ \pi^{\mu\nu} $ are dissipative effects. In an ideal fluid, all the dissipative terms vanish, i.e. $ \Pi = 0 $ and $ \pi^{\mu\nu} = 0 $.
The second part for the energy-momentum tensor comes from the electromagnetic fields,
where
Inserting Eq. (1) into the above equation yields,
The charge conservation equation reads,
The charge current $ j^{\mu} $ can be decomposed as,
where $ n $ is the electric charge density and $ \sigma $ is the electric conductivity.
For simplicity, we consider a charge neutral fluid, i.e.
In order to simplify the calculations, we will use the ideal MHD limit. The ideal MHD limit means that the electric conductivity is infinite, i.e.
To avoid the divergence in $ \sigma E^{\mu} $ term in Eq. (16), the four vector form of $ E^{\mu} $ must vanish,
In the ideal limit, i.e. $ E^\mu = 0 $, the energy-momentum tensor in Eq. (14) reduces to,
Now, we will discuss the covariant form of Maxwell's equation,
In ideal MHD, the covariant equations for magnetic fields are given by
In the local rest frame, i.e. $ u^{\mu} = (1,{\bf{0}}) $, the above equation reduces to
Contracting Eq. (22) with $ u_{\mu} $, yields,
which gives
in the local rest frame. Contracting Eq. (22) with $ B_{\mu} $, yields,
The thermodynamical relations reads,
where $ T,\; s,\; \mu,\; n $ are temperature, entropy density, chemical potential and charge number density, respectively. In our case, since we have considered the charge neutral fluid $ n = 0 $, the thermodynamical relation becomes,
Its differential form is given by