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2 The Hydrodynamic Theory for Dielectrics
2.1 Lorentz Transformation and Thermodynamics
The starting point of every hydrodynamic theory is the thermodynamic theory.
The energy density ɛ tot is a function of all the other conserved quantities, the
entropy density s and the field variables B and D. For a two-component liquid
we have, in the rest frame,
dɛ tot =dε + c 2 dρ = T ds +(µ + c 2 )dρ
+µc dρc + H · dB + E · dD , (1)
where the constraints
∇·B =0, ∇·D = ρel (2)
are satisfied in equilibrium. (ρel is the charge density.) Note that ɛ tot is the total
energy including the rest mass. The mass densities, ρ and ρc, are connected
to the particle numbers n1 and n2 by
ρ = m1 n1 + m2 n2 , ρc = m2 n2, (3)
where m1 and m2 denote the respective masses. The choice of µ + c 2 as the
conjugate variable to ρ renders the expansion in the small parameter ε/ρ c 2
simple. (At most densities, the rest mass is certainly the by far dominating
contribution.)
The equilibrium fluxes of energy and momentum in the rest frame are [5],
Q = c E × H , (4)
Πij =(Ts+ µρ+ µc ρc + E · D + H · B − ε) δij
− 1
2 [Hi Bj + Ei Dj +(i ↔ j))] . (5)
These variables and their fluxes constitute the energy-momentum 4-tensor
Π µν ,
Π 00 = ɛ tot , Π 0k =Π k0 = Qk/c = cg tot
k ,
Π ik =Πik =Πki . (6)
The local conservation laws ensure that Π µν satisfies
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