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energy flux, this leads to a third order term.) Then the lab-frame expressions
for the energy, momentum and the fluxes are
where
g tot =(Ts+ µρ+ µc ρc + ρc 2 ) v/c 2 + E × H/c , (14)
Q = c 2 g tot + vl gl v , (15)
Πij =(Ts+ µρ+ µc ρc + v · g
+E · D + H · B − ε) δij
+ 1
2 [gi vj − Ei Dj − Hi Bj +(i ↔ j))] , (16)
g = g tot − D × B/c . (17)
The other thermodynamic variables also need to be transformed. First, the
four fields:
B 0 = B − v × E/c , D 0 = D + v × H/c , (18)
H 0 = H − v × D/c , E 0 = E + v × B/c . (19)
Both chemical potentials, µ + c 2 and µc, obey analogous formulas:
µ + c 2 =[1− v 2 /(2 c 2 )] (µ 0 + c 2 ) , (20)
µc =[1− v 2 /(2 c 2 )] µ 0 c . (21)
However, because of c 2 , only µ is altered
µ = µ 0 − v 2 /2 , (22)
while µc = µ 0 c remains invariant in linear order of v.
The quantities s, ρc
c
and T are also invariant to this order, s = s0 , ρc = ρ0 c and T = T 0 .Inthe
combination ρc2 ,wehave
ρ =[1+v 2 /(2 c 2 )] ρ 0 ; (23)
otherwise, it is ρ = ρ 0 .
Because ε ≪ ρc 2 , the nonrelativistic expression for g tot is
g tot = ρ v + E × H/c . (24)
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