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(E, B) µν ⎛
⎞
⎜ 0 −Ex −Ey −Ez ⎟
⎜
⎟
⎜
⎟
⎜ Ex 0 −Bz By ⎟
= ⎜
⎟
⎜
⎟ .
⎜ Ey Bz 0 −Bx
⎟
⎝
⎠
(40)
Ez −By Bx 0
• The equilibrium quantities (s, s v/c), (ρ, ρ v/c), (ρc,ρc v/c) and (ρel c, ρel v)
are 4-vectors. The same applies to the nonequilibrium quantities (s D c, −f D ),
(ρ D c, −j D ), (ρ D c c, −j D c ) and (ρ D el c, j D el).
The dissipative contributions of the variables s D , ρ D , ρ D c , g tot,D , B D , D D and
ρ D el vanish in the local rest frame, v = 0, because this is where the nonrelativistic
and relativistic physics overlap. Here, according to the concept of local
equilibrium, the variables only contain equilibrium information. As a result,
their form in an arbitrary frame is given as:
B D = v × E D /c , D D = −v × H D /c , (41)
ρ D el = v · j D el/c 2 , s D = −v · f D /c 2 , (42)
ρ D = −v · j D /c 2 , ρ D c = −v · j D c /c 2 , (43)
g tot,D
i = −Π D ij vj/c 2 , (44)
and because of the symmetry of the energy-momentum 4-tensor,
with
yet an order higher.
Q D i = c 2 g tot,D
i = −vj Π D ij, (45)
ε D = O
v 2 /c 2
(46)
To actually obtain the dissipative currents, we substitute ∂tɛ tot in Eq(36) for
the expressions of Eq(25),
T∂ts +(µ + c 2 ) ∂tρ + µc ∂tρc + v · ∂tg + H · ∂tB
+E · ∂tD + ∂tε D = −∇ · (Q + Q D ) , (47)
insert the appropriate equations of motion, Eqs(30 - 35), excluding terms of
order higher than v
c , and arrive at two expressions, the energy flux QD i and
the entropy production R. The energy flux is
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