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Putting all this together, the Π 00 -component yields
dε =dɛ tot − c 2 dρ = T ds + µ dρ + µc dρc + v · dg
+H · dB + E · dD . (25)
The isotropy of the system, ie the invariance of the energy density under an
infinitesimal rotation implies:
v × g + H × B + E × D =0. (26)
2.2 The Equilibrium Conditions
Maximizing the integrated entropy d 3 rs under the constrains of various
conservation laws and Eqs(2), the resulting Euler-Lagrange equations, or the
equilibrium conditions are [7]
where
∂tT =0, ∂tµ =0, ∂tµc =0,
∂tH 0 =0, ∂tE 0 =0, vij =0,
∇µ + ∂tv + µ∂tv/c 2 =0,
∇ 0 T − 1
µ + c2 T ∇0 µ =0,
∇ 0 µc − 1
µ + c2 µc ∇ 0 µ =0,
∇ 0 × H 0 + 1
µ + c2 H0 ×∇ 0 µ =0,
∇ 0 × E 0 + 1
µ + c2 E0 ×∇ 0 µ = 0 (27)
∇ 0 = ∇ + v
c 2 ∂t , vij = 1
2 (∇ivj + ∇jvi) . (28)
Note the preponderance of rest frame quantities. Except for the chemical potential,
all spatial derivatives are small. If the charges are eventually able to
move around, the total entropy may be further maximized if
E 0 =0. (29)
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