4.3 Moments as four-fluxes and the vector potential

Just like in the non-relativistic case the most plausible – and popular – choice of the four-fluxes FA in relativistic thermodynamics is moments of the phase density f(x, p,t) of the atoms, viz.
∫ F Aα = pApαf dP , (α = 1, 2,...n), (A = 0,1,2,3 ) (67 )
This is formally identical to the non-relativistic case that was treated in Section 34. There are essential differences, however

Both are important differences. But many results from the non-relativistic theory will remain formally valid.

Thus for instance in the relativistic case we still have

∫ ′A A h = p F (χ)dP (68 )
with
( − χ ) F (χ) = ∓ky ln 1 ± e k , (69 )
just like (33View Equation) and (39View Equation). We conclude that the vector potential ′A h is not generally in the class of moments. However, in the non-degenerate limit, where − χ∕k e ≪ 1 holds, we obtain from (69View Equation) (see also (41View Equation))
− χ F (χ) = − kye k or F = − kf. (70 )
Therefore h ′A for a non-degenerate gas reads
∫ h′A = − k pAf dP (71 )
and that is in the class of moments. In fact h ′A is equal to the four-velocity U A of the gas to within a factor. We have
′A nk- A h = − c U , (72 )
where n is the number density of atoms in the rest frame of the gas.

We recall the discussion – in Section 4.2 – of the important role played by h′A in ensuring symmetric hyperbolicity of the field equations: Symmetric hyperbolicity was due to the concavity of ( ) h¯ζ F ζ¯ in the privileged frame moving with the four-velocity ′A cζ¯A = − c∘-h----- h′Ah ′A. Now we see from (72View Equation) that – for the non-degenerate gas – we have ¯A A cζ = U so that the privileged frame is the local rest frame of the gas. This is quite satisfactory, since the rest frame is naturally privileged. [There remains the question of why the rest frame is not the privileged one for a degenerate gas. This point is open and invites investigation.]


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