Moments as four-fluxes and the vector potential

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.

This is formally identical to the non-relativistic case that was treated in Section 3⁴ . There are essential differences, however

$A p$ is now the Lorentz vector of atomic four-momentum with $0 p > 0$ and $A 2 2 p pA = m c$ . Accordingly $pα$ now stands for polynomials in the components of four-momentum. Thus instead of (27 ) we have $( || 1 |||| pB1 |{ B1 B2 pα = | p p |||| ... ||( B1 B2 BN p p ...p .$
The element $dP$ of phase space is now equal to $dp ∕p0$ instead of $dp$ .

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

with

just like (33

) and (39

). 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 (69

) (see also (41

))

Therefore $h ′A$ for a non-degenerate gas reads

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

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 (72 ) 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.]