4.4 Upper and Lower Bounds 4 Finite Speeds in Relativistic 4.2 Symmetric Hyperbolicity

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 tex2html_wrap_inline3361 in relativistic thermodynamics is moments of the phase density tex2html_wrap_inline3767 of the atoms, viz.

  equation1043

This is formally identical to the non-relativistic case that was treated in Chapter  3Popup Footnote . 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

  equation1071

with

  equation1076

just like (33Popup Equation) and (39Popup Equation). We conclude that the vector potential tex2html_wrap_inline3687 is not generally in the class of moments. However, in the non-degenerate limit, where tex2html_wrap_inline4191 holds, we obtain from (69Popup Equation) (see also (41Popup Equation))

  equation1088

Therefore tex2html_wrap_inline3687 for a non-degenerate gas reads

  equation1095

and that is in the class of moments. In fact tex2html_wrap_inline3687 is equal to the four-velocity tex2html_wrap_inline4199 of the gas to within a factor. We have

  equation1103

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 tex2html_wrap_inline3687 in ensuring symmetric hyperbolicity of the field equations: Symmetric hyperbolicity was due to the concavity of tex2html_wrap_inline4025 in the privileged frame moving with the four-velocity tex2html_wrap_inline4207 . Now we see from (72Popup Equation) that - for the non-degenerate gas - we have tex2html_wrap_inline4209 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.]



4.4 Upper and Lower Bounds 4 Finite Speeds in Relativistic 4.2 Symmetric Hyperbolicity

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
http://www.livingreviews.org/lrr-1999-1
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de