4.3 Moments as Four-Fluxes and 4 Finite Speeds in Relativistic 4.1 Concavity of a Privileged

4.2 Symmetric Hyperbolicity 

The transformation tex2html_wrap_inline4071 helps us to recognize the structure of the field equations. Obviously with tex2html_wrap_inline3481 as the field vector, instead of tex2html_wrap_inline3995 we may rephrase the field equations (10Popup Equation) as

  equation954

or, by (60Popup Equation):

  equation964

We observe that the coefficient matrices are Hessian matrices and therefore symmetric.

By the definition of symmetric hyperbolicity due to Friedrichs [17] the system is symmetric hyperbolic, if there exists at least one co-vector tex2html_wrap_inline3467 for which

  equation976

In our case - with the concavity (5Popup Equation) of the entropy density tex2html_wrap_inline3999 for tex2html_wrap_inline4119 - it is clear that such a co-vector exists. It is tex2html_wrap_inline4033 itself! Indeed we have

  equation991

by (62Popup Equation) and (58Popup Equation). Thus symmetric hyperbolicity is implied by the concavity of the entropy density both in the relativistic and the non-relativistic case.

It is true that in the relativistic case we have to rely on the privileged co-vector tex2html_wrap_inline4119 in this context and therefore on a privileged Lorentz frame whose entropy density tex2html_wrap_inline3999 is concave in tex2html_wrap_inline3995 . The significance of this choice is not really understood . Indeed, we might have preferred the privileged frame to be the local rest frame of the body. In that respect it is reassuring that tex2html_wrap_inline4141 is often co-linear to the four-velocity tex2html_wrap_inline4143 as we shall see in Section  4.3 below; but not always! A better understanding is needed.

Note that in the non-relativistic case the only time-like co-vector is tex2html_wrap_inline4145, a constant vector. In that case all the above-mentioned complications are absent: Concavity of the one and only entropy density tex2html_wrap_inline3415 is equivalent to symmetric hyperbolicity, see Chapter  3 above.

Also note that the requirement (65Popup Equation) of symmetric hyperbolicity ensures finite characteristic speeds, not necessarily speeds smaller than c as we might have wished. [In this respect we may be tempted to replace Friedrich's definition of symmetric hyperbolicity by one of our own making, which might require (65Popup Equation) to be true for all time-like co-vectors tex2html_wrap_inline4151 - instead of at least one . If we did that, we should anticipate the whole problem of speeds greater than c . Indeed, we recall the characteristic equation (15Popup Equation) which - for our system (64Popup Equation) - reads

displaymath1026

If (65Popup Equation) were to hold for all time-like co-vectors tex2html_wrap_inline3467, we could now conclude that tex2html_wrap_inline4161 is space-like, or light-like, so that tex2html_wrap_inline4163 holds. Thus (12Popup Equation) would imply tex2html_wrap_inline4165 . This is a clear case of assuming the desired result in a disguise and we do not follow this path.]



4.3 Moments as Four-Fluxes and 4 Finite Speeds in Relativistic 4.1 Concavity of a Privileged

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