Relativistic Rankine-Hugoniot Equations

Abstract

In Part I of this paper the stress energy tensor and the mean velocity vector of a simple gas are expressed in terms of the Maxwell-Boltzman distribution function. The rest density ${\rho }^0$, pressure, $p$, and internal energy per unit rest mass $\epsilon$ are defined in terms of invariants formed from these tensor quantities. It is shown that $\epsilon$ cannot be an arbitrary function of $p$ and ${\rho }^0$ but must satisfy a certain inequality. Thus $\epsilon =\frac{\left(\frac{1}{\gamma -1\mathrm{}}\right)p}{{\rho }^0}$ for $\gamma >\frac{5}{3}$ is impossible. It is known that if $\epsilon$ is given by this relation and $\gamma >2\mathrm{}$, then sound velocity in the medium may be greater than that of light in vacuum. This difficulty is now removed by the inequality mentioned above. In Part II of this paper the relativistic form of the Rankine-Hugoniot equations are derived and it is shown that as a consequence of the inequality mentioned earlier that the shock wave velocity is always less than that of light in vacuum for sufficiently strong shocks.