Relativistic shock formation in the presence of radial entropy gradients

The nonlinear steepening of relativistic acoustic waves is investigated. The nonlinear evolution of a planar wave in a homentropic flow field is understood well through relativistic simple waves. However, in situations where the wave is nonplanar and the flow field is nonhomentropic, the concept of simple waves cannot be used. In the present paper, effect of entropy gradients on the nonlinear distortion of a spherical wave is analyzed using the wave front expansion technique. It is shown that the behavior of a relativistic wave in nonhomentropic environment is slightly different from the nonrelativistic wave. A closed form solution is obtained for the slope at the wave front. A general criterion for a compression wave to steepen into a shock is obtained. The distortion of compression and rarefaction waves is examined for some well known equations of state. However, the method used in this paper is general and can be easily extended to analyze shock formation in any fluid. Also, expressions for time and location of shock formation are obtained. The effects of gravity or self-gravity are not taken into account in this paper