Relativistic Self-Similar Flows

Abstract

A mathematical framework analyzing a class of spherical self-similar hydrodynamical flows on a class of homothetic spherically symmetric background geometries is presented. The analysis covers relativistic flows propagating in the following class of spacetimes: (i) Minkowski spacetime, (ii) the homothetic Bondi–Tolman spacetime, (iii) the spatially flat Friedman–Robertson–Walker cosmologies. After the introduction of the mathematical framework, there follows a detailed investigation of special relativistic flows with the pressurePand densityρobeyingP=kρ,ka constant 0⩽k⩽1. An extensive analysis of the singular and critical points of the differential equations determining the flow is presented. By a combination of analytical and numerical techniques several classes of flows have been constructed. We have found flows describing an oscillation of the medium, as well as flows representing relativistic detonation waves propagating on a medium at rest. In addition, two classes of analytical closed form solutions are given. By invoking the relativistic Rankine–Hugoniot conditions we briefly discuss a way by which two self-similar solutions can be joined across an expanding (or contracting) shock wave.