Introduction

In the previous chapter we introduced the concept of propagating singular surfaces (or singular hypersurfaces in space-time) and derived the compatibility relations among the jumps of the field variables. When only derivatives of the field variables can be discontinuous across a propagating singular surface, we are dealing with weak discontinuities. When the fields themselves can be discontinuous we have strong discontinuities (among which shock waves are of paramount importance).

Whereas shock waves can be produced from an initially smooth solution as a consequence of nonlinear steepening and breaking, weak discontinuities can only be produced as a result of discontinuities in initial or boundary conditions. For instance, in gas dynamics, a jump in the derivatives of the velocity can appear in a flow along a solid obstacle with angles. Also, a jump in the derivatives of the pressure can appear (among other discontinuities like shocks and contact discontinuities) when the initial condition consists of two adjoining masses of gas compressed to different pressures (Landau and Lifshitz, 1959a). Although conceptually these two examples remain valid also for relativistic fluid motion, for relativistic flow in an astrophysical context only the latter is meaningful. For instance, the case of a cloud moving relativistically in a jet has been considered by Blandford and Königl (1979) in the context of models for the rapid variations in extragalactic radio sources' emissions.

An extremely important application of the concept of weak discontinuity waves is to the study of impulsive gravitational waves.

Relativistic fluid models are of considerable interest in several areas of astrophysics, plasma physics, and nuclear physics. Here we will mention briefly some of these areas and emphasize the problems which form the physical motivations for the theories expounded in this book.

Theories of gravitational collapse and models of supernova explosions (Van Riper, 1979; Chevalier, 1981; Shapiro and Teukolsky, 1983) are based on a relativistic fluid model for the star. In most models a key feature is the occurrence of an outward propagating relativistic shock. The precise conditions under which the shock forms at some point with exactly the necessary strength to expel the bulk of the star but still leave behind a remnant remain to be studied in detail and are the subject of current investigation. The effects of deviations from spherical symmetry due to an initial angular momentum and magnetic field must also be assessed. This requires the use of relativistic magneto-fluid dynamical models (Yodzsis, 1971; Maeda and Oohara 1982; Sloan and Smarr, 1986). The problem of the shock stability when traversing regions where the equation of state softens could be of interest for supernova models.

In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium (Peebles, 1980). Other components consisting of collisionless particles (such as massive neutrinos or photinos) are usually treated within a kinetic framework (Peebles, 1980).

Introduction

The stability of relativistic shock waves is a fundamental problem of relativistic fluid dynamics and its resolution might have interesting consequences for astrophysics, plasma physics, and nuclear physics. In particular, in models of gravitational collapse and supernova explosions, the bounce shock traverses regions of varying state equations. The stability of this shock is essential for the overall validity of the model. Present numerical codes, being restricted to spherical symmetry, are not capable of detecting these kinds of instabilities. In high-energy heavy ion collisions, the formation of a quark-gluon plasma might occur through a relativistic shock and its instability might signal the transition (e.g., by observing shock splitting; Barz et al., 1985).

In nonrelativistic fluid dynamics the stability of plane shock fronts has been investigated by D'Yakov (1956) and Erpenbeck (1962), using linear stability methods.

The problem has also been studied by Gardner (1963) in the framework of shock splitting methods and more recently by Fowels (1981, and references therein) who considered the stability of the front against an impinging acoustic wave from behind. By requiring that the perturbations decay sufficiently fast at infinity these authors obtained restrictions on the equation of state of a fluid in order for it to sustain plane fronted shock waves.

The linear stability analysis has been extended by Gardner and Kruskal (1964) to the fast magnetoacoustic shock and by Lessen and Deshpande (1967) to the slow magnetoacoustic shock.

Introduction

Simple waves are exact solutions of quasi-linear equations representing traveling waves. They are the most natural nonlinear analog of the plane traveling waves of the linear theory. Although they correspond to special initial conditions, they are still sufficiently general to be of physical interest (as witnessed by Friedrichs' theorem, which, loosely speaking, states that any one dimensional smooth solution neighboring a constant state must be a simple wave). Simple waves are exact analytical solutions which show clearly some of the main features of nonlinear wave propagation in general, such as steepening, breaking, and shock formation. They are also sufficiently complex to be useful as benchmarks against which to test numerical codes.

In classical fluid dynamics simple waves are of paramount importance for several reasons. Simple waves are the basic ingredients for constructing analytical solutions to the Riemann problem (or shock-tube problem: the evolution of an initial state corresponding to two adjacent fluids at different pressures). These solutions are among the standard tests for numerical hydrodynamical codes (Sod, 1978) because they comprise some of the key features of general hydrodynamical behavior (nonlinear steepening and the occurrence of discontinuities). Also, analytical solutions to the Riemann problem can be used in order to construct sophisticated numerical codes able to deal very accurately with shock front tracking (Plohr, Glimm, and McBryan, 1983). Simple waves are also used in order to construct approximate analytical solutions to the problem of weak shock decay (Landau and Lifshitz, 1959a; Whitham, 1974; Courant and Friedrichs, 1976) and in Whitham's theory of geometric shock dynamics (Whitham, 1974).

Most of the matter in the universe can, in some form or other, be treated as a fluid and in several instances (supernova explosions, jets in extragalactic radio sources, accretion onto neutron stars and black holes, high-energy particle beams, high-energy nuclear collisions, etc.) undergoes relativistic motion. This consideration alone should be sufficient to motivate research in relativistic fluids. In the past most of the results on relativistic fluids have been obtained in a piecemeal way, in relation to a particular problem under consideration and using ad hoc techniques. Although this approach is perfectly legitimate in the process of research in the various areas of applications (astrophysics, plasma physics, nuclear physics), in the long run it is unsatisfactory because it tends to obscure the underlying unity of the subject and of the relevant techniques. In fact, a problem tailored approach (instead of a systematic and general one) necessarily precludes utilizing, in a particular area, results obtained in another area, and therefore hinders the cross fertilization of various techniques, a method which has been fruitful in several areas of science.

In 1967 the French mathematician André Lichnerowicz published a masterful monograph on relativistic fluid dynamics and magneto-fluid dynamics, which covered mainly existence and uniqueness results. Since then there has been no other attempt at a systematic development of the subject, although there have been several important developments in the field (particularly in shock wave theory).