The relativistic Riemann invariants and the propagation of initial data surfaces
- 43 Downloads
A method is described by which the relativistic Riemann invariants can be found for a fluid with an arbitrary equation of state, undergoing dissipation and moving in a general metric. Specific formulae are derived for a spherically symmetric system. Limiting cases defined by relativistic and non-relativistic gases, both warm, cold, fast and slow are examined. We prove that the invariants do exist, and a necessary and sufficient condition for their determination is the solution of a differential equation with the structure of an exterior one form of two components. The common parameter of these components is the characteristic space-time direction which is also derived in the process of determining the invariants. The characteristic surfaces, being the surfaces over which initial data is carried, all coalesce to the forward light cone in the extreme relativistic limit. Relativistic fluids emanating from receding sources appear to increase their internal kinetic energy as they decelerate.
A non-linear distance-velocity relation for these waves is evident in the differential equations which are found. Their full meaning remains to be explored.
KeywordsVirial Theorem Riemann Invariant Relativistic Fluid Classical Wave Equation Adiabatic System
Unable to display preview. Download preview PDF.
- Allen, W. (1954).Astrophysical Quantities, Chapter 14, p. 242. Athlone Press, London.Google Scholar
- Courant, R. and Hilbert, D. (1937).Methods of Mathematical Physics, German edition, Vol. II, Chapter 50, p. 313. Springer-Verlag, Berlin.Google Scholar
- Forsyth, A. W. (1959).Theory of Differential Equations, Vol. I, Chapter I, p. 11. Dover Pub. Co., New York.Google Scholar
- Schwartz, L. (1951).Theorie des Distributions, Actualities Scientifiques et Industrielles, Vol. II, No. 1122, p. 137 (see also Vol. I, No. 1245,1957);Mathematics for the Physical Sciences (1966). Paris-Hermann.Google Scholar
- Yamanouchi, T. (1970).Mathematics Applied to Physics (edited by G. A. Deschampset al.), LC-78-79553, Chapter X, p. 595. Springer-Verlag, New York.Google Scholar