General Relativistic Hydrodynamics: The Comoving, Eulerian, And Velocity Potential Formalism
L. Smarr, C. Taub, and J. R. Wilson, in Essays in General Relativity: A Festschrift for Abraham Taub, ed. F. Tipler, (Academic Press) (1980).
Abstract
We compare the three major approaches to general relativistic hydrodynamics: comoving. Eulerian. or mixed Euler-Lagrange. and velocity potential. We show that the comoving gauge has three major disadvantages: (1) shocks cause discontinuities in the spacetime metric; (2) the formation of black holes can lead to time-slicing singularities; (3) coordinate lines can become tangled in nonspherical flows. The mixed Euler-Lagrange scheme we use solves these three problems. We show how the comoving limit is obtained from our general equations. The circulation of a fluid flow in the presence of shocks is discussed and illustrated by numerical example. We suggest a possible relevance for this to supernova ejection mechanisms. Finally, the velocity potential formalism is discussed and compared to the comoving and Eulerian approaches. Taub’s contributions are stressed throughout the discussion.