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Explore Spacetimes, Metrics & Symmetries in Relativity Theory
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[QUOTE="vanhees71, post: 6860912, member: 260864"] What I mean is the treatment of relativistic hydro as shown in my manuscript for SR [URL]https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf[/URL] I got this description from the great book by D. E. Soper, Classical field theory, Dover Publications, Minneola, New York (2008). It's a not so well-known approach, which I find however very illuminating, because it treats the fluid analogous to the mechanics of point particles. Of course at the end it's equivalent with the standard approach, leading directly to the equations of motion by using the (local) conservation of energy-momentum and particle number (or conserved charges). The idea is to start with a "Lagrangian description", i.e., you define the fluid as the motion of "fluid elements". In a "standard configuration". For a fluid it's most simple to take the initial positions ##(\xi^A)##, with ##A \in \{1,2,3 \}## of the fluid elements. You can take the ##\xi^A## to denote the Cartesian coordinates for the position with respect to an arbitrary (inertial) reference frame. Then the fluid in the initial state is described by a particle-number density ##\tilde{n}(\vec{\xi})##. The motion of the fluid elements is then simply described by ##x^{\mu}(s,\vec{\xi})##, where ##s## is an arbitrary world-line parameter, which can be conveniently chosen as the proper time of the fluid element, i.e., $$\partial_s x^{\mu} \partial_s x_{\nu}=u^{\mu} u_{\mu}=1.$$ Of course also the coordinate time ##t## can be used instead of ##s##, and thus one can define the fluid motion as well by the Eulerian description, i.e., you define the ##\xi^A## as functions of ##x=(x^{\mu})##: ##\xi^{A}(x)## then are the ##\xi^{A}## of the fluid element in the standard configuration, which at time ##t## is at position ##\vec{x}##. Now the ##\xi^{A}(x)## are three scalar fields. For the further treatment of fluid dynamics, using this "kinematics" in the Lagrangian action principle, see the above quoted manuscript (Sect. 3.5). [/QUOTE]
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