Perfect fluids from high power sigma-models (Q2891195)

A perfect fluid on a space-time \((M,g)\) is a triple \((U,\rho,p)\) such that: (i) \(U\) is a time-like future-pointing unit vector field on \(M\), called the flow vector field, (ii) \(\rho, p:M\to\mathbb R\) are the mass (energy) density and pressure, respectively, (iii) \(\text{div}\,T=0\), where \(T\) is the stress-energy tensor given by \(T=p\,g+(p+\rho)\,\omega\otimes\omega\), with the \(1\)-form \(\omega=U^\flat\) metrically equivalent to \(U\).NEWLINENEWLINEThe fact that perfect fluid dynamics admits a Lagrangian formulation is widely known. The Lagrangian is given in terms of a dual variable \(\varphi\) that is a submersion defined on the space-time such that the fluid velocity four-vector spans its vertical foliation. It is interesting to notice that the sextic Lagrangian is exactly the extension of the Skyrme model. In fact, the equations for all perfect fluids with cosmological equation of state are generated in a similar way by raising to a proper power this sextic Lagrangian.NEWLINENEWLINEIn this paper, the author presents these constructions in the standard differential geometric setup. The following topics are addressed: the geometric interpretation of the Euler equations for the fluid equations for the field, the effect of conformal change of metrics, the irrational case, the sheer-free case in relation with the harmonic morphisms theory, and the coupling with gravity.