Stationary tensor fields in relativistic continuum mechanics

Summary

A tensor field in relativistic continuum mechanics is said to be stationary iff its Jaumann derivative with respect to the timelike unit flow vector vanishes. The gravitational potentials, geodesic flow are always stationary. Ifg _ab is the metric tensor, then the 3-dimensional projection operatorg _ab −u _a u _b is found to be stationary when and only when the first curvature of the streamline vanishes. The necessary and sufficient conditions for the stationary character of the 2-dimensional projection operatorg _ab −u _a u _b +P _a P _b are found to beK ₂+(1/2)(_γ123−_γ123)=0,_γ124=_γ123, whereP _a is the unit acceleration vector field,K ₂ is the second curvature or torsion and_γABC are the Ricci-coefficients of rotation formed from the relativistic Serret-Frenet formulae for the stream line of a particle in the continuum. The nonstationary character of the relativistic Serret-Frenet tetrad is also established. Further it is shown thatJ _u J _u (g _ab −u _a u _b )=0 iffK ₁=0,_γ124, whereJ _u is the Jaumann transport operator.