Non-relativistic limit analysis of the Chandrasekhar-Thorne relativistic Euler equations with physical vacuum

Our results provide a first step to make the formal analysis rigorous in terms of 1 c2 proposed by Chandrasekhar [S. Chandrasekhar, The post-Newtonian equations of hydrodynamics in general relativity, Astrophys. J. 142 (1965) 1488-1512; S. Chandrasekhar, post-Newtonian equations of hydrodynamics and the stability of gaseous masses in general relativity, Phys. Rev. Lett. 14 (1965) 241-244], motivated by the methods of Einstein, Infeld and Hoffmann, see Thorne [K. S. Thorne, The general-relativistic theory of stellar structure and dynamics, in Proc. Int. School of Physics "Enrico Fermi," Course XXXV, at Varenna, Italy, July 12-24, 1965, ed. L. Gratton (Academic Press, 1966), pp. 166-280]. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light c and suitably small time T, we obtain uniform, with respect to c, "a priori" estimates for the local smooth solutions. Moreover, the smooth solutions of the cylindrically symmetric relativistic Euler equations converge to the solutions of the classical compressible Euler equation at the rate of order 1 c2.