General relativistic hydrodynamics in multiple coordinate systems

In this paper, the general relativistic hydrodynamic equations of a thermally conducting, viscous and compressible fluid in multiple coordinate systems are deduced in terms of the scheme developed by Damour, Sofel, and Xu (DSX scheme). Our paper is the first one to describe the hydrodynamic equations of a nonperfect fluid in every local coordinate system at the first post-Newtonian approximation of Einstein’s theory of gravity. The hydrodynamic equations in local coordinate systems are useful for calculating multipole moments of post-Newtonian N-body problems in the DSX scheme. Therefore, this paper is a supplement to the DSX scheme in some meaning. The corresponding PN thermodynamic equations in local coordinate systems are also represented. Lastly, some remarks on the possible applications are mentioned.