Post-Newtonian Methods and Conservation Laws

Abstract

The present paper is concerned with the conservation laws in general relativity as expressed in terms of the Landau-Lifshitz complex and their role in the development of the successive post-Newtonian approximations to the equations of general relativity of a perfect fluid. In § I, the known conservation laws of Newtonian hydrodynamics are formulated in a language that the relativistic laws appear as natural generalizations. In § II, the same laws are considered in the framework of general relativity. In particular the conserved energy is identified as the difference between the (0,0)-component of the Landau-Lifshitz complex and the conserved rest-mass energy ($={{c}^{2}}p{{u}^{0}}\sqrt{-g}$). In § III, the development of the first and the second post-Newtonian approximations to the equations of relativistic hydrodynamics are described and illustrated. And finally in § IV, the manner in which one can obtain the equations of the 2 1/2 — post-Newtonian approximation is described. In this approximation all terms inclusive of O(c⁻⁵) beyond the Newtonian are retained; it is in this approximation that terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance. It is shown how the derived radiation-reaction terms of O(c⁻⁵) contribute to the dissipation of energy and angular momentum in agreement with the predictions of the linearized theory of gravitational radiation.