The 2½-post-Newtonian equations of hydrodynamics and radiation reaction in general relativity

Abstract

In this paper the equations of hydrodynamics in the 2½-post-Newtonian approximation to general relativity are derived. In this approximation all terms of O(c^-5) are retained consistently with Einstein's field equations; it is also the approximation in which terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance. The paper is in four parts. In Part I (by S. C.) the lowest-order terms in the metric coefficients are derived which are consequences of the imposition of the Sommerfeld radiation-condition at infinity. It is shown (following an early investigation of Trautman) that these terms are of O(c^-5) in g₀₀, of O(c^-6) in g_0α, and of O(c^-5) in g_αβ. Unique expressions are obtained for these terms. They are found to be purely of Newtonian origin. In Part II (by S. C. and F. P. E.) the equations of motion governing the fluid in the 2½-post-Newtonian approximation are derived. In addition to the coefficients already determined, these equations depend on a knowledge of the term of O(c^-7) in g_oo. This term is determined by an explicit appeal to the field equation. It is further shown that this approximation brings no change to the density (c²ρμ⁰√-g) and the linear momentum (π_α) that are conserved in the second post-Newtonian approximation. In Part III (by S. C.) it is shown that the terms of O(c^-5) in the equations of motion contribute principally to the dissipation of the energy and the angular momentum conserved in the second post-Newtonian approximation. The rates of dissipation of energy and of angular momentum that are predicted are in exact agreement with the expectations of the linearized theory of gravitational radiation. Finally, in Part IV (by S. C. and F. P. E.) the energy, θ⁰⁰-c²ρμ⁰√-g, to be associated with the 2½-post-Newtonian approximation is derived by evaluating the (0, 0)-component of the Landau-Lifshitz complex and the conserved density in the 3½-post-Newtonian approximation.