General-relativistic rotation laws in rotating fluid bodies

Abstract

We formulate new general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids. They have been used to re-derive, in the first post-Newtonian approximation, the well known geometric dragging of frames. We derive two other general-relativistic weak-field effects within rotating tori: the recently discovered dynamic anti-dragging and a new effect that measures the deviation from the Keplerian motion and/or the contribution of the fluids selfgravity. One can use the rotation laws to study the uniqueness and the convergence of the post-Newtonian approximations, and the existence of the post-Newtonian limits.

General-relativistic Rotation Laws in Rotating Fluid Bodies: Constant Linear Velocity

“…We found in [5] that the consistency of the formalism and the uniqueness of those solutions that possess the 1st post-Newtonian limit, lead to well defined numerical values of the coefficients in the rotation law (1). It is interesting that the definition (1) is rigid-there is no any parametric freedom left for those solutions that have the 1st post-Newtonian limit.…”

“…It was proven in [5] that if c 0 is the Newtonian hydrodynamic energy per unit mass, then the exact solution satisfies the first post-Newtonian (1PN) equations. We shall outline here the calculation.…”

“…The Keplerian rotation is related to the choice of δ = −1/3 and w 2 = GM , where M is a mass. The particular form of the expression (1 − 3δ)/(1 + δ) in the denominator of (5) follows from the condition that a rotating infinitely thin disk made of weightless dust in a Schwarzschild space-time satisfies exactly the Bernoulli equation and the Keplerian rotation law [5].…”

“…The general-relativistic rotation law employed in [5] equates the angular momentum per unit mass, given by (3), to a specific function j(Ω):…”

“…Quite recently two of us have found general-relativistic rotation curves j = j(Ω) [5] that in the nonrelativistic limit exactly coincide with all monomial rotation laws Ω 0 = w/r 2/(1−δ) , with the exception of the constant linear velocity case (−∞ ≤ δ ≤ 0, δ = −1, w = const). We obtained in particular the general-relativistic Keplerian rotation law that possesses the first post-Newtonian limit (1PN) and exactly encompasses the solution corresponding to the massless disk of dust in the Schwarzschild spacetime.…”