Erratum: Thermodynamics of binary black holes and neutron stars [Phys. Rev. D65, 064035 (2002)]

Abstract

We consider compact binary systems, modeled in general relativity as vacuum or perfect-fluid spacetimes with a helical Killing vector k α , heuristically, the generator of time-translations in a corotating frame. Systems that are stationary in this sense are not asymptotically flat, but have asymptotic behavior corresponding to equal amounts of ingoing and outgoing radiation. For blackhole binaries, a rigidity theorem implies that the Killing vector lies along the horizon's generators, and from this one can deduce the zeroth law (constant surface gravity of the horizon). Remarkably, although the mass and angular momentum of such a system are not defined, there is an exact first law, relating the change in the asymptotic Noether charge to the changes in the vorticity, baryon mass, and entropy of the fluid, and in the area of black holes.Binary systems with M Ω small have an approximate asymptopia in which one can write the first law in terms of the asymptotic mass and angular momentum. Asymptotic flatness is precise in two classes of solutions used to model binary systems: spacetimes satisfying the post-Newtonian equations, and solutions to a modified set of field equations that have a spatially conformally flat metric. (The spatial conformal flatness formalism with helical symmetry, however, is consistent with maximal slicing only if replaces the extrinsic curvature in the field equations by an artificially tracefree expression in terms of the shift vector.) For these spacetimes, nearby equilibria whose stars have the same vorticity obey the relation δM = ΩδJ, from which one can obtain a turning point criterion that governs the stability of orbits.

Circular orbits and spin in black-hole initial data

“…The EP method was originally applied only to configurations in which the direction and magnitude of the spin was also held constant along sequences However, when we consider sequences of corotating binaries, the magnitude of the spins of the black holes are no longer held constant and we must reconsider how we will fix the mass freedom. From Friedman et al [29] and the results shown above, is seems clear that we should hold the irreducible mass (or the area of the apparent horizon) fixed along sequences of configurations. We will define the total mass m and reduced mass as…”

“…Although corotation is not considered to be an astrophysically realistic state for black-hole binaries, it is an important test case because it is the one configuration for black-hole binaries that is compatible with a true helical Killing vector [29,37,38]. The thermodynamic relations obtained by Friedman et al [29],…”

“…Refs. [7,29] and references therein). These arguments show that a system in quasiequilibrium necessarily satisfies the Komar-mass ansatz.…”

“…(32) [29,[37][38][39][40][41]. A serious fault with such spacetimes is that they cannot be asymptotically flat since they contain a balancing amount of incoming and outgoing radiation for all time.…”

Sources of Gravitational Waves: Theory and Observations

“…The redshift has been computed numerically through a GSF calculation as a function of the (gauge invariant) orbital frequency and it has been compared to analytical predictions [146,150,151] and used to extract yet unknown higher-order PN terms beyond the test-particle limit [146,147,149,259,260]. The latter was possible through the first-law of binary BH dynamics [147,260,261]. Quite importantly, the redshift factor has been shown to be simply related to the binding energy and angular momentum of circular-orbit binaries [260].…”