Effects Of Gravitational Radiation Reaction Research Articles

Post-Newtonian dynamics in dense star clusters: Binary black holes in the LISA band

The dynamical processing of black holes in the dense cores of globular clusters (GCs), makes them efficient factories for producing binary black holes (BBHs). Here we explore the population of BBHs that form dynamically in GCs and may be observable at mHz frequencies or higher with LISA. We use our Monte Carlo stellar dynamics code, which includes gravitational radiation reaction effects for all BH encounters. By creating a representative local universe of GCs, we show that up to dozens of these systems may be resolvable by LISA with signal-to-noise ratios of at least 5. Approximately one third of these binaries will have measurable eccentricities ($e > 10^{-3}$) in the LISA band and a small number ($\lesssim 5$) may evolve from the LISA band to the LIGO band during the LISA mission.

Authenticating the Presence of a Relativistic Massive Black Hole Binary in OJ 287 Using Its General Relativity Centenary Flare: Improved Orbital Parameters

Abstract Results from regular monitoring of relativistic compact binaries like PSR 1913+16 are consistent with the dominant (quadrupole) order emission of gravitational waves (GWs). We show that observations associated with the binary black hole (BBH) central engine of blazar OJ 287 demand the inclusion of gravitational radiation reaction effects beyond the quadrupolar order. It turns out that even the effects of certain hereditary contributions to GW emission are required to predict impact flare timings of OJ 287. We develop an approach that incorporates this effect into the BBH model for OJ 287. This allows us to demonstrate an excellent agreement between the observed impact flare timings and those predicted from ten orbital cycles of the BBH central engine model. The deduced rate of orbital period decay is nine orders of magnitude higher than the observed rate in PSR 1913+16, demonstrating again the relativistic nature of OJ 287's central engine. Finally, we argue that precise timing of the predicted 2019 impact flare should allow a test of the celebrated black hole “no-hair theorem” at the 10% level.

Binary black hole coalescence in semianalytic puncture evolution

Binary black hole coalescence is treated semianalytically by a novel approach. Our prescription employs the conservative Skeleton Hamiltonian that describes orbiting Brill-Lindquist wormholes (termed punctures in numerical relativity) within a waveless truncation to the Einstein field equations [G. Faye, P. Jaranowski, and G. Sch\"afer, Phys. Rev. D 69, 124029 (2004)]. We incorporate, in a transparent Hamiltonian way and in Burke-Thorne gauge structure, the effects of gravitational radiation reaction into the above Skeleton dynamics with the help of 3.5PN accurate angular momentum flux for compact binaries in quasicircular orbits to obtain a semianalytic puncture evolution to model merging black hole binaries. With the help of the TaylorT4 approximant at 3.5PN order, we perform a first-order comparison between gravitational-wave phase evolutions in numerical relativity and our approach for equal-mass binary black holes. This comparison reveals that a modified Skeletonian reactive dynamics that employs flexible parameters will be required to prevent the dephasing between our scheme and numerical relativity, similar to what is pursued in the effective one-body approach. A rough estimate for the gravitational waveform associated with the binary black hole coalescence in our approach is also provided.

Evolution of the Carter constant for inspirals into a black hole: Effect of the black hole quadrupole

We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment $Q$ to linear order in $Q$, to the leading post-Newtonian order, and to linear order in the mass ratio. This system admits three constants of the motion in absence of radiation reaction: energy, angular momentum along the symmetry axis, and a third constant analogous to the Carter constant. We compute instantaneous and time-averaged rates of change of these three constants. For a point particle orbiting a black hole, Ryan has computed the leading order evolution of the orbit's Carter constant, which is linear in the spin. Our result, when combined with an interaction quadratic in the spin (the coupling of the black hole's spin to its own radiation reaction field), gives the next to leading order evolution. The effect of the quadrupole, like that of the linear spin term, is to circularize eccentric orbits and to drive the orbital plane towards antialignment with the symmetry axis. In addition we consider a system of two point masses where one body has a single mass multipole or current multipole of order $l$. To linear order in the mass ratio, to linear order in the multipole, and to the leading post-Newtonian order, we show that there does not exist an analog of the Carter constant for such a system (except for the cases of an $l=1$ current moment and an $l=2$ mass moment). Thus, the existence of the Carter constant in Kerr depends on interaction effects between the different multipoles. With mild additional assumptions, this result falsifies the conjecture that all vacuum, axisymmetric spacetimes possess a third constant of the motion for geodesic motion.

Three‐Body Dynamics with Gravitational Wave Emission

We present numerical three-body experiments that include the effects of gravitational radiation reaction by using equations of motion that include the 2.5-order post-Newtonian force terms, which are the leading order terms of energy loss from gravitational waves. We simulate binary-single interactions and show that close approach cross sections for three 1 solar mass objects are unchanged from the purely Newtonian dynamics except for close approaches smaller than 1.0e-5 times the initial semimajor axis of the binary. We also present cross sections for mergers resulting from gravitational radiation during three-body encounters for a range of binary semimajor axes and mass ratios including those of interest for intermediate-mass black holes (IMBHs). Building on previous work, we simulate sequences of high-mass-ratio three-body encounters that include the effects of gravitational radiation. The simulations show that the binaries merge with extremely high eccentricity such that when the gravitational waves are detectable by LISA, most of the binaries will have eccentricities e > 0.9 though all will have circularized by the time they are detectable by LIGO. We also investigate the implications for the formation and growth of IMBHs and find that the inclusion of gravitational waves during the encounter results in roughly half as many black holes ejected from the host cluster for each black hole accreted onto the growing IMBH.

Motion and gravitational radiation of a binary system consisting of an oscillating and rotating coplanar dusty disk and a point-like object

A binary system composed of an oscillating and rotating coplanar dusty disk and a point mass is considered. The conservative dynamics is treated on the Newtonian level. The effects of gravitational radiation reaction and wave emission are studied to leading quadrupole order. The related waveforms are given. The dynamical evolution of the system is determined semi-analytically exploiting the Hamiltonian equations of motion which comprise the effects both of the Newtonian tidal interaction and the radiation reaction on the motion of the binary system in elliptic orbits. Tidal resonance effects between orbital and oscillatory motions are considered in the presence of radiation damping.

Resonance behaviour and partial averaging in a three-body system with gravitational radiation damping

In a previous investigation, a model of three-body motion was developed which included the effects of gravitational radiation reaction. The aim was to describe the motion of a relativistic binary pulsar that is perturbed by a third mass and look for resonances between the binary and third mass orbits. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. These $(m:n)$ resonances are defined for the present purposes by the resonance condition, $m\omega=2n\Omega$, where $m$ and $n$ are relatively prime integers and $\omega$ and $\Omega$ are the angular frequencies of the binary orbit and third mass orbit, respectively. The resonance condition consequently fixes a value for the semimajor axis $a$ of the binary orbit for the duration of the resonance because of the Kepler relationship $\omega=a^{-3/2}$. This paper outlines a method of averaging developed by Chicone, Mashhoon, and Retzloff which renders a nonlinear system that undergoes resonance capture into a mathematically amenable form. This method is applied to the present system and one arrives at an analytical solution that describes the average motion during resonance. Furthermore, prominent features of the full nonlinear system, such as the frequency of oscillation and antidamping, accord with their analytically derived formulae.

Gravitational radiation damping and the three-body problem

A model of three-body motion is developed which includes the effects of gravitational radiation reaction. The radiation reaction due to the emission of gravitational waves is the only post-Newtonian effect that is included here. For simplicity, all of the motion is taken to be planar. Two of the masses are viewed as a binary system and the third mass, whose motion will be a fixed orbit around the center-of-mass of the binary system, is viewed as a perturbation. This model aims to describe the motion of a relativistic binary pulsar that is perturbed by a third mass. Numerical integration of this simplified model reveals that given the right initial conditions and parameters one can see resonances. These (m,n) resonances are defined by the resonance condition, $m\omega=2n\Omega$, where $m$ and $n$ are relatively prime integers and $\omega$ and $\Omega$ are the angular frequencies of the binary orbit and third mass orbit, respectively. The resonance condition consequently fixes a value for the semimajor axis of the binary orbit for the duration of the resonance; therefore, the binary energy remains constant on the average while its angular momentum changes during the resonance.

Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. II. Two-body equations of motion to second post-Newtonian order, and radiation reaction to 3.5 post-Newtonian order

We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in terms of multipole moments of the system and their time derivatives. All potentials are well defined and free of divergences associated with integrating quantities over all space. Using a model of the bodies as spherical, non-rotating fluid balls whose characteristic size s is small compared to the bodies' separation r, we develop a method for carefully extracting only terms that are independent of the parameter s, thereby ignoring tidal interactions, spin effects, and internal self-gravity effects. Through 2.5PN order, the resulting equations agree completely with those obtained by other methods; the new 3.5PN back-reaction results are shown to be consistent with the loss of energy and angular momentum via radiation to infinity.

Gravitational radiation reaction to a particle motion

In this paper, we discuss the leading order correction to the equation of motion of the particle, which presumably describes the effect of gravitational radiation reaction. We derive the equation of motion in two different ways. The first one is an extension of the well-known formalism by DeWitt and Brehme developed for deriving the equation of motion of an electrically charged particle. In contrast to the electromagnetic case, in which there are two different charges, i.e., the electric charge and the mass, the gravitational counterpart has only one charge. This fact prevents us from using the same renormalization scheme that was used in the electromagnetic case. To make clear the subtlety in the first approach, we then consider the asymptotic matching of two different schemes, i.e., the internal scheme in which the small particle is represented by a spherically symmetric black hole with tidal perturbations and the external scheme in which the metric is given by small perturbations on the given background geometry. The equation of motion is obtained from the consistency condition of the matching. We find that in both ways the same equation of motion is obtained. The resulting equation of motion is analogous to that derived in the electromagnetic case. We discuss implications of this equation of motion.

Erratum: Gravitational waves from inspiralling compact binaries: Energy loss and waveform to second-post-Newtonian order

Gravitational waves generated by inspiralling compact binaries are investigated to the second--post-Newtonian (2PN) approximation of general relativity. Using a recently developed 2PN-accurate wave generation formalism, we compute the gravitational waveform and associated energy loss rate from a binary system of point-masses moving on a quasi-circular orbit. The crucial new input is our computation of the 2PN-accurate ``source'' quadrupole moment of the binary. Tails in both the waveform and energy loss rate at infinity are explicitly computed. Gravitational radiation reaction effects on the orbital frequency and phase of the binary are deduced from the energy loss. In the limiting case of a very small mass ratio between the two bodies we recover the results obtained by black hole perturbation methods. We find that finite mass ratio effects are very significant as they increase the 2PN contribution to the phase by up to 52\%. The results of this paper should be of use when deciphering the signals observed by the future LIGO/VIRGO network of gravitational-wave detectors.

Effect of gravitational radiation reaction on nonequatorial orbits around a Kerr black hole.

The effect of gravitational radiation reaction on orbits around a spinning black hole is analyzed. Such orbits possess three constants of motion: $\ensuremath{\iota}$, $e$, and $a$, which correspond, in the Newtonian limit of the orbit being an ellipse, to the inclination angle of the orbital plane to the hole's equatorial plane, the eccentricity, and the semimajor axis length, respectively. First, it is argued that circular orbits ($e=0$) remain circular under gravitational radiation reaction. Second, for elliptical orbits (removing the restriction of $e=0$), the evolution of $\ensuremath{\iota}$, $e$, and $a$ is computed to leading order in $S$ (the magnitude of the spin angular momentum of the hole) and in $\frac{M}{a}$, where $M$ is the mass of the black hole. As $a$ decreases, $\ensuremath{\iota}$ increases and $e$ decreases.

Binary-Single Encounters of 10 M⊙ Black Holes Including the Effects of Gravitational Radiation

We compute the outcomes of close encounters between a binary and a single black holes including the effects of gravitational radiation reaction. All masses of individual black holes are assumed to be 1 M⊙. We found that merger of two black holes takes place during the encounters in some cases. Thus the gravitational radiation can act as a mechanism for the dissipation of energy of a cluster mainly composed of 10 M⊙ black holes which are produced by the evolution of high mass stars. The merger probability depends on many parameters in a complex way. Our preliminary calculations show that about 10% of the strong encounters (i.e., rp ∼ a) between a binary of hardness 100 and a single lead to mergers of two black holes in the stellar system of one-dimensional velocity σ = 100 km/s.

Effect of gravitational radiation reaction on circular orbits around a spinning black hole.

The effect of gravitational radiation reaction on circular orbits around a spinning (Kerr) black hole is computed to leading order in S (the magnitude of the spin angular momentum of the hole) and in the strength of gravity M/r (where M is the mass of the black hole, r is the orbital radius, and G=c=1). The radiation reaction makes the orbit shrink but leaves it circular and drives the orbital plane very slowly toward antialignment with the spin of the hole: tan(\ensuremath{\iota}/2)=tan(${\mathrm{\ensuremath{\iota}}}_{0}$/2)[1+(61/72)(S/${\mathit{M}}^{2}$)(M/r${)}^{3/2}$], where \ensuremath{\iota} is the angle between the normal to the orbital plane and the spin direction, and ${\mathrm{\ensuremath{\iota}}}_{0}$ is the initial value of \ensuremath{\iota}, when r is very large.

Gravitational waves from inspiralling compact binaries: Energy loss and waveform to second-post-Newtonian order.

Gravitational waves generated by inspiralling compact binaries are investigated to the second-post-Newtonian (2PN) approximation of general relativity. Using a recently developed 2PN-accurate wave generation formalism, we compute the gravitational waveform and associated energy loss rate from a binary system of point masses moving on a quasicircular orbit. The crucial new input is our computation of the 2PN-accurate ``source'' quadrupole moment of the binary. Tails in both the waveform and energy loss rate at infinity are explicitly computed. Gravitational radiation reaction effects on the orbital frequency and phase of the binary are deduced from the energy loss. In the limiting case of a very small mass ratio between the two bodies we recover the results obtained by black hole perturbation methods. We find that finite mass ratio effects are very significant as they increase the 2PN contribution to the phase by up to 52%. The results of this paper should be of use when deciphering the signals observed by the future LIGO-VIRGO network of gravitational-wave detectors.

Time-asymmetric structure of gravitational radiation.

Gravitational radiation reaction effects in the dynamics of an isolated system arise from the use of retarded potentials for the radiation field, satisfying time-asymmetric boundary conditions imposed at past-null infinity. Part one of this paper investigates the ``antisymmetric'' component, a solution of the wave equation of the type retarded minus advanced, of the linearized gravitational field generated by an isolated system in the exterior region of the system. At linearized order such a component is well defined and is ``time odd'' in the usual post-Newtonian (PN) sense. We introduce a new linearized coordinate system which generalizes the Burke and Thorne coordinate system both in its space-time domain of validity, which is no longer limited to the near zone of the source, and in the post-Newtonian smallness of the linear antisymmetric (``time-odd'') component of the metric, for all multipolarities of antisymmetric waves. These waves (as viewed in the near zone) define a generalized radiation reaction four-tensor potential ${\mathit{V}}_{\mathrm{react}}^{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\beta}}}$ of the linear theory.At the 2.5 post-Newtonian approximation, the tensor potential reduces to the standard Burke-Thorne scalar potential of the lowest-order local radiation reaction force. At the 3.5 PN approximation, the potential involves scalar (${\mathit{V}}_{\mathrm{react}}^{00}$) and vector (${\mathit{V}}_{\mathrm{react}}^{0\mathit{i}}$) components which are associated with subdominant radiation reaction effects such as the recoil effect. At the higher-order PN approximations, the potential is intrinsically tensorial. A nonlinear exterior metric is iteratively constructed from the new linearized metric by the method of a previous work. Part two of this paper is devoted to the near-zone reexpansion of the nonlinear iterations of the exterior metric. We use a very convenient decomposition of the integral of the retarded potentials into a particular solution involving only ``instantaneous'' potentials, and a homogeneous solution of the antisymmetric type. The former particular solution is ``even'' in the sense that it explicitly contains only even powers of ${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$. The latter homogeneous solution defines a component of the exterior metric which is associated with radiation reaction effects of nonlinear origin. This decomposition of the retarded integral enables us to control the occurrence and the magnitude of ``odd'' terms in any nonlinear iterations of the metric, and to compute explicitly the radiation reaction potential of the nonlinear theory up to the 3.5 PN approximation. Finally we recover and complete a previous work concerning the hereditary modification, of quadratic nonlinear origin, of the radiation reaction potential at the 4 PN approximation.

Nonaxisymmetric normal modes and secular instabilities of rotating stars

A variational principle is used to study the normal modes of oscillation of rotating polytropes. Sequences of uniformly rotating stars are modelled by polytropes in adiabatic equilibrium with polytropic indices of l.0 and 1.5. The effects of rapid uniform rotation are fully accounted for. The stability analysis is performed for nonaxisymmetric normal modes with values of the azimuthal wavenumber ranging from zero to five. The frequencies for these modes are presented, and the eigenfunctions are described. The dissipative time scales due to gravitational radiation reaction (GRR) are presented, and the effect of GRR on the critical angular velocities for the onset of secular instability in the presence of viscosity is discussed.

An Eulerian variational principle and a criterion for the occurrence of nonaxisymmetric neutral modes along rotating axisymmetric sequences

view Abstract Citations (36) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS An Eulerian variational principle and a criterion for the occurrence of nonaxisymmetric neutral modes along rotating axisymmetric sequences Ipser, J. R. ; Managan, R. A. Abstract In the theory of self-gravitating fluid configurations, an important development is related to the discovery that general-relativistic effects of gravitational radiation reaction induce secular instability at certain nonaxisymmetric points of bifurcation along Newtonian sequences of axisymmetric rotating equilibria. Such a point of bifurcation is a point of neutral stability. The present investigation has the objective to develop a new method for isolating nonaxisymmetric bifurcation points. The basis is a derivation of an 'Eulerian' variational principle for neutral, or zero-frequency, nonaxisymmetric modes of oscillation of axisymmetric configurations. The equations governing fluid configurations are discussed, taking into account the equations governing the unperturbed axisymmetric equilibrium configuration, and the equations governing the perturbed configuration. Attention is given to the variational principle, the method for locating Dedekind bifurcation points, and Dedekind and Jacobi bar modes of uniformly rotating configurations. Publication: The Astrophysical Journal Pub Date: May 1985 DOI: 10.1086/163185 Bibcode: 1985ApJ...292..517I Keywords: Computational Astrophysics; Stellar Oscillations; Stellar Rotation; Variational Principles; Branching (Mathematics); Inviscid Flow; Perturbation Theory; Points (Mathematics); Astrophysics full text sources ADS |

On the secular instabilities of the Maclaurin spheroids

The combined effects of gravitational radiation reaction and of viscosity on the stability of the Maclaurin spheroids are discussed. Each of these dissipative effects is known to induce a secular instability in the Maclaurin sequence past the point of bifurcation of the Jacobi and the Dedekind sequences. We find, however, that when both effects are considered together, these instabilities tend to cancel each other. The sequence of stable Maclaurin spheroids therefore reaches past the bifurcation point to a new point determined by the ratio of the strengths of the viscous and the radiative forces.

Gravitational radiation-reaction effects in a star-planet system

We consider a star-planet system in which the planet's circular orbit shows a "precession of the nodes" caused by a mass quadrupole moment in the star. We calculate the radiation-reaction effects of the gravitational waves emitted by the system. Besides the well-known gradual fall of the planet towards the star, we find a change in the angle between the equatorial plane of the star and the orbital plane of the planet. For oblate stars the angle decreases to zero; for prolate stars it tends toward $\frac{\ensuremath{\pi}}{2}$.