Newtonian gravitational potential sourced by a homogeneous circular ring in arbitrary dimensional Euclidean space takes a simple form if the spatial dimension is even. In contrast, if the spatial dimension is odd, it is given in a form that includes complete elliptic integrals. In this paper, we analyze the dynamics of a freely falling massive particle in its Newtonian potential. Focusing on circular orbits on the symmetric plane where the ring is placed, we observe that they are unstable in 4D space and above, while they are stable in 3D space. The sequence of stable circular orbits disappears at $1.6095\cdots$ times the radius of the ring, which corresponds to the innermost stable circular orbit (ISCO). On the axis of symmetry of the ring, there are no circular orbits in 3D space but more than in 4D space. In particular, the circular orbits are stable between the ISCO and infinity in 4D space and between the ISCO and the outermost stable circular orbit in 5D space. There exist no stable circular orbits in 6D space and above.
The dynamics of the planar circular restricted three-body problem with Kerr-like primaries in the context of a beyond-Newtonian approximation is studied. The beyond-Newtonian potential is developed by using the Fodor-Hoenselaers-Perjés procedure. An expansion in the Kerr potential is performed and terms up-to the first non-Newtonian contribution of both the mass and spin effects are included. With this potential, a model for a test particle of infinitesimal mass orbiting in the equatorial plane of the two primaries is examined. The introduction of a parameter, $ε$, allows examination of the sy…
We study the dynamics of the planar circular restricted three-body problem in the context of a pseudo-Newtonian approximation. By using the Fodor-Hoenselaers-Perjés procedure, we perform an expansion in the mass potential of a static massive spherical source up to the first non-Newtonian term, giving place to a gravitational potential that includes first-order general relativistic effects. With this result, we model a system composed by two pseudo-Newtonian primaries describing circular orbits around their common center of mass, and a test particle orbiting the system in the equatorial plane. …
In Newtonian physics or in general relativity, energy dissipation causes observers moving along circular orbits to slowly spiral towards the source of the gravitational field. We show that the loss of energy has the same effect in any theory of gravity respecting the weak equivalence principle, by exhibiting an intimate relation between the energy of a massive test particle and the stability of its orbit. Ultimately, massive particles either plunge, or are driven towards minima of the generalized Newtonian potential, where they become static. In addition, we construct a toy metric which displa…
The increasing sophistication and accuracy of numerical simulations of compact binaries (especially binary black holes) presents the opportunity to test the regime in which post-Newtonian (PN) predictions for the emitted gravitational waves are accurate. In order to confront numerical results with those of post-Newtonian theory, it is convenient to compare multipolar decompositions of the two waveforms. It is pointed out here that the individual modes can be computed to higher post-Newtonian order by examining the radiative multipole moments of the system, rather than by decomposing the 2.5PN …
Using the post-Newtonian (PN) expansion technique of the gravitational wave perturbation around a Schwarzschild black hole, we calculate analytically the energy flux of gravitational waves induced by a particle in circular orbits up to the 5.5PN order, i.e. $O(v^{11})$ beyond Newtonian. By comparing the formula with numerical data, we find that the error of the 5.5PN formula is about 4% when the particle is on the last stable circular orbit. We also estimate the error $ΔN$ in the total cycle of gravitational waves from coalescing compact binaries in a laser interferometer's band produced by us…
The equations of motion of two point masses have recently been derived at the 3PN approximation of general relativity. From that work we determine the location of the innermost circular orbit or ICO, defined by the minimum of the binary's 3PN energy as a function of the orbital frequency for circular orbits. We find that the post-Newtonian series converges well for equal masses. Spin effects appropriate to corotational black-hole binaries are included. We compare the result with a recent numerical calculation of the ICO in the case of two black holes moving on exactly circular orbits (helical …
Post-Newtonian expansion of gravitational waves from a particle in circular orbits around a rotating black hole: Up to $O(v^8)$ beyond the quadrupole formula
本文推广了Sasaki在Schwarzschild情况下和Shibata，Sasaki，Tagoshi和Tanaka在Kerr情况下提出的方法，计算了从绕旋转黑洞的圆形轨道上的测试粒子到四极公式之外的$O(v^8)$的引力波光度的后牛顿展开。试验粒子的轨道被限制在赤道平面上。我们发现，自旋相关的条款出现在每个后牛顿的顺序，并在$O(v^6)$，他们有一个显着的影响轨道相的演变合并紧凑的双星。通过比较后牛顿公式和数值计算结果发现，对于$30M\lesssim r \lesssim 100M$，$O(v^6)$和$O(v^7)$处的自旋相关项显著提高了后牛顿公式的精度，而$O(v^8)$处的自旋相关项没有提高.
This paper investigates the coplanar and circular three-body problem in the parametrized post-Newtonian (PPN) formalism, for which we focus on a class of fully conservative theories characterized by the Eddington-Robertson parameters $β$ and $γ$. It is shown that there can still exist a collinear equilibrium configuration and a triangular one, each of which is a generalization of the post-Newtonian equilibrium configuration in general relativity. The collinear configuration can exist for arbitrary mass ratio, $β$, and $γ$. On the other hand, the PPN triangular configuration depends on the nonl…
A particle of mass $μ$ moves on a circular orbit around a nonrotating black hole of mass $M$. Under the assumption $μ\ll M$ the gravitational waves emitted by such a binary system can be calculated exactly numerically using black-hole perturbation theory. If, further, the particle is slowly moving, then the waves can be calculated approximately analytically, and expressed in the form of a post-Newtonian expansion. We determine the accuracy of this expansion in a quantitative way by calculating the reduction in signal-to-noise ratio incurred when matched filtering the exact signal with a nonopt…
In the present paper, using the first-order approximation of the $n$-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass/distance and velocity/speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to a good numerical conservation of the Jac…
The innermost stable circular orbits (ISCO) of coalescing neutron star-black hole binary are studied taking into account both the tidal and relativistic effects. We adopt the generalized pseudo-Newtonian potential to mimic the general relativistic effects of gravitation. It is found that the separation of the neutron star and the black hole at the ISCO is greater than that obtained by using either the Newtonian potential or the second post-Newtonian equation of motion of point mass. In equal mass cases, it is found that for $\bar{a}/m_1 > 3.5$ with $\bar{a}$ and $m_1$ being the mean radius and…
We compute the quadrupole mode of the gravitational waveform of inspiralling compact binaries at the third and a half post-Newtonian (3.5PN) approximation of general relativity. The computation is performed using the multipolar post-Newtonian formalism, and restricted to binaries without spins moving on quasi-circular orbits. The new inputs mainly include the 3.5PN terms in the mass quadrupole moment of the source, and the control of required subdominant corrections to the contributions of hereditary integrals (tails and non-linear memory effect). The result is given in the form of the quadrup…
We present a generalized Newtonian description of particle dynamics valid for any spherically symmetric, static black hole spacetime. This approach is derived from the geodesic motion of test particles in the low-energy limit. It reproduces exactly the location of the marginally stable, marginally bound, and photon circular orbits; the radial dependence of the energy and angular momentum of circular orbits; parabolic motion; pericentre shift; and the spatial projection of general trajectories. As explicit examples of the new prescription, we apply it to the Schwarzschild, Schwarzschild-de Sitt…
We extend our previous results of the 14th post-Newtonian (PN) order expansion of gravitational waves for a test particle in circular orbits around a Schwarzschild black hole to the 22PN order, i.e. $v^{44}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. Comparing our 22PN formula for the energy flux with high precision numerical results, we find that the relative error of the 22PN flux at the innermost stable circular orbit is about $10^{-5}$. We also estimate the phase difference between the 22PN waveforms and numerical waveforms after a two-…
We compute the non-linear memory contributions to the gravitational-wave amplitudes for compact binaries in eccentric orbits at the third post-Newtonian (3PN) order in general relativity. These contributions are hereditary in nature as they are sourced by gravitational waves emitted during the binary's entire dynamical past. Combining these with already available instantaneous and tail contributions we get the complete 3PN accurate gravitational waveform.
Gravitational radiation can be expressed in terms of an infinite series of radiative, symmetric trace-free (STF) multipole moments which can be connected to the behavior of the source. We consider a truncated model for gravitational radiation from binary systems in which each STF mass and current moment of order l is given by the lowest-order, Newtonian-like l-pole moment of the orbiting masses; we neglect post-Newtonian corrections to each STF moment. Specializing to orbits which are circular (apart from the radiation-induced inspiral), we find an explicit infinite series for the energy flux …
We report on the first calculations of fully relativistic binary circular orbits to span a range of separation distances from the innermost stable circular orbit (ISCO), deeply inside the strong field regime, to a distance ($\sim$ 200 km) where the system is accurately described by Newtonian dynamics. We consider a binary system composed of two identical corotating neutron stars, with 1.43 $M_\odot$ gravitational mass each in isolation. Using a conformally flat spatial metric we find solutions to the initial value equations that correspond to semi-stable circular orbits. At large distance, our…
We study how the neutron-star equation of state affects the onset of the dynamical instability in the equations of motion for inspiraling neutron-star binaries near coalescence. A combination of relativistic effects and Newtonian tidal effects cause the stars to begin their final, rapid, and dynamically-unstable plunge to merger when the stars are still well separated and the orbital frequency is $\approx$ 500 cycles/sec (i.e. the gravitational wave frequency is approximately 1000 Hz). The orbital frequency at which the dynamical instability occurs (i.e. the orbital frequency at the innermost …
In this article the quasi-Keplerian parameterisation for the case that spins and orbital angular momentum in a compact binary system are aligned or anti-aligned with the orbital angular momentum vector is extended to 3PN point-mass, next-to-next-to-leading order spin-orbit, next-to-next-to-leading order spin(1)-spin(2), and next-to-leading order spin-squared dynamics in the conservative regime. In a further step, we use the expressions for the radiative multipole moments with spin to leading order linear and quadratic in both spins to compute radiation losses of the orbital binding energy and …
The problem of a compact binary system whose components move on circular orbits is addressed using two different approximation techniques in general relativity. The post-Newtonian (PN) approximation involves an expansion in powers of v/c<<1, and is most appropriate for small orbital velocities v. The perturbative self-force (SF) analysis requires an extreme mass ratio m1/m2<<1 for the components of the binary. A particular coordinate-invariant observable is determined as a function of the orbital frequency of the system using these two different approximations. The post-Newtonian calculation i…
When a particle moves around a Kerr black hole, it radiates gravitational waves.Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass mu in a circular orbit on an equatorial plane around a Kerr black hole of mass M. We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/c)^8) compared to …
We study the accuracy of the post-Newtonian (PN) approximation and its formal region of validity, by investigating its optimal asymptotic expansion for the quasi-circular, adiabatic inspiral of a point particle into a Schwarzschild black hole. By comparing the PN expansion of the energy flux to numerical calculations in the perturbative Teukolsky formalism, we show that (i) the inclusion of higher multipoles is necessary to establish the accuracy of high-order PN terms, and (ii) the region of validity of PN theory is largest at relative O(1/c^6) (3PN order). The latter result suggests that the…
A Fokker action for post-Minkowski approximation with the first post-Newtonian correction is introduced in our previous paper, and a solution for the helically symmetric circular orbit is obtained. We present supplemental results for the circular solution of two unequal mass point-particles. Circular solutions for selected mass ratios are found numerically, and analytic formulas in the extreme mass ratio limit are derived. The leading terms of the analytic formulas agree with the first post-Newtonian formulas in this limit.
Recent breakthroughs in numerical relativity enable one to examine the validity of the post-Newtonian expansion in the late stages of inspiral. For the comparison between post-Newtonian (PN) expansion and numerical simulations, the waveforms in terms of the spin-weighted spherical harmonics are more useful than the plus and cross polarizations, which are used for data analysis of gravitational waves. Factorized resummed waveforms achieve better agreement with numerical results than the conventional Taylor expanded post-Newtonian waveforms. In this paper, we revisit the post-Newtonian expansion…
We consider the effective theory of perturbative quantum gravity coupled to a point particle, quantizing fluctuations of both the gravitational field and the particle's position around flat space. Using a recent relational approach to construct gauge-invariant observables, we compute one-loop graviton corrections to the invariant metric perturbation, whose time-time component gives the Newtonian gravitational potential. The resulting quantum correction consists of two parts: the first stems from graviton loops and agrees with the correction derived by other methods, while the second one is sou…
We compare different methods of computing the orbital eccentricity of quasi-circular binary black hole systems using the orbital variables and gravitational wave phase and frequency. For eccentricities of about a per cent, most methods work satisfactorily. For small eccentricity, however, the gravitational wave phase allows a particularly clean and reliable measurement of the eccentricity. Furthermore, we measure the decay of the orbital eccentricity during the inspiral and find reasonable agreement with post-Newtonian results. Finally, we measure the periastron advance of non-spinning binary …
The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. In the test-mass limit, well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes. In the finite-mass case, there are a large variety of ways to define an ISCO in a post-Newtonian (PN) context. Here I generalize the gauge-invariant ISCO condition of Blanchet & Iyer (2003) to the case of spinning (nonprecessing) binaries. The Blanchet-Iyer ISCO condition has two desirable and unexpected properties: (1) it exactly reproduces the Schwarzschild ISCO in …
The gravitational waveform (GWF) generated by inspiralling compact binaries moving in quasi-circular orbits is computed at the third post-Newtonian (3PN) approximation to general relativity. Our motivation is two-fold: (i) To provide accurate templates for the data analysis of gravitational wave inspiral signals in laser interferometric detectors; (ii) To provide the associated spin-weighted spherical harmonic decomposition to facilitate comparison and match of the high post-Newtonian prediction for the inspiral waveform to the numerically-generated waveforms for the merger and ringdown. This …
In this paper, we revisit and complete our preceding work on the Fokker Lagrangian describing the dynamics of compact binary systems at the fourth post-Newtonian (4PN) order in harmonic coordinates. We clarify the impact of the non-local character of the Fokker Lagrangian or the associated Hamiltonian on both the conserved energy and the relativistic periastron precession for circular orbits. We show that the non-locality of the action, due to the presence of the tail effect at the 4PN order, gives rise to an extra contribution to the conserved integral of energy with respect to the Hamiltonia…
We compute the energy flux of the gravitational waves radiated by a particle of mass $\m$ in circular orbits around a rotating black hole of mass $M$ up to the 11th post-Newtonian order (11PN), i.e. $v^{22}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of the particle. By comparing the PN results for the energy flux with high precision numerical results in black hole perturbation theory, we find the region of validity in the PN approximation becomes larger with increasing PN orders. If one requires the relative error of the energy flux in the PN approximation to…
[abridged] Barack & Sago have recently computed the shift of the innermost stable circular orbit (ISCO) due to the conservative self-force that arises from the finite-mass of an orbiting test-particle. This is one of the first concrete results of the self-force program, and provides an exact point of comparison with approximate post-Newtonian (PN) computations of the ISCO. Here this exact ISCO shift is compared with nearly all known PN-based methods. These include both "nonresummed" and "resummed" approaches (the latter reproduce the test-particle limit by construction). The best agreement wit…
We complete the computation of the gravitational waveform amplitude for non-spinning compact binaries to the third and a half post-Newtonian (PN) order in the quasi-circular case. This is done by performing a spin-weighted spherical harmonics decomposition of the amplitude. This computation is achieved using the post-Newtonian-Multipolar-post-Minkowskian formalism. The amplitude modes are written in a PN expanded way as well as a factorized form suitable for EOB template building. Combining this work with previous ones, we provide in the supplemental materials the expressions of the modes incl…
Standard choices of quasi-circular orbit parameters for black-hole binary evolutions result in eccentric inspiral. We introduce a conceptually simple method, which is to integrate the post-Newtonian equations of motion through hundreds of orbits, and read off the values of the momenta at the separation at which we wish to start a fully general relativistic numerical evolution. For the particular case of non-spinning equal-mass inspiral with an initial coordinate separation of $D = 11M$ we show that this approach reduces the eccentricity by at least a factor of five to $e < 0.002$ as compared t…
The article delivers the only still unknown coefficient in the 4th post-Newtonian energy expression for binary point masses on circular orbits as function of orbital angular frequency. Apart from a single coefficient, which is known solely numerically, all the coefficients are given as exact numbers. The shown Hamiltonian is presented in the center-of-mass frame and out of its 57 coefficients 51 are given fully explicitly. Those coefficients are six coefficients more than previously achieved [Jaranowski/Schäfer, Phys. Rev. D 86, 061503(R) (2012)]. The local divergences in the point-mass model …
We examine isothermal dark matter halos in hydrostatic equilibrium with a cosmological constant Lambda =Omega_Λrho_{crit}c^2, where Omega_Λ=0.7, and rho_{crit} is the present value of the critical density with h=0.65. The Newtonian limit of General Relativity yields equilibrium equations that are different from those arising by merely coupling an ``isothermal sphere'' to the Lambda-field within a Newtonian framework. The conditions for the existence and stability of circular geodesic orbits show the existence of (I) an ``isothermal region'' (0<r<r_2), circular orbits are stable and all variabl…
We continue a previous work on the comparison between the post-Newtonian (PN) approximation and the gravitational self-force (SF) analysis of circular orbits in a Schwarzschild background. We show that the numerical SF data contain physical information corresponding to extremely high PN approximations. We find that knowing analytically determined appropriate PN parameters helps tremendously in allowing the numerical data to be used to obtain higher order PN coefficients. Using standard PN theory we compute analytically the leading 4PN and the next-to-leading 5PN logarithmic terms in the conser…
We discuss the analytical determination of the location of the Last Stable Orbit (LSO) in circular general relativistic orbits of two point masses. We use several different ``resummation methods'' (including new ones) based on the consideration of gauge-invariant functions, and compare the results they give at the third post-Newtonian (3PN) approximation of general relativity. Our treatment is based on the 3PN Hamiltonian of Jaranowski and Schäfer. One of the new methods we introduce is based on the consideration of the (invariant) function linking the angular momentum and the angular frequenc…
We apply standard post-Newtonian methods in general relativity to locate the innermost circular orbit (ICO) of irrotational and corotational binary black-hole systems. We find that the post-Newtonian series converges well when the two masses are comparable. We argue that the result for the ICO which is predicted by the third post-Newtonian (3PN) approximation is likely to be very close to the ``exact'' solution, within 1% of fractional accuracy or better. The 3PN result is also in remarkable agreement with a numerical calculation of the ICO in the case of two corotating black holes moving on e…
Convergence of post-Newtonian for quasi-circular non-precessing comparable mass ratios BBHs
California Institute of Technology（加州理工学院）；Radboud University（拉德堡德大学）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；University of Mississippi（密西西比大学）
Two incorrect arguments against MOND in elliptical galaxies could be that the equivalent circular velocity curves tend to become flat at much larger accelerations than in spiral galaxies, and that the Newtonian dark matter halos are more concentrated than in spirals. Here, we compare published scaling relations for the dark halos of elliptical galaxies to the scaling relations expected for MONDian phantom halos. We represent the baryonic content of galaxies by spherical profiles, and their corresponding MONDian phantom halos by logarithmic halos. We then derive the surface densities, central d…
We calculate the perihelion precession delta for nearly circular orbits in a central potential V(r). Differently from other approaches to this problem, we do not assume that the potential is close to the Newtonian one. The main idea in the deduction is to apply the underlying symmetries of the system to show that delta must be a function of r V''(r)/V'(r), and to use the transformation behaviour of delta in a rotating system of reference. This is equivalent to say, that the effective potential can be written in a one-parameter set of possibilities as sum of centrifugal potential and potential …
We investigate the (conservative) dynamics of binary black holes using the Hamiltonian formulation of the post-Newtonian (PN) equations of motion. The Hamiltonian we use includes spin-orbit coupling, spin-spin coupling, and mass monopole/spin-induced quadrupole interaction terms. In the case of both quasi-circular and eccentric orbits, we search for the presence of chaos (using the method of Lyapunov exponents) for a large variety of initial conditions. For quasi-circular orbits, we find no chaotic behavior for black holes with total mass 10 - 40 solar masses when initially at a separation cor…
We analyze a class of physical properties, forming the content of the so-called von Zeipel theorem, which characterizes stationary, axisymmetric, non-selfgravitating perfect fluids in circular motion in the gravitational field of a compact object. We consider the extension of the theorem to the magnetohydrodynamic regime, under the assumption of an infinitely conductive fluid, both in the Newtonian and in the relativistic framework. When the magnetic field is toroidal, the conditions required by the theorem are equivalent to integrability conditions, as it is the case for purely hydrodynamic f…
We generalize to Kerr spacetime previous gravitational self-force results on gyroscope precession along circular orbits in the Schwarzschild spacetime. In particular we present high order post- Newtonian expansions for the gauge invariant precession function along circular geodesics valid for arbitrary Kerr spin parameter and show agreement between these results and those derived from the full post-Newtonian conservative dynamics. Finally we present strong field numerical data for a range of the Kerr spin parameter, showing agreement with the GSF-PN results, and the expected lightring divergen…
Motivated by applications to the study of self-force effects in scalar-tensor theories of gravity, we calculate the self-force exerted on a scalar charge in a circular orbit about a Reissner-Nordström black hole. We obtain the self-force via a mode-sum calculation, and find that our results differ from recent post-Newtonian calculations even in the slow-motion regime. We compute the radiative fluxes towards infinity and down the black hole, and verify that they are balanced by energy dissipated through the local self-force - in contrast to the reported post-Newtonian results. The self-force an…
In this paper, we present results obtained from our recent studies on the location of the innermost stable circular orbit (ISCO) for binary neutron stars (BNSs) in several levels of post Newtonian (PN) approximations. We reach the following conclusion at present: (1) even in the Newtonian case, there exists the ISCO for binary of sufficiently stiff equation of state (EOS). If the mass and the radius of each star are fixed, the angular velocity at the ISCO $Ω_{ISCO}$ is larger for softer EOS: (2) when we include the first PN correction, there appear roughly two kinds of effects. One is the effe…
Using an energy variational method, we calculate quasi-equilibrium configurations of binary neutron stars modeled as compressible triaxial ellipsoids obeying a polytropic equation of state. Our energy functional includes terms both for the internal hydrodynamics of the stars and for the external orbital motion. We add the leading post-Newtonian (PN) corrections to the internal and gravitational energies of the stars, and adopt hybrid orbital terms which are fully relativistic in the test-mass limit and always accurate to PN order. The total energy functional is varied to find quasi-equilibrium…
We consider a general class of quantum gravity-inspired, modified gravity theories, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy. This class of theories includes Einstein-Dilaton-Gauss-Bonnet and Chern-Simons modified gravity as special cases. We analytically derive and solve the coupled field equations in the post-Newtonian approximation, assuming a comparable-mass, spinning black hole binary source in a quasi-circular, weak-field/slow-motion orbit. We find that a naive s…
We prescribe a pseudo-Newtonian vector potential for studying accretion disks around Kerr black holes. The potential is useful to study the inner properties of disk not confined to the equatorial plane where general relativistic effect is indispensable. Therefore, we incorporate the essential properties of the metric at the inner radii through the pseudo-Newtonian potential derived from the general Kerr spacetime. The potential, reproducing most of the salient features of the general-relativity, is valid for entire regime of Kerr parameter. It reproduces the last stable circular orbit exactly …
