post-Newtonian Theory Research Articles

Residual eccentricity of inspiralling orbits at the gravitational-wave detection threshold: Accurate estimates using post-Newtonian theory

We use equations of motion containing gravitational radiation-reaction terms through 4.5 post-Newtonian order to calculate the late-time eccentricities of inspiraling binary systems of non-spinning compact bodies as they cross the detection threshold of ground-based gravitational-wave interferometers. The initial eccentricities can be as large as 0.999. We find that the final eccentricities are systematically smaller than those predicted by the leading quadrupole approximation, by as much as 30 percent for a 300 solar mass binary crossing the LIGO/Virgo detection threshold at 10 Hz, or eight percent smaller for a 60 solar mass binary. We find an analytic formula for the late-time eccentricity that accurately accounts for the higher-order post-Newtonian effects, generalizing a formula derived by Peters and Mathews in the 1960s. We also find that the final eccentricities are independent of the ratio of the masses of the two compact bodies to better than two percent.

Universal features of gravitational waves emitted by superkick binary black hole systems

We use numerical relativity to study the merger and ringdown stages of ``superkick'' binary black hole systems (those with equal mass and antiparallel spins). We find a universal way to describe the mass and current quadrupole gravitational waves emitted by these systems during the merger and ringdown stage: (i) The time evolutions of these waves are insensitive to the progenitor's parameters (spins) after being normalized by their own peak values. (ii) The peak values, which encode all the spin information of the progenitor, can be consistently fitted to formulas inspired by post-Newtonian theory. We find that the universal evolution of the mass quadrupole wave can be accurately modeled by the so-called Backwards One-Body (BOB) model. However, the BOB model, in its present form, leads to a lower waveform match and a significant parameter-estimation bias for the current quadrupole wave. We also decompose the ringdown signal into seven overtones, and study the dependence of mode amplitudes on the progenitor's parameters. Such dependence is found to be insensitive to the overtone index (up to a scaling factor). Finally, we use the Fisher matrix technique to investigate how the ringdown waveform can be at least as important for parameter estimation as the inspiral stage. Assuming the Cosmic Explorer, we find the contribution of ringdown portion dominates as the total mass exceeds $\ensuremath{\sim}250\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$. For massive binary black hole (BBH) systems, the accuracy of parameter measurement is improved by incorporating the information of ringdown---the ringdown sector gives rise to a different parameter correlation from inspiral stage; hence, the overall parameter correlation is reduced in full signal.

Jeans instability from post-Newtonian Boltzmann equation

Jeans instability within the framework of post-Newtonian Boltzmann and Poisson equations are analyzed. The components of the energy–momentum tensor are calculated from a post-Newtonian Maxwell-Jüttner distribution function. The perturbations of the distribution function and gravitational potentials from their background states with the representation of the perturbations as plane waves lead to a dispersion relation with post-Newtonian corrections. The influence of the post-Newtonian approximation on the Jeans mass is determined and it was shown that the mass necessary for an overdensity to begin the gravitational collapse in the post-Newtonian theory is smaller than the one in the Newtonian theory.

Compact body in a tidal environment: New types of relativistic Love numbers, and a post-Newtonian operational definition for tidally induced multipole moments

We examine the tidal deformation of a nonrotating compact body (material body or black hole) in general relativity. The body's exterior metric is calculated in a simultaneous expansion in powers of the ratio between the distance to the body and three distinct length scales: the radius of curvature of the external spacetime in which the body is inserted, the scale of spatial inhomogeneity of the curvature, and the scale of temporal variation. The metric is valid in the body's immediate neighborhood, which excludes the external matter responsible for the tidal environment. The body's tidal response is encapsulated in four types of relativistic Love numbers: $k_\ell$, the familiar Love number that measures the linear response to a static tidal field, $p_\ell$, which measures the quadratic response to the tidal field, $\dot{k}_\ell$ and $\ddot{k}_\ell$, associated with first and second time derivatives of the tidal field, respectively. The Love numbers acquire an operational meaning through the definition of tidally induced multipole moments. Previously proposed definitions for the moments suffer from ambiguities associated with the subtraction of a "pure tidal field" from the full metric. A robust operational definition is proposed here. It relies on inserting the body's local metric within a global metric constructed in post-Newtonian theory; the global metric includes the external matter responsible for the tidal environment. When viewed in the post-Newtonian spacetime, the compact body appears as a skeletonized object with a specific multipole structure. The tidally induced multipole moments provide a description of this structure. They manifest themselves, for example, in the body's tidal acceleration, which is nonlinear in the tidal field. At leading order in the tidal interaction, the acceleration is proportional to the $k_2$ Love number as calculated in full general relativity.

Post-Newtonian kinetic theory

A kinetic theory for relativistic gases in the presence of gravitational fields is developed in the second post-Newtonian approximation. The corresponding Boltzmann equation is determined from the evolution of the one-particle distribution function with respect to the proper time along the world line of the particle. From the knowledge of the equilibrium Maxwell–Jüttner distribution function in the second post-Newtonian approximation the components of the particle four-flow and energy–momentum tensor are obtained. The Eulerian hydrodynamic equations for the mass density, mass-energy density and momentum density in the second post-Newtonian approximation are determined from the Boltzmann equation. It is shown that the combination of the hydrodynamic equations of mass and mass-energy densities leads to the hydrodynamic equation for the internal energy density in the first post-Newtonian approximation.

Post-Newtonian Dynamics with State-Dependent Delay

The retarded potentials appearing in electrodynamics are an example of systems with state-dependent delay. By analogy, Newton’s law of gravitation may be generalized using the concept of a retarded gravitational potential. For systems with state-dependent delay (SDD), delay and state or observations are implicitly related. Such an SDD-approach provides an approximation to general relativity for weak (gravitational) fields and slow speeds. While it has been shown that solutions to Einstein’s field equations always exist, an existence theorem says nothing about the computation. So a semi-relativistic approximation is beneficial: One where space-time may still be considered as flat, but where Newton’s equations (where gravity acts instantaneously) are replaced by a post-Newtonian theory, involving propagation of gravity at the speed of light. In this paper some direct effects (gravitational entrainment, dynamic friction, and orbital velocity for binaries) of such a post-Newtonian theory are described. We show also that a naive delay correction in Newton’s gravitation law cannot be correct.

Analytic post-Newtonian expansion of the energy and angular momentum radiated to infinity by eccentric-orbit nonspinning extreme-mass-ratio inspirals to the 19th order

We develop new high-order results for the post-Newtonian (PN) expansions of the energy and angular momentum fluxes at infinity for eccentric-orbit extreme-mass-ratio inspirals (EMRIs) on a Schwarzschild background. The series are derived through direct expansion of the MST solutions within the RWZ formalism for first-order black hole perturbation theory (BHPT). By utilizing factorization and a few computational simplifications, we are able to compute the fluxes to 19PN, with each PN term calculated as a power series in (Darwin) eccentricity to $e^{10}$. This compares favorably with the numeric fitting approach used in previous work. We also compute PN terms to $e^{20}$ through 10PN. Then, we analyze the convergence properties of the composite energy flux expansion by checking against numeric data for several orbits, both for the full flux and also for the individual 220 mode, with various resummation schemes tried for each. The match between the high-order series and numerical calculations is generally strong, maintaining relative error better than $10^{-5}$ except when $p$ (the semi-latus rectum) is small and $e$ is large. However, the full-flux expansion demonstrates superior fidelity (particularly at high $e$), as it is able to incorporate additional information from PN theory. For the orbit $(p=10, e=1/2)$, the full flux achieves a best error near $10^{-5}$, while the 220 mode exhibits error worse than $1\%$. Finally, we describe a procedure for transforming these expansions to the harmonic gauge of PN theory by analyzing Schwarzschild geodesic motion in harmonic coordinates. This will facilitate future comparisons between BHPT and PN theory.

Evolution of inspiralling neutron star binaries: Effects of tidal interactions and orbital eccentricities

Neutron star (NS) binaries formed dynamically may have significant eccentricities while emitting gravitational waves (GWs) in the LIGO/VIRGO band. We study tidal effects in such eccentric inspiralling NS binaries using a set of hybrid equations. The NS is modelled as a compressible ellipsoid, which can deform nonlinearly due to tidal forces, while the orbit evolution is treated with the post-Newtonian (PN) theory up to 2.5-PN order. We find that in general, the tidal interaction can accelerate the inspiral, and cause orbital frequency and phase shifts. For circular inspirals, our calculations reproduce previous linear result at large binary separations, but incorporate the dynamical response of the NS at small separations. For eccentric inspirals, the frequency and phase shifts oscillate considerably near pericenter passages, and the oscillating amplitudes increase with eccentricities. As a result, the GW phase is also significantly influenced by the tidal effect. At merger, the cumulative GW phase shift can reach more than 10 radians (for typical NS mass $1.4M_\odot$ and radius 11.6 km), much larger than the circular inspiral case. Although the event rate of eccentric NS mergers is likely low, the detection of such mergers could provide a useful constraint on the NS equation of state.

Inertial Navigation Method of Spacecraft Based on Geodesic Deviation Equation

A new method of spacecraft inertial navigation is proposed in paper. The method uses the earth gravitational field model and the on-board accelerometer to measure data, so the real-time deviation of the spacecraft relative to the geodetic orbit is calculated. Based on the post-Newtonian approximate gravity theory of general relativity, the high precision geodesic deviation equation around the earth is derived, and the equation of the spacecraft relative to the geodesic orbit is transformed into orbital element space. The position of the spacecraft is determined by calculating the deviation between the spacecraft and the nearest geodetic orbit. Because the reference points are selected different in the adjacent following inertia system during navigation, the accumulation error of the traditional inertial navigation is avoided, and the navigation precision can be effectively improved.

Gravitomagnetic Love tensor of a slowly rotating body: Post-Newtonian theory

The gravitomagnetic tidal Love number of a slowly rotating body was calculated previously under the assumption that the velocity perturbation created by the tidal field consists of an induction piece proportional to the vector potential, and a rotational piece that scales with $\Omega$, the body's angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like $\Omega^0 = 1$. The previous calculations are therefore incorrect, and the purpose of this paper is to repair the mistake. To keep the technical difficulties to a minimum, the treatment here is restricted to a post-Newtonian expansion carried out to leading order -- previous calculations of the gravitomagnetic Love number were performed in full general relativity. On the other hand, the computation presented here is not restricted to a stationary tidal field. I show that the correct scaling of the velocity perturbation with $\Omega$ leads to the promotion of the Love number to a Love tensor $k_{jk}^{\ \ pq}$, a four-index object that relates the body's current quadrupole moment $S_{jk}$ to the gravitomagnetic tidal moment ${\cal B}_{pq}$. The tensorial nature of this quantity has to do with the fact that each $e^{im\phi}$ piece of the tidal force gives rise to an $m$-specific velocity perturbation, and therefore to a Love number that depends on $m$. The collection of these $m$-specific Love numbers makes up the Love tensor $k_{jk}^{\ \ pq}$.

Can we use next-generation gravitational wave detectors for terrestrial precision measurements of Shapiro delay?

Shapiro time delay is one of the fundamental tests of general relativity and post-Newtonian theories of gravity. Consequently, its measurements can be used to probe the parameter γ which is related to spacetime curvature produced by a unit mass in the post-Newtonian formalism of gravity. To date all measurements of time delay have been conducted on astronomical scales. It was asserted in 2010 that gravitational wave detectors on Earth could be used to measure Shapiro delay on a terrestrial scale via massive rotating systems. Building on that work, we consider how measurements of Shapiro delay can be made using next-generation gravitational wave detectors. We perform an analysis for measuring Shapiro delay with the next-generation gravitational wave detectors Cosmic Explorer and Einstein Telescope to determine how precisely the effect can be measured. Using a rotating mass unit design, we find that Cosmic Explorer and Einstein Telescope can measure the Shapiro delay signal with amplitude signal to noise ratios upwards of ∼28 and ∼43 in 1 year of integration time, respectively. By measuring Shapiro delay with this technique, next-generation interferometers will allow for terrestrial measurements of γ in the paramaterized post-Newtonian formalism of gravity with sub-percent precision.

Determination of new coefficients in the angular momentum and energy fluxes at infinity to 9PN order for eccentric Schwarzschild extreme-mass-ratio inspirals using mode-by-mode fitting

We present an extension of work in an earlier paper showing high precision comparisons between black hole perturbation theory and post-Newtonian (PN) theory in their region of overlapping validity for bound, eccentric-orbit, Schwarzschild extreme-mass-ratio inspirals. As before we apply a numerical fitting scheme to extract eccentricity coefficients in the PN expansion of the gravitational wave fluxes, which are then converted to exact analytic form using an integer-relation algorithm. In this work, however, we fit to individual $lmn$ modes to exploit simplifying factorizations that lie therein. Since the previous paper focused solely on the energy flux, here we concentrate initially on analyzing the angular momentum flux to infinity. A first step involves finding convenient forms for hereditary contributions to the flux at low-PN order, analogous to similar terms worked out previously for the energy flux. We then apply the upgraded techniques to find new PN terms through 9PN order and (at many PN orders) to $e^{30}$ in the power series in eccentricity. With the new approach applied to angular momentum fluxes, we return to the energy fluxes at infinity to extend those previous results. Like before, the underlying method uses a \textsc{Mathematica} code based on use of the Mano-Suzuki-Takasugi (MST) function expansion formalism to represent gravitational perturbations and spectral source integration (SSI) to find numerical results at arbitrarily high precision.

First survey of spinning eccentric black hole mergers: Numerical relativity simulations, hybrid waveforms, and parameter estimation

We analyze a new numerical relativity dataset of spinning but nonprecessing binary black holes on eccentric orbits, with eccentricities from approximately $0.1$ to $0.5$, with dimensionless spins up to $0.75$ included at mass ratios $q=m_1/m_2 = (1, 2)$, and further nonspinning binaries at mass ratios $q = (1.5, 3, 4)$. A comparison of the final mass and spin of these simulations with noneccentric data extends previous results in the literature on circularization of eccentric binaries to the spinning case. For the $(l,m)=(2,2)$ spin-weighted spherical harmonic mode we construct eccentric hybrid waveforms that connect the numerical relativity data to a post-Newtonian description for the inspiral, and we discuss the limitations in the current knowledge about post-Newtonian theory which complicate the generation of eccentric hybrid waveforms. We also perform a Bayesian parameter estimation study, quantifying the parameter biases introduced when using three different quasicircular waveform models to estimate the parameters of highly eccentric binary systems. We find that the used aligned-spin quasicircular model including higher order modes produces lower bias in certain parameters than the nonprecessing quasicircular model without higher order modes and the quasicircular precessing model.

Comparison of post-Newtonian mode amplitudes with numerical relativity simulations of binary black holes

Gravitational waves from the coalescence of two black holes carry the signature of the strong field dynamics of binary black holes. In this work we have used numerical relativity simulations and post-Newtonian theory to investigate this dynamics. Post-Newtonian theory is a low-velocity expansion that assumes the companion bodies to be point-particles, while numerical relativity treats black holes as extended objects with horizons and fully captures their dynamics. There is a priori no reason for the waveforms computed using these disparate methods to agree with each other, especially at late times when the black holes move close to the speed of light. We find, remarkably, that the leading order amplitudes in post-Newtonian theory agree well with the full general relativity solution for a large set of spherical harmonic modes, even in the most dynamical part of the binary evolution, with only some modes showing distinctly different behavior than that found by numerical relativity simulations. In particular, modes with spherical harmonic indices as well as are least modified from their dominant post-Newtonian behavior. Understanding the nature of these modes in terms of the post-Newtonian description will aid in formulating better models of the emitted waveforms in the strong field regime of the dynamics.

Black hole spin axis in numerical relativity

Colliding black holes are systems of profound interest in both gravitational-wave astronomy and in gravitation theory, and a variety of methods have been developed for modeling their dynamics in detail. The features of these dynamics are determined by the masses of the holes and by the magnitudes and axes of their spins. While masses and spin magnitudes can be defined in reasonably unambiguous ways, the spin axis is a concept that, despite great physical importance, is seriously undermined by the coordinate freedom of general relativity. Despite a great wealth of detailed numerical simulations of generic spinning black hole collisions, very little attention has gone into defining or justifying the definitions of the spin axis used in the numerical relativity literature. In this paper, we summarize and contrast the various spin direction measures available in the spec code, including a comparison with a method common in other codes, we explain why these measures have shown qualitatively different nutation features than one would expect from post-Newtonian theory, and we derive and implement new measures that give much better agreement.

A complete analytic gravitational wave model for undergraduates

Gravitational waves are produced by orbiting massive binary objects, such as black holes and neutron stars, and propagate as ripples in the very fabric of space-time. As the waves carry off orbital energy, the two bodies spiral into each other and eventually merge. They are described by Einstein’s equations of general relativity. For the early phase of the orbit, called the inspiral, Einstein equations can be linearised and solved through analytical approximations, while for the late phase, near the merger, we need to solve the fully nonlinear Einstein’s equations on supercomputers. In order to recover the gravitational wave for the entire evolution of the binary, a match is required between the inspiral and the merger waveforms. Our objectives are to establish an educational oriented toy model for a streamlined matching method, that will allow an analytical calculation of the complete gravitational waveform, while developing a gravitational wave modelling tutorial for undergraduate physics students. We use post-Newtonian (PN) theory for the inspiral phase, which offers an excellent training ground for students, and rely on Mathematica for our calculations, a tool easily accessible to undergraduates. For the merger phase we bypass Einstein’s equations by using a simple analytic toy model named the implicit rotating source. After building the inspiral and merger waveforms, we construct our matching method and validate it by comparing our results with the waveforms for the first detection, GW150914, available as open source. Several future projects can be developed based from this project: building complete waveforms for all the detected signals, extending the PN model to take into account non-zero eccentricity, employing and testing a more realistic analytic model for the merger, building a separate model for the ringdown, and optimising the matching technique.

Simple procedures to reduce eccentricity of binary black hole simulations

We present simple procedures to construct quasicircular initial data for numerical evolutions of binary black hole spacetimes. Our method consists of using the post-Newtonian (PN) theory in three ways: first to provide an initial guess for the initial momenta at 3.5PN order that implies low residual eccentricity, second to measure the resulting eccentricity, and third to calculate corrections to the momenta or initial separation that further reduce the eccentricity. Regarding the initial guess, we compare numerical evolutions in post-Newtonian theory to the postcircular and post-postcircular analytical approximations to quasicircular data. We discuss a robust fitting procedure to measure eccentricity from numerical simulations using the orbital frequency $\mathrm{\ensuremath{\Omega}}$, and derive from the quasi-Keplerian parametrization at 1PN order the correction factors for the tangential and radial momentum components required to reduce the measured eccentricity to zero. We first test our procedure integrating PN equations of motion at 3.5PN where low eccentric initial data are easily obtained, and then apply our method to sets of binary black hole numerical relativity simulations with different mass ratios ($q={m}_{2}/{m}_{1}=1,2,\dots{},8$), spin configurations, and separations. Our set of simulations contains nonspinning, spin-aligned, and precessing simulations. We observe that the iterative procedure produces low eccentric simulations with eccentricities of the order $\mathcal{O}({10}^{\ensuremath{-}4})$ with only one iteration. The simplicity of the procedure allows one to obtain low eccentric numerical relativity simulations easily and save computational resources. Moreover, the analytical PN formulas derived in this paper will be useful to generate eccentric hybrid waveforms.

The eccentric behavior of inspiralling compact binaries

Even if there is no gauge invariant definition of eccentricity, it has an important impact on the observed gravitational wave signal of such systems, generating power in all possible harmonics of the orbital period. We here clarify the possible discrepancies between different eccentricity parameters used to describe the orbital dynamics of binary systems across different approximations, specifically the post-Newtonian approximation, the self-force approximation, and numerical relativity. To this end, we highlight disparities between the typically used orbit averaged method of evolving binary systems under radiation reaction, and more direct techniques of solving the two-body problem in post-Newtonian theory. We show, both numerically and analytically, that the orbit averaged method breaks down in the late inspiral, failing to capture a strong secular growth in the Keplerian eccentricity parameter and producing a orbital de-phasing relative to direct integration of the two-body equations of motion. We show that the secular growth and de-phasing affect the observed gravitational wave signal, which could bias how accurately we may recover parameters for systems with signal-to-noise ratios . We further develop a frequency domain post-adiabatic waveform model to capture these effects, and study the precision to which we may estimate parameters with this model through a Fisher information matrix analysis.

Post-Newtonian Magnetohydrodynamics

In this paper, we derive the post-Newtonian equations of the ideal Magnetohydrodynamics. To do so, we use the modern approach to post-Newtonian theory, where the harmonic gauge is used instead of the standard post-Newtonian gauge, and find the post-Newtonian metric in the presence of the electromagnetic fields. We show that although the electric field does not contribute in the metric and curvature of the spacetime, the magnetic field appears in the time-time component of the metric. The appearance of the magnetic field, in principle, leads to new relativistic contributions to the magnetohydrodynamic governing equations. Therefore, using the post-Newtonian metric, we find the relativistic corrections to the magnetohydrodynamic equations up to the first post-Newtonian order. In addition, as usage of this derivation, we obtain a complete set of equations by which the behavior of a self-gravitating plasma can be determined in post-Newtonian gravity.

Post-Newtonian Corrections to Toomre's Criterion

Abstract The gravitational stability of a two-dimensional self-gravitating and differentially rotating gaseous disk in the context of post-Newtonian (PN) theory is studied. Using the perturbative method and applying the second iterated equations of PN approximation, the relativistic version of the dispersion relation for the propagation of small perturbations is found. We obtain the PN version of Toomre’s local stability criterion by utilizing this PN dispersion relation. In other words, we find relativistic corrections to Toomre’s criterion in the first PN approximation. Two stability parameters, η and μ, related to gravity and pressure are introduced. We illustrate how these parameters determine the stability of the Newtonian and PN systems. Moreover, we show that, in general, the differentially rotating fluid disk is more stable in the context of PN theory relative to the Newtonian one. Also, we explicitly show that although the relativistic PN corrections destabilize nonrotating systems, they have the stabilizing role in the rotating thin disks. Finally, we apply the results to the relativistic disks around hypermassive neutron stars and find that although Newtonian description predicts the occurrence of local fragmentations, PN theory remains in agreement with the relevant simulations and rules out the existence of local fragmentations.