Relativistic Two-body Problem Research Articles

Post-Minkowskian theory meets the spinning effective-one-body approach for two-body scattering

Effective-one-body (EOB) waveforms employed by the LIGO-Virgo-KAGRA Collaboration have primarily been developed by resumming the post-Newtonian expansion of the relativistic two-body problem. Given the recent significant advancements in post-Minkowskian (PM) theory and gravitational self-force formalism, there is considerable interest in creating waveform models that integrate information from various perturbative methods in innovative ways. This becomes particularly crucial when tackling the accuracy challenge posed by upcoming ground-based detectors (such as the Einstein Telescope and Cosmic Explorer) and space-based detectors (such as LISA, TianQin, or Taiji) expected to operate in the next decade. In this context, we present the derivation of the first spinning EOB (SEOB) Hamiltonian that incorporates PM results up to three-loop order: the SEOB-PM model. The model accounts for the complete hyperbolic motion, encompassing nonlocal-in-time tails. To evaluate its accuracy, we compare its predictions for the conservative scattering angle, augmented with dissipative contributions, against numerical-relativity data of nonspinning and spinning equal-mass black holes. We observe very good agreement, comparable, and in some cases slightly better to the recently proposed wEOB-potential model, of which the SEOB-PM model is a resummation around the probe limit. Indeed, in the probe limit, the SEOB-PM Hamiltonian and scattering angles reduce to the one of a test mass in Kerr spacetime. Once complemented with nonlocal-in-time contributions for bound orbits, the SEOB-PM Hamiltonian can be utilized to generate waveform models for spinning black holes on quasicircular orbits. Published by the American Physical Society 2024

Bootstrapping the relativistic two-body problem

We describe the formalism to compute gravitational-wave observables for compact binaries via the effective field theory framework in combination with modern tools from collider physics. We put particular emphasis on solving the ‘multi-loop’ integration problem via the methodology of differential equations and expansion by regions. This allows us to bootstrap the two-body relativistic dynamics in the Post-Minkowskian (PM) expansion from boundary data evaluated in the near-static (soft) limit. We illustrate the procedure with the derivation of the total spacetime impulse in the scattering of non-spinning bodies to 4PM (three-loop) order, i.e. mathcal{O} (G4), including conservative and dissipative effects.

Success of the small mass-ratio approximation during the final orbits of binary black hole simulations

Recent studies have shown the surprising effectiveness of the small mass-ratio approximation (SMR) in modeling the relativistic two-body problem even at comparable masses. Up to now this effectiveness has been demonstrated only during inspiral, before the binary transitions into plunge and merger. Here we examine the binding energy of nonspinning binary black hole simulations with mass ratios from $20\ensuremath{\mathbin:}1$ to equal mass. We show for the first time that the binaries undergo a transition to plunge as predicted by analytic theory, and estimate the size of the transition region, which is $\ensuremath{\sim}10$ gravitational wave cycles for equal mass binaries. By including transition, the SMR expansion of the binding energy is accurate until the last cycle of gravitational wave emission. This is true even for comparable mass binaries such as those observed by current gravitational wave detectors, where the transition often makes up much of the observed signal. Our work provides further evidence that the SMR approximation can be directly applied to current gravitational wave observations.

A comparison of explicit Runge-Kutta methods

Recent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here. doi: 10.1017/S1446181122000141

Eccentricity or spin precession? Distinguishing subdominant effects in gravitational-wave data

ABSTRACT Eccentricity and spin precession are key observables in gravitational-wave astronomy, encoding precious information about the astrophysical formation of compact binaries together with fine details of the relativistic two-body problem. However, the two effects can mimic each other in the emitted signals, raising issues around their distinguishability. Since inferring the existence of both eccentricity and spin precession simultaneously is – at present – not possible, current state-of-the-art analyses assume that either one of the effects may be present in the data. In such a setup, what are the conditions required for a confident identification of either effect? We present simulated parameter inference studies in realistic LIGO/Virgo noise, studying events consistent with either spin precessing or eccentric binary black hole coalescences and recovering under the assumption that either of the two effects may be at play. We quantify how the distinguishability of eccentricity and spin precession increases with the number of visible orbital cycles, confirming that the signal must be sufficiently long for the two effects to be separable. The threshold depends on the injected source, with inclination, eccentricity, and effective spin playing crucial roles. In particular, for injections similar to GW190521, we find that it is impossible to confidently distinguish eccentricity from spin precession.

The General Relativistic Two-body Problem at 5PN

The General Relativistic Two-body Problem at 5PN

A COMPARISON OF EXPLICIT RUNGE–KUTTA METHODS

AbstractRecent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here.

Momentum recoil in the relativistic two-body problem: Higher-order tails

In the description of the relativistic two-body interaction, together with the effects of energy and angular momentum losses due to the emission of gravitational radiation, one has to take into account also the loss of linear momentum, which is responsible for the recoil of the center-of-mass of the system. We compute higher-order tail (i.e., tail-of-tail and tail-squared) contributions to the linear momentum flux for a nonspinning binary system either along hyperboliclike or ellipticlike orbits. The corresponding orbital averages are evaluated at their leading post-Newtonian approximation, using harmonic coordinates and working in the Fourier domain. The final expressions are given in a large-eccentricity (or large-angular momentum) expansion along hyperboliclike orbits and in a small-eccentricity expansion along ellipticlike orbits. We thus complete a previous analysis focusing on both energy and angular momentum losses [Phys. Rev. D \textbf{104}, no.10, 104020 (2021)], providing brick-type results which will be useful, e.g., in the high-accurate determination of the radiated impulses of the two bodies undergoing a scattering process.

Conservative and radiative dynamics in classical relativistic scattering and bound systems

As recent work continues to demonstrate, the study of relativistic scattering processes leads to valuable insights and computational tools applicable to the relativistic bound-orbit two-body problem. This is particularly relevant in the post-Minkowskian approach to the gravitational two-body problem, where the field has only recently reached a full description of certain physical observables for scattering orbits, including radiative effects, at the third post-Minkowskian (3PM) order. As an historically instructive simpler example, we consider here the analogous problem in electromagnetism in flat spacetime. We compute the changes in linear momentum of each particle and the total radiated linear momentum, in the relativistic classical scattering of two point-charges, at sixth order in the charges (analogous to 3PM order in gravity). We accomplish this here via direct iteration of the classical equations of motion, while making comparisons where possible to results from quantum scattering amplitudes, with the aim of contributing to the elucidation of conceptual issues and scalability on both sides. We also discuss further extensions to radiative quantities of recently established relations, which analytically continue certain observables from the scattering regime to the regime of bound orbits, applicable for both the electromagnetic and gravitational cases.

Gravitational spin-orbit and aligned spin1−spin2 couplings through third-subleading post-Newtonian orders

The study of scattering encounters continues to provide new insights into the general relativistic two-body problem. The local-in-time conservative dynamics of an aligned-spin binary, for both unbound and bound orbits, is fully encoded in the gauge-invariant scattering-angle function, which is most naturally expressed in a post-Minkowskian (PM) expansion, and which exhibits a remarkably simple dependence on the masses of the two bodies (in terms of appropriate geometric variables). This dependence links the PM and small-mass-ratio approximations, allowing gravitational self-force results to determine new post-Newtonian (PN) information to all orders in the mass ratio. In this paper, we exploit this interplay between relativistic scattering and self-force theory to obtain the third-subleading (4.5PN) spin-orbit dynamics for generic spins, and the third-subleading (5PN) spin$_1$-spin$_2$ dynamics for aligned spins. We further implement these novel PN results in an effective-one-body framework, and demonstrate the improvement in accuracy by comparing against numerical-relativity simulations.

Classical Yang-Mills observables from amplitudes

The double copy suggests that the basis of the dynamics of general relativity is Yang-Mills theory. Motivated by the importance of the relativistic two-body problem, we study the classical dynamics of colour-charged particle scattering from the perspective of amplitudes, rather than equations of motion. We explain how to compute the change of colour, and the radiation of colour, during a classical collision. We apply our formalism at next-to-leading order for the colour change and at leading order for colour radiation.

Hamilton-Jacobi equation for spinning particles near black holes

A compact stellar-mass object inspiralling onto a massive black hole deviates from geodesic motion due to radiation-reaction forces as well as finite-size effects. Such post-geodesic deviations need to be included with sufficient precision into wave-form models for the upcoming space-based gravitational-wave detector LISA. I present the formulation and solution of the Hamilton-Jacobi equation of geodesics near Kerr black holes perturbed by the so-called spin-curvature coupling, the leading order finite-size effect. In return, this solution allows to compute a number of observables such as the turning points of the orbits as well as the fundamental frequencies of motion. This result provides one of the necessary ingredients for waveform models for LISA and an important contribution useful for the relativistic two-body problem in general.

Fundamentals of numerical relativity for gravitational wave sources.

Einstein's theory of general relativity affords an enormously successful description of gravity. The theory encodes the gravitational interaction in the metric, a tensor field on spacetime that satisfies partial differential equations known as the Einstein equations. This review introduces some of the fundamental concepts of numerical relativity-solving the Einstein equations on the computer-in simple terms. As a primary example, we consider the solution of the general relativistic two-body problem, which features prominently in the new field of gravitational wave astronomy.

High-energy gravitational scattering and the general relativistic two-body problem

A technique for translating the classical scattering function of two gravitationally interacting bodies into a corresponding (effective one-body) Hamiltonian description has been recently introduced [Phys.\ Rev.\ D {\bf 94}, 104015 (2016)]. Using this technique, we derive, for the first time, to second-order in Newton's constant (i.e. one classical loop) the Hamiltonian of two point masses having an arbitrary (possibly relativistic) relative velocity. The resulting (second post-Minkowskian) Hamiltonian is found to have a tame high-energy structure which we relate both to gravitational self-force studies of large mass-ratio binary systems, and to the ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano. We derive several consequences of our second post-Minkowskian Hamiltonian: (i) the need to use special phase-space gauges to get a tame high-energy limit; and (ii) predictions about a (rest-mass independent) linear Regge trajectory behavior of high-angular-momenta, high-energy circular orbits. Ways of testing these predictions by dedicated numerical simulations are indicated. We finally indicate a way to connect our classical results to the quantum gravitational scattering amplitude of two particles, and we urge amplitude experts to use their novel techniques to compute the 2-loop scattering amplitude of scalar masses, from which one could deduce the third post-Minkowskian effective one-body Hamiltonian.

Pair states in one-dimensional Dirac systems

Analytic solutions of the quantum relativistic two-body problem are obtained for an interaction potential modeled as a one-dimensional smooth square well. Both stationary and moving pairs are considered and the limit of the {\delta}-function interaction is studied in depth. Our result can be utilized for understanding excitonic states in narrow-gap carbon nanotubes. We also show the existence of bound states within the gap for a pair of particles of the same charge.

Solitary solutions to a relativistic two-body problem

Necessary and sufficient conditions for the existence of solitary solutions to a generalized model of a two-body problem perturbed by small post-Newtonian relativistic term are derived in this paper. It is demonstrated that kink, bright and dark solitary solutions exist in the model, when the relativistic effects are treated as higher order perturbations. Numerical experiments are used to verify theoretical results.

Relativistic Two-Boson System in Presence of Electromagnetic Plane Wave

The relativistic two-body problem is considered for spinless particles subject to an external macroscopic electromagnetic field. When this field is made of the monochromatic superposition of two counter-propagating plane waves (and provided the mutual interaction between particles is known), it is possible to write down explicitly a pair of coupled wave equations (corresponding to a pair of mass-shell constraints) which takes into account also the field contribution. These equations are manifestly covariant; constants of the motion are exhibited, so one ends up with a reduced problem involving five degrees of freedom.

Numerical integration of a relativistic two-body problem via a multiple scales method

We offer an analytical study on the dynamics of a two-body problem perturbed by small post-Newtonian relativistic term. We prove that, while the angular momentum is not conserved, the motion is planar. We also show that the energy is subject to small changes due to the relativistic effect. We also offer a periodic solution to this problem, obtained by a method based of separation of timescales. We demonstrate that our solution is more general than the method developed in the book by Brumberg (1991). The practical applicability of this model may be studies of the long-term evolution of relativistic binaries (neutron stars or black holes).

High-order half-integral conservative post-Newtonian coefficients in the redshift factor of black hole binaries

The post-Newtonian approximation is still the most widely used approach to obtaining explicit solutions in general relativity, especially for the relativistic two-body problem with arbitrary mass ratio. Within many of its applications, it is often required to use a regularization procedure. Though frequently misunderstood, the regularization is essential for waveform generation without reference to the internal structure of orbiting bodies. In recent years, direct comparison with the self-force approach, constructed specifically for highly relativistic particles in the extreme mass ratio limit, has enabled preliminary confirmation of the foundations of both computational methods, including their very independent regularization procedures, with high numerical precision. In this paper, we build upon earlier work to carry this comparison still further, by examining next-to-next-to-leading order contributions beyond the half integral 5.5PN conservative effect, which arise from terms to cubic and higher orders in the metric and its multipole moments, thus extending scrutiny of the post-Newtonian methods to one of the highest orders yet achieved. We do this by explicitly constructing tail-of-tail terms at 6.5PN and 7.5PN order, computing the redshift factor for compact binaries in the small mass ratio limit, and comparing directly with numerically and analytically computed terms in the self-force approach, obtained using solutions for metric perturbations in the Schwarzschild space-time, and a combination of exact series representations possibly with more typical PN expansions. While self-force results may be relativistic but with restricted mass ratio, our methods, valid primarily in the weak-field slowly-moving regime, are nevertheless in principle applicable for arbitrary mass ratios.

Resonantly enhanced and diminished strong-field gravitational-wave fluxes

The inspiral of a stellar mass ($1\ensuremath{-}100{M}_{\ensuremath{\bigodot}}$) compact body into a massive ($1{0}^{5}\ensuremath{-}1{0}^{7}{M}_{\ensuremath{\bigodot}}$) black hole has been a focus of much effort, both for the promise of such systems as astrophysical sources of gravitational waves, and because they are a clean limit of the general relativistic two-body problem. Our understanding of this problem has advanced significantly in recent years, with much progress in modeling the ``self-force'' arising from the small body's interaction with its own spacetime deformation. Recent work has shown that this self-interaction is especially interesting when the frequencies associated with the orbit's $\ensuremath{\theta}$ and $r$ motions are in an integer ratio: ${\mathrm{\ensuremath{\Omega}}}_{\ensuremath{\theta}}/{\mathrm{\ensuremath{\Omega}}}_{r}={\ensuremath{\beta}}_{\ensuremath{\theta}}/{\ensuremath{\beta}}_{r}$, with ${\ensuremath{\beta}}_{\ensuremath{\theta}}$ and ${\ensuremath{\beta}}_{r}$ both integers. In this paper, we show that key aspects of the self-interaction for such ``resonant'' orbits can be understood with a relatively simple Teukolsky-equation-based calculation of gravitational-wave fluxes. We show that fluxes from resonant orbits depend on the relative phase of radial and angular motions. The purpose of this paper is to illustrate in simple terms how this phase dependence arises using tools that are good for strong-field orbits, and to present a first study of how strongly the fluxes vary as a function of this phase and other orbital parameters. Future work will use the full dissipative self-force to examine resonant and near resonant strong-field effects in greater depth, which will be needed to characterize how a binary evolves through orbital resonances.