A new gravitational wave event reveals a binary black hole merger with total mass 190-265 solar masses, indicating black holes can form via gravitational-wave driven mergers beyond standard stellar channels.
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Post-Newtonian Theory for Gravitational Waves
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abstract
To be observed and analyzed by the network of current gravitational wave detectors (LIGO, Virgo, KAGRA), and in anticipation of future third generation ground based (Einstein Telescope, Cosmic Explorer) and space borne (LISA) detectors, inspiralling compact binaries -- binary star systems composed of neutron stars and/or black holes in their late stage of evolution prior the final coalescence -- require high-accuracy predictions from general relativity. The orbital dynamics and emitted gravitational waves of these very relativistic systems can be accurately modelled using state-of-the-art post-Newtonian theory. In this article we review the Multipolar-Post-Minkowskian approximation scheme, merged to the standard Post-Newtonian expansion into a single formalism valid for general isolated matter system. This cocktail of approximation methods (called MPM-PN) has been successfully applied to compact binary systems, producing equations of motion up to the fourth-post-Newtonian (4PN) level, and gravitational waveform and flux to 4.5PN order beyond the Einstein quadrupole formula. We describe the dimensional regularization at work in such high post-Newtonian calculations, for curing both ultra-violet and infra-red divergences. Several landmark results are detailed: the definition of multipole moments, the gravitational radiation reaction, the conservative dynamics of circular orbits, the first law of compact binary mechanics, and the non-linear effects in the gravitational wave propagation (tails, iterated tails and non-linear memory). We also discuss the case of compact binaries moving on eccentric orbits, and the effects of spins (both spin-orbit and spin-spin) on the equations of motion and gravitational wave energy flux and waveform.
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representative citing papers
The first informative astrophysical calibration of gravitational-wave detectors is reported using GW240925 and GW250207.
Numerical relativity simulations of black hole scattering in Einstein-scalar-Gauss-Bonnet gravity agree closely with effective-one-body analytic predictions.
Spin-induced tidal dissipation in compact binaries enters the gravitational-wave phase at 2.5 post-Newtonian order with a logarithmic frequency dependence.
Future microhertz detections combined with nanohertz pulsar terms can serve as gravity echoes to measure supermassive black hole binary inspiral rates from hundreds to thousands of years in the past.
Derives the 4PN conservative map between constants of motion and fundamental frequencies for eccentric orbits, resummed over eccentricity and validated against circular-orbit and self-force results.
At 5PM-1SF order, Calabi-Yau three-fold periods emerge in radiation-reacted observables for classical black hole scattering computed with worldline QFT and advanced IBP/DE methods.
Ultra-relativistic black hole flybys can radiate over 65% of their energy in gravitational waves via irregular waveforms caused by radiation trapping and lensing, without coalescence.
Numerical simulations of equal-mass boson-star mergers reveal larger waveform deviations from black-hole binaries in late inspiral and merger, plus odd multipole excitations for certain scalar-field phases, with some signals degenerate until IMR consistency tests are applied.
Derives large-eccentricity asymptotics for post-Newtonian eccentric waveform Fourier modes and builds a fast endpoint-constrained analytic approximation with error under 10^{-3} valid to p=200.
No three-body encounter signatures detected in GW170817, GW190814, and GW230627_015337, constraining intermediate-mass black holes above 100 solar masses within roughly 0.1 AU of these binaries.
The gwNRHME framework constructs a multi-modal non-spinning eccentric gravitational waveform surrogate by modulating quasi-circular models with universal eccentric functions, achieving median mismatches of ~9e-5 against 156 NR waveforms.
Polynomial models for the (2,2) post-merger waveform amplitudes of eccentric non-spinning binary black holes are constructed from numerical-relativity data as functions of symmetric mass ratio and two merger-time dynamical parameters.
Closed-form Sommerfeld factor via EFT connection matrix with analytic O(G^10) magnitude and phase for l=0,1,2 waves, plus a new RG equation for radiative multipole moments that improves waveform resummation beyond tail logarithms.
Horizon multipole moments of a Kerr black hole are computed in closed form from two definitions, yielding different values for l >= 1 at nonzero spin and sharing parity and small-spin scaling with field multipoles.
Self-force theory is extended to compute merger and ringdown waveforms in beyond-GR black hole binaries under the extreme mass-ratio approximation, with first calculations of self-force corrections to the merger waveform.
A unified framework for the perturbed Kepler problem derives modified eccentric orbits and gravitational wave imprints from a general perturbed potential, offering a source-specific alternative to post-Newtonian expansions.
New non-orbit-averaged 2.5PN equations for eccentric non-spinning black-hole binaries derived via energy-momentum mappings, showing Peters 1964 orbit-averaged equations break at first pericenter.
Derives NNLO post-Newtonian tidal contributions to conservative dynamics and ten conserved quantities in massless scalar-tensor theories for spinless sources, with extension to Einstein-scalar-Gauss-Bonnet gravity.
For orbital velocities below 0.45, PN energy flux agreement with NR improves up to incomplete 6PN with non-monotonic behavior, but convergence is lost near v approximately 0.5.
Numerical relativity simulations of triple black hole systems reveal redshift effects and gravitational lensing in ringdown signals from head-on mergers, with no additional black hole formation from amplified waves.
The gravitational impulse for ultrarelativistic massive scalars is resummed to all orders in G_N at fixed G_N s/mb, recovering post-Minkowski results and predicting the leading high-energy behavior to eleventh post-Minkowski order.
Constrained polarization model for Kerr ringdown modes enables inclination inference from two-detector data for non-precessing mergers but introduces biases when applied to precessing systems.
The NLO gravitational spin-orbit Hamiltonian for N spinning bodies is computed via PN-EFT, with only three-body diagrams new beyond the binary case, and the result matches the known ADM Hamiltonian up to canonical transformation.
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Spin-induced tidal dissipation in compact binaries enters the gravitational-wave phase at 2.5 post-Newtonian order with a logarithmic frequency dependence.
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Gravity Echoes from Supermassive Black Hole Binaries
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Constants of motion and fundamental frequencies for elliptic orbits at fourth post-Newtonian order
Derives the 4PN conservative map between constants of motion and fundamental frequencies for eccentric orbits, resummed over eccentricity and validated against circular-orbit and self-force results.
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Emergence of Calabi-Yau manifolds in high-precision black hole scattering
At 5PM-1SF order, Calabi-Yau three-fold periods emerge in radiation-reacted observables for classical black hole scattering computed with worldline QFT and advanced IBP/DE methods.
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Trapping, Irregular Waveforms, and Efficient Radiation in Ultra-relativistic Black Hole Encounters
Ultra-relativistic black hole flybys can radiate over 65% of their energy in gravitational waves via irregular waveforms caused by radiation trapping and lensing, without coalescence.
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Lessons from binary dynamics of inspiralling equal-mass boson-star mergers
Numerical simulations of equal-mass boson-star mergers reveal larger waveform deviations from black-hole binaries in late inspiral and merger, plus odd multipole excitations for certain scalar-field phases, with some signals degenerate until IMR consistency tests are applied.
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Large-Eccentricity Asymptotics and Fast Analytic Approximation for Fourier modes of Post-Newtonian Eccentric Waveforms
Derives large-eccentricity asymptotics for post-Newtonian eccentric waveform Fourier modes and builds a fast endpoint-constrained analytic approximation with error under 10^{-3} valid to p=200.
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How lonely are the Binary Compact Objects Detected by the LIGO-Virgo-KAGRA Collaboration?
No three-body encounter signatures detected in GW170817, GW190814, and GW230627_015337, constraining intermediate-mass black holes above 100 solar masses within roughly 0.1 AU of these binaries.
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Including higher-order modes in a quadrupolar eccentric numerical relativity surrogate using universal eccentric modulation functions
The gwNRHME framework constructs a multi-modal non-spinning eccentric gravitational waveform surrogate by modulating quasi-circular models with universal eccentric functions, achieving median mismatches of ~9e-5 against 156 NR waveforms.
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Highly eccentric non-spinning binary black hole mergers: quadrupolar post-merger waveforms
Polynomial models for the (2,2) post-merger waveform amplitudes of eccentric non-spinning binary black holes are constructed from numerical-relativity data as functions of symmetric mass ratio and two merger-time dynamical parameters.
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Gravitational Sommerfeld Effects: Formalism, Renormalization, and Perturbation to $O(G^{10})$
Closed-form Sommerfeld factor via EFT connection matrix with analytic O(G^10) magnitude and phase for l=0,1,2 waves, plus a new RG equation for radiative multipole moments that improves waveform resummation beyond tail logarithms.
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Horizon Multipole Moments of a Kerr Black Hole
Horizon multipole moments of a Kerr black hole are computed in closed form from two definitions, yielding different values for l >= 1 at nonzero spin and sharing parity and small-spin scaling with field multipoles.
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Black hole mergers beyond general relativity: a self-force approach
Self-force theory is extended to compute merger and ringdown waveforms in beyond-GR black hole binaries under the extreme mass-ratio approximation, with first calculations of self-force corrections to the merger waveform.
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Generalized Perturbed Kepler Problem: Gravitational Wave Imprints from Eccentric Compact Binaries
A unified framework for the perturbed Kepler problem derives modified eccentric orbits and gravitational wave imprints from a general perturbed potential, offering a source-specific alternative to post-Newtonian expansions.
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Non-adiabatic dynamics of eccentric black-hole binaries in post-Newtonian theory
New non-orbit-averaged 2.5PN equations for eccentric non-spinning black-hole binaries derived via energy-momentum mappings, showing Peters 1964 orbit-averaged equations break at first pericenter.
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Tidal effects up to next-to-next-to leading post-Newtonian order in massless scalar-tensor theories
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Convergence of post-Newtonian for quasi-circular non-precessing comparable mass ratios BBHs
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The third wheel: ringdown and lensing of triple systems
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The Classical Gravitational Impulse at High Energies
The gravitational impulse for ultrarelativistic massive scalars is resummed to all orders in G_N at fixed G_N s/mb, recovering post-Minkowski results and predicting the leading high-energy behavior to eleventh post-Minkowski order.
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Polarization Analysis of Ringdown Signals
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N-body next-to-leading order gravitational spin-orbit interaction via effective field theory
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A Runway to Dissipation of Angular Momentum via Worldline Quantum Field Theory
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The Good, the Bad, and the Subtle: Relativistic mode sums for neutron-star tidal response
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Black Hole Response Theory and its Exact Shockwave Limit
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All-order structure of static gravitational interactions and the seventh post-Newtonian potential
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Heterotic Footprints in Classical Gravity: PM dynamics from On-Shell soft amplitudes at one loop
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Black Hole Spectroscopy and Tests of General Relativity with GW250114
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Tidal effects in the total flux and waveform in massless scalar-tensor theories to, respectively, relative 2PN and 1.5PN orders
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Gravitational wave surrogate model for spinning, intermediate mass ratio binaries based on perturbation theory and numerical relativity
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Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA
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Worldline effective field theory of inspiralling black hole binaries in presence of dark photon and axionic dark matter
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Closed-form solutions of spinning, eccentric binary black holes at 1.5 post-Newtonian order
Two closed-form solutions for the 1.5PN Hamiltonian of spinning eccentric binary black holes are derived by combining prior methods, implemented in the public BBHpnToolkit package, and verified against numerical integration.
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Tests of General Relativity with the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1
Binary black hole signals in GWTC-1 are consistent with general relativity predictions, with an improved graviton mass bound of mg ≤ 4.7 × 10^{-23} eV/c² at 90% credible level.
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A Legendre transform establishes an exact duality between the classical Polchinski equation and the authors' classical RG equation for the gravitational effective action.
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The Impact of Spin Priors on Parameterized Tests of General Relativity
Spin prior choices propagate into tests of GR via the 1.5PN deviation parameter δφ̂3 in a non-trivial, event-dependent way, with stronger effects for short-inspiral events and partial degeneracy with χ_eff when the deviation is included.
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Quantum description of gravitational waves generated by a classical source
Quantum expectation value of gravitational waves from a classical source reproduces the classical retarded solution with Poisson graviton emission indicating a coherent state.
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Energy conditions in static, spherically symmetric spacetimes and effective geometries
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Dynamical Tidal Response of Non-rotating Black Holes: Connecting the MST Formalism and Worldline EFT
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Persistence of post-Newtonian amplitude structure in binary black hole mergers
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