post-Newtonian expansion
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| post-Newtonian expansion | |
|---|---|
| Name | Post-Newtonian expansion |
| Caption | Approximation method in general relativity |
| Field | Theoretical physics |
| Known for | Approximation scheme for weak-field slow-motion systems in Albert Einstein's General relativity framework |
post-Newtonian expansion The post-Newtonian expansion is an approximation scheme used to compute corrections to Isaac Newton's laws within Albert Einstein's General relativity for systems where gravitational fields are relatively weak and velocities are small compared with the speed of light. It provides a systematic series of terms that refine Newtonian dynamics, enabling precise modeling of compact binaries, solar-system tests, and timing of pulsars. The method connects analytic approaches developed in the early 20th century to modern computational programs in gravitational-wave observatories such as LIGO and Virgo.
Introduction
The expansion treats deviations from Newtonian gravity as successive corrections ordered by powers of v/c, where v is a characteristic velocity and c is the speed of light. It is central to interpreting experimental tests associated with Mercury's perihelion precession, timing observations of PSR B1913+16 in pulsar binaries, and waveform modeling for detections by LIGO Scientific Collaboration and KAGRA. Historically, key figures such as Albert Einstein, Arthur Eddington, Chandrasekhar, Richard Feynman, and Thibault Damour contributed methods or applications that shaped the expansion's formulation. Post-Newtonian results feed into templates used by initiatives like LIGO, Virgo, and the planned Laser Interferometer Space Antenna.
Mathematical Formalism
Mathematically, the scheme expands the metric tensor and equations of motion of General relativity around a flat Minkowski background, organizing terms by powers of (v/c). The formalism employs tensor calculus introduced by Bernhard Riemann and coordinate conditions akin to the harmonic gauge advocated by Albert Einstein and refined by André Lichnerowicz. Perturbative techniques from Sir Arthur Eddington's work and methods in multipolar expansions trace to approaches used by Hermann Weyl and Tullio Levi-Civita. Conservative dynamics and radiation-reaction effects are separated using techniques associated with Lars Onsager-style asymptotics and matched asymptotic expansions developed in studies related to Paul Dirac's radiation reaction. Field redefinitions and regularization methods adapted from Richard Feynman and Kenneth Wilson help manage divergences when modeling compact objects like Neutron stars and Black holes.
Orders and Parameterization
Orders are labeled as nPN where n indicates powers of (v/c)^{2n} beyond Newtonian order; for example, 1PN, 2PN, 3PN, and higher. Each order introduces parameters analogous to the parametrized post-Newtonian (PPN) framework championed by Kenneth Nordtvedt, Clifford Will, and Robert Dicke to quantify deviations from Einsteinian predictions. Terms at 1PN incorporate effects studied by Simon Newcomb and Percival Lowell in classical tests, while 2.5PN and higher include radiation-reaction influences relevant to analyses by Thibault Damour and Luc Blanchet. Spin-dependent couplings, treated by methods inspired by David R. Nelson and later by Eanna Flanagan, introduce additional structure at half-integer PN orders, affecting systems involving members of the Binary pulsar population such as PSR J0737−3039A/B.
Applications in Gravitational-Wave Physics
Post-Newtonian templates underpin matched-filter searches executed by the LIGO Scientific Collaboration and Virgo Collaboration during detections like GW150914 and GW170817. Waveform families blending PN results with phenomenological models were developed by groups around Frans Pretorius, Alessandra Buonanno, and Luc Blanchet. The expansion provides phase evolution essential for parameter estimation of source properties—masses and spins—used in analyses by teams including Nicolás Yunes and Bela Sathyaprakash. In multimessenger events involving NGC 4993, PN-informed models enabled electromagnetic follow-up strategies coordinated with observatories such as NASA's Fermi Gamma-ray Space Telescope.
Relation to Numerical Relativity and Effective Field Theory
Post-Newtonian methods complement full numerical-relativity simulations pioneered by Frans Pretorius, Maya Kaul, and James M. Centrella that solve Einstein's equations without perturbative truncation. Hybrid waveforms stitch PN inspiral results to numerical-relativity merger and ringdown outputs used by collaborations such as NRAR (Numerical Relativity–Analytical Relativity) and SXS (Simulating eXtreme Spacetimes). Effective field theory (EFT) approaches introduced by Walter Goldberger and Ira Rothstein recast PN calculations using Feynman-diagram techniques, enabling cross-fertilization with particle-physics methods promoted by Richard Feynman and Kenneth Wilson.
Limitations and Validity
The expansion is valid when gravitational potentials are weak and velocities are small; it breaks down near strong-field, high-velocity regimes around late-stage mergers of Black hole binaries and in environments like the vicinity of the Event Horizon Telescope targets. Regularization ambiguities challenged early 3PN calculations addressed by contributors including Luc Blanchet, Thibault Damour, and Bala Iyer. Applicability is also limited in systems with extreme mass ratios where methods such as gravitational self-force approaches developed by Leor Barack and Steven Detweiler are more appropriate.
Historical Development and Key Contributions
Foundational ideas trace from Albert Einstein's early perturbative work and Arthur Eddington's observational tests to systematic PN formulations by Tullio Levi-Civita and later formalizations by Chandrasekhar and Kip Thorne. Modern high-order computations were advanced by Luc Blanchet, Thibault Damour, Bala Iyer, Gerhard Schäfer, and teams within the LIGO Scientific Collaboration and Virgo Collaboration. Cross-disciplinary inputs from particle physics and EFT came via Walter Goldberger and Ira Rothstein, while numerical breakthroughs by Frans Pretorius and collaborations like SXS enabled the synthesis of analytic PN results with full numerical solutions for robust waveform modeling.