Gravitational Waves
Post-Newtonian Expansion: Building an Inspiral Waveform Order by Order in v/c
In the final second before two neutron stars collide, they whirl around each other roughly 100 times, sweeping through the LIGO band as their orbital speed climbs past 30% of the speed of light. To track those thousands of wave cycles well enough to detect them, physicists need a formula for the emitted gravitational wave accurate to better than a fraction of a radian over the whole inspiral. The Post-Newtonian (PN) expansion is the machinery that delivers it: a systematic Taylor series of Einstein's equations in the small parameter v/c, where v is the orbital velocity and c the speed of light.
Rather than solving the full nonlinear field equations at once, the PN expansion writes the orbital dynamics, energy loss, and waveform phase as a power series in (v/c)² ≈ GM/(rc²). Each successive term — labeled 1PN, 1.5PN, 2PN, and so on — is a correction of relative order (v/c)² beyond Newtonian gravity, capturing ever-subtler relativistic effects that shape the chirp signal detectors actually record.
- TypePerturbative approximation to general relativity
- RegimeSlow-motion, weak-field inspiral (v/c ≲ 0.3)
- Expansion parameter(v/c)² ≈ GM/(rc²)
- State of the art4.5PN phasing, i.e. order (v/c)^9
- Key relationdE/dt = −(32/5)(G⁴/c⁵)(m₁²m₂²M)/r⁵
- Observed inGW150914, GW170817, and all LIGO–Virgo–KAGRA detections
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What the Post-Newtonian Expansion Is
General relativity's field equations are nonlinear and have no closed-form solution for two orbiting bodies. The Post-Newtonian expansion tames this by exploiting the fact that during the inspiral the bodies move slowly compared with light and their mutual gravity is weak. The controlling small quantity is
- ε ≈ (v/c)² ≈ GM/(rc²), where v is orbital speed, M the total mass, and r the separation.
By the virial theorem the kinetic term v²/c² and the potential term GM/(rc²) are the same order, so a single bookkeeping parameter suffices. Every observable — the equations of motion, the binding energy E(v), and the gravitational-wave luminosity — is written as a series: a Newtonian leading term plus corrections of relative order ε, ε^{3/2}, ε², ε^{5/2}, and so on. A term of relative order ε^n is called nPN. Half-integer orders (1.5PN, 2.5PN) are not sloppiness: they encode genuinely distinct physics such as radiation-reaction tails and spin effects that scale with odd powers of v/c.
How the Waveform Is Built Order by Order
The engine of the chirp is the energy balance equation: the orbit's binding energy is carried away by gravitational waves, so dE/dt = −F, where F is the flux. At leading (Newtonian) order the flux is the Einstein quadrupole formula,
- F = (32/5)(G⁴/c⁵)(m₁²m₂²M)/r⁵, giving dE/dt ∝ −v¹⁰/c⁵.
Both E(v) and F(v) are then computed as PN series. Combining them yields the orbital frequency's evolution dφ/dt and, crucially, the accumulated phase φ(t). Because a matched-filter search correlates against φ, getting the phase right to a small fraction of a cycle over ~10⁴ cycles is what makes detection possible. Modern techniques include the stationary phase approximation (SPA), producing the frequency-domain TaylorF2 waveform in which the phase is an explicit polynomial in the dimensionless frequency variable x = (πMf)^{2/3}. Each PN coefficient in that polynomial is a known function of the mass ratio and spins.
Key Quantities and a Worked Example
The natural mass parameters are the total mass M = m₁ + m₂ and the chirp mass ℳ = (m₁m₂)^{3/5}/M^{1/5}, which dominates the leading phasing. The symmetric mass ratio η = m₁m₂/M² (0 < η ≤ 0.25) controls the higher-order coefficients.
- For a binary neutron star with m₁ = m₂ = 1.4 M_sun: M = 2.8 M_sun, η = 0.25, ℳ ≈ 1.22 M_sun.
- The innermost stable circular orbit (ISCO) for a test particle sits at r = 6GM/c², where v/c ≈ 0.41 — the PN series is straining there.
- The gravitational-wave frequency at ISCO is roughly f_ISCO ≈ 4400 (M_sun/M) Hz ≈ 1570 Hz for this binary.
The number of cycles from a low frequency f is enormous: N ≈ (1/32π)(c³/Gℳ)^{5/3}(πf)^{-5/3}, giving on the order of 10⁴ cycles from 10 Hz for a BNS. A phase error of even one radian accumulated over those cycles would ruin the matched-filter overlap, which is why 3.5PN and beyond matter.
Where It Is Observed and Detected
PN waveforms are the backbone of the template banks used by LIGO, Virgo, and KAGRA. The inspiral portion of every compact-binary signal is fit against PN-based models. Landmark cases:
- GW150914 (2015): a 36 + 29 M_sun black-hole merger — but at such high mass the observable band was dominated by the merger and ringdown, where PN alone breaks down.
- GW170817 (2017): a binary neutron star inspiral lasting ~100 s in band with thousands of cycles — the archetypal PN success, letting the chirp mass be measured to ℳ ≈ 1.188 M_sun with exquisite precision.
PN is also the analytic input for tests of general relativity: LIGO collaborations let each PN phase coefficient float and check that the data agree with GR's predicted values, constraining hypothetical deviations (including a graviton mass and dipole radiation). And PN results anchor the effective-one-body (EOB) and surrogate models, and calibrate numerical-relativity–tuned phenomenological waveforms.
Comparison with Related Regimes and Methods
The PN expansion is one of three complementary tools for the two-body problem, each valid in a different regime:
- Post-Newtonian — weak field, slow motion (v/c ≲ 0.3, r ≫ GM/c²). Governs the long inspiral. Fails as v/c → 0.4 near merger.
- Numerical relativity (NR) — solves the full Einstein equations on a supercomputer; the only method valid through the violent merger, but expensive and limited to the last few tens of orbits.
- Black-hole perturbation theory / self-force — expands in the mass ratio (small m₂/m₁) rather than v/c, ideal for extreme-mass-ratio inspirals but not comparable-mass systems.
A distinct but related scheme, the Post-Minkowskian (PM) expansion, expands in G (weak field) while keeping velocities relativistic — useful for scattering and increasingly computed with quantum-field-theory amplitude methods. The effective-one-body framework resums PN information into a form that stays accurate closer to merger, and hybrid IMR (inspiral–merger–ringdown) waveforms stitch PN, NR, and ringdown together.
Significance, Open Questions, and Famous Milestones
The PN program is a 100-year arc: Einstein's 1918 quadrupole formula, the Peters–Mathews (1963) orbital-decay formula that predicted the Hulse–Taylor binary pulsar's inspiral to better than 0.2% (Nobel Prize 1993), and the modern high-order phasing derived by groups led by Luc Blanchet, Thibault Damour, Gerhard Schäfer, and others. The non-spinning inspiral phase is now known to 4.5PN order, i.e. (v/c)⁹.
Open and active questions:
- Pushing spin, spin–spin, and tidal (finite-size, Love-number) terms to matching high order — tidal effects enter formally at 5PN but are numerically important for neutron stars and were pivotal in constraining the nuclear equation of state via GW170817.
- The PN series is asymptotic, not convergent; how best to resum it (Padé, EOB) to extend validity toward ISCO remains debated.
- Preparing for LISA and next-generation ground detectors, where even higher accuracy and the extreme-mass-ratio regime demand tighter control of systematic waveform error.
| PN order | Relative order in v/c | Physics first appearing | First derived |
|---|---|---|---|
| Newtonian (0PN) | (v/c)⁰ | Quadrupole radiation, leading chirp | Einstein 1918 / Peters–Mathews 1963 |
| 1PN | (v/c)² | Relativistic orbital corrections, periastron advance | 1970s–1980s |
| 1.5PN | (v/c)³ | Leading gravitational-wave tail (backscatter off curvature) | Blanchet–Damour 1992 |
| 2.5PN | (v/c)⁵ | Leading radiation-reaction contribution to phase | Iyer–Will 1993 |
| 3.5PN | (v/c)⁷ | Higher tails, near-complete non-spin phase | Blanchet et al. ~2004 |
| 4.5PN | (v/c)⁹ | Current state-of-the-art non-spin phasing | 2010s–2020s |
Frequently asked questions
What is the small parameter in the post-Newtonian expansion?
It is (v/c)², where v is the orbital velocity and c the speed of light. By the virial theorem this equals GM/(rc²), so orbital speed and gravitational potential are the same order. A term of relative order (v/c)^{2n} is called nPN, and each higher order adds a smaller relativistic correction.
Why are there half-integer PN orders like 1.5PN and 2.5PN?
Half-integer orders arise from physics that scales with odd powers of v/c. The 1.5PN term is the leading gravitational-wave 'tail,' produced when radiation backscatters off the binary's own spacetime curvature, and it also carries the leading spin-orbit coupling; the leading radiation-reaction contribution enters at 2.5PN. They are not errors; they encode genuinely distinct effects.
Why does getting the phase right matter so much?
Detectors use matched filtering, correlating the data against a template's phase evolution over up to ~10,000 cycles. Because signal-to-noise builds by keeping the template in step with the true wave, even a fraction-of-a-radian phase error accumulated over the inspiral degrades the overlap and can cause a real signal to be missed.
How far does the PN expansion stay valid?
It works well while the binary is widely separated and slow, roughly v/c ≲ 0.3. As the bodies approach the innermost stable circular orbit at r = 6GM/c² (v/c ≈ 0.41), the series is stretched to its limit, and through the merger only numerical relativity is trustworthy. Resummation schemes like EOB extend PN accuracy somewhat closer to merger.
What is the chirp mass and why is it central to PN phasing?
The chirp mass is ℳ = (m₁m₂)^{3/5}/(m₁+m₂)^{1/5}. It is the single combination of masses that controls the leading (Newtonian) rate of frequency sweep, so it is by far the best-measured parameter from an inspiral. GW170817's chirp mass was pinned to about 1.188 M_sun.
How is the post-Newtonian expansion different from the post-Minkowskian one?
The PN expansion assumes both weak field and slow motion, expanding in v/c, and is tailored to bound inspirals. The post-Minkowskian (PM) expansion expands only in Newton's constant G (weak field) while allowing relativistic velocities, making it natural for scattering problems and increasingly computed with particle-physics amplitude techniques.