Relativity: Gravitational Waves
Relativity » Gravitational Waves
Gravitational Waves
Gravitational waves are propagating ripples in spacetime curvature, predicted by Einstein in 1916, inferred from binary-pulsar orbital decay in the 1970s–80s, and first detected directly on 14 September 2015. This page develops them from the linearized field equations through gauge fixing and the two physical polarizations, the quadrupole formula for generation and energy loss, the inspiral–merger–ringdown of compact binaries, and interferometric detection. It closes with the observational record through the end of the fourth LIGO–Virgo–KAGRA observing run (O4) in November 2025, and the detectors planned for the next two decades.
It assumes General Relativity and the differential geometry on Tensor Formalism.
Conventions. Geometric units $G = c = 1$ except where $G$ and $c$ are shown explicitly. Signature $(-,+,+,+)$. Greek indices run over $0$–$3$, Latin over spatial $1$–$3$; repeated indices are summed. $\Box \equiv \eta^{\mu\nu}\partial_\mu\partial_\nu = -\partial_t^2 + \nabla^2$ is the flat-space d’Alembertian.
Linearized Gravity
Between source and detector, spacetime is very nearly flat, so the gravitational field can be treated as a small perturbation of Minkowski space. Linearizing the Einstein equations turns them into a wave equation with the same structure as electromagnetism in Lorenz gauge.
The weak-field metric
Write
\[g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \qquad |h_{\mu\nu}| \ll 1\]and keep only terms linear in $h_{\mu\nu}$. Indices are raised and lowered with $\eta_{\mu\nu}$, since using $g_{\mu\nu}$ would only add second-order terms. To this order the Christoffel symbols and Ricci tensor are
\[\Gamma^\lambda_{\mu\nu} = \frac{1}{2}\eta^{\lambda\sigma}\left(\partial_\mu h_{\sigma\nu} + \partial_\nu h_{\sigma\mu} - \partial_\sigma h_{\mu\nu}\right)\] \[R_{\mu\nu} = \frac{1}{2}\left(\partial_\sigma\partial_\mu h^\sigma{}_\nu + \partial_\sigma\partial_\nu h^\sigma{}_\mu - \Box h_{\mu\nu} - \partial_\mu\partial_\nu h\right)\]where $h \equiv \eta^{\mu\nu}h_{\mu\nu}$ is the trace.
The trace-reversed perturbation
Define the trace-reversed perturbation
\[\bar{h}_{\mu\nu} \equiv h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu} h\]so called because $\bar{h} = -h$; the inverse is $h_{\mu\nu} = \bar{h}{\mu\nu} - \tfrac{1}{2}\eta{\mu\nu}\bar{h}$. The linearized Einstein tensor becomes
\[G_{\mu\nu} = -\frac{1}{2}\Box \bar{h}_{\mu\nu} + \partial_\alpha\partial_{(\mu}\bar{h}^\alpha{}_{\nu)} - \frac{1}{2}\eta_{\mu\nu}\partial_\alpha\partial_\beta \bar{h}^{\alpha\beta}\]The last two terms involve only the divergence $\partial_\alpha\bar{h}^{\alpha\beta}$, which a coordinate choice can remove.
Gauge freedom and the Lorenz gauge
An infinitesimal coordinate change $x^\mu \to x^\mu + \xi^\mu$ leaves $\eta_{\mu\nu}$ unchanged but shifts the perturbation:
\[h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu\]This is the gauge freedom of linearized gravity, the analogue of $A_\mu \to A_\mu - \partial_\mu\chi$ in electromagnetism; only gauge-invariant quantities, such as the linearized Riemann tensor, are physical. Under this change $\partial_\mu\bar{h}^{\mu\nu} \to \partial_\mu\bar{h}^{\mu\nu} - \Box\xi^\nu$, so solving $\Box\xi^\nu = \partial_\mu\bar{h}^{\mu\nu}$ reaches the Lorenz (harmonic, de Donder) gauge
\[\partial_\mu \bar{h}^{\mu\nu} = 0\]in which the linearized field equations become a set of decoupled, sourced wave equations:
\[\Box \bar{h}_{\mu\nu} = -16\pi G\, T_{\mu\nu}\]Compare Maxwell’s $\Box A^\mu = -4\pi J^\mu$ in Gaussian units. Disturbances propagate at the speed of light; in vacuum, $\Box \bar{h}_{\mu\nu} = 0$.
The Lorenz condition does not use up all the freedom: any further $\xi^\mu$ with $\Box\xi^\mu = 0$ preserves it. This residual freedom is used next to remove the non-radiative components.
The Transverse–Traceless Gauge
Definition
Take a vacuum plane wave $\bar{h}{\mu\nu} = A{\mu\nu}e^{ik_\alpha x^\alpha}$. The wave equation forces $k_\alpha k^\alpha = 0$ (the wave vector is null, so the wave travels at $c$), and the Lorenz condition forces $k^\mu A_{\mu\nu} = 0$. The residual gauge freedom can then impose
\[h_{0\mu} = 0, \qquad h^i{}_i = 0, \qquad \partial^i h_{ij} = 0\]— the perturbation is purely spatial, traceless, and transverse to the direction of propagation. This is the transverse–traceless (TT) gauge. Because the wave is traceless, $\bar{h}{ij} = h{ij}$, written $h^{TT}_{ij}$.
Two polarizations
A symmetric $4\times4$ tensor has 10 components. The Lorenz condition removes 4 and the residual gauge freedom removes 4 more, leaving 2 physical degrees of freedom. For a wave travelling along $z$:
\[h^{TT}_{ij} = \begin{pmatrix} h_+ & h_\times & 0 \\ h_\times & -h_+ & 0 \\ 0 & 0 & 0 \end{pmatrix}\cos\left[\omega(t - z)\right]\]The amplitudes $h_+$ (“plus”) and $h_\times$ (“cross”) are the two polarizations. Under a rotation by angle $\psi$ about the propagation axis they mix with angle $2\psi$, the signature of a helicity-2 (spin-2) field: the two linear polarizations are $45^\circ$ apart, compared with $90^\circ$ for the helicity-1 photon. Generic metric theories of gravity allow up to six polarizations (adding two vector and two scalar modes); detector networks constrain these, and data so far are consistent with GR’s two tensor modes.
Effect on free test masses
In TT coordinates, freely falling masses keep fixed coordinates, but the proper distance between them oscillates. For masses separated by $L$ along unit vector $\hat n$ in the transverse plane,
\[\frac{\delta L}{L} = \frac{1}{2}\,h^{TT}_{ij}\,n^i n^j\]A $+$ wave stretches a ring of particles along $x$ while squeezing it along $y$, then reverses; a $\times$ wave does the same along axes rotated by $45^\circ$. The dimensionless strain $h$ is thus a fractional length change. Typical waves reaching Earth have $h \sim 10^{-21}$; over a 4 km arm that is $\delta L \sim 4\times10^{-18}$ m, about $1/200$ of a proton radius.
graph TD
A["g = eta + h<br/>(weak-field metric)"] --> B["Lorenz gauge<br/>(divergence of h-bar = 0)"]
B --> C["Wave equation<br/>box h-bar = -16 pi G T"]
C --> D["Vacuum: box h-bar = 0<br/>null wave vector, speed c"]
D --> E["TT gauge<br/>spatial, transverse, traceless"]
E --> F["Two polarizations<br/>h-plus and h-cross"]
F --> G["Measured as strain<br/>delta L / L"]
Generation: The Quadrupole Formula
The Lorenz-gauge wave equation is solved with the retarded Green’s function, as in electromagnetism:
\[\bar{h}_{\mu\nu}(t,\mathbf{x}) = 4G \int \frac{T_{\mu\nu}(t - |\mathbf{x}-\mathbf{x}'|,\, \mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}\, d^3x'\]Far from a slowly moving, compact source ($v \ll c$, size much smaller than the wavelength), the leading term is the quadrupole formula.
Why there is no dipole radiation
In electromagnetism the leading radiation is electric dipole, $\propto \ddot{\mathbf{d}}$. The gravitational analogues vanish by conservation laws:
- The mass dipole is $\int \rho\,\mathbf{x}\,d^3x = M\mathbf{X}{\rm cm}$; momentum conservation gives $\ddot{\mathbf{X}}{\rm cm} = 0$.
- The “magnetic dipole” analogue is the total angular momentum, which is conserved.
The leading radiation is therefore quadrupolar, which is one reason gravitational radiation is so weak.
The reduced quadrupole moment
Define the trace-free (reduced) mass quadrupole moment
\[\mathcal{I}_{ij} = \int \rho\left(x_i x_j - \frac{1}{3}\delta_{ij} r^2\right)d^3x\]The radiation field at distance $r$ is
\[h^{TT}_{ij} = \frac{2G}{c^4 r}\,\ddot{\mathcal{I}}^{TT}_{ij}(t - r/c)\]where TT denotes projection transverse to the line of sight and removal of the trace. The factor $G/c^4 \approx 8\times10^{-45}\ \text{s}^2\,\text{kg}^{-1}\,\text{m}^{-1}$ is why only astrophysical masses moving at a significant fraction of $c$ produce measurable strain.
Radiated energy
Gravitational-wave energy cannot be localized at a point; it is defined by averaging over several wavelengths (the Isaacson effective stress–energy tensor). The resulting luminosity is
\[L_{\rm GW} = \frac{G}{5c^5}\left\langle \dddot{\mathcal{I}}_{ij}\, \dddot{\mathcal{I}}_{ij} \right\rangle\]The natural luminosity scale is $c^5/G \approx 3.6\times10^{52}$ W. A laboratory source — two 1 kg masses on a 1 m rod spinning at 1 kHz — radiates of order $10^{-30}$ W. A binary black-hole merger radiates a few percent of $c^5/G$ at peak: GW150914 briefly emitted about $3.6\times10^{49}$ W, more than the combined electromagnetic output of all stars in the observable universe.
Compact Binaries
Two neutron stars or black holes in orbit are the dominant sources for ground-based detectors. Radiation drains orbital energy, the orbit shrinks, the orbital frequency rises, and the emission strengthens: a runaway that ends in merger.
Orbital decay
For a circular binary with masses $m_1, m_2$, total mass $M$, reduced mass $\mu = m_1 m_2/M$, and separation $a$, the waves are emitted at twice the orbital frequency, $f_{\rm GW} = 2f_{\rm orb}$. Equating the quadrupole luminosity to the loss of Newtonian orbital energy $E = -GM\mu/2a$ gives the Peters decay rate
\[\frac{da}{dt} = -\frac{64}{5}\frac{G^3}{c^5}\frac{\mu M^2}{a^3}\]and a finite time to coalescence from initial separation $a_0$:
\[t_{\rm c} = \frac{5}{256}\frac{c^5\, a_0^4}{G^3\, \mu M^2}\]Eccentric orbits decay faster and circularize as they shrink (Peters 1964), which is why most binaries are nearly circular by the time they enter the ground-based band. For the Hulse–Taylor pulsar the remaining lifetime is about 300 million years.
The chirp mass
In the inspiral, the frequency evolution depends on the masses only through the chirp mass
\[\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}} = \mu^{3/5} M^{2/5}\]via
\[\frac{df_{\rm GW}}{dt} = \frac{96}{5}\pi^{8/3}\left(\frac{G\mathcal{M}}{c^3}\right)^{5/3} f_{\rm GW}^{11/3}\]Integrating gives $f_{\rm GW} \propto (t_c - t)^{-3/8}$, and the strain amplitude grows as $f_{\rm GW}^{2/3}$. Measuring the rate at which the frequency sweeps upward yields $\mathcal{M}$ directly, and it is usually the best-measured parameter of an event. The individual masses, spins, and tidal deformability enter at higher post-Newtonian order and are less precisely determined.
Standard sirens. The amplitude falls as $1/d_L$ and depends on $\mathcal{M}$, which the phase evolution measures independently. A compact binary therefore gives its own luminosity distance without any calibration ladder. Combined with a redshift from an identified host galaxy (a “bright siren”) or statistically from galaxy catalogues (“dark sirens”), this measures the Hubble constant; see Cosmology — The Hubble tension.
The three phases: inspiral, merger, ringdown
| Phase | Physics | Modelling |
|---|---|---|
| Inspiral | Well-separated bodies, $v/c \lesssim 0.3$; slowly rising frequency and amplitude | Post-Newtonian expansion in $v/c$ (known to 4.5PN order for the phase of non-spinning binaries) |
| Merger | Strong-field, highly nonlinear; peak amplitude | Numerical relativity, routine since the 2005 breakthroughs |
| Ringdown | Remnant Kerr black hole sheds its distortions as damped quasinormal modes | Black-hole perturbation theory; mode frequencies depend only on final mass and spin |
Search pipelines and parameter estimation use waveform families that stitch these regimes together and are calibrated to numerical relativity: effective-one-body models (SEOBNR), phenomenological models (IMRPhenom), and surrogate models interpolating numerical-relativity simulations directly.
The binary pulsar
The first evidence came before any detector. PSR B1913+16, discovered by Hulse and Taylor in 1974, is a pulsar orbiting a neutron-star companion with a 7.75-hour period. Its orbital period decreases by about $76$ microseconds per year, matching the quadrupole-formula prediction to within about 0.2%. The 1993 Nobel Prize in Physics recognized the discovery. The Double Pulsar PSR J0737−3039A/B, in which both neutron stars are observed as pulsars, now confirms the prediction to about $10^{-4}$.
Detection
Laser interferometers
LIGO (Hanford and Livingston, USA, 4 km arms), Virgo (Italy, 3 km), and KAGRA (Japan, 3 km, underground and cryogenic) are Michelson interferometers whose end mirrors are suspended test masses. A wave with its $+$ polarization aligned to the arms lengthens one arm while shortening the other, giving a differential change
\[\Delta L = L_x - L_y = h L\]The interferometer is held near a dark fringe, so a differential length change produces a change in light power at the output photodetector.
graph LR
L["Laser"] --> PR["Power-recycling<br/>mirror"]
PR --> BS["Beam splitter"]
BS <--> X["X-arm Fabry-Perot<br/>cavity, 4 km"]
BS <--> Y["Y-arm Fabry-Perot<br/>cavity, 4 km"]
BS --> SR["Signal-recycling<br/>mirror"]
SR --> PD["Output photodetector<br/>(dark port)"]
SQ["Squeezed-vacuum<br/>source"] -.-> SR
GW["Gravitational wave"] -.->|"stretches X,<br/>squeezes Y"| X
GW -.-> Y
Reaching $\Delta L \sim 10^{-19}$ m requires suppressing every noise source:
- Fabry–Pérot arm cavities fold the light hundreds of times, increasing the phase shift per unit strain.
- Power recycling raises the circulating arm power to several hundred kilowatts, reducing relative photon shot noise; signal recycling shapes the frequency response.
- Seismic isolation: active platforms and multi-stage pendulum suspensions with fused-silica fibres attenuate ground motion above a few hertz.
- Squeezed light: injecting squeezed vacuum reduces quantum noise. Since O4, LIGO uses frequency-dependent squeezing via a 300 m filter cavity, reducing shot noise at high frequencies without increasing radiation-pressure noise at low frequencies.
- Ultra-high vacuum in the beam tubes removes phase noise from residual gas.
Seismic and Newtonian (gravity-gradient) noise dominate below about 10 Hz, thermal noise of suspensions and mirror coatings in the mid band, and quantum shot noise at high frequencies. The resulting band, roughly 10 Hz to a few kHz, matches stellar-mass compact binaries. Sensitivity is often quoted as the binary-neutron-star range, the sky-averaged distance at which a $1.4 + 1.4\,M_\odot$ merger is detectable; during O4 the LIGO detectors typically reached about 150–170 Mpc.
A network of detectors is essential. Coincidence between widely separated sites rejects local noise transients (“glitches”), and arrival-time differences (up to about 10 ms between the LIGO sites) plus relative amplitudes localize the source on the sky for electromagnetic follow-up.
Frequency bands and detector types
| Band | Detectors | Principal sources |
|---|---|---|
| ~10 Hz – few kHz | LIGO, Virgo, KAGRA; planned LIGO-India, Einstein Telescope, Cosmic Explorer | Stellar-mass black-hole and neutron-star mergers; core-collapse supernovae; spinning neutron stars |
| ~0.1 mHz – 0.1 Hz | LISA (ESA with NASA; adopted January 2024, launch planned for about 2035) | Massive black-hole mergers ($10^4$–$10^7\,M_\odot$), extreme-mass-ratio inspirals, galactic white-dwarf binaries |
| ~1–100 nHz | Pulsar timing arrays: NANOGrav, EPTA/InPTA, PPTA, CPTA, MeerKAT PTA (combined as the IPTA) | Supermassive black-hole binaries; possible early-universe sources |
| ~$10^{-18}$–$10^{-16}$ Hz | CMB B-mode polarization (BICEP/Keck, Simons Observatory, future CMB experiments) | Primordial gravitational waves from inflation (not yet detected) |
LISA will be three spacecraft in a triangle with 2.5-million-km arms, trailing Earth around the Sun. Pulsar timing arrays use millisecond pulsars as clocks: a passing nanohertz wave shifts pulse arrival times, and a gravitational-wave background produces a characteristic angular correlation between pulsar pairs, the Hellings–Downs curve.
Observational Record
Landmark events
| Event | Date | What it was | Significance |
|---|---|---|---|
| GW150914 | 14 Sep 2015 | Binary black hole, about $36 + 29\,M_\odot$ → $62\,M_\odot$, at about 440 Mpc | First direct detection; about $3\,M_\odot c^2$ radiated in ~0.2 s. 2017 Nobel Prize (Weiss, Barish, Thorne) |
| GW170817 | 17 Aug 2017 | Binary neutron star, 40 Mpc, in NGC 4993 | Gamma-ray burst 1.7 s later; kilonova observed across the spectrum; $\lvert v_{\rm GW} - c\rvert/c \lesssim 10^{-15}$; first standard-siren $H_0$; r-process nucleosynthesis site |
| GW190521 | 21 May 2019 | About $85 + 66\,M_\odot$ → $142\,M_\odot$ | First intermediate-mass black-hole remnant; primary in the pair-instability mass gap |
| GW230529 | 29 May 2023 | Primary of about $2.5$–$4.5\,M_\odot$ with a neutron-star companion | Object in the “lower mass gap” between the heaviest neutron stars and lightest black holes |
| GW231123 | 23 Nov 2023 | About $137 + 103\,M_\odot$, both rapidly spinning; remnant about $225\,M_\odot$ | Most massive binary observed; high masses and spins suggest hierarchical (repeated) mergers |
| GW250114 | 14 Jan 2025 | About $34 + 32\,M_\odot$ → $63\,M_\odot$ | Loudest event to date (network SNR about 80). Confirmed Hawking’s area theorem at high confidence and measured the fundamental ringdown mode and first overtone, consistent with a Kerr black hole |
Catalogues and populations
The LIGO–Virgo–KAGRA Gravitational-Wave Transient Catalog (GWTC) collects confident detections. GWTC-3 (2021) closed out runs O1–O3 with 90 candidates. GWTC-4.0 (August 2025) added 128 new candidates from the first part of O4 (O4a, May 2023 – January 2024), roughly doubling the catalogue. O4 as a whole ran from 24 May 2023 to 18 November 2025 and issued hundreds of public alerts; catalogues for its later parts are in preparation.
Population analyses of these catalogues show that most detections are binary black holes; the black-hole mass distribution has a peak near $35\,M_\odot$ and a tail extending into the pair-instability gap; spins are typically small; and the merger rate increases with redshift, roughly tracking star formation. Neutron-star–black-hole binaries (first seen in 2020 as GW200105 and GW200115) and binary neutron stars remain comparatively rare.
Pulsar-timing evidence for a nanohertz background
In June 2023 NANOGrav (15-year data set), EPTA/InPTA, PPTA, and CPTA jointly reported evidence for a stochastic nanohertz background, including the Hellings–Downs correlation at about $3$–$4\sigma$ in the individual data sets. The amplitude is consistent with, and somewhat above typical predictions for, a cosmic population of supermassive black-hole binaries; more exotic early-universe origins are not excluded. A combined IPTA analysis is expected to strengthen the detection.
Tests of general relativity
Each loud event tests GR in the strong-field, dynamical regime:
- Waveform consistency. The inspiral and post-merger portions of a signal independently predict consistent remnant masses and spins; post-Newtonian coefficients agree with GR.
- Propagation. GW170817 bounds the speed of gravity; the absence of dispersion bounds the graviton mass at about $m_g \lesssim 10^{-23}\ \text{eV}/c^2$.
- Polarizations. Network analyses favour pure tensor polarizations over pure vector or scalar alternatives.
- Ringdown spectroscopy and the area theorem. GW250114 measured multiple quasinormal modes consistent with a single Kerr black hole (the no-hair theorem) and showed the final horizon area exceeds the sum of the initial areas, as Hawking’s 1971 area theorem requires.
No deviation from GR has been found.
Future Detectors
timeline
title Gravitational-wave detectors, observed and planned
2015-2020 : O1-O3 (Advanced LIGO and Virgo, KAGRA from 2020)
2023-2025 : O4 (LIGO, Virgo, KAGRA)
2026-2027 : Short interim run, then upgrades
Late 2020s : O5 (LIGO A+, upgraded Virgo)
~2030s : LIGO-India
~2035 : LISA launch
Late 2030s : Einstein Telescope and Cosmic Explorer (proposed)
- O5 and A+. The LIGO A+ upgrade (larger test masses, improved coatings, further squeezing) aims to roughly double the O4 range. LIGO has indicated an interim observing period after O4 and the start of O5 in the late 2020s; dates remain subject to commissioning and funding.
- LIGO-India, a third 4 km LIGO detector, approved by the Indian government in 2023, will substantially improve sky localization.
- Third-generation ground detectors. The Einstein Telescope (Europe; underground, 10 km arms, cryogenic) and Cosmic Explorer (USA; 40 km and 20 km arms) would reach binary black-hole mergers throughout essentially the whole observable universe and detect hundreds of thousands of events per year.
- LISA will open the millihertz band, where massive black-hole mergers are expected to be detectable with very high signal-to-noise ratio.
See Also
- General Relativity — the field equations whose wave solutions these are.
- Tensor Formalism & the Field Equations — curvature and the linearized field equations.
- Black Holes — Kerr black holes, quasinormal modes, the no-hair and area theorems.
- Relativistic Cosmology — standard sirens, the Hubble tension, and primordial gravitational waves from inflation.
- Special Relativity — the Minkowski background.
- Graduate Topics Hub — how the deep-dive pages fit together.
- Computational Physics — numerical relativity and waveform modelling.
- Quantum Field Theory — the graviton as a massless spin-2 field.
- Physics Hub — all physics topics.