Gravitational wave

A gravitational wave is a propagating disturbance of spacetime curvature produced by changing mass-energy multipole moments. In general relativity, its two transverse gravitational wave polarizations alternately stretch and squeeze freely falling separations. The leading radiation from a slowly moving source is governed by its mass quadrupole moment, rather than by its total mass alone.

The experimental breakthrough was GW150914: observed on 14 September 2015 and published on 11 February 2016. The LIGO Scientific Collaboration and Virgo Collaboration reported a merging pair of black holes, using the two LIGO detectors at Hanford and Livingston; Virgo was being upgraded and did not observe this event.^[ref]

The 2017 Nobel Prize in Physics recognized Rainer Weiss, Barry C. Barish and Kip S. Thorne for contributions to LIGO and gravitational wave observation. Rainer Weiss developed interferometric detection and its noise analysis; Barry C. Barish led the project into construction and operation; Kip S. Thorne developed its theoretical and astrophysical foundations. The experiment depended on a large collaboration.^[ref]

GW150914 is the first directly detected gravitational wave, interpreted as a binary black hole inspiral, merger and ringdown. Its discovery paper reports a peak projected gravitational wave strain of approximately $10^{-21}$ and a combined matched filter signal-to-noise ratio of 24. The inferred source-frame component masses were $36_{-4}^{+5}$ and $29_{-4}^{+4}$ solar masses, with 90 percent credible intervals. These are model-dependent estimates from a detector network.^[ref]

The interesting measurement is a chain: curvature changes relative light travel times, wave interference turns those changes into a photodetector signal, calibrated records from separate sites constrain a common source, and statistical inference tests competing explanations. The following original models explain that chain; the plotted time series are stipulated calculations, never measured GW150914 data.

Each LIGO instrument is a Michelson interferometer with approximately $4\mathrm{km}$ arms containing Fabry–Pérot interferometers. A stabilized $1064\mathrm{nm}$ laser illuminates suspended $40\mathrm{kg}$ fused-silica mirrors in vacuum. Four-stage suspensions and active isolation suppress ground motion; low-loss materials reduce thermal noise. Power recycling increases circulating power; signal recycling shapes the response to the gravitational wave.^[ref]

Why measure two perpendicular arms? In the long-wavelength limit, take a gravitational wave with plus polarization normally incident on ideal perpendicular arms. Its transverse-traceless tensor has $h_{xx}=h_{+}$ and $h_{yy}=-h_{+}$. Expanding each proper length to first order givesHere $h_{+}$ is dimensionless, whereas $\Delta L$ is a length. A stipulated $h_{+}=10^{-21}$ and $L=4000\mathrm{m}$ imply $\Delta L=4\times 10^{-18}\mathrm{m}$, with individual arm changes of opposite sign and magnitude $2\times 10^{-18}\mathrm{m}$. These are equivalent differential arm-length changes, not a photograph of mirrors moving against a rigid external ruler. The observable compares returning light phases; coordinate systems can distribute the description between mirror coordinates and light propagation.

For unit arm directions $\mathbf{u},\mathbf{v}$, define the detector tensor $D^{ij}=(u^i{u}^j-v^i{v}^j)/2$. If $e_{ij}^{+},e_{ij}^{\times }$ are the transverse polarization tensors, thenThe dimensionless factors $F_A$ encode sky direction and the polarization basis. Thus even identical sources can have different observed amplitudes and wave phases. Motion changing both arms equally cancels in this ideal differential observable, although real asymmetries can couple common disturbances into it.

Suspension supplies an inertial reference above its resonances. For example, a single ideal harmonic oscillator with support motion $y$ obeys $\ddot{x}+{\omega }_0^2(x-y)=0$. Its response at angular frequency $\omega$ is $x/y={\omega }_0^2/({\omega }_0^2-{\omega }^2)$, which decreases in magnitude as $({\omega }_0/\omega {)}^2$ well above resonance. The mirror follows slow support drift but approximately free motion in the high-frequency limit. Real multistage suspensions also require damping and alignment control; resonant frequencies are not regions of good isolation.

Rainer Weiss's 1972 report proposed suspended masses, multiple-pass optical delay lines, and a servo holding an interference fringe. Its noise analysis separately treats photon-counting fluctuations, laser frequency instability, mechanical thermal noise, radiation pressure and ground motion. Those delay lines should be distinguished from the resonant arm cavities of Advanced LIGO.^[ref]

An ideal single-round-trip Michelson interferometer converts differential length into phase difference $\delta \varphi =4\pi \Delta L/\lambda$. If its dark-port power scale is $P_0$ and its static phase offset is ${\varphi }_0$, thenExactly at a dark fringe, ${\varphi }_0=0$, power is quadratic in small displacement and loses its sign. A deliberate offset supplies an optical reference field: interference with the small signal field produces a linear term. Equivalently, $|E_{\mathrm{ref}}+E_{\mathrm{sig}}{|}^2$ contains $2\mathrm{Re}(E_{\mathrm{ref}}^{\ast }{E}_{\mathrm{sig}})$. The photodetector measures power, not a signed displacement directly.

Advanced LIGO uses this DC homodyne principle with a small differential arm offset and an output mode cleaner. Its feedback control maintains optical resonance and alignment, using actuation distributed over the suspension stages.^[ref] The simple formula above omits resonant storage: arm and recycling cavities give a frequency-dependent optical gain and delay, so no constant pass-count multiplier describes the entire observing band.

To see why a locked detector can still measure a disturbance, use a linear feedback control model. Let $x_{\mathrm{ext}}$ be the external differential displacement, $d_{\mathrm{err}}$ the residual readout, $d_{\mathrm{ctrl}}$ the recorded control command, $C(f)$ the sensing response and $A(f)$ the actuation response. With the convention $x_{\mathrm{res}}=x_{\mathrm{ext}}-A{d}_{\mathrm{ctrl}}$, reconstruction givesHere tildes denote Fourier transforms; $C$ has readout units per metre and $A$ metres per command unit. Feedback suppresses residual motion but records the compensating command. Omitting that command would suppress the reconstructed low-frequency signal too. In the actual GW150914 experiment, a modulated auxiliary laser exerted calibrated radiation pressure on a mirror to measure the response; periodic excitations monitored calibration.^[ref]

The modeled matched filter search found an arrival difference of about $7\mathrm{ms}$, within the roughly $10\mathrm{ms}$ intersite light travel time. Background events were obtained by shifting one detector's timestamps far beyond that time and repeating the coincidence analysis. The binary search bounded the false alarm rate below one event per $203000$ years. That is an empirically estimated background rate, not a probability that the source interpretation is false.^[ref]

The logic is to destroy a common astronomical arrival while retaining each instrument's real glitches. A Poisson distribution approximation would turn a background rate $r$ into $1-e^{-rT}$ for at least one equally strong noise event during live time $T$, only after accounting for the search's ranking and trials. Repeated time slides reuse data; their accumulated background duration is not an equally long series of independent new observations. Correlated disturbances also require separate controls because shifting timestamps can erase them along with a true signal.

The dedicated noise study inspected auxiliary channels and deliberately applied magnetic, acoustic and mechanical disturbances to measure their coupling into strain. For GW150914, checks included environmental coincidences, instrument state, and exclusion of an intentional hardware injection. It also examined potentially correlated global magnetic disturbances, which time slides alone cannot exclude.^[ref] The point of an injection test is quantitative: a quiet environmental monitor is informative only if its sensitivity and coupling bound the disturbance needed to explain the candidate.

For detector $k$, model the calibrated data as $d_k(t)=h_k(t;\theta )+n_k(t)$, with a shared source parameter vector $\theta$, separate detector responses, propagation delays and noise $n_k$. Under an independent stationary Gaussian process approximation with one-sided power spectral density $S_k(f)$, define the noise-weighted inner product and likelihood functionStrain power spectral density has units ${\mathrm{Hz}}^{-1}$; the displayed inner product is dimensionless when the Fourier transform integrates over seconds. A matched filter rewards agreement over the full evolving wave phase, weighting frequency bands by their noise. One conspicuous peak is insufficient: a candidate should also have the right distribution of signal across frequency bands and compatible projections in the network.

The parameter paper used coherent Bayesian inference, numerical relativity-informed inspiral–merger–ringdown models and explicit calibration uncertainties. It found a strong distance–inclination correlation; source-frame masses additionally require converting the detector-frame masses by a cosmological redshift.^[ref] Conceptually, multiply the likelihood function by a prior distribution to obtain a posterior distribution. Changes in waveform families, residual structure after subtraction, and checks with a generic transient reconstruction test parts of the modeling assumptions. Detection significance and parameter uncertainty answer different questions.

A Newtonian binary chirp is a leading-order model in which loss of orbital energy to gravitational radiation makes an almost circular binary system shrink and its gravitational-wave frequency rise. Here is a derivation with explicit SI units. Let the constant component masses $m_1,m_2$ be in kilograms, $M=m_1+m_2$, $\mu =m_1{m}_2/M$, separation $a$ in metres, and orbital angular frequency $\Omega$ in radians per second. Newtonian gravity, circular motion and the quadrupole formula giveThe factor $\pi$ rather than $2\pi$ appears because the dominant mass quadrupole moment repeats twice per orbit. Since $dE/da>0$, radiative energy loss implies $\dot{a}<0$, henceThe quantity $G\mathcal{M}/c^3$ is a time in seconds. Thus $K$ has units ${\mathrm{s}}^{5/3}$ and $K{f}^{11/3}$ has units ${\mathrm{s}}^{-2}$, as required for frequency change per second. This model assumes weak gravity, slow relative motion, negligible eccentricity of a conic, no spin corrections and adiabatic evolution over an orbit. It does not describe merger or ringdown.

Separating variables gives a finite formal coalescence time $t_c$ and accumulated wave phase $\Phi$:Indeed $d\Phi /dt=2\pi f$; the formal divergence at $t_c$ signals extrapolation outside the approximation. Over a finite frequency interval, the number of gravitational wave cycles is $N=3(f_0^{-5/3}-f_1^{-5/3})/(5K)$. This accumulated phase is why a chirp mass can be constrained even when distance and inclination remain entangled.

The chirp mass is the combination of two component masses controlling the leading Newtonian binary chirp. Inverting its frequency equation givesUse $f$ in hertz and $\dot{f}$ in hertz per second to obtain kilograms. For equal component masses $m$, $\mathcal{M}=m/2^{1/5}$. A stipulated $20+20$ solar mass system therefore has $\mathcal{M}\approx 17.411$ solar masses. In a cosmological observation the time and frequency measured at the detector instead yield $(1+z){\mathcal{M}}_{\mathrm{source}}$, where $z$ is redshift; phase evolution alone does not independently determine the source mass and redshift.

To construct an original example rather than imitate GW150914, set $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, and adopt the rounded solar mass $M_{\odot }=1.98847\times 10^{30}\mathrm{kg}$. Compare $10+10$ and $20+20$ solar mass binaries starting at $20\mathrm{Hz}$. Stop each at the stipulated orbital compactness thresholdThis is a conservative pedagogical stopping rule, not a claimed error bound: the relative speed already reaches $\sqrt{0.10}c$. Higher-order corrections accumulate wave phase well before merger, and precision inference requires better models. The two cutoffs are approximately $102.2$ and $51.1\mathrm{Hz}$ respectively; doubling the chirp mass multiplies $\dot{f}$ at fixed $f$ by $2^{5/3}$.

For a nearby stipulated source at distance $D$, neglect redshift and define inclination $\iota$ as the angle between the orbital angular momentum and the line of sight. The leading quadrupole formula yieldsThe numerator of $A$ has units ${\mathrm{m}}^5{\mathrm{s}}^{-4}$, matching $c^{4}D$, so both polarizations are dimensionless. Its $f^{2/3}$ growth and the rising frequency produce the tightening, strengthening oscillations. For face-on viewing the polarization components have equal magnitudes in phase quadrature; for edge-on viewing the cross component vanishes, agreeing with circular-binary quadrupole radiation.

Shifting an arrival time aligns the chirp evolution, but different polarization projections need not overlay after only a sign flip. The last panel divides the ideal projected records by their known envelope factors to expose this phase difference. A physical source model must jointly explain those differences, not erase them by arbitrary rescaling. Independent integration of the separation equation, the radiated power and $d\Phi /dt=2\pi f$ checks the analytic chirp against orbital energy balance over the entire stipulated interval.

In the weak-field, short-wavelength averaging regime, a transverse-traceless tensor strain $H_{ij}$ carries the displayed effective stress-energy tensor in units $c=1$. The average spans several wave periods while remaining small compared with the background curvature scale. The stress is second order in amplitude, so it is absent from a first-order vacuum Linearized Einstein equations calculation. For a plane wave, summing the two equal transverse diagonal contributions gives flux $(16\pi G{)}^{-1}⟨{\dot{H}}_{+}^2+{\dot{H}}_{\times }^2⟩$. This effective averaged stress is not an arbitrary pointwise gravitational energy tensor.

An exact axial Killing vector field makes Komar angular momentum independent of the surrounding vacuum surface. Therefore gravitational waves in a configuration with exact axisymmetry carry no net angular momentum about that symmetry axis, although they can carry energy. This is specific to axial angular momentum; it is not an absence of radiation.

The gravitational-wave frequency is the temporal oscillation frequency of a radiative metric perturbation. A rigid source rotating with angular frequency $\omega$ and invariant under a half-turn radiates dominantly at angular frequency $2\omega$.

The trace-free mass quadrupole moment of a mass distribution isIt is the leading time-dependent mass multipole that can radiate in general relativity: mass conservation removes monopole radiation, while momentum conservation removes mass-dipole radiation.

For a symmetric stress-energy tensor with compact support and ${\partial }_b{T}^{ab}=0$ in Minkowski spacetime, let $I^{ij}=\int T^{00}{x}^i{x}^j{d}^{3}x$. Two integrations by parts give ${\dot{I}}^{ij}=\int (T^{0i}{x}^j+T^{0j}{x}^i)d^{3}x$ and the displayed identity. Compact support removes the boundary fluxes. The support should stay bounded over the time interval used when differentiating the integrals.

The second mass moment tensor records the quadratic spatial distribution of mass. Its trace-free part is the mass quadrupole moment $Q_{ij}=I_{ij}-{\delta }_{ij}{I}_{kk}/3$. The mechanical moment-of-inertia tensor instead equals $J_{ij}={\delta }_{ij}{I}_{kk}-I_{ij}$. A rigid rotation $L$ transforms the second mass moment tensor as $I(t)=L(t)I(0)L(t{)}^T$.

The scalar second mass moment is the trace of the second mass moment tensor. The moment of inertia about a specified axis instead integrates squared distance from that axis. For an isotropic distribution each axial moment of inertia is $2I/3$; the sum over three orthogonal axes is $2I$.

For a slowly moving isolated source, the leading gravitational-wave luminosity iswhere $Q_{ij}$ is the trace-free mass quadrupole moment.

In units $c=1$, a slowly moving isolated source has trace-free mass quadrupole moment $Q^{ij}=\int T^{00}(x^i{x}^j-{\delta }^{ij}|\mathbf{x}{|}^2/3)d^{3}x$. The retarded linearized gravity solution and the stress-energy quadrupole identity give the displayed physical spatial strain, with $\mathbf{n}=\mathbf{x}/r$ and the transverse-traceless projector $\Lambda$. If the spatial metric is written $g_{ij}=-{\delta }_{ij}-2E_{ij}$ for metric signature $(+---)$, its trace-free Lorenz-gauge coefficient before the transverse projection is $E_{ij}=G{\ddot{Q}}_{ij}(t-r)/r$. Thus $H^{\mathrm{TT}}=2E^{\mathrm{TT}}$. Far distance alone does not permit dropping source-dependent retardation: the quadrupole approximation also requires the source size to be small compared with the radiation wavelength.

For a weakly self-gravitating bound source of size $R$, mass $M$ and nonspherical dynamical mass quadrupole moment $Q=O(M{R}^2)$, the virial theorem gives $v^2=O(GM/R)=O(ϵ)$. Its time scale is $R/v$, so $\ddot{Q}=O(M{v}^2)$. The far-zone quadrupole strain then has a spatial half-strain coefficient $E=O(GM{v}^2/r)=O({ϵ}^{2}R/r)$ in units $c=1$. The physical transverse-traceless strain is $2E^{\mathrm{TT}}$. Spherical or stationary sources can have zero radiation, so this is a characteristic scale for a changing nonspherical quadrupole, not a positive lower bound. The radiation zone also requires $r\gg R/v$.

Two equal masses $m$ orbiting their centre of mass at radius $R$ have separation $2R$. Newton's law of universal gravitation gives the displayed orbital frequency. Their mass quadrupole moment varies at twice that frequency, producing gravitational radiation with two gravitational wave polarizations. For inclination $\iota$, its leading plus and cross amplitudes are proportional to $1+{\cos }^2\iota$ and $2\cos \iota$, respectively. Face-on radiation is circularly polarized; edge-on radiation is linearly polarized.

The dimensionless ratio of gravitational radius to orbital scale measures how relativistic a bound orbit is. In an equal-mass circular binary with individual orbital radius $R$, the leading quadrupole formula gives $P=(2/5)(c^5/G)[GM/(R{c}^2){]}^5$. Compact objects permit smaller noncontact orbits than extended stars and hence larger orbital compactness; near merger the Newtonian approximation must be replaced.

For two masses $M$ at opposite points of a circular orbit, each at radius $R$ from the centre of mass, ${\Omega }^2=GM/(4R^3)$. The trace-free mass quadrupole moment oscillates at twice the orbital frequency, and its third-derivative squared contraction is $128M^2{R}^4{\Omega }^6$. The quadrupole formula gives the displayed phase-independent luminosity. Using separation $2R$ rather than individual radius $R$ avoids a factor-of-32 error.

For two equal point masses at $\pm z$ falling from rest at $\pm z_0$ under Newtonian gravity, in geometrized units one has $\ddot{z}=-m/(4z^2)$ and $\dot{z}=-\sqrt{m(1/z-1/z_0)/2}$. The only nonzero component of the second mass moment tensor is $I_{zz}=2m{z}^2$, and its third derivative is $-m^2\dot{z}/z^2$. The trace-free mass quadrupole moment and quadrupole formula give the displayed power. From infinity, integrating down to $z=z_c$ gives $E=2\sqrt{2}{m}^{9/2}/(105z_c^{7/2})$. Relative to the total initial mass $2m$ the fraction is $\sqrt{2}(m/z_c{)}^{7/2}/105$. The weak-field slow-motion approximation must be distinguished from its extrapolation to a compact endpoint.

For a point mass moving as $z=L\cos \omega t$ along one fixed axis, the trace-free mass quadrupole moment is $m{L}^2{\cos }^2\omega t\mathrm{diag}(-1/3,-1/3,2/3)$. Its radiating component has twice the source angular frequency. The quadrupole formula gives $⟨P⟩=16G{m}^2{L}^4{\omega }^6/(15c^5)$ for this contribution in the leading slow-motion weak-field approximation. Any physical driver or support has its own quadrupole contribution; this expression is for the prescribed mass alone.

For a slowly evolving circular binary star with separation $a$, reduced mass $\mu$ and orbital angular frequency $\Omega$, the leading gravitational wave losses satisfy $\dot{E}=\Omega \dot{J}$. One direct check uses the standard quadrupole angular-momentum loss lawwhere $Q$ is the mass quadrupole moment and ${ϵ}_{ijk}$ is the Levi-Civita symbol. For $\mathbf{r}=a(\cos \Omega t,\sin \Omega t,0)$, put $A=\mu a^2$; then $Q_{xx}=A(1/6+\cos (2\Omega t)/2)$, $Q_{yy}=A(1/6-\cos (2\Omega t)/2)$, $Q_{xy}=A\sin (2\Omega t)/2$, and $Q_{zz}=-A/3$. Substitution gives ${\dot{J}}_z=-32G{A}^2{\Omega }^5/(5c^5)$ and the quadrupole formula gives $\dot{E}=-32G{A}^2{\Omega }^6/(5c^5)$. Their ratio proves the balance relation. The same relation follows from the rotating wave's angular harmonic number two and frequency $2\Omega$. The additional angular-momentum flux law, not the energy flux alone, is needed to justify preservation of secular circularity.

For a Newtonian Kepler orbit, let $M=M_1+M_2$ be the total mass and $\mu =M_1{M}_2/M$ the reduced mass; put $k=GM$, $\mathbf{n}=\mathbf{r}/r$, $u=\dot{r}$ and ${\mathbf{v}}_{\perp }=\dot{\mathbf{r}}-u\mathbf{n}$, with $v^2=\dot{\mathbf{r}}\cdot \dot{\mathbf{r}}$. In the convention $q_{ij}=\mu (3r_i{r}_j-r^2{\delta }_{ij})/2$, differentiating the acceleration $\ddot{\mathbf{r}}=-k\mathbf{r}/r^3$ givesThe two terms have zero cross tensor contraction, with squared norms $6u^2$ and $72v_{\perp }^2$. Therefore ${\stackrel{...}{q}}_{ij}{\stackrel{...}{q}}_{ij}=72{\mu }^2{k}^2(v^2-11u^2/12)/r^4$. Substitution into the quadrupole formula in this convention, $L=4G{\stackrel{...}{q}}_{ij}{\stackrel{...}{q}}_{ij}/(45c^5)$, proves the luminosity. It is nonnegative since $v^2-11u^2/12=v_{\perp }^2+u^2/12$. The formula uses a weak gravitational field, slow motion and leading radiation order; orbital averaging gives a secular luminosity.

For a rigid body rotating around its third principal axis, the anisotropic part of its second mass moment tensor oscillates at angular frequency $2\Omega$. The quadrupole formula gives the displayed luminosity. Equal transverse principal moments remove this rotating quadrupole and its leading gravitational wave emission.

In the radiation zone, the time-averaged energy flux of a weak wave in transverse-traceless gauge isCombining this with the far-zone quadrupole field and integrating over directions gives the quadrupole formula.

The Newman--Penrose formalism resolves spacetime curvature into scalars by contracting the Weyl tensor with a complex null tetrad.

A complex null tetrad consists of two real null vectors and a complex-conjugate transverse pair, normalized so that the metric is reconstructed from their pairwise products.

For a null tetrad $(\ell ,k,m,\bar{m})$, one common curvature-sign convention defines ${\Psi }_4=C_{\mu \nu \rho \sigma }{k}^{\mu }{\bar{m}}^{\nu }{k}^{\rho }{\bar{m}}^{\sigma }$. In an outgoing wave zone it encodes the two transverse gravitational wave polarizations as one complex scalar.

Forthe only potentially nonzero Ricci component isThe vacuum equation is therefore the transverse Laplace equation.

For the plane-fronted gravitational wave $d{s}^2=2dudv+d{\mathbf{x}}^2+{\mathbf{x}}^{T}M(u)\mathbf{x}d{u}^2$, synchronized neighboring geodesics parametrized by $u$ have transverse separation $\mathbf{\xi }$ satisfying the displayed equation. Along the central timelike geodesic $(u,v,\mathbf{x})=(\tau ,-\tau /2,0)$ the coordinate basis is parallel transport, so this is also the transverse Jacobi field equation in a transported frame. Vacuum requires $\mathrm{tr}M=0$, producing equal and opposite instantaneous transverse tidal acceleration eigenvalues. The ordinary coordinate equation ${\ddot{\xi }}^v=0$ holds on that central reference, but generally fails on off-axis reference curves.

A homogeneous quadratic polynomial in the transverse coordinates $x,y$ is a harmonic function exactly when the sum of its $x^2$ and $y^2$ coefficients is zero. Thus it has the displayed form. For the Minkowski metric $dudv+d{x}^2+d{y}^2$, a metric perturbation $h=\psi d{u}^2$ has zero trace, zero divergence and $\square h=0$, since ${\partial }_v\psi =0$ and ${\partial }_x^2\psi +{\partial }_y^2\psi =0$. The two coefficient functions describe the two gravitational wave polarizations.

For $d{s}^2=-2dudv+d{x}^2+d{y}^2-2H(u,x,y)d{u}^2$, the nonzero connections are ${\Gamma }_{uu}^v=H_u$, ${\Gamma }_{ux}^v=H_x$, ${\Gamma }_{uy}^v=H_y$, ${\Gamma }_{uu}^x=H_x$, and ${\Gamma }_{uu}^y=H_y$, with symmetric lower indices. Their traces and quadratic Ricci products vanish, leaving only the displayed Ricci component. Hence the vacuum equations require a harmonic transverse profile. The sign is tied to the metric's $-2Hd{u}^2$ convention.

For a Brinkmann plane wave whose transverse profile is quadratic, Killing fields obtained from solutions of transverse oscillator equations act transitively on every constant-retarded-time wavefront. Each wavefront is therefore a homogeneous null hypersurface.

Rosen coordinates put a plane wave into the transverse-metric formFor diagonal $g_{AB}=\mathrm{diag}(f^2,g^2)$, the only potentially nonzero Ricci tensor component is $R_{uu}=-f^{\prime \prime }/f-g^{\prime \prime }/g$.

For $g=-d{t}^2+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2+d{z}^2$, $u=t-z$, with nowhere-zero smooth $A,B$, take $e^0=dt$, $e^1=Adx$, $e^2=Bdy$, $e^3=dz$, $\ell =du$. In signature $(-,+,+,+)$, Cartan's first structure equation and metric compatibility give the lower-index connection 1-forms ${\omega }_{01}={\omega }_{13}=-(A^{\prime }/A)e^1$ and ${\omega }_{02}={\omega }_{23}=-(B^{\prime }/B)e^2$. Cartan's second structure equation gives ${\Theta }_{01}={\Theta }_{13}=-(A^{\prime \prime }/A)\ell \wedge e^1$ and ${\Theta }_{02}={\Theta }_{23}=-(B^{\prime \prime }/B)\ell \wedge e^2$. All other independent curvature 2-forms vanish: in particular the $12$ component cancels between the time and longitudinal terms. Contracting $R^a{}_{bad}$ yields $\mathrm{Ric}=-(A^{\prime \prime }/A+B^{\prime \prime }/B)du\otimes du$. Thus the Vacuum Einstein equations are $A^{\prime \prime }/A+B^{\prime \prime }/B=0$, while nonzero $A^{\prime \prime }/A=-B^{\prime \prime }/B$ gives opposite transverse tidal curvatures and plus polarization relative to these axes.

A gravitational-wave pulse has curvature supported in a bounded interval of retarded time. Spacetime can be flat both before and after the pulse even though freely falling bodies retain a displacement or relative velocity produced while the pulse passed.

The gravitational-wave memory effect is a lasting change in the relative configuration of freely falling test bodies after a transient gravitational wave has passed.

Displacement memory is a persistent change in the separation of freely falling bodies after a gravitational-wave pulse.

Velocity memory is a persistent change in the relative velocity of freely falling bodies after a gravitational-wave pulse.