Gravitational Wave Memory Effect

• Gravitational Wave Memory Effect (GWME) is a permanent, nonoscillatory shift in detector strain after a burst, distinguishing linear (ordinary) and nonlinear (null) contributions in GR.
• It connects key theoretical frameworks such as Bondi–Sachs asymptotics, BMS balance laws, and soft-graviton relations, and serves as a probe for gravitational dynamics.
• Advanced waveform modeling and numerical methods quantify GWME, offering actionable insights for testing GR and exploring modifications in extended gravity theories.

Searching arXiv for recent and foundational papers on gravitational-wave memory to ground the article in current literature. Searching arXiv for recent and foundational papers on gravitational-wave memory to ground the article in current literature. Gravitational-wave memory effect (GWME) denotes a permanent, nonoscillatory change in the gravitational-wave strain after a burst of radiation has passed a detector. In an ideal freely falling interferometer, the late-time strain differs from the early-time strain by a finite offset, so the separation of test masses does not return to its original value. In general relativity (GR), this includes ordinary memory sourced by unbound matter or radiation and null, or Christodoulou, memory sourced by energy flux to future null infinity, including the gravitational waves themselves; exact plane-wave analyses also exhibit persistent velocity memory. Contemporary work places GWME at the intersection of Bondi–Sachs asymptotics, BMS balance laws, soft-graviton relations, numerical-relativity waveform modeling, and tests of gravity beyond GR (Heisenberg et al., 2023, Favata, 2010).

1. Definitions, taxonomy, and observables

A standard operational definition writes the memory in each polarization as

$$\Delta h_{+,\times}^{\mathrm{mem}} \equiv \lim_{t\to+\infty} h_{+,\times}(t)-\lim_{t\to-\infty} h_{+,\times}(t),$$

so that, in the weak-field detector frame,

$$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$

Equivalently, in Fermi normal coordinates one may describe the lasting displacement through geodesic deviation,

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$

with $A_{ij}$ the radiative transverse-traceless tensor built from the physical metric perturbation (Heisenberg et al., 2023, Kilicarslan et al., 2018).

The standard classification separates ordinary, or linear, memory from null, or nonlinear/Christodoulou, memory. Ordinary memory is generated by unbound massive matter or radiation, such as scattering, recoils, anisotropic neutrino emission, or ejected matter, and in asymptotic language is associated with changes in the Bondi mass aspect and related source terms that do not themselves reach null infinity. Null memory is sourced by null energy flux escaping to future null infinity, notably the energy carried by the gravitational waves themselves, and therefore directly manifests the nonlinearity of GR (Heisenberg et al., 2023, Favata, 2010).

A related but distinct effect is velocity memory. In exact pp-wave or plane-wave spacetimes, initially comoving particles can acquire a nonzero constant relative velocity after the pulse, so that the separation grows linearly with retarded time rather than saturating to a finite offset. Exact analyses of square, ramp, Gaussian, and related pulse profiles in Brinkmann coordinates repeatedly find monotonic separation growth together with a final nonzero constant transverse drift velocity, which is retained after the pulse dies out (Datta et al., 2023, Chakraborty et al., 2022).

The broader memory family also includes spin memory, center-of-mass memory, and refraction or superboost memories. These are associated with subleading BMS charges and different observables. The dominant compact-binary target in present waveform modeling and detector studies remains tensor displacement memory, especially the $(\ell,m)=(2,0)$ mode for nonprecessing binaries (Heisenberg et al., 2023, Sun et al., 2022).

2. Null infinity, Bondi–Sachs structure, and the GR memory formula

In asymptotically flat spacetimes, memory is naturally formulated at future null infinity $\mathscr I^+$. Depending on convention, the Bondi news is written as either $N_{AB}\equiv -\dot C_{AB}$ or $N_{AB}=\partial_u C_{AB}$, with $C_{AB}$ the Bondi shear and $u=t-r$ retarded time. The permanent memory is the net change in the shear,

$$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$0

and correspondingly in the TT strain (Heisenberg et al., 2023, Gasparotto, 8 May 2026).

For high-frequency TT waves in GR, the Isaacson stress-energy tensor is

$$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$1

so that the energy flux per unit solid angle at null infinity is

$$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$2

The directional Christodoulou memory then takes the form

$$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$3

with the TT-projected kernel fixing the angular response (Heisenberg et al., 2023).

Mode decompositions make this expression practical for waveform calculations. Writing the complex strain $$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$4 in spin-weighted spherical harmonics, one obtains memory modes from quadratic products of time derivatives of oscillatory modes,

$$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$5

or equivalently through angular-coupling coefficients $$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$6 constructed from spin-weighted harmonic integrals (Talbot et al., 2018, Inchauspé et al., 2024). For nonprecessing compact binaries, the dominant oscillatory power in $$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$7 implies that the dominant memory mode is $$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$8, with $$\Delta x^i \simeq \frac{1}{2}\,\Delta h_{ij}^{\rm TT}\,x^j.$$9 subleading and higher $$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$0 modes smaller or symmetry-suppressed (Inchauspé et al., 2024).

In quasi-circular binaries, the leading plus-polarized memory may be written

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$1

while $$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$2 vanishes in the standard nonprecessing source frame at leading order (Favata, 2010). This angular pattern differs from the oscillatory quadrupolar pattern: it is strongest for edge-on systems and vanishes for face-on orientations.

The same asymptotic structure also connects memory to BMS supertranslations and to soft theorems. In GR, displacement memory corresponds to transitions between BMS supertranslation vacua, and the flux-balance identity relating $$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$3 to integrated news is the asymptotic expression of Weinberg’s soft graviton relation (Heisenberg et al., 2023, Gasparotto, 8 May 2026).

3. Exact plane-wave and pp-wave realizations

Exact plane-wave and pp-wave spacetimes provide analytically solvable models of local memory. In Brinkmann coordinates, the pp-wave metric is

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$4

with vacuum condition $$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$5 and polarization decomposition

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$6

The nonvanishing curvature components are

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$7

and the transverse geodesic equations reduce to linear ODEs with time-dependent coefficients (Datta et al., 2023).

For a ramp pulse,

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$8

the plus-polarized equations become Airy equations inside the pulse,

$$\ddot s_i=-R_{i0j0}s_j,\qquad \Delta s_i\simeq \tfrac{1}{2}\,\Delta A_{ij}\,s_j,$$9

Matching the Airy-function solution to free motion before and after the pulse yields constant nonzero post-pulse velocities $A_{ij}$0 and $A_{ij}$1, so that

$A_{ij}$2

The separation increases monotonically from an initial constant value, and the relative velocity grows from zero and settles to a final nonzero constant value. The resulting memory is therefore predominantly velocity memory, accompanied by monotonic growth of displacement (Datta et al., 2023).

Square-pulse sandwich waves admit similarly explicit solutions. For a plus-polarized square pulse $A_{ij}$3, the Region II equations reduce to oscillator and inverted-oscillator systems,

$A_{ij}$4

with piecewise solutions

$A_{ij}$5

inside the pulse and linear motion afterward. The post-pulse velocities

$A_{ij}$6

are generically nonzero, and pairwise geodesic separations can focus to caustics at finite $A_{ij}$7 (Chakraborty et al., 2022).

A more invariant characterization is given by four “memory tensors,” three independent, which summarize how a sandwich wave maps initial transverse displacement and velocity data into final displacement and velocity data. In the notation of Harte et al., these are $A_{ij}$8, $A_{ij}$9, $(\ell,m)=(2,0)$0, and $(\ell,m)=(2,0)$1, with exact scattering matrix

$(\ell,m)=(2,0)$2

These tensors also control null-geodesic deflections, scalar-field scattering, and the spin Hall dynamics of massless spinning particles. In the weak-field limit, they are curvature moments of $(\ell,m)=(2,0)$3, and similar constructions are expected to apply locally to small systems via the Penrose limit (Harte et al., 2024).

4. Structural robustness and modifications beyond general relativity

A central recent result is that tensor null memory is structurally robust across a broad class of viable extensions of GR. Using Isaacson averaging, one splits the metric into slow and fast parts, $(\ell,m)=(2,0)$4, and obtains separate equations for high-frequency propagation and low-frequency backreaction,

$(\ell,m)=(2,0)$5

If the linearized massless gravitational fields obey decoupled wave equations, then the low-frequency memory perturbation satisfies

$(\ell,m)=(2,0)$6

with $(\ell,m)=(2,0)$7 a conserved, gauge-invariant averaged stress-energy built from quadratic derivatives of the massless waves. The tensor null-memory kernel and $(\ell,m)=(2,0)$8 scaling are therefore unchanged; modifications arise only through additive flux terms from extra massless radiative fields (Heisenberg et al., 2023).

For scalar–vector–tensor theories with second-order equations and vanishing potentials, a canonical field redefinition diagonalizes the kinetic sector,

$(\ell,m)=(2,0)$9

and the averaged stress-energy takes the explicit form

$\mathscr I^+$0

The resulting tensor memory is the GR expression with $\mathscr I^+$1 replaced by the total radiative energy density, so extra massless scalar or vector channels additively enhance memory (Heisenberg et al., 2023).

Brans–Dicke theory illustrates this mechanism in a fully asymptotic framework. The BMS supermomentum balance law acquires an extra scalar-flux term,

$\mathscr I^+$2

and in the PN regime the tensor displacement memory acquires a logarithmic dependence on the PN parameter because the scalar dipole flux enters at $\mathscr I^+$3PN order. The tensor spin memory analogously receives a relative $\mathscr I^+$4PN correction. These effects are small in the approximation of weak scalar radiation, but they modify both amplitude and sky pattern relative to GR (Tahura et al., 2021).

Scalar Gauss–Bonnet and related quadratic-gravity models occupy an intermediate regime. At asymptotic null infinity, the higher-curvature terms do not alter the tensor-memory evolution equation at leading order; the scalar contributes only through its canonical radiative flux. Full inspiral–merger–ringdown calculations in scalar Gauss–Bonnet gravity find percent-level deviations from GR memory, driven predominantly by modified merger dynamics rather than by direct scalar-induced tensor memory. In the systems studied, the scalar radiation is dominated by the isotropic monopole, so its contribution to tensor memory is strongly suppressed; representative final memory amplitudes differ from GR by about $\mathscr I^+$5, and detector-oriented comparisons give up to $\mathscr I^+$6 relative difference with the best-fit GR template (Gasparotto, 8 May 2026, Gasparotto et al., 10 Apr 2026).

Not all theories preserve the GR kernel. In Fierz–Pauli massive gravity, the memory is exponentially Yukawa-suppressed,

$\mathscr I^+$7

and the ordinary memory is additionally reduced by a vDVZ-type numerical factor that survives in the massless limit. With the strongest bounds quoted in the cited analysis, the memory is essentially wiped out for sources at distances above $\mathscr I^+$8 Mpc (Kilicarslan et al., 2018).

More speculative modifications appear in quantum-gravity-inspired echo models. If post-merger horizons or exotic compact-object surfaces are partially reflective, the echo strain generates an additional delayed electric memory,

$\mathscr I^+$9

which is a delayed secondary step in the memory signal. Because it depends primarily on integrated echo energy flux rather than detailed echo morphology, this contribution is described as model-independent in shape relative to direct echo searches (Deppe et al., 27 Feb 2025).

5. Waveform modeling, numerical methods, and inferred memory observables

The first quantitative coalescence-wide models of binary-black-hole memory were built by Favata. In the minimal waveform model, the inspiral radiative quadrupole is matched at a time $N_{AB}\equiv -\dot C_{AB}$0 to a sum of quasi-normal modes,

$N_{AB}\equiv -\dot C_{AB}$1

yielding a memory that rises slowly in inspiral, grows rapidly near merger, and saturates within a few damping times. In the corresponding EOB model calibrated to numerical relativity, the memory plateau is smaller than in the bare minimal model, and a multiplicative factor $N_{AB}\equiv -\dot C_{AB}$2 applied to the latter reproduces the late-time EOB plateau (0811.3451).

A different line of development treats memory as a post-processing functional of a generic oscillatory waveform. The GWMemory method computes

$N_{AB}\equiv -\dot C_{AB}$3

with

$N_{AB}\equiv -\dot C_{AB}$4

Using NRSur7dq2 oscillatory modes up to $N_{AB}\equiv -\dot C_{AB}$5, this framework shows that higher harmonics modify the late-time memory amplitude at the $N_{AB}\equiv -\dot C_{AB}$6 level and generate nonzero cross-polarized memory, especially for unequal-mass or precessing systems. It also quantifies the “memory of the memory,” which produces an $N_{AB}\equiv -\dot C_{AB}$7 correction to the primary memory waveform (Talbot et al., 2018).

A distinct BMS-based approach computes memory directly from known non-memory waveforms without weak-field or slow-motion assumptions. In that formalism, each memory mode is reconstructed from the oscillatory $N_{AB}\equiv -\dot C_{AB}$8 modes, plus the Bondi mass aspect term $N_{AB}\equiv -\dot C_{AB}$9, through an exact mode-balance identity. Applied to SXS non-memory waveforms for spin-aligned binaries, this method confirms the preliminary numerical-relativity memory waveform and yields compact empirical fits for the total $N_{AB}=\partial_u C_{AB}$0 memory amplitude,

$N_{AB}=\partial_u C_{AB}$1

with $N_{AB}=\partial_u C_{AB}$2 (Liu et al., 2023).

Recent work has separated the final memory offset itself as an explicit waveform-model ingredient. For nonspinning, quasicircular binaries, the final $N_{AB}=\partial_u C_{AB}$3 offset is fit as

$N_{AB}=\partial_u C_{AB}$4

and the linear-in-$N_{AB}=\partial_u C_{AB}$5 coefficient is fixed by an analytic EMRI calculation using 22PN test-mass waveforms. The EMRI-inspired value

$N_{AB}=\partial_u C_{AB}$6

anchors interpolation between the comparable-mass and extreme-mass-ratio regimes (Elhashash et al., 2024).

Memory has also been promoted to an inferred observable derived from oscillatory-waveform posteriors. Using public GWTC-1 posterior samples and the BMS supermomentum balance law, inferred posterior distributions for memory modes such as $N_{AB}=\partial_u C_{AB}$7, $N_{AB}=\partial_u C_{AB}$8, and $N_{AB}=\partial_u C_{AB}$9 were constructed for IMRPhenomPv2 and SEOBNRv3 waveforms. The most pronounced discrepancies appeared in the $C_{AB}$0 memory mode, suggesting that inferred memory posteriors can diagnose waveform systematics not evident in conventional intrinsic-parameter posteriors (Khera et al., 2020).

6. Detection, detector response, and stochastic memory backgrounds

Memory is concentrated at very low frequencies. For a net strain step $C_{AB}$1, its Fourier transform obeys the zero-frequency limit

$C_{AB}$2

so $C_{AB}$3. This is favorable for detectors with low-frequency sensitivity and problematic for instruments limited by seismic or control noise at low frequency (Gasparotto, 8 May 2026, Inchauspé et al., 2024).

Ground-based expectations have therefore emphasized either rare nearby events or stacking. Early estimates gave Advanced LIGO only a “slim chance” for nearby heavy binary black holes, whereas LISA was already promising for supermassive binaries (0811.3451). More recent beyond-GR studies quantify the single-event memory signal-to-noise ratio as

$C_{AB}$4

and argue that third-generation detectors could detect percent-level memory deviations for favorable low-mass, near-equal-mass systems out to $C_{AB}$5, especially when memory is included jointly with the oscillatory inspiral–merger–ringdown signal (Gasparotto, 8 May 2026).

LISA is particularly well suited because of both sensitivity and source class. For massive black-hole binaries, the memory SNR peaks at somewhat lower masses than the dominant $C_{AB}$6 mode, and time-domain simulations of the full second-generation TDI response show that memory appears as a burst-like feature near merger rather than as an actual DC offset. In the low-frequency limit, the Michelson-like TDI combinations behave schematically as time derivatives of the strain, with

$C_{AB}$7

so the measured memory signal is strongly high-pass filtered. Even so, realistic end-to-end simulations predict that several events in a four-year mission should have detectable memory, with the most favorable astrophysical populations corresponding to heavy seeds and short delays (Inchauspé et al., 2024).

TianQin forecasts are more modest. Using BMS-balance-law memory built from IMRPhenomXHM waveforms, single-event displacement-memory detection with $C_{AB}$8 is expected at the level of about $C_{AB}$9 events over five years, depending on the massive-black-hole population model, while spin memory is essentially negligible for single-event detection. The study also finds that the parameter-space region in which memory is detectable closely tracks the region in which omitting memory induces waveform-model mismatch above threshold (Sun et al., 2022).

At the population level, many step-like memory bursts generate a stochastic gravitational-wave memory background. If the observation time exceeds the mean interval between events, the summed detector response behaves as Brownian motion with variance growth

$u=t-r$0

and power spectral density

$u=t-r$1

The corresponding characteristic strain scales as $u=t-r$2, while $u=t-r$3. For supermassive-binary populations, this stochastic memory background lies in the mHz band and is therefore particularly relevant for space-based detectors such as LISA, Taiji, and TianQin (Zhao et al., 2021).

The experimental significance of GWME is thus twofold. First, a direct detection would probe the nonlinear energy-carrying character of gravity and its asymptotic conservation laws. Second, because the tensor memory kernel is highly robust in many viable beyond-GR theories while extra radiative sectors contribute additively through energy flux, any measured excess or deficit relative to the GR prediction built from the observed oscillatory waveform would point not to a modified kernel but to modified radiation content, new dynamical degrees of freedom, or, in more radical scenarios, departures from the massless asymptotically flat framework itself (Heisenberg et al., 2023, Gasparotto, 8 May 2026).