Gravitational-Wave Memory

A permanent dent in spacetime

A gravitational wave is conventionally pictured as a transverse oscillation that propagates through spacetime: a ring of freely-falling test particles stretches one way, squeezes the other, then returns to a circle. That picture is correct for the oscillatory piece of the waveform — but it is incomplete. A more careful calculation in general relativity reveals that the ring does not quite come back to a circle. After the wave has fully passed, the test masses settle into a configuration permanently displaced from the original. The strain h(t), instead of returning to zero, asymptotes to a non-zero value h_∞.

This residual is the gravitational-wave memory effect. It is a real, physical, in-principle-observable feature of every gravitational-wave source. The size of the effect is modest — for a binary black-hole merger like GW150914, the memory step is roughly five percent of the peak oscillatory strain — but it is qualitatively distinct: the waveform contains a piece that does not decay, a step function rather than a ringing.

The concept was first identified by Yakov Zel'dovich and Andrei Polnarev in 1974, in the context of stellar encounters and supernova neutrino emission, and given its full general-relativistic treatment by Demetrios Christodoulou in 1991. Three decades later it has become one of the most theoretically rich corners of gravitational-wave physics, with deep connections to asymptotic symmetries, soft theorems in scattering amplitudes, and — most provocatively — the black-hole information paradox.

The strain that does not relax

Take two freely-falling masses separated by a baseline L₀, far from any source. A gravitational wave passes. During the passage, the fractional separation ΔL/L₀ tracks the metric strain h(t). For an oscillatory wave with no memory, h(t) starts at zero, oscillates with some envelope, and ends at zero. The masses return to their original separation L₀.

For the same wave with memory, h(t) starts at zero but ends at h_∞ ≠ 0. Long after the burst, ΔL/L₀ = h_∞: the masses are now permanently displaced by h_∞ × L₀. There is no physical force pushing them apart any more — they are just freely-falling on a slightly different geodesic than before. The wave has rearranged the local inertial frame.

h(t → −∞) = 0
h(t → +∞) = h_∞ ≠ 0
Δh = h_∞ − 0 = h_∞    (the "memory step")

Spectrally, the memory is the f → 0 limit of the strain. The Fourier transform h̃(f) of a step function is 1/f, so the memory contributes a divergent low-frequency tail. Real waveforms have a finite rise time (set by the merger dynamics) and so the memory is a smoothed step over the merger timescale rather than a true Heaviside — but the integral of h(t) is non-zero, which is the hallmark.

Linear memory and Christodoulou memory

Two physically distinct mechanisms contribute to the memory.

Linear memory (Zel'dovich–Polnarev 1974). Any source that emits matter or radiation with a non-zero asymptotic velocity contributes a memory term proportional to the second time-derivative of the quadrupole moment integrated against the velocity field of the unbound material. Concretely:

h_lin^TT = (2G/c⁴ R) Σ_n  M_n β_n β_n / (1 − β_n · n̂) |^TT
                       n = 1..N (unbound particles)

The vertical bars denote the transverse-traceless projection along the direction to the observer n̂. Sources of linear memory include core-collapse supernova neutrino bursts (the bulk of supernova energy is radiated as neutrinos with asymmetric angular distribution), hyperbolic stellar flybys, and asymmetric jets from gamma-ray bursts. In each case, the unbound particles fly off to infinity and the metric retains the imprint of their changed momenta.

Nonlinear (Christodoulou) memory (1991). The gravitational-wave field itself carries energy-momentum. By the equivalence principle, that energy-momentum sources further curvature — and in the radiation zone, it sources further gravitational waves. The non-linear memory is the part of the metric strain that comes from this 'gravitons sourcing gravitons' effect. Christodoulou's key insight was that the energy flux of an outgoing gravitational wave, integrated over the duration of the burst, contributes a permanent strain at infinity proportional to the radiated energy:

h_nl^TT(t, R, Ω) = (4G/c⁴ R) ∫₋∞^t dt' ∫ dΩ' (dE_GW/dt'dΩ')
                   × [angular kernel(Ω, Ω')]

Crucially, every gravitational-wave source produces nonlinear memory, even sources with no unbound matter at all. A binary black-hole merger in vacuum — like GW150914 — has zero linear memory (the holes merge, no matter leaves the system) but a substantial nonlinear memory, because the merger radiates roughly 3 M☉c² of energy in gravitational waves and that radiated energy itself sources a permanent residual strain.

How big is the memory?

For a binary black-hole merger, dimensional analysis already gives the scale. The peak strain is

h_peak ~ G M_total / (c² R)   (at order unity)
       ~ 10⁻²¹    for a 60 M☉ merger at 400 Mpc

The memory step is suppressed relative to this by roughly the ratio of radiated energy to total mass, multiplied by an order-unity geometric factor:

h_mem / h_peak ~ E_radiated / (M_total c²) × O(1)
              ~ 5% × O(1)
              ~ 0.05    for typical BBH mergers

So h_mem ~ 5 × 10⁻²³ for GW150914 — below the detector noise floor at the frequencies where the step lives. The fraction depends on geometry: a face-on equal-mass binary gives the largest memory because radiation is most directional, while an edge-on system has reduced memory because of cancellations in the angular kernel.

Source class	h_peak (typical)	h_mem / h_peak	Memory type	Best detector
Stellar BBH merger	10⁻²¹	~5 %	Nonlinear dominant	LIGO O5
Binary neutron-star merger	10⁻²²	~3 %	Nonlinear + ejecta linear	LIGO O5
Massive BH merger (10⁶ M☉)	10⁻¹⁷ at 1 Gpc	~5 %	Nonlinear	LISA
Core-collapse supernova ν-burst	10⁻²¹ at 10 kpc	~50 % at f → 0	Linear (neutrinos)	LIGO O5 (galactic)
Asymmetric hypernova jet	10⁻²³ at 100 Mpc	order unity	Linear	LISA / DECIGO
EMRI plunge (LISA)	10⁻¹⁸	few %	Nonlinear	LISA

Statistical detection: stacking events

A single binary-black-hole merger's memory is buried below detector noise — for LIGO at design sensitivity, by roughly an order of magnitude. The standard route to detection is therefore not to look for memory in a single event but to combine many events. Each event has a predicted memory waveform (from numerical relativity templates) and contributes an independent measurement; coherent stacking improves SNR as √N for N events.

Hubner, Talbot, Lasky and Thrane (2020) carried out the most-cited modern projection. With 10 binary black-hole detections at advanced-LIGO sensitivity — the kind of catalogue available at O3 — the cumulative SNR for memory is below 1 and the effect is statistically indistinguishable from noise. They estimate that 100s of high-SNR events are required to reach SNR ~ 3 for the memory channel. With observing-run O4 already producing more than 100 mergers and O5 expected to reach the thousands, the first credible 'memory detection' announcement is on the late-2020s horizon.

LISA, scheduled to launch in the 2030s, will detect a handful of massive black-hole mergers per year at SNR thousands. For such individual events the memory amplitude is well above LISA's noise floor, and a single merger should suffice for a confident detection of the memory waveform. LISA's mHz band is also far better matched to the spectral shape of the memory step.

Worked example: GW150914's memory

The first detected binary black-hole merger, GW150914, had source-frame masses 36 + 29 M☉ (final 62 M☉, so 3 M☉ radiated as gravitational waves), distance 410 Mpc, and peak strain h_peak ≈ 1.0 × 10⁻²¹. The predicted nonlinear memory step is

h_mem ≈ (E_radiated / M_total c²) × h_peak × geometric factor
     ≈ (3 / 65) × 1.0 × 10⁻²¹ × ~1.2
     ≈ 5.5 × 10⁻²³

The frequency content of this step (rise time ~5 ms = the merger duration) lives near 100 Hz. The Advanced LIGO noise floor at 100 Hz was ~3 × 10⁻²³ Hz^(-1/2) at O1 — formally a small fraction of the memory's contribution at f ≈ 100 Hz, but the memory's energy is integrated over a much shorter time window than the chirp, so the actual matched-filter SNR contribution is dominated by the noise. For GW150914 alone the memory remained undetectable. Stacking it with the ~80 BBH detections of O3 plus the next observing runs is what eventually moves the integrated SNR above threshold.

BMS symmetries — why memory is geometry, not just a number

The deepest aspect of the memory effect is its identification with the asymptotic-symmetry structure of general relativity at null infinity. Sources of gravitational radiation isolated in asymptotically flat spacetime are characterised at null infinity by the Bondi-Sachs framework. The symmetry group preserving the structure of null infinity turns out not to be the familiar Poincaré group (10-dimensional) but a much larger group — the Bondi-Metzner-Sachs (BMS) group — which contains, in addition to translations, rotations and boosts, an infinite-dimensional family of supertranslations. A supertranslation shifts retarded time by an arbitrary angle-dependent function on the celestial sphere.

The remarkable fact is that a supertranslation acts on the asymptotic metric exactly as a constant shift in the shear tensor C_AB at null infinity — and this shear shift is the same gauge-invariant quantity as the gravitational-wave memory. In other words, the memory effect is a finite supertranslation between the past and future state of the system. The act of emitting gravitational waves with a non-trivial energy flux pattern across the celestial sphere implements a non-trivial supertranslation; the memory is the macroscopic signature of that supertranslation.

This was made explicit through the soft-graviton theorem of Weinberg (1965), which states that the amplitude for any scattering process to emit one additional graviton of momentum k → 0 has a universal 1/ω pole with a coefficient determined by the asymptotic momenta of the hard particles. Andrew Strominger and collaborators showed in the mid-2010s that Weinberg's soft theorem, the BMS supertranslation Ward identity, and the gravitational-wave memory formula are three equivalent statements of the same physics. Memory is the classical, finite-amplitude shadow of soft gravitons.

Soft hair and the information paradox

If asymptotic-symmetry charges are physical observables that can be measured (via the memory), then they are physically real labels on quantum states. Stephen Hawking, Malcolm Perry and Andrew Strominger pursued this thought in 2016 ("Soft hair on black holes", Phys. Rev. Lett. 116, 231301). They argued that black holes carry an infinite-dimensional family of supertranslation charges — soft hair — that classical no-hair theorems missed because those theorems considered only finite-dimensional symmetry groups.

The Hawking-Perry-Strominger proposal does not claim to fully resolve the black-hole information paradox. But it does claim that the assumption underlying Hawking's original 1976 calculation — that black holes are featureless beyond mass, charge, and angular momentum — is wrong. Soft hair gives black holes infinitely many additional quantum numbers, with which information about infalling matter could in principle be encoded and retrieved through correlations in the outgoing Hawking radiation. A gravitational-wave memory observation directly confirms that supertranslation charges are physical — it would be the first experimental hint that the conceptual scaffolding of the soft-hair programme is on solid ground.

Velocity memory, spin memory, and higher-order effects

The 'displacement memory' described so far — a permanent shift in the relative position of test masses — is the leading-order effect. General relativity also predicts subleading memory effects, each tied to a different asymptotic symmetry.

• Velocity memory. In addition to a permanent displacement, freely-falling masses acquire a small permanent relative velocity after the wave passes. The associated symmetry is super-Lorentz (sometimes called the 'extended BMS' superrotation). Detectable in principle but with amplitude even smaller than the displacement memory.
• Spin memory. Pasterski, Strominger and Zhiboedov (2016) showed that a rotating gravitational-wave burst leaves a 'time delay' between counter-orbiting test particles that depends on the direction of orbit — a permanent imprint of angular-momentum flux. The associated soft theorem is the subleading soft graviton theorem of Cachazo and Strominger.
• Center-of-mass memory. A subset of spin memory tied to boost-like supertranslations. Distinct again in waveform.
• Higher harmonics. Numerical relativity simulations show that beyond the dominant ℓ = 2 quadrupole memory, higher multipoles contribute at the ~10 % level for asymmetric mergers (unequal mass, high spin).

These subleading effects are far smaller than the displacement memory and will require third-generation ground-based detectors (Cosmic Explorer, Einstein Telescope) or LISA to test.

Adjacent concepts

• Electromagnetic memory. The same mathematical structure exists in electromagnetism (Bieri & Garfinkle 2013): a passing electromagnetic pulse leaves a 'velocity kick' on charged particles. Sources are bursts from accelerating charges, e.g. a beta decay. Linked to large-gauge transformations of QED in the same way GW memory is linked to BMS supertranslations.
• Yang-Mills memory. The non-Abelian generalisation. Predicted for QCD pulses in the early universe and gluon scattering; experimentally inaccessible but theoretically important for completing the 'asymptotic symmetries ↔ memory ↔ soft theorems' triangle in all gauge theories.
• Antenna pattern of the Earth-LIGO baselines. Memory waveforms have a different angular sensitivity from the oscillatory chirp; in particular, a network of detectors gives a cleaner localisation of memory because of its 'DC' character. Improvements to LIGO's low-frequency noise (suspension upgrades, radiation-pressure suppression) directly improve the memory channel.

Common pitfalls

• Confusing memory with detector zero-drift. Real interferometers drift on long timescales for mundane reasons (thermal expansion, laser power fluctuation). The memory step has a specific, calculable waveform tied to the merger and a specific spectral shape; it is not a generic DC offset and is distinguishable from instrumental drift by matched filtering on the chirp + memory combined template.
• Treating linear and nonlinear memory as alternatives. They add. Most sources have both; only vacuum BBH mergers have only nonlinear memory. Binary neutron-star mergers, with their kilonova ejecta, have both.
• Confusing 'memory' with hysteresis. The system is not hysteretic — there is no friction or dissipation. The masses end up displaced because the underlying spacetime geometry has been altered in a way that affects geodesics, not because anything in the masses themselves changed.
• Believing memory has been detected. As of 2026, no claimed detection of gravitational-wave memory in any single event or stacked-event analysis has reached confidence comparable to the original GW150914 discovery. The community's working assumption is that the first credible detection is still a few years away.
• Reducing the BMS connection to a slogan. The statement 'memory = supertranslation' is rigorous but technical; it requires a careful definition of the asymptotic-symmetry group and the gauge fixing at null infinity. The pop-science version is fine, but care is needed when stating quantitative claims about soft-hair charges and the information paradox.