Displacement Memory Effect in Gravitational Waves

• Displacement Memory Effect is the permanent change in the relative separation of detectors after a burst of gravitational radiation, marked by a jump in radiative shear.
• It is analyzed using geodesic deviation equations and asymptotic Bondi-Sachs frameworks, with distinctions made from velocity, spin, and other memory effects.
• Practical insights from numerical relativity and plane-wave models inform waveform modeling, testing gravitational theories and enhancing detection prospects.

Searching arXiv for recent and foundational papers on displacement memory effect. arxiv_search(query="gravitational displacement memory effect plane wave Christodoulou numerical relativity", max_results=10, sort_by="relevance") arxiv_search(query="displacement memory effect gravitational waves", max_results=10, sort_by="relevance") The displacement memory effect is the permanent change in the relative separation of freely falling test particles or detector masses after a burst of gravitational radiation has passed. In the broad memory literature, it is the position-space counterpart of a persistent tidal interaction: in asymptotically flat formulations it is encoded by the jump of the radiative shear or strain between early and late non-radiative regimes, while in plane-wave models it is often defined more restrictively by a nonzero final displacement together with vanishing final relative velocity. This places it alongside, but distinct from, velocity memory, spin memory, center-of-mass memory, and more recent proposals such as angular momentum memory (Bieri et al., 2024, An et al., 2024, Zhang et al., 2024).

1. Kinematic definition and basic observables

At the detector level, displacement memory is read from the geodesic deviation equation,

$$\frac{D^{2}s^{\mu}}{d\tau^{2}}=-R^{\mu}{}_{\delta\sigma\lambda}T^{\delta}T^{\lambda}s^{\sigma},$$

or, in large-distance weak-field approximations,

$$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$

The defining statement is that the post-burst separation does not return to its pre-burst value. In asymptotically flat settings this is commonly expressed as a permanent change in detector arm length, while in plane-wave analyses one frequently imposes the stronger condition that the final relative velocity vanish, so that the after-zone motion becomes constant in time (Kumar, 2021, Hou, 2020, Zhang et al., 2024).

A useful minimal distinction among memory observables is the following.

Observable	Defining asymptotic criterion	Representative expression
Displacement memory	Permanent offset in separation	$\Delta X^i=X^i(+\infty)-X^i(-\infty)$
Velocity memory	Nonzero residual relative velocity	$\Delta \dot X^i\neq 0$
Spin memory	Persistent loop time delay	$$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$

In the nonlinear Bondi-Sachs description, displacement memory is tied to the asymptotic shear. One representative detector formula is

$$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$

so the observable is directly proportional to the jump of the asymptotic shear. In related gravito-electromagnetic formulations, kick memory is associated with one time integration of the radiative field, whereas displacement memory is associated with two; equivalently, kick memory is tied to a first Mellin moment and displacement memory to a second Mellin moment (An et al., 2024, Seraj et al., 2022).

2. Exact plane waves, sandwich waves, and flyby profiles

Exact plane-wave spacetimes provide the sharpest arena for separating displacement memory from velocity memory. In Brinkmann form, a linearly polarized plane wave may be written as

$$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$

and the transverse geodesic equation reduces to a time-dependent oscillator equation such as

$\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$

For exact sandwich plane waves, a central exact result is that the asymptotic relative velocity is generically constant but not zero, so the generic memory effect is velocity memory rather than permanent static displacement. The same analysis links the after-zone behavior to Bondi-Pirani caustics and shows that the late-time transverse motion is linear in the null time $u$ (Zhang et al., 2017).

Later plane-wave work sharpened the distinction by identifying special, fine-tuned situations in which pure displacement memory does occur. For particles initially at rest in a sandwich wave, a convenient criterion is

$\dot X(U=-\infty)=0=\dot X(U=\infty).$

For Gaussian and Pöschl-Teller profiles, pure displacement memory appears only at exceptional wave amplitudes. In the Gaussian example, numerical critical values

$$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$0

correspond to trajectories composed of an integer number of approximate standing half-waves; for the Pöschl-Teller profile,

$$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$1

the exact displacement-memory solutions are Legendre polynomials,

$$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$2

with $$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$3 (Zhang et al., 2024).

An analogous conclusion holds for flyby profiles. With the derivative-of-Gaussian profile proposed by Gibbons and Hawking,

$$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$4

generic amplitudes produce velocity memory, whereas fine-tuned amplitudes

$$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$5

produce pure displacement memory. A derivative Pöschl-Teller approximation yields the same qualitative mechanism and links it again to half-wave counting and Legendre-type structure (Zhang et al., 2024).

A more systematic classification has been given for both pulse and step profiles. In that framework, the outcome depends jointly on the wave profile and on the initial relative motion. For pulse profiles, velocity memory is generic, velocity-memory zero is a tuned intermediate case, and displacement memory requires $$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$6. For step profiles, one must first tune away asymptotic acceleration before classifying the late-time behavior; after that, velocity memory remains generic and displacement memory is again exceptional (Achour et al., 2024).

Other profile studies broaden the admissible pulse shapes without changing the qualitative distinction. For pp-wave spacetimes with profiles such as $$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$7, $$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$8, and $$\ddot S^{\hat A}\approx-R_{u\hat Bu}{}^{\hat A}S^{\hat B}.$$9, the relative separation after the pulse remains nonzero; the displacement memory either increases or decreases monotonically, whereas the velocity memory reaches saturation after an initial rise or drop (Datta et al., 2022). A later supersymmetric treatment recast the transverse geodesic equation as a Sturm-Liouville problem, unified the Pöschl-Teller and Scarf profiles through a superpotential, and derived a compact displacement-memory formula,

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$0

thereby making the critical amplitudes appear as a SUSY quantization condition (Catak et al., 7 Apr 2025).

3. Asymptotically flat sources and the Bondi-Sachs/BMS formulation

In the asymptotically flat, source-based formulation of general relativity, displacement memory is the classical Christodoulou-type permanent detector displacement produced by the passage of radiation through future null infinity. The geometry is encoded in the asymptotic shear, and the memory is the difference between its late- and early-time limits. In one formulation,

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$1

while in Newman-Penrose notation the same effect appears as

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$2

The observable meaning is explicit: the shear jump is proportional to the permanent displacement of test masses (An et al., 2024).

A complementary formulation separates the displacement memory into an ordinary and a null contribution. With

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$3

one has

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$4

together with the magnetic-parity companion equation

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$5

In this language, $\Delta X^i=X^i(+\infty)-X^i(-\infty)$6 controls velocity memory and $\Delta X^i=X^i(+\infty)-X^i(-\infty)$7 controls displacement memory: $\Delta X^i=X^i(+\infty)-X^i(-\infty)$8

$\Delta X^i=X^i(+\infty)-X^i(-\infty)$9

For the source classes emphasized there, the memory is purely electric parity (Bieri et al., 2024).

Compact binary coalescence provides the standard astrophysical realization. There the nonlinear memory effect leaves a non-vanishing final strain in the after-zone,

$\Delta \dot X^i\neq 0$0

which in turn produces a permanent relative displacement of freely falling detector masses. The memory strain is sourced by the hereditary energy-flux integral,

$\Delta \dot X^i\neq 0$1

with

$\Delta \dot X^i\neq 0$2

In the treatment of compact binaries, the velocity-independent terms $\Delta \dot X^i\neq 0$3 are precisely the classic displacement memory, while additional first-order velocity-memory terms appear only if the detector masses do not start exactly at rest (Divakarla et al., 2021).

4. Variants in alternative theories and non-Minkowskian asymptotics

Alternative theories modify both the field content and the symmetry interpretation of displacement memory. In Brans-Dicke theory, the permanent detector displacement contains both a tensor-sector and a scalar-sector contribution: $\Delta \dot X^i\neq 0$4 The tensor displacement memory is due to a vacuum transition caused by null energy fluxes penetrating null infinity and is tied to supertranslations, whereas the scalar-sector displacement memory is associated with Lorentz transformations and angular-momentum fluxes. The tensor memory is constrained by a supermomentum balance law, while the scalar memory satisfies a distinct angular-momentum-based constraint (Hou, 2020).

In de Sitter spacetime, quadrupolar radiation from a localized source below the Hubble scale also produces a displacement memory effect, but the asymptotic structure differs sharply from the $\Delta \dot X^i\neq 0$5 case. Near future infinity, the leading boundary metric $\Delta \dot X^i\neq 0$6 itself changes under the passage of radiation, and the even-parity sector of the memory originates from a $\Delta \dot X^i\neq 0$7-BMS transition between two non-radiative regions of $\Delta \dot X^i\neq 0$8. The odd-parity or current-quadrupole contribution still gives memory, but not one removable by a residual $\Delta \dot X^i\neq 0$9-BMS transformation. In that sense, the de Sitter effect is a leading-order boundary-geometry memory rather than only a subleading shear memory (Compère et al., 2023).

These extensions indicate that the term “displacement memory” is stable at the detector level but not unique at the symmetry level. In general relativity it is linked to the Bondi shear and BMS supertranslations; in Brans-Dicke theory it splits into tensor and scalar sectors with different asymptotic symmetries; in de Sitter it is partly governed by $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$0-BMS data. This suggests that the observable concept is broader than any single asymptotic construction, even though the underlying vacuum-transition mechanism depends on the theory and on the asymptotic geometry.

5. Curved wave backgrounds, horizons, and compact-object geometries

Displacement memory also appears in exact radiative spacetimes that are not asymptotically Minkowskian in the standard Bondi sense. In Kundt wave spacetimes with transverse curvature, the late-time behavior depends strongly on the sign of that curvature. For $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$1 with constant negative curvature, the pulse leaves a permanent change in geodesic separation but no residual velocity: displacement memory is present and velocity memory is absent. For the positively curved Plebański-Hacyan case $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$2, the $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$3-direction shows both displacement and velocity memory, whereas the $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$4-direction exhibits a distinct “frequency memory effect,” namely persistent post-pulse oscillations of the separation. Variable-curvature cases with fixed sign reproduce the same qualitative pattern (Chakraborty et al., 2020).

In Brans-Dicke Kundt geometries, displacement memory can be extracted both from direct geodesic integration and from the geodesic deviation equation. The localized wave profile $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$5 and, when present, the gyraton profile $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$6 both contribute. A particularly transparent exactly solvable case occurs at $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$7, where the Ricci scalar is $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$8 and the transverse geodesic can be solved analytically; for other values of $$\tau=\frac{1}{r|\partial D|}\int du\iint_D Q\,dx^1\wedge dx^2$$9, numerical geodesics and deviation analyses continue to show permanent displacement memory, with qualitative differences from the general-relativistic Kundt results (Siddhant et al., 2020).

Near black hole horizons, displacement memory acquires a near-horizon asymptotic-symmetry interpretation. In Gaussian null or related coordinates, the shift in the separation vector is governed by the horizon data $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$0, $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$1, and $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$2. Representative expressions take the form

$$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$3

and the shifts are induced by near-horizon extended-BMS generators. Related soldering constructions across null shells on Schwarzschild and extreme Reissner-Nordström horizons show that the shell data $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$4 encode the memory directly, sometimes as a pure tangential displacement and sometimes with an out-of-surface component $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$5 (Bhattacharjee et al., 2020, Kumar, 2021).

Exotic compact objects exhibit model-dependent displacement-memory signatures. In generalized Ellis-Bronnikov wormholes, a pulse

$$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$6

produces permanent changes in $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$7 and $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$8, with the radial/tortoise separation larger than the angular one; the effect increases with the steepness parameter $$(x^A_{(B)})^+-(x^A_{(B)})^-=-\frac{d_0}{r}\left(\theta^+_{AB}-\theta^-_{AB}\right),$$9 and is enhanced as the throat radius $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$0 decreases (Bhattacharya et al., 5 Feb 2025). In regular black hole spacetimes modeled by a restricted Bondi-Sachs line element with

$$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$1

the net displacement memory depends on the regularization parameter $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$2, the pulse height, and the specific regular black hole family. In the Neves-Saa family, the Schwarzschild limit $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$3 yields the largest memory, while increasing $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$4 suppresses it; different regular black holes, including Bardeen and Hayward, produce quantitatively different asymptotic separations (Acharyya et al., 17 Feb 2026).

6. Numerical relativity, detectability, and interpretive status

The transition from formal asymptotics to waveform-level observables has been made explicit in numerical relativity. Spectral simulations using the SXS Collaboration’s $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$5 code, followed by Cauchy-characteristic extraction with $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$6, produced waveforms that resolve both displacement and spin memory at future null infinity. In that framework,

$$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$7

The simulations resolve the traditional $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$8 memory modes and some $$ds^2=\delta_{ij}\,dX^i dX^j+2\,dU\,dV+\frac12\,\mathcal A(U)\big((X^1)^2-(X^2)^2\big)\,dU^2,$$9 oscillatory memory modes, show that the magnetic memory is zero up to numerical error, and decompose the strain into a slowly accumulating memory contribution plus quasinormal-mode ringdown. For the GW150914-like run SXS:BBH:0305, the final $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$0 strain mode is $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$1, while the flux-balance prediction gives $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$2; the optimized signal-to-noise ratios for $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$3 are about $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$4 in Advanced LIGO, $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$5 in ET-B, and $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$6 in LISA (Mitman et al., 2020).

Space-based detection forecasts make the displacement memory effect an observational target rather than only an asymptotic invariant. For TianQin, direct single-event detection is predicted to be difficult but not excluded: over a five-year mission, about $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$7 detectable massive-black-hole-binary signals may contain displacement memory with signal-to-noise ratio greater than $\frac{d^2X}{dU^2}+\frac12 H(U)\,X=0.$8, whereas the spin-memory detection chance is negligible. The same analysis concludes that displacement memory is the only memory component TianQin has a realistic chance to detect directly and that, in parts of parameter space, it is significant enough to matter for waveform modeling (Sun et al., 2022).

A recurrent interpretive issue is that “displacement memory effect” is used in partially different senses across subliteratures. In source-based Bondi-Sachs analyses it refers to the permanent detector offset associated with a shear or strain jump. In exact plane-wave analyses it often denotes the stricter condition of zero final relative velocity with nonzero final displacement. Exact sandwich plane waves show that velocity memory is generic, while tuned profiles and amplitudes can realize pure displacement memory. A plausible implication is that the central invariant is the permanent post-radiative imprint on relative motion, but its preferred parametrization—shear jump, final detector offset, or zero-final-velocity trajectory—depends on the geometric regime and on the chosen asymptotic structure (Zhang et al., 2017, Zhang et al., 2024, Achour et al., 2024).