Memory, Soft Structure, and Balance Laws

Gravitational-wave memory turns a transient radiative history into a persistent or time-integrated observable. The simplest effect is a permanent electric-parity displacement. The same asymptotic constraints also support nonlinear memory sourced by radiative energy, magnetic-parity spin memory sourced by angular-momentum transport, and an electric-parity center-of-mass time moment. Soft-graviton and Ward-identity descriptions reorganize these classical balance laws under additional scattering assumptions. This chapter keeps the observables distinct while showing their common Bondi–Sachs origin.

Helpful background. Einstein Equations supplies the asymptotic mass and angular constraints. Waves, Green Functions, and Geometric Optics supplies retarded integrals and zero-frequency limits. Gravitational Radiation and Waveforms supplies strain, news, fluxes, modes, and frames. Causality and Global Structure supplies future null infinity, BMS transformations, and stationary endpoint cuts.

One shear, several memories

Use the volume-wide null-infinity dictionary

$$h_{AB}=q_{AB}+\frac{C_{AB}}r+O(r^{-2}), \qquad N_{AB}=\partial_uC_{AB},$$ $$H=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-i h_\times), \qquad \mathcal N=\partial_uH,$$

with $m^A\bar m_A=1$. The displacement endpoint is

$$\Delta C_{AB}=\int_{u_i}^{u_f}N_{AB}\,du.$$

Decompose the trace-free shear into sphere parity potentials:

$$C_{AB} =\left(D_AD_B-\frac12q_{AB}D^2\right)\Phi +\epsilon_{C(A}D_{B)}D^C\Psi.$$

This decomposition separates observables that a single plus/cross rotation cannot:

Effect	Sphere parity and time operation	Principal source
Ordinary displacement	Electric endpoint $\Delta\Phi$	Change in massive/Coulombic asymptotic state
Nonlinear displacement	Electric endpoint $\Delta\Phi$	Energy flux of gravitational waves
Spin memory	Magnetic time integral $\int\Psi\,du$	Angular-momentum flux and superspin charge change
Center-of-mass memory	Subtracted electric time moment	Center-of-mass angular-momentum flux and super-CM charge change

Ordinary and nonlinear memory can contribute to the same final displacement tensor, but their sources differ. Spin memory is not merely cross polarization, and center-of-mass memory is not merely displacement memory integrated for an arbitrary time.

Favata gives a convention-explicit displacement-memory treatment in Favata 2010, §§2–6, pp. 3–9, PDF. The spin and center-of-mass observables and their separate parity/charge structures are developed in Nichols 2017, §§II–V, PDF and Nichols 2018, §§II–III, PDF.

The displacement balance law

The Bondi mass-aspect constraint is

$$\partial_um =\frac14D_AD_BN^{AB} -\frac18N_{AB}N^{AB} -4\pi G\mathcal T_{uu}.$$

After integration,

$$D_AD_B\Delta C^{AB} =4\Delta m +16\pi G(F_{\mathrm{GW}}+F_{\mathrm{matter}}),$$

where

$$F_{\mathrm{GW}} =\frac{1}{32\pi G} \int N_{AB}N^{AB}\,du.$$

The change $\Delta m$ contains the Coulombic endpoint information associated with ordinary memory. The flux contains null memory; its gravitational part is nonlinear because it is quadratic in first-order news. Projecting onto $\ell\ge2$ and inverting the electric sphere operator produces the tensor memory. The $\ell=0,1$ kernel must be fixed by charge and frame data rather than divided out.

The positive energy flux does not make every strain component positive. It first undergoes low-mode projection and angular differentiation, producing a trace-free pattern.

Angular-momentum memories

The angular-momentum-aspect constraint contains charge changes and flux terms bilinear in shear and news. Its magnetic-parity curl gives the spin-memory balance; its electric-parity part and first retarded-time moment give the center-of-mass balance.

For spin memory,

$$\Delta\Sigma=\int_{u_i}^{u_f}\Psi\,du.$$

A closed-contour circulation of $D^BC_{AB}$ removes the electric gradient and retains the magnetic rotated gradient. This is the asymptotic basis for a counter-propagating Sagnac time-delay observable.

For center-of-mass memory, separate the null displacement potential $\Phi^{(n)}$, set $X=\Phi-\Phi^{(n)}$, and define

$$\Delta\mathcal K =\int_{u_i}^{u_f}X\,du-\left.uX\right|_{u_i}^{u_f}.$$

The endpoint subtraction prevents settled ordinary memory from generating a quantity that grows with the arbitrary duration of the final quiet era. Defining $\Phi^{(n)}$ requires the full energy flux, including that carried by magnetic-parity waves; it cannot be inferred from the instantaneous electric shear alone.

The ordinary and null pieces require a common Bondi frame and charge convention. Independence from the lengths of quiet endpoint intervals does not imply independence from the clock origin: under $u'=u+a$ with the angular pairing held fixed, $\Delta\mathcal K'=\Delta\mathcal K-a\Delta X$. Charge comparisons that change the BMS generator are a separate operation, discussed on the center-of-mass-memory page.

The soft triangle—and its hypotheses

With

$$\widetilde f(\omega)=\int du\,e^{i\omega u}f(u),$$

the zero-frequency news is

$$\widetilde{\mathcal N}(0)=\Delta H.$$

In asymptotically flat scattering, the leading $1/\omega$ soft-graviton factor, the supertranslation Ward identity, and this displacement-memory endpoint can encode one conservation law. Strominger presents the associated null-infinity fields, matching, and infrared triangle in Strominger 2017, §§5.1–5.2, pp. 64–71, and §6, pp. 78–82, PDF.

The equivalence requires compatible falloffs, past/future matching, retarded/advanced boundary choices, a treatment of massive particles and infrared states, and a common polarization/Fourier convention. Subleading angular-momentum soft structure relates to spin and center-of-mass balances only with further qualifications about symmetry generators, charge improvements, loops, and boundary terms. A finite-radius or finite-duration waveform is not automatically an $S$-matrix soft limit.

Reading route

1. Ordinary Displacement Memory derives the unbound-particle endpoint formula and permanent geodesic separation.
2. Nonlinear Christodoulou Memory derives the news-squared source and electric sphere inversion.
3. Spin Memory constructs the magnetic time integral and closed-contour diagnostic.
4. Center-of-Mass Memory constructs the subtracted electric time moment and distinguishes the combined observable from its frame-dependent charge and flux pieces.
5. Soft Graviton Relations states when amplitude, Ward-identity, and classical memory descriptions are equivalent.
6. Memory in Waveform Models supplies a pinned endpoint benchmark, convergence controls, and failure diagnostics for finite data.

The first five pages define the physics. The final page asks whether a sampled waveform preserves it.

Modeling requirements

A finite memory waveform must record:

• the memory type and source term included;
• the early cut and integration constant;
• Bondi/inertial frame and $h_+-i h_\times$ harmonic convention;
• sample rate, duration, taper, and endpoint stationarity;
• angular-mode cutoff and quadratic mode couplings;
• curvature/news integration method and low-frequency filtering;
• independent zero-memory and injected-offset controls;
• convergence under duration, resolution, and window changes.

Memory is especially sensitive to processing that is benign for oscillatory signals. A high-pass filter, polynomial drift removal, or arbitrary zero-mean condition can remove the physical endpoint. Conversely, integration drift can imitate one. The computational page demonstrates why duration convergence can matter more than grid refinement.

Scope and observation boundary

This chapter develops asymptotic classical observables and their waveform construction. Instrument transfer functions, calibration, search statistics, and the status of current memory measurements belong to Gravitational-Wave Astronomy and dated Research records. No detection follows merely from a nonzero theoretical offset.

Review the chapter

1. Classify the observable. Match each expression to its memory type:

$$\Delta\Phi,\qquad \int\Psi\,du,\qquad \int X\,du-[uX]_{u_i}^{u_f}.$$

Solution

$\Delta\Phi$ is electric-parity displacement memory; its source can be ordinary or null/nonlinear. The integral $\int\Psi\,du$ is magnetic-parity spin memory. The subtracted electric time moment is center-of-mass memory. Parity plus the time operation is needed for classification: “permanent,” “integrated,” or “cross-like” alone is insufficient.

2. Isotropic flux. An isolated burst radiates nonzero but exactly isotropic null energy and has no $\ell\ge2$ change in mass aspect. What displacement-memory tensor follows?

Solution

The isotropic flux has only $\ell=0$. The memory sphere equation is inverted after projecting onto $\ell\ge2$, so its tensor source vanishes. Total Bondi energy decreases, but no trace-free angular displacement pattern is produced. Nonzero total energy loss and nonzero displacement memory are therefore distinct statements.

3. Finite-window warning. A numerical strain is forced to have zero mean after a high-pass filter. Can its final-minus-initial average be used as an unbiased memory estimate?

Solution

Not without an independent reconstruction. The processing explicitly suppresses zero and low frequencies and chooses an integration constant, both of which can erase a genuine endpoint. One must characterize the filter transfer function, recover the lost baseline from news/flux or another extraction, and demonstrate duration and window convergence with an injected-memory benchmark.

References

• Favata, Marc. “The Gravitational-Wave Memory Effect.” Classical and Quantum Gravity 27 (2010): 084036. DOI. Open PDF.
• Nichols, David A. “Center-of-Mass Angular Momentum and Memory Effect in Asymptotically Flat Spacetimes.” Physical Review D 98 (2018): 064032. DOI. Open PDF.
• Nichols, David A. “Spin Memory Effect for Compact Binaries in the Post-Newtonian Approximation.” Physical Review D 95 (2017): 084048. DOI. Open PDF.
• Strominger, Andrew. Lectures on the Infrared Structure of Gravity and Gauge Theory. Princeton, NJ: Princeton University Press, 2018. Open PDF, arXiv:1703.05448.