Gravitational Memory Theorems

Gravitational memory is the permanent change in the asymptotic shear after a finite burst of radiation. For freely falling detectors it appears as a lasting relative displacement, even though the news vanishes before and after the burst. Mathematically, memory is an integrated null constraint: changes in Coulombic data and energy flux determine the electric-parity part of the shear difference between two nonradiative cuts.

Required background. Analysis for Mathematical Relativity supplies convergence of retarded-time integrals, and exterior algebra and differential forms supplies sphere operators and flux orientations.

Helpful background. The Einstein equation supplies the null constraint; Causality and Global Structure supplies $\mathscr I^+$; PDEs and Green Functions for Gravitation supplies asymptotic evolution; and Gravitational Radiation and Waveforms supplies detector response.

Shear difference and detector displacement

Use the Bondi convention of the preceding page,

$$g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad N_{AB}=\partial_u C_{AB}, \qquad q^{AB}C_{AB}=0.$$

Assume $N_{AB}$ is integrable in retarded time and tends to zero as $u\to u_\pm$, where the endpoint cuts lie in nonradiative regimes. The memory tensor is

$$\Delta C_{AB} =C_{AB}(u_+)-C_{AB}(u_-) =\int_{u_-}^{u_+}N_{AB}\,du.$$

For two nearby freely falling test masses at large areal radius $r$, let $X^{\hat A}$ be their initial physical transverse separation in an orthonormal screen basis, with negligible initial relative velocity. For a burst short compared with the background curvature scale, geodesic deviation gives the leading change

$$\Delta X^{\hat A} =\frac{1}{2r}\Delta C^{\hat A}{}_{\hat B}X^{\hat B}+O(r^{-2}).$$

Here $C_{\hat A\hat B}$ is projected with an orthonormal dyad of the unit sphere. This is a measured change of separation, not a claim that the Bondi angular coordinates of the particles must change; a coordinate choice can keep their angles fixed while the metric distance changes.

The factor $1/2$ follows from the metric convention above; it moves if shear is normalized differently. A time-independent BMS supertranslation changes both endpoint representatives by the same inhomogeneous shear, so $\Delta C_{AB}$ is unchanged when the two cuts are compared in one Bondi frame. Comparing independently refixed frames would obscure this invariant content; the relation between this endpoint shear change, BMS supertranslations, and the displacement observable is made explicit in Strominger and Zhiboedov 2016, §§2–3.

Integrating the null constraint

With matter coefficient $T_{uu}=r^{-2}\mathcal T_{uu}+o(r^{-2})$, the mass-aspect equation is

$$\partial_u m =\frac14D_A D_BN^{AB} -\frac18N_{AB}N^{AB} -4\pi G\mathcal T_{uu}.$$

Integrating between the endpoint cuts yields

$$D_A D_B\Delta C^{AB} =4\Delta m +\frac12\int_{u_-}^{u_+}N_{AB}N^{AB}\,du +16\pi G\int_{u_-}^{u_+}\mathcal T_{uu}\,du.$$

The sphere average of this equation is precisely the Bondi mass-loss law. Memory is obtained from its $\ell\ge2$ part. Decompose the electric-parity shear as

$$\Delta C^{E}_{AB} =\left(D_A D_B-\frac12q_{AB}D^2\right)\Delta\Phi.$$

On the unit sphere,

$$D^A D^B\Delta C^{E}_{AB} =\frac12D^2(D^2+2)\Delta\Phi.$$

The operator annihilates the $\ell=0,1$ harmonics, which represent no tensor memory. After projecting the right-hand side to $\ell\ge2$, it can be inverted mode by mode. This is the central derivation: the permanent displacement is fixed by a change in the mass aspect plus the angular distribution of gravitational and matter energy radiated to null infinity.

Ordinary and null memory

The term involving $\Delta m$ is commonly called ordinary memory: it records changes in the Coulombic field, for example when massive bodies change asymptotic velocities. The retarded-time flux terms give null memory: energy carried all the way to $\mathscr I^+$ by gravitational waves, electromagnetic radiation, neutrinos, or other massless fields leaves a permanent effect. “Linear” and “nonlinear” are less reliable names because matter flux can contribute at leading order and gravitational energy is intrinsically quadratic Bieri, Garfinkle, and Yau 2015, §§2–4, pp. 5–23.

Christodoulou isolated the vacuum gravitational-flux contribution in Christodoulou 1991, eqs. (5)–(7), pp. 1487–1488. Bieri, Chen, and Yau prove that the Maxwell field contributes at the same leading order in Einstein–Maxwell developments Bieri, Chen, and Yau 2011, Theorems 6–7 and §4. These claims share the integrated structure above but have different global hypotheses and decay estimates; the Bondi calculation alone does not establish that an arbitrary initial-data development reaches the required asymptotic regime.

Magnetic-parity memory and spin memory obey related curl constraints and probe different charges. They are not contained in the scalar electric-parity inversion written here. The limitation is important: calling every lasting timing or displacement change “the Christodoulou memory” conflates distinct observables and source sectors.

Independent checks

If the spacetime is stationary at both ends, no energy crosses $\mathscr I^+$, and the $\ell\ge2$ part of $\Delta m$ vanishes, the source of $\Delta\Phi$ is zero. Inverting on $\ell\ge2$ gives no electric-parity memory. For a single spherical harmonic $Y_{\ell m}$, $D^2Y_{\ell m}=-\ell(\ell+1)Y_{\ell m}$, so the inversion denominator is

$$\frac12(\ell-1)\ell(\ell+1)(\ell+2),$$

which is positive for every radiative mode and vanishes exactly at $\ell=0,1$.

The angular projection is required by conservation, not merely a convenient way to invert a singular operator. The $\ell=0$ and $\ell=1$ components of the entire source must vanish when the energy and momentum balance laws are satisfied. Dropping those components from arbitrary inconsistent source data would conceal a failed constraint.

For the quadrupolar pulse $C_{AB}=\varepsilon\tau\tanh(u/\tau)Q_{AB}$, the memory is $2\varepsilon\tau Q_{AB}$ while the energy loss is quadratic in $\varepsilon$. Replacing the pulse by a profile with equal positive and negative news can cancel this signed integral while leaving a positive integral of the squared news. The full angular mass-aspect constraint still determines which endpoint data accompany either profile.

Suppose the projected source is $S_{20}Y_{20}$. Find the potential coefficient $\Delta\Phi_{20}$.

Solution

For $\ell=2$, the eigenvalue of $\tfrac12D^2(D^2+2)$ is $\tfrac12(1)(2)(3)(4)=12$. Therefore $\Delta\Phi_{20}=S_{20}/12$. Applying the tracefree Hessian then gives the tensor memory. The sign agrees with the integrated constraint because both Laplacians contribute negative eigenvalues before the $+2$ shift is taken.

References

• Bieri, Lydia, David Garfinkle, and Shing-Tung Yau. “Gravitational Waves and Their Memory in General Relativity.” Surveys in Differential Geometry 20 (2015): 1–34. arXiv.
• Bieri, Lydia, Po-Ning Chen, and Shing-Tung Yau. “Null Asymptotics of Solutions of the Einstein–Maxwell Equations in General Relativity and Gravitational Radiation.” Advances in Theoretical and Mathematical Physics 15 (2011): 1085–1113. arXiv.
• Christodoulou, Demetrios. “Nonlinear Nature of Gravitation and Gravitational-Wave Experiments.” Physical Review Letters 67 (1991): 1486–1489. DOI.
• Strominger, Andrew, and Alexander Zhiboedov. “Gravitational Memory, BMS Supertranslations and Soft Theorems.” Journal of High Energy Physics 2016, no. 1 (2016): 86. DOI.