Memory, Soft Relations, and Asymptotic Charges

Gravitational memory, BMS charge balance, and the leading soft-graviton relation are three descriptions of one asymptotic structure under a shared set of boundary and scattering assumptions. Memory is a classical change of shear, charge balance is a conservation statement across null infinity, and the soft relation is a low-frequency statement about quantum scattering. The correspondence is powerful, but it is not an identity between a detector displacement, a charge, and an amplitude without the maps that connect them.

Required background. Variational and Symplectic Geometry supplies charges and Ward identities. Helpful background. Lorentzian Metrics and Causal Cones and Geodesics and Jacobi Fields supply null boundaries and detector deviation; Causality and Global Structure supplies $\mathscr I$; Formulations of General Relativity develops charge construction; Waves, Green Functions, and Geometric Optics supplies radiative limits.

Memory as a shear transition

In a Bondi frame at future null infinity, the news is

$$N_{AB}=\partial_uC_{AB}.$$

If radiation is confined to a finite retarded-time interval and early and late eras are nonradiative, the integrated news gives a permanent shear change

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

For the electric-parity displacement sector, the vacuum shear can be expressed through a sphere potential. The early and late vacua then differ by a supertranslation. The transition is physical when related to the asymptotic limit of geodesic deviation: two freely falling detectors retain a relative displacement after the wave has passed.

The boundary formula alone does not give a finite-radius detector signal. One must match Bondi shear to radiative strain, fix polarization and distance conventions, control subleading corrections, and remove ordinary detector motion. Those tasks belong to Ordinary Displacement Memory and Nonlinear Christodoulou Memory.

Different moments and parity sectors produce spin and center-of-mass memory. They are paired with subleading asymptotic structures and soft expansions, not obtained by mechanically reusing the leading displacement formula.

Charge balance and the soft piece

For a supertranslation parameter $f(x^A)$, split the asymptotic charge into a part depending on boundary shear zero modes and a part associated with hard energy-carrying fields. The null constraint gives a balance law schematically of the form

$$Q_f^{\rm late}-Q_f^{\rm early} =-\mathcal F_f^{\rm hard}-\mathcal F_f^{\rm soft}.$$

The hard flux includes matter energy and the quadratic news contribution. The soft term is linear in the zero-frequency or integrated-news mode and can be rewritten in terms of $\Delta C_{AB}$. Integrating by parts on the sphere makes the relation between an angle-dependent energy flux and the memory potential explicit.

For example, integrate the Bondi constraint between two nonradiative cuts and define the angle-dependent integrated hard flux

$$F(x^A)=\int_{u_1}^{u_2} \left(\frac18N_{AB}N^{AB}+4\pi G T_{uu}^{(2)}\right)du.$$

It has the same length dimension as $m$ and $C_{AB}$. The constraint is then

$$D_AD_B\Delta C^{AB}=4(\Delta m+F).$$

In the electric-parity sector write $\Delta C_{AB}=-2D_AD_B\Delta C+q_{AB}D^2\Delta C$. Commuting derivatives on the unit sphere gives

$$D^2(D^2+2)\Delta C=-4(\Delta m+F).$$

On a scalar harmonic of order $\ell$, the left operator has eigenvalue $\ell(\ell+1)[\ell(\ell+1)-2]$. It therefore determines the $\ell\geq2$ memory potential from the corresponding hard flux and change in the Coulombic mass aspect. Its $\ell=0,1$ kernel supplies the energy–momentum balance conditions, not additional displacement memory. This also explains why flux alone is insufficient unless the early and late mass aspects are controlled Compère and Fiorucci 2019, §3.3, Eqs. (3.34)–(3.36).

For $f=1$, the balance reduces to Bondi energy loss. For the $\ell=1$ harmonics it gives momentum balance. Higher harmonics are supermomentum conservation laws. A localized flux distribution can leave total energy balance unchanged in form while producing nontrivial higher-mode memory.

The statement requires smooth compact cuts, adequate decay at the endpoints of $\mathscr I$, and a charge prescription whose flux is consistent with the chosen presymplectic form. Massive particles and long-range fields introduce contributions through timelike infinity and spatial matching. Compère and Fiorucci develop the BMS, memory, and matching structure in Compère and Fiorucci 2019, §§3.2–3.3, pp. 75–84.

Ward identity and the leading soft theorem

After matching the relevant symmetry between $\mathscr I^-$ and $\mathscr I^+$, conservation becomes an operator statement for the scattering matrix:

$$\langle\mathrm{out}\rvert \left(Q_f^+S-SQ_f^-\right) \lvert\mathrm{in}\rangle=0.$$

Decompose $Q_f=Q_f^{\rm hard}+Q_f^{\rm soft}$. The hard charge acts on energetic external states according to their momenta, while the soft charge creates or annihilates a zero-frequency graviton mode. Moving the hard action to the other side produces the leading soft-graviton factorization.

For a graviton of momentum $q=\omega\widehat q$ and polarization tensor $\varepsilon_{\mu\nu}$, the tree-level schematic relation is

$$\mathcal M_{n+1}(q;p_1,\ldots,p_n) \underset{\omega\to0}{=} \frac{\kappa}{2} \sum_{k=1}^n\eta_k \frac{p_k^\mu p_k^\nu\varepsilon_{\mu\nu}} {p_k\mathbin{\cdot}q} \mathcal M_n(p_1,\ldots,p_n) +O(\omega^0),$$

where $\kappa=\sqrt{32\pi G}$, all hard momenta $p_k$ are future directed, and $\eta_k=+1$ for outgoing and $-1$ for incoming legs. The emitted graviton is outgoing, and the displayed amplitude convention fixes the overall sign. Sphere smearing with $f$ maps this factor to the BMS Ward identity. Strominger gives the symmetry–soft–memory chain with its scattering assumptions in Strominger 2017, Chapters 5–7.

This is an interface statement. Detailed amplitude normalization, loop corrections, infrared dressing, and soft limits belong to Soft Graviton Relations and to the quantum treatment of Soft Charges and Asymptotic Observables.

Assumptions behind the triangle

The three corners align only after several choices agree:

• the same asymptotically flat phase space and Bondi normalization are used;
• past and future BMS parameters are matched through spatial infinity;
• endpoint news and massive-particle contributions are controlled;
• the classical zero-frequency limit is related to the quantum soft insertion with a declared Fourier convention;
• infrared-divergent Fock-space amplitudes are regulated or replaced by suitable inclusive/dressed statements; and
• the order of zero-frequency, large-radius, and late-time limits is specified.

At loop level, subleading soft statements can acquire corrections, while the leading relation has its own infrared interpretation. In four dimensions long-range interactions complicate naive scattering states. A formal Ward identity written on undressed Fock states is not by itself a finite observable.

The classical correspondence is also directional. Given a suitable flux history, the constraint determines memory up to homogeneous/vacuum data. Measuring one component of detector memory does not reconstruct the full BMS charge distribution or the scattering amplitude. The equivalence concerns a shared zero-mode relation, not equality of all data.

Checks and limits of the memory–charge relation

Three checks keep the structural relation honest. With no news or matter flux and identical boundary vacua, memory and the charge difference vanish. An isotropic energy flux contributes to the time-translation charge but has no trace-free $\ell\geq2$ displacement pattern. A proper gauge transformation has zero charge and cannot produce a detector memory observable.

The causal and conformal boundary, BMS transformations, and charge-balance relation provide the geometric starting point. Memory, Soft Structure, and Balance Laws develops detector observables, waveform models, radiative flux, and detailed memory classes. Quantum-gravity and QFT treatments develop infrared Hilbert spaces and scattering amplitudes. These additional ingredients are needed to connect a boundary symmetry to a detection or a quantum theorem.

Exercise

Assume $N_{AB}=0$ outside $u_1\leq u\leq u_2$ and no matter flux. If $C_{AB}(u_2)=C_{AB}(u_1)$, what can be concluded about displacement memory and total radiated energy?

Solution

The shear difference is

$$\Delta C_{AB}=\int_{u_1}^{u_2}N_{AB}\,du=0,$$

so the leading displacement-memory tensor vanishes. This does not imply that no energy was radiated. The energy flux is quadratic:

$$\Delta M_B =-\frac{1}{32\pi G} \int_{u_1}^{u_2}du\int_{S^2}N_{AB}N^{AB}d\Omega,$$

which can be negative even when the signed time integral of $N_{AB}$ vanishes. Memory probes a zero-frequency integral; radiated energy probes a positive quadratic norm.

The next chapter returns to the boundary geometry itself with Spacelike Conformal Infinity in de Sitter Geometry.

References

• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§3.2–3.3, pp. 75–84, Springer, 2019.
• Strominger, Andrew, Lectures on the Infrared Structure of Gravity and Gauge Theory, Chapters 5–7, Princeton University Press, 2018.