BMS, Memory, and Soft Sectors as Holographic Data

At leading order in four-dimensional gravity, one physical statement can be written three ways: a BMS supertranslation Ward identity, Weinberg’s soft-graviton theorem, or displacement memory. Their equivalence is powerful but conditional. It assumes specified falloffs, matching between past and future null infinity, a perturbative scattering regime, and the treatment of massive and infrared sectors.

Required background. Null-Infinity Radiative Data as Candidate Boundary Data defines shear and charges, while Soft Theorems supplies the momentum-space factorization theorem.

Helpful background. Soft Limits as On-Shell Constraints supplies amplitude kinematics; What Is a Symmetry of a QFT? distinguishes charged symmetries from redundancy; and Asymptotic Symmetry, Soft Limits, and the Boundary Interface develops the general boundary mechanism.

Hard and soft BMS charges

A supertranslation is labeled by a function $f(x^A)$ on the celestial sphere. Its charge splits into hard and soft parts,

$$Q_f=Q_f^{\rm hard}+Q_f^{\rm soft}.$$

The hard part measures the flux of finite-frequency matter and gravitons weighted by $f$. The soft part is linear in the zero-frequency boundary graviton and changes the vacuum shear. With antipodal matching across spatial infinity, conservation gives the scattering Ward identity

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

Inserting the soft charge and Fourier-transforming the news isolates the $\omega\to0$ graviton mode. The resulting factor multiplying the hard amplitude is Weinberg’s leading soft factor. Thus the soft theorem realizes the asymptotic symmetry on scattering data.

Memory as the classical observable

A compact radiation burst changes the shear by

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

For freely falling detectors at large radius, geodesic deviation produces a permanent relative displacement proportional to $\Delta C_{AB}$. The integrated Bondi constraint relates this memory to hard energy flux plus a change in Coulombic charge. The zero-frequency soft mode, the vacuum transition generated by $Q_f^{\rm soft}$, and the memory tensor therefore encode the same leading sector under the stated boundary conditions.

First application. Derive the leading charge Ward identity for a radiative scattering process and connect its soft insertion to a measurable memory displacement. Start from the charge-flux balance law, use matching to move the past charge through $S$, and identify the soft mode with the $\omega^{-1}$ pole. Fourier inversion then gives $\Delta C_{AB}$ for the detector response.

Where the triangle changes

Massive external particles contribute through timelike infinity, modifying the simplest null matching argument. Extended BMS or superrotation charges need stronger control of sphere singularities and angular momentum. Subleading soft relations receive loop and logarithmic effects that are absent from the leading theorem. Memory also has distinct displacement, spin, and center-of-mass versions; they should not be merged into one observable.

Adversarial control. Change the falloff so a proposed superrotation charge diverges, include a massive particle while retaining a purely null hard charge, or apply the leading equivalence to a loop-corrected subleading soft factor. In each case, one corner of the triangle loses its stated definition or acquires an extra term. The leading supertranslation result survives within its original domain.

Evidence ceiling

The BMS–soft–memory relation is a precise perturbative bridge among symmetry, scattering, and classical detection. It does not supply all infrared-dressed states, determine a local celestial dynamics, or prove that asymptotic charges form a complete set of quantum-gravity observables.

The leading supertranslation Ward identity and Weinberg soft theorem are related under the stated scattering assumptions by He et al. 2015, and the associated displacement memory relation is developed by Strominger and Zhiboedov 2016.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• He, Temple, Vyacheslav Lysov, Prahar Mitra, and Andrew Strominger. “BMS Supertranslations and Weinberg’s Soft Graviton Theorem.” Journal of High Energy Physics 2015, no. 5 (2015): 151. DOI; Open PDF.
• Strominger, Andrew, and Alexander Zhiboedov. “Gravitational Memory, BMS Supertranslations and Soft Theorems.” Journal of High Energy Physics 2016, no. 1 (2016): 086. DOI; Open PDF.
• Weinberg, Steven. “Infrared Photons and Gravitons.” Physical Review 140 (1965): B516–B524. DOI.