Soft Charges and Asymptotic Observables

Asymptotically flat gravity has transformations that act nontrivially at null infinity. Their charges split into hard flux carried by finite-energy particles or radiation and soft pieces carried by zero-frequency metric modes. Charge conservation becomes a Ward identity for the gravitational scattering operator and reproduces the universal leading soft-graviton factor. This is a powerful exact infrared constraint; it does not determine the full hard amplitude or prove that soft modes encode every detail of an interior state.

Required background. Variational and Symplectic Geometry supplies boundary charges and fluxes. Equivalence Principles and Gravitational Measurement supplies the operational meaning of asymptotic detectors.

Helpful background. Flat-Spacetime Quantum Field Theory supplies scattering amplitudes and infrared limits, Einstein Equations supplies asymptotic dynamics, Quantum Information supplies distinctions between charges and complete state data, and Quantum Gravity supplies the evidence hierarchy.

Asymptotic phase space and charges

Near future null infinity $\mathscr I^+$, use retarded Bondi coordinates $(u,r,z,\bar z)$ and impose asymptotically flat falloffs. The radiative data are contained in the Bondi shear $C_{AB}(u,z,\bar z)$, with news

$$N_{AB}=\partial_u C_{AB}.$$

A BMS supertranslation is labelled by a function $f(z,\bar z)$ on the celestial sphere. Its charge has the schematic decomposition

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

The hard part measures energy flux weighted by $f$; the soft part is linear in the zero-frequency boundary graviton. Matching suitable data across spatial infinity relates charges on $\mathscr I^-$ and $\mathscr I^+$. The quantum conservation law is the Ward identity

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

The phase space, matching rule, and allowed falloffs are part of the statement. Changing them can change the asymptotic symmetry algebra or add corner terms. Strominger derives the equivalence between diagonal BMS supertranslations and the leading soft theorem under the standard asymptotically flat scattering assumptions Strominger 2014, §§2–6.

Results and limitations

This discussion reflects results available through 2026-08-20. These are analytic infrared results, not empirical detections of individual soft gravitons.

Result or formulation	Observable and evidence	Assumptions and regime	Scope or limitation
Leading soft-graviton theorem	Universal pole multiplying any hard amplitude Weinberg 1965, pp. B516–B524	On-shell scattering, soft momentum taken to zero, conserved hard momentum, standard asymptotic states	Fixes the leading soft behavior, not the finite-frequency amplitude
BMS supertranslation Ward identity	Conservation of soft plus hard asymptotic charge constrains the scattering operator	Bondi falloffs, matching across spatial infinity, declared massless or massive sectors	Charge algebra and matching depend on boundary conditions; infrared Hilbert spaces require care
Displacement memory	Permanent relative displacement after a radiation burst is related to integrated news and energy flux	Asymptotically flat radiative spacetime and idealized early/late stationary eras	A classical observable linked to the same infrared structure; not by itself evidence of quantized geometry
Subleading soft and extended symmetries	Additional frequency orders relate angular momentum operators and proposed asymptotic charges	Loop order, matter content, regularity, and symmetry prescription matter	Universality is weaker; anomalies, logarithms, and boundary choices can modify simple tree-level statements
Soft-hair information claims	Soft charge sectors label infrared data and constrain transitions	Choice of dressed states, boundary algebra, and finite-time access	Charges are coarse data and do not generally determine all hard-state amplitudes or correlations

The conservative position treats the leading theorem, Ward identity, and memory relation as three representations of the same infrared conservation law. It assigns no additional microscopic information capacity without an explicit injective map from hard states to measurable soft data. The permissive position treats the enlarged asymptotic algebra as a structural clue that boundary degrees of freedom organize quantum gravity more deeply. These competing interpretations depend on the infrared representation, matching conditions, finite-time accessibility, and the distinction between a charge sector and a complete state.

The equivalence is controlled at leading order, whereas the status of broader symmetry extensions and information claims remains unresolved. Parallel symmetry language should not obscure the hierarchy between a proven amplitude factorization and a conjectured reconstruction mechanism.

The leading soft factor

Let a graviton of outgoing momentum $q$ and polarization $\varepsilon_{\mu\nu}$ become soft. With all hard momenta $p_k$ taken future-directed and $\eta_k=+1$ for outgoing and $-1$ for incoming legs, the amplitude behaves as

$$\mathcal M_{n+1}(q;p_1,\ldots,p_n) =S^{(0)}(q)\mathcal M_n(p_1,\ldots,p_n)+O(q^0),$$

where

$$S^{(0)}(q)=\frac{\kappa}{2} \sum_{k=1}^{n}\eta_k \frac{p_k^\mu p_k^\nu\varepsilon_{\mu\nu}} {p_k\cdot q}.$$

The $1/(p_k\cdot q)$ pole comes from attaching the soft graviton to each external hard leg. Internal attachments are less singular. Universality follows because gravity couples to energy–momentum. With the site’s Fourier convention, a positive-frequency outgoing mode carries $e^{-i\omega u}$; reversing mode conventions conjugates corresponding formulas but not the Ward identity.

A decisive check replaces the polarization by a pure-gauge tensor $\varepsilon_{\mu\nu}=q_\mu\lambda_\nu+q_\nu\lambda_\mu$. Then

$$S^{(0)}_{\mathrm{gauge}} =\kappa\sum_k\eta_k p_k\cdot\lambda =\kappa\lambda_\mu\sum_k\eta_kp_k^\mu=0$$

by hard momentum conservation. Thus the leading soft amplitude obeys the graviton Ward identity. If momentum can leak through an unaccounted boundary or background, the last equality fails because the scattering problem was incompletely specified.

Memory and zero-frequency data

Integrating the news gives

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

This shear change produces a permanent relative displacement of widely separated inertial detectors after a burst. Nonlinear gravitational-wave energy itself contributes to the memory, as established by Christodoulou Christodoulou 1991, pp. 1486–1489. The soft theorem, asymptotic charge conservation, and memory effect are often called an infrared triangle: amplitude, symmetry, and classical observable encode closely related boundary data.

The relationship has limits. Actual detectors observe finite frequencies and finite times, not an exact $\omega=0$ graviton. Displacement memory includes ordinary memory from changes in massive sources and null memory from energy escaping to infinity; the gravitational-wave contribution to null memory is nonlinear. These are classical effects, and detecting any of them would not alone demonstrate graviton quantization. A quantum measurement needs a state, detector response, infrared-safe observable, and inclusive treatment of unresolved radiation.

Infrared states and information bounds

Fock states with a finite number of charged or massive particles are not adequate asymptotic states in theories with long-range forces: arbitrarily soft radiation produces infrared divergences. Inclusive probabilities or coherently dressed asymptotic states yield infrared-safe questions. The soft charge then helps label the dressing sector, but different hard configurations can share the same finite collection of charges.

Consequently, conservation constrains allowed transitions without making the scattering operator unique. If $[Q,\mathcal S]=0$, the matrix is block diagonal in charge sectors; each block can still contain many independent amplitudes. Recovering a full interior state would require a complete set of observables and a demonstrated measurement protocol, not simply infinitely many formal labels whose independent accessibility is unproved.

Adverse checks include massive external particles, alternative spatial-infinity matching, loop corrections, logarithmic soft terms, finite retarded-time windows, and different infrared dressings. Claims that survive only one idealization should retain that boundary in their wording.

Exercises

Exercise. Apply the pure-gauge replacement $\varepsilon_{\mu\nu}=q_\mu\lambda_\nu+q_\nu\lambda_\mu$ to the leading soft factor. Show that it vanishes for a momentum-conserving scattering process. What does a nonzero remainder diagnose?

Solution

The numerator becomes

$$p_k^\mu p_k^\nu\varepsilon_{\mu\nu} =2(p_k\cdot q)(p_k\cdot\lambda).$$

It cancels the denominator, giving

$$S^{(0)}_{\mathrm{gauge}} =\kappa\lambda\cdot\sum_k\eta_kp_k=0.$$

A nonzero remainder means that the hard momentum listed in the amplitude is not conserved. This can signal a missing external leg, momentum flux through an unmodelled boundary or background, or an inconsistent sign convention. It is not evidence for a physical longitudinal graviton.

What could change these conclusions

The conclusions could change when an independent derivation changes the soft–charge equivalence, a loop or boundary analysis changes the claimed symmetry algebra, a finite-time protocol establishes new measurable soft information, or evidence after 2026-08-20 changes the information-theoretic interpretation.

References

• Christodoulou, Demetrios. “Nonlinear Nature of Gravitation and Gravitational-Wave Experiments.” Physical Review Letters 67 (1991): 1486–1489. DOI.
• Strominger, Andrew. “On BMS Invariance of Gravitational Scattering.” Journal of High Energy Physics 2014, no. 7 (2014): 152. DOI. Open PDF.
• Weinberg, Steven. “Infrared Photons and Gravitons.” Physical Review 140 (1965): B516–B524. DOI.