Soft Graviton Relations

The leading soft-graviton theorem, a BMS supertranslation Ward identity, and displacement memory can encode the same asymptotic conservation law in three languages: scattering amplitudes, charges at null infinity, and classical waveform endpoints. The relation is powerful precisely because it is conditional. It requires asymptotically flat scattering, compatible past/future boundary data, controlled zero-frequency limits, and a common normalization. It is not an identity between arbitrary finite-radius waveforms, cosmological signals, or finite observation windows.

Helpful background. Einstein Equations supplies the constraints behind asymptotic charge conservation. Waves, Green Functions, and Geometric Optics supplies retarded solutions and Fourier zero-frequency limits. Gravitational Radiation and Waveforms supplies memory, news, and polarization conventions. Causality and Global Structure supplies past and future null infinity, BMS matching, and asymptotic scattering data.

The classical zero-frequency limit

Fix the retarded-time transform

$$\widetilde f(\omega) =\int_{-\infty}^{\infty}du\,e^{i\omega u}f(u), \qquad f(u)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi} e^{-i\omega u}\widetilde f(\omega).$$

This inverse transform matches the site’s $e^{-i\omega t}$ convention. For the complex Bondi strain $H$ and news $\mathcal N=\partial_uH$,

$$\widetilde{\mathcal N}(0) =\int_{-\infty}^{\infty}\mathcal N\,du =H(+\infty)-H(-\infty) =\Delta H.$$

Thus displacement memory is the zero-frequency news. In the strain transform it appears as a distributional or $1/\omega$ low-frequency term because integrating a step-like endpoint difference requires division by frequency. A waveform can radiate substantial oscillatory energy while having $\Delta H=0$; the soft limit selects an endpoint conservation statement, not the entire signal.

For example, the ideal step $H(u)=H_i+\Delta H\,\Theta(u)$ has

$$\widetilde H(\omega) =\pi(H_i+H_f)\delta(\omega) +i\Delta H\,\operatorname{PV}\frac1\omega,$$

where $\operatorname{PV}$ is the Cauchy principal value and $H_f=H_i+\Delta H$. Multiplication by $-i\omega$ removes the constant-offset delta function and yields $\widetilde{\mathcal N}=\Delta H$. A smooth transition changes the finite-frequency profile, while the same endpoint controls its leading soft term. This is why the unknown constant strain and the measurable endpoint difference must not be conflated.

For asymptotically inertial particles, the classical soft field reproduces

$$\Delta h^{\mathrm{TT}}_{ij} =\frac{4G}{r}\, \Delta\!\left[ \sum_A \frac{p_{Ai}p_{Aj}} {E_A(1-\boldsymbol n\cdot\boldsymbol v_A)} \right]^{\mathrm{TT}}.$$

This is the ordinary memory formula in momentum notation. Favata reviews this classical endpoint relation and its separation from nonlinear memory in Favata 2010, §§2–4, pp. 3–6, PDF.

Weinberg’s leading soft factor

Let a graviton of momentum $q^\mu=\omega(1,\boldsymbol n)$ and polarization $\varepsilon_{\mu\nu}$ become soft in an amplitude with hard external momenta $p_k^\mu$. At leading order,

$$\mathcal M_{n+1}(q;p_k) =S^{(0)}(q)\,\mathcal M_n(p_k)+O(\omega^0),$$ $$S^{(0)}(q) =\frac{\kappa}{2} \sum_k\eta_k \frac{p_k^\mu p_k^\nu\varepsilon_{\mu\nu}} {p_k\cdot q}, \qquad \kappa=\sqrt{32\pi G}.$$

Here $p_k$ is future directed, $\eta_k=+1$ for an outgoing hard leg and $-1$ for an incoming leg, and $\varepsilon_{\mu\nu}$ is the polarization tensor appearing in the outgoing-soft-leg amplitude. With the site’s signature, $p_k\cdot q=-\omega E_k(1-\boldsymbol n\cdot\boldsymbol v_k)$; using a positive denominator instead requires translating the overall sign. Since $p_k\cdot q=O(\omega)$, the factor is $O(\omega^{-1})$. This normalization has $\kappa/2=\sqrt{8\pi G}$, matching Strominger 2017, p. 71, eq. (5.2.22), PDF. Taking the classical limit and Fourier transforming relates the incoming/outgoing momentum difference to the memory field.

Every sign in this comparison depends on the common choice of all-outgoing versus physical incoming momenta, polarization dyad, $h_+-i h_\times$, Fourier transform, and retarded versus advanced solution. Matching one denominator while ignoring those other choices is not a convention translation.

The Ward-identity language

Split an asymptotic supertranslation charge into hard and soft pieces,

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

where $f(x^A)$ is a function on the celestial sphere. The hard term measures angle-dependent matter and radiative energy flow; the soft term is linear in the zero-frequency boundary graviton. With suitable matching between past and future null infinity, the Ward identity is

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

Moving the hard charge action to one side and inserting the soft mode on the other yields the leading soft factor. Taking the classical limit and integrating the Bondi constraint yields the displacement-memory balance. The relevant gravitational derivations are Strominger 2017, §§5.1–5.2, pp. 64–71, and §6, pp. 78–82, PDF.

The three statements are equivalent only after specifying:

• asymptotic flatness and falloffs that make the charges finite;
• retarded/advanced boundary conditions and matching through spatial infinity;
• stationary or sufficiently decaying early and late data;
• the treatment of massive particles reaching timelike rather than null infinity;
• infrared states or dressing appropriate to long-range gravity;
• the perturbative order and any loop or anomaly qualifications.

The classical balance law remains meaningful in regimes where a simple Fock-space scattering amplitude is infrared singular, but then the word “equivalent” requires an appropriately dressed or inclusive formulation.

Subleading soft structure

The subleading soft-graviton factor contains the angular-momentum operator of each hard leg. Schematically,

$$S^{(1)}(q) \propto \sum_k\eta_k \frac{p_k^{(\mu}q_\rho J_k^{\nu)\rho} \varepsilon_{\mu\nu}} {p_k\cdot q}.$$

Its asymptotic-charge interpretation involves Lorentz or extended superrotation generators; the tree-level result and its angular-momentum consistency check are in Strominger 2017, §5.3.4, pp. 76–78, eqs. (5.3.24)–(5.3.29), PDF. In the classical time domain, the magnetic-parity angular-momentum-flux relation underlies spin memory, while electric-parity center-of-mass charge/flux relations underlie center-of-mass memory. These mappings are more delicate than the leading triangle: smooth versus singular symmetry generators, boundary terms, loop corrections, charge improvements, and low-harmonic kernels all matter.

It is therefore accurate to say that subleading soft structure organizes the angular-momentum memory relations under matched hypotheses. It is too strong to say that any measured time integral of cross polarization “is the subleading soft theorem.”

Where the equivalence stops

At finite radius, near-zone and tail fields contaminate a naive zero-frequency strain. In an expanding cosmology, null infinity and the scattering problem differ from the asymptotically flat construction. A finite data segment replaces the true $\omega\to0$ limit by a windowed estimate with leakage and unknown prehistory. None of those cases is disproved by the soft theorem; they simply require a different boundary-value and observational analysis.

Current searches and empirical conclusions belong to Gravitational-Wave Astronomy and dated Research records. This page supplies the asymptotic theory that such analyses would have to translate, not a claim about their outcome.

Common pitfalls

Identifying zero frequency with a small but finite Fourier bin. The theorem concerns a controlled limit and asymptotic states. Windowing, tapering, and finite duration can dominate the lowest numerical bins.

Equating the three corners without boundary assumptions. Memory, a Ward identity, and a soft amplitude live in different mathematical settings. The matching map is part of the claim.

Extending the leading triangle unchanged to subleading order. Angular-momentum charges, symmetry extensions, loop effects, and boundary improvements introduce additional qualifications.

Exercises

1. Zero-frequency news. Let $\mathcal N(u)$ have compact support and $H(-\infty)=0$. Show that $\widetilde{\mathcal N}(0)=H(+\infty)$, and state what happens when the memory vanishes.

Solution

By definition,

$$\widetilde{\mathcal N}(0) =\int_{-\infty}^{\infty}\partial_uH\,du =H(+\infty)-H(-\infty) =H(+\infty).$$

If the memory vanishes, the zero-frequency news is zero even though $\widetilde{\mathcal N}(\omega)$ can be nonzero at finite frequency. Oscillatory radiation is therefore not excluded by a zero soft endpoint.

2. Soft scaling. Why is $S^{(0)}$ proportional to $\omega^{-1}$ for fixed hard momenta and observation direction?

Solution

The soft momentum is $q^\mu=\omega(1,\boldsymbol n)$, so each denominator is

$$p_k\cdot q=\omega\,p_k\cdot(1,\boldsymbol n).$$

The numerator contains only fixed hard momenta and the normalized polarization tensor, so it is $O(\omega^0)$. Every generic term therefore scales as $1/\omega$. Collinear or vanishing denominators require their own limiting prescription and do not change the generic soft power counting.

References

• Favata, Marc. “The Gravitational-Wave Memory Effect.” Classical and Quantum Gravity 27 (2010): 084036. 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.