An asymptotic proof of the classical log soft graviton theorem

We present a derivation of the classical log soft graviton theorem within the asymptotic framework of Compère, Gralla, and Wei. The proof relies solely on Einstein equations near timelike, spatial, and null infinity, together with matching properties across these regions. The approach is fully covariant under time reversal and incorporates contributions from incoming soft radiation. In the absence of incoming memory one recovers the standard log soft factor, which features an asymmetry between future and past hard components. From an asymptotic perspective, the origin of this asymmetry lies in a long-known discontinuity of the gravitational field at spatial infinity.

💡 Research Summary

The paper provides a fully covariant proof of the classical logarithmic soft graviton theorem within the asymptotic framework developed by Compère, Gralla, and Wei (CGW). The authors work exclusively with the Einstein equations in the neighborhoods of timelike infinity (future H⁺ and past H⁻), spatial infinity (H⁰), and null infinity (future I⁺ and past I⁻), together with the matching conditions that relate the fields across these regions. By treating all five asymptotic boundaries on an equal footing, they avoid any reliance on a fixed background metric and keep the analysis manifestly covariant under time reversal.

A central technical tool is the “log deviation vector” c^μ(V), which appears as the coefficient of the logarithmic term in the large‑affine‑parameter expansion of asymptotic geodesics. Logarithmic diffeomorphisms ξ^μ = log|s| L^μ act on this vector as δ_L c^μ = –L^μ, so the choice of a “log frame” (future radiative, past radiative, or harmonic) amounts to fixing the values of c^μ at I⁺ and I⁻. The authors emphasize that the difference between the future and past values of c^μ is not free but is fixed by the total spacetime momentum P^μ through the well‑known discontinuity at spatial infinity: c^μ_{I⁺} – c^μ_{I⁻} = 4 G P^μ. This relation (eq. 2.17) underlies the asymptotic matching condition (eq. 2.16) that identifies the log deviation vectors on timelike, spatial, and null infinities.

The classical soft theorems are then expressed in terms of the radiative metric perturbation h_out^{μν}(u, n̂) = lim_{r→∞} r(g^{μν} – η^{μν}) and its Fourier transform. The leading 1/ω term reproduces Weinberg’s memory effect, while the logarithmic term is split into two contributions: (i) a “divergent angular momentum” piece h_div^{μν} that depends on the log‑divergent part of each particle’s angular momentum J_div^{μν}=c^{