An asymptotic proof of the classical log soft graviton theorem

TL;DR

This paper provides an asymptotic derivation of the classical log soft graviton theorem using Einstein equations near infinity, revealing the origin of asymmetry between future and past components.

Contribution

It offers a covariant, Einstein equation-based proof of the log soft graviton theorem, including contributions from incoming radiation and clarifying the asymmetry origin.

Findings

Recovers the standard log soft factor in absence of incoming memory

Identifies the gravitational field discontinuity at spatial infinity as the asymmetry source

Provides a fully covariant derivation relying on Einstein equations near infinity

Abstract

We present a derivation of the classical log soft graviton theorem within the asymptotic framework of Comp\`ere, 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.