An asymptotic proof of the classical log soft graviton theorem

Abstract

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.

Figures (2)

• Figure 1: Left: Penrose diagram of an asymptotically flat spacetime. Right: A "democratic" representations of null, timelike and spatial infinities. In both cases the null infinities $\mathcal{I}^\pm$ are described by cylinders. The timelike $\mathcal{H}^\pm$ and spatial $\mathcal{H}^0$ infinities on the right diagram are hyperboloids parametrizing the directions of asymptotic geodesics at the points $i^\pm$ and $i^0$ on the left diagram.
• Figure 2: Simplified drawing of the five infinities and their boundaries, together with a null ray at infinity that connects asymptotic past and future null directions. The hyperbolic spaces $\mathcal{H}^\pm$ are depicted as disks, and the de Sitter space $\mathcal{H}^0$ as a cylinder. The blue dots represent points at each of the eight boundaries. There are four blue links indicating their matching across adjacent boundaries. The red vertical lines are null generators at $\mathcal{I}^\pm$ . The curved red lines represent the evolution of null rays across $\mathcal{H}^0$.