Transverse expansion of the metric at null infinity

TL;DR

This paper investigates the conformal Einstein equations at null infinity across all spacetime dimensions and topologies, revealing how free data determines the geometry and establishing existence and uniqueness results for asymptotically flat spacetimes.

Contribution

It provides a coordinate-free analysis of null infinity, characterizes free data, and proves an existence theorem for solutions to the Einstein equations to infinite order.

Findings

Null infinity geometry is constrained by free data.

Any two spacetimes with the same free data are isometric to infinite order.

Existence of asymptotically flat solutions matching prescribed data.

Abstract

In this paper we analyze the conformal Einstein equations to all orders at null infinity without imposing any restriction on the spacetime dimension, the topology of $\mathscr{I}$, or fall-off conditions for the Weyl tensor. In particular, we study how the equations constrain the geometry of null infinity when it is assumed to be foliated by cross-sections, not necessarily spheres. Our approach is coordinate-free and treats the conformal factor $\Omega$ as a dynamical variable. After identifying the free data at $\mathscr{I}$, we show that any two asymptotically flat spacetimes sharing the same free data at null infinity are necessarily isometric to infinite order. In addition, we provide a detached definition of null infinity and prove an existence theorem for asymptotically flat spacetimes solving the field equations to infinite order at $\mathscr{I}$ realizing the prescribed initial data.