Transverse expansion of the metric at null infinity

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.