Conformal geodesics and the hyperboloid at timelike infinity

Abstract

A description of scattering in asymptotically flat spacetimes that includes both radiation and massive asymptotic states requires control of the relation between null and timelike infinity. Motivated by this problem, we use the properties of conformal geodesics to construct an extended representation of timelike infinity in Minkowski spacetime. The construction exploits the fact that every non-null metric geodesic in an Einstein spacetime is, up to an explicit local reparametrisation, a conformal geodesic. It replaces the point $i^+$ by a hyperboloidal boundary $\mathcal{H}^+$ meeting future null infinity $\mathscr{I}^+$ at a corner $\mathcal{C}^{+}$, while fixing the location of $\mathscr{I}^+$ in the associated conformal Gaussian gauge. In this gauge, we derive the equations governing massless scalar and spin-$2$ fields. In particular, we obtain a transport equation for the scalar field along future null infinity and analyse the asymptotic behaviour of its spherical-harmonic modes near timelike infinity. We also derive the explicit massless spin-$2$ equations in this gauge. Finally, we examine the relation to projective extensions and to the hyperbolic base of the extended Carrollian space $\mathrm{Ti}$. Our results provide a geometric framework for studying the connection between late-time and null asymptotics which is a prerequisite for scattering constructions relevant to flat space holography.