Logarithmic matching between past infinity and future infinity: The massless scalar field

Abstract

Matching conditions relating the fields at the future of past null infinity with the fields at the past of future null infinity play a central role in the analysis of asymptotic symmetries and conservation laws in asymptotically flat spacetimes. These matching conditions can be derived from initial data given on a Cauchy hypersurface by integrating forward and backward in time the field equations to leading order in an asymptotic expansion, all the way to future and past null infinities. The standard matching conditions considered in the literature are valid only in the case when the expansion near null infinity (which is generically polylogarithmic) has no dominant logarithmic term. This paper is the first in a series in which we derive the matching conditions for a massless scalar field with initial conditions leading to dominant logarithms at null infinity. We prove that these involve the opposite sign with respect to the usual matching conditions. We also analyse the matching of the angle-dependent conserved charges that follow from the asymptotic decay and Lorentz invariance. We show in particular that these are well defined and finite at null infinity even in the presence of leading logarithmic terms provided one uses the correct definitions. The free massless scalar field has the virtue of presenting the polylogarithmic features in a particularly clear setting that shows their inevitability, since there is no subtle gauge fixing issue or nonlinear intrincacies involved in the problem. We also consider the case of higher spacetime dimensions where fractional powers of $r$ (odd spacetime dimensions) or subdominant logarithmic terms (even spacetime dimensions) are present. Mixed matching conditions are then relevant. In subsequent papers, we will extend the analysis to the electromagnetic and the gravitational fields.