Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

TL;DR

This paper rigorously derives second-order asymptotics with logarithmic corrections for the radiation field of wave equations on black hole backgrounds, confirming and extending previous heuristic and numerical results.

Contribution

It provides the first rigorous derivation of logarithmic correction terms in the asymptotic expansion for radiation fields on black hole spacetimes, including cases with vanishing Newman-Penrose constant.

Findings

Confirmed the existence of logarithmic corrections for non-zero Newman-Penrose constant cases.

Explicitly derived new logarithmic correction terms for compactly supported initial data.

Validated previous heuristic and numerical predictions in a rigorous mathematical framework.

Abstract

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner-Nordstrom families of black holes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections were heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman-Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for compactly supported initial data (and hence with a vanishing Newman-Penrose constant) explicitly obtained here appear to be new.