The Maxwell field on the Schwarzschild spacetime: behaviour near spatial infinity

TL;DR

This paper analyzes the behavior of the Maxwell field near spatial infinity in Schwarzschild spacetime using transport equations, revealing conditions for regularity and the emergence of logarithmic singularities at critical sets.

Contribution

It introduces a regularity condition on initial data that ensures smooth solutions at certain orders, and identifies when singularities appear, advancing understanding of fields near spatial infinity.

Findings

Regularity condition ensures smooth solutions at orders p and p+1.

Solutions at order p+2 generally contain logarithmic singularities.

Methodology may extend to Einstein equations and rigidity conjectures.

Abstract

The behaviour of the Maxwell field near one of the spatial infinities of the Schwarzschild solution is analysed by means of the transport equations implied by the Maxwell equations on the cylinder at spatial infinity. Initial data for the Maxwell equations will be assumed to be expandable in terms of powers of a coordinate $\rho$ measuring the geodesic distance to spatial infinity (in the conformal picture) and such that the highest possible spherical harmonics at order $p$ are $2^p$-polar ones. It is shown that if the $2^p$-polar harmonics at order $p$ in the initial data satisfy a certain regularity condition then the solutions to the transport equations at orders $p$ and $p+1$ are completely regular at the critical sets where null infinity touches spatial infinity. On the other hand, the solutions to the transport equations of order $p+2$ contain, in general, logarithmic singularities at the critical sets. It is expected that the ideas and techniques developed in the analysis of this problem can be employed to discuss the more challenging case of the transport equations implied by the conformal Einstein equations, and thus provide a way of constructing a proof of a certain rigidity conjecture concerning the developments of conformally flat initial data sets.