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

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.

18 pages

References in corpus (4)

• Spin-2 fields on Minkowski space near space-like and null infinity
• A new class of obstructions to the smoothness of null infinity
• Time asymmetric spacetimes near null and spatial infinity. I. Expansions of developments of conformally flat data
• Regularity at space-like and null infinity

Cited by in corpus (8)

• Peeling of Dirac and Maxwell fields on a Schwarzschild background
• A rigidity property of asymptotically simple spacetimes arising from conformally flat data
• Gravitational metamaterials from optical properties of spacetime media
• The Maxwell-scalar field system near spatial infinity
• Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background
• Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. III: Nonspherical Schwarzschild waves and singularities at null infinity
• Snell's law for the Schwarzschild black hole
• The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild background