Decay of solutions to the Maxwell equation on the Schwarzschild background

TL;DR

This paper establishes a new decay estimate for Maxwell fields on Schwarzschild black holes using a novel superenergy tensor, leading to insights into electromagnetic field behavior in curved spacetime.

Contribution

Introduces a new superenergy tensor and proves a Morawetz estimate for Maxwell fields on Schwarzschild backgrounds, advancing understanding of electromagnetic decay in curved spacetime.

Findings

Proves decay estimates for Maxwell fields on Schwarzschild spacetime.

Defines a new superenergy tensor that aids in energy estimates.

Demonstrates decay towards Coulomb solutions in the exterior region.

Abstract

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor $H_{ab}$ defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current $H_{ab} (\partial_t)^b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field to the Coulomb solution on the Schwarzschild exterior.