Almost Price's law in Schwarzschild and decay estimates in Kerr for Maxwell field
Siyuan Ma
TL;DR
<3-5 sentence high-level summary> The paper develops a comprehensive framework for the decay of Maxwell fields on Kerr and Schwarzschild spacetimes by linking robust energy–Morawetz control (BEAM) to basic energy decay and then to pointwise decay via a basic energy γ-decay condition. It introduces a spin-weighted $r^p$ hierarchy and a mode-decomposed Teukolsky–type analysis to obtain uniform exterior decay, with refinements in slowly rotating Kerr leading to a total decay power of $-7/2$. On Schwarzschild, the leading late-time tails are shown to be almost Price’s law, with the leading rate determined by Newman–Penrose constants; vanishing constants yield improved rates. Collectively, the results connect mode stability, red-shift effects, and NP constants to precise tail behavior of Maxwell fields in black-hole backgrounds, and they extend the BEAM-decay program to spin $\pm1$ fields in the full subextremal Kerr regime (under suitable assumptions).
Abstract
We consider in this work the asymptotics of a Maxwell field in Schwarzschild and Kerr spacetimes. In any subextremal Kerr spacetime, we show energy and pointwise decay estimates for all components under an assumption of a basic energy and Morawetz estimate for spin $\pm 1$ components. If restricted to slowly rotating Kerr, we utilize the basic energy and Morawetz estimates proven in an earlier work to further improve these decay estimates such that the total power of decay for all components of Maxwell field is $-7/2$. In the end, depending on if the Newman--Penrose constant vanishes or not, we prove almost sharp Price's law decay $τ^{-5+}$ (or $τ^{-4+}$) for Maxwell field and $τ^{-\ell -4+}$ (or $τ^{-\ell -3+}$) for any $\ell$ mode of the field towards a static solution on a Schwarzschild background. All estimates are uniform in the exterior of the black hole.