Decay of the Maxwell field on the Schwarzschild manifold

2007/10/22 by Pieter Blue, Blue, P. · 2 citations
Physics and Astronomy · #35Q75 #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #Relativity and Gravitational Theory

paper · pdf · doi:10.48550/arxiv.0710.4102

openalex publication_date 2007/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate r ranges over 2M < r₁ < r < r₂, we obtain a decay rate of t^-1 for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition. In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, r_*>εt, we obtain decay for the null components with rates of |ϕ₊| ∼ |α| < C r^-5/2, |ϕ₀| ∼ |ρ| + |σ| < C r^-2 |t-r_*|^-1/2, and |ϕ_-1| ∼ |\underlineα| < C r^-1 |t-r_*|^-1. Along the event horizon and in ingoing regions, where r_*<0, and when t+r_*1, all components (normalized with respect to an ingoing null basis) decay at a rate of C \uout^-1 with \uout=t+r_* in the exterior region.

Citations

• Stability of a Schwarzschild Singularity

Cited by

• The Maxwell equations on full sub-extremal and extremal Kerr spacetimes
• Mathematical general relativity: a sampler

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