[Paper Review] Local energy decay for Maxwell fields part I: Spherically symmetric black-hole backgrounds

Jacob Sterbenz, Daniel Tataru|arXiv (Cornell University)|May 22, 2013

Advanced Mathematical Physics Problems13 references4 citations

TL;DR

This paper establishes local energy decay estimates for Maxwell fields on spherically symmetric, stationary black-hole spacetimes with non-degenerate trapping and a red-shift effect. Using a first-order formulation and reduction to spin-zero wave equations, the authors prove weighted $L^2$ decay for solutions under minimal assumptions, extending dispersive estimates to the Maxwell system in curved backgrounds with applications to stability and radiation fields.

ABSTRACT

We prove local energy decay estimates for solutions to the inhomogeneous Maxwell system on a generic class of spherically symmetric black holes.

Motivation & Objective

• To establish local energy decay for Maxwell fields in spherically symmetric, stationary black-hole spacetimes with non-degenerate trapping and a red-shift effect.
• To reduce the Maxwell system to a first-order system and further to a spin-zero wave equation to leverage known local energy decay estimates.
• To prove uniform decay bounds for the Maxwell field in weighted $L^2$ norms, even in the presence of trapped null geodesics.
• To extend dispersive estimates to the Maxwell field in curved spacetimes beyond the Einstein vacuum case, under minimal geometric assumptions.
• To provide a foundation for long-term stability and radiation field analysis in black hole spacetimes via energy decay estimates.

Proposed method

• Formulate the Maxwell equations as a first-order symmetric hyperbolic system using the Hodge star and divergence conditions.
• Reduce the Maxwell system to a spin-zero wave equation via the use of the curvature potential and gauge-invariant variables.
• Apply the multiplier method to the spin-zero wave operator $\Box^0$ to derive energy estimates in the presence of trapping.
• Use dyadic decomposition and weighted $L^2$ norms with weights $\langle x \rangle^{-s}$ to localize and control decay in spatial frequency.
• Employ Hardy and Poincaré-type inequalities to control lower-order terms and achieve sharp decay rates.
• Apply Young’s convolution inequality and interpolation to sum dyadic estimates and derive global decay bounds in $\ell^p L^2$-type norms.

Experimental results

Research questions

• RQ1 Can local energy decay estimates be established for Maxwell fields in spherically symmetric black-hole spacetimes with trapped null geodesics?
• RQ2 How does the presence of a red-shift effect at the horizon influence the decay of Maxwell fields?
• RQ3 To what extent can the Maxwell system be reduced to a spin-zero wave equation to leverage existing decay theory?
• RQ4 What weighted $L^2$ norms are necessary and sufficient to control the decay of Maxwell fields in the exterior region?
• RQ5 How do the assumptions of non-degenerate hyperbolicity and strict trapping affect the long-term behavior of Maxwell fields?

Key findings

• The paper proves a local energy decay estimate for Maxwell fields in spherically symmetric, stationary black-hole spacetimes with non-degenerate trapping and a red-shift effect.
• The decay is quantified via weighted $L^2$ norms: $\| \langle x \rangle^{-s} F \|_{\ell^p L^2} \lesssim \| \chi_0 F \|_{L^2} + \| \langle x \rangle^{1-s} \partial_x F \|_{\ell^\infty L^2} $, with $s > \frac{1}{2}$, ensuring integrability.
• The main result is achieved by reducing the Maxwell system to a spin-zero wave equation $\Box^0$, for which local energy decay is established via the multiplier method.
• A key technical component is the use of dyadic decomposition and weighted estimates to control the spatial decay of solutions across different frequency scales.
• The uniform bound $\| G(t) \|_{L^2} \lesssim \| (T^{-1/2}G, r^{-1/2}G, r^{1/2} \partial_t G) \|_{L^2(dt \, dr \, dV_{\mathbb{S}^2})}$ is proven, ensuring uniform control in time.
• The analysis confirms the absence of super-radiance and establishes that trapped null geodesics do not obstruct decay, due to the normally hyperbolic trapping condition.