A note on energy currents and decay for the wave equation on a Schwarzschild background
Abstract
In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_g\phi=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $\phi\le Cv_+^{1}$, which holds throughout the domain of outer communications, where $v$ is an advanced EddingtonFinkelstein coordinate, $v_+=\max\{v,1\}$, and $C$ is a constant depending on a Sobolev norm of initial data. A crucial estimate in the proof required a decomposition into spherical harmonics. We here give an alternative proof of this estimate not requiring such a decomposition.
 Publication:

arXiv eprints
 Pub Date:
 September 2007
 arXiv:
 arXiv:0710.0171
 Bibcode:
 2007arXiv0710.0171D
 Keywords:

 Mathematics  Analysis of PDEs;
 General Relativity and Quantum Cosmology;
 Mathematics  Differential Geometry
 EPrint:
 10 pages