Improved decay for solutions to the linear wave equation on a Schwarzschild black hole

Abstract

We prove that sufficiently regular solutions to the wave equation ${\square }_g\varphi =0$ on the exterior of the Schwarzschild black hole obey the estimates $|\varphi |\le C_{\delta }{v}_{+}^{-3/2+\delta }$ and $|{\partial }_t\varphi |\le C_{\delta }{v}_{+}^{-2+\delta }$ on a compact region of $r$ and along the event horizon. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.

Remarks and References Added

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