A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole

Abstract

We prove that sufficiently regular solutions to the wave equation ${\square }_{g_K}\Phi =0$ on the exterior of a sufficiently slowly rotating Kerr black hole obey the estimates $\left|\Phi \right|\le C{\left(t^{\ast }\right)}^{-3∕2+\eta }$ on a compact region of $r$. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski and Schwarzschild spacetime. This result improves the known robust decay rates that are proved using the vector field method in the region of finite $r$ and along the event horizon.