Price's law for scalar waves on 1+4 dimensional subextremal Myers-Perry spacetimes

Abstract

We consider Myers--Perry spacetimes $(M,g_{\mathbf{a},\mathbf{b},\mathfrak{m}})$ in $1+4$ dimensions with two spin parameters $(\mathbf{a},\mathbf{b})\in\mathbb{R}^2$, which are a direct generalization of the well known Kerr spacetime in $1+3$ dimensions. We prove that the Hamiltonian flow associated to $g_{\mathbf{a},\mathbf{b},\mathfrak{m}}^{-1}$ is normally hyperbolically trapped for the full subextremal range \[ (|\mathbf{a}|+|\mathbf{b}|)^2<2\mathfrak{m}. \] As an application, we prove a Price law with explicit leading order terms for solutions $ϕ(t,x)$ of the scalar wave equation \[ \Box_{g_{\mathbf{a},\mathbf{b},\mathfrak{m}}}ϕ=0 \] under the assumption of real mode stability. Our precise asymptotics imply that $ϕ(t,x)=ct^{-3}+\mathcal{O}(t^{-4+})$, where $c\in\mathbb{C}$ is an explicitly computable constant.