Note about late-time wave tails on a dynamical background

Consider a spherically symmetric spacetime generated by a self-gravitating massless scalar field $\varphi$ and let $\psi$ be a test (nonspherical) massless scalar field propagating on this dynamical background. Gundlach, Price, and Pullin [Phys. Rev. D 49, 890 (1994).] computed numerically the late-time tails for different multipoles of the field $\psi$ and suggested that solutions with compactly supported initial data decay in accord with Price’s law as $t^{-(2\ell +3)}$ at timelike infinity. We show that in the case of the time-dependent background dispersing to Minkowski spacetime Price’s law holds only for $\ell =0$ while for each $\ell \ge 1$ the tail decays as $t^{-(2\ell +2)}$.

Figure 1

Left: The difference between the local power index and the theoretical prediction: $n(r=1,t)-3$ for $\ell =0$ and $n(r=1,t)-2(\ell +2)$ for $\ell =1$, 2, 3, as a function of $1/t$. Right: The log-log plots of $|\psi (t,r=1)|$ for $\ell =0$, 1, 2, 3. Both panels correspond to initial data generated by $a(x)=\exp (-x^2)/\sqrt{2\pi }$ (for ${\varphi }_1$) and $b(x)=x^2\exp (-x^2)$ (for ${\psi }_0$), with $\epsilon =2^{-8}$.