Let's consider the wave equation $\partial_t^2 \phi(t, x) - \Delta\phi(t, x)=0$ in $\mathbb{R} \times \mathbb{R^3}$ with smooth and compactly supported initial data. Suppose that $\psi = \psi(t, |x|)$ is a solution to such a wave equation and it is spherically symmetric (we can obtain it e.g. by considering spherically symmetric initial data).
It is known that
- The function $\gamma(t, r) = r \psi(t, r)$ satisfies the wave equation in $\mathbb{R}\times \mathbb{R}$, and
- The solutions to the wave equation in dimension $1+n$ decay as $O(t^{-\frac{n-1}{2}})$.
Thus, our spherical symmetric solution $\psi=\frac{\gamma}{r}$ decays as $O\left(\frac{1}{r}\right)$, using that $\gamma$ is a 1-D solution and therefore is bounded by a constant. However, $\psi$ is a 3-D solution, so we expect that it decays as $O\left(\frac{1}{t}\right)$. We have that $r^{-1} < t^{-1}$ only outside of the light cone, so we can prove this fact only there.
Question: How can I see that $\frac{\gamma(t, r)}{r}$ decays as $\frac{1}{t}$ in the entire space?