 Shrinking cone energy argument

For the homogeneous speed-one wave equation, integrate its wave energy density over $B(x_{\ast },T-t)$. The moving boundary contributes $-e$, while the usual flux contributes $u_t{\partial }_{n}u$. Their sum is $-\frac{1}{2}(u_t-{\partial }_{n}u{)}^2-\frac{1}{2}|{\nabla }_{\tan }u{|}^2\le 0$. Zero Cauchy data in the initial ball therefore force zero solution in its backward light cone.