  We prove exterior energy lower bounds for (nonradial) solutions to the
energy-critical nonlinear wave equation in space dimensions $3 \le d \le 5$,
with compactly supported initial data. In particular, it is shown that
nontrivial global solutions with compact spatial support must be radiative in
the sense that at least one of the following is true: (1) $\int_{|x|> |t|}
\left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge \eta_1(u) > 0, \
\mathrm{for} \ \mathrm{all} \ t \ge 0 \ \mathrm{or} \ \mathrm{all} \ t \le 0,$
(2) $\int_{|x|> -\varepsilon +|t|} \left( |\partial_t u|^2 + |\nabla u|^2
\right) \mathrm{d}x \ge \eta_2(\varepsilon, u) > 0, \ \mathrm{for} \
\mathrm{all} \ t \in \mathbb{R}, \varepsilon > 0.$ In space dimensions 3 and 4,
a nontrivial soliton background is also considered. As an application, we
obtain partial results on the rigidity conjecture concerning solutions with the
compactness property, including a new proof for the global existence of such
solutions.
