  In this paper we consider a semi-linear, energy-critical, shifted wave
equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$:
  \[
  \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = \zeta |u|^{4/(n-2)} u,
\quad (x,t)\in {\mathbb H}^n \times {\mathbb R}.
  \]
  Here $\zeta = \pm 1$ and $\rho = (n-1)/2$ are constants. We introduce a
family of Strichartz estimates compatible with initial data in the energy space
$H^{0,1} \times L^2 ({\mathbb H}^n)$ and then establish a local theory with
these initial data. In addition, we prove a Morawetz-type inequality
  \[
  \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{\rho (\cosh |x|)
|u(x,t)|^{2n/(n-2)}}{\sinh |x|} d\mu(x) dt \leq n {\mathcal E},
  \]
  in the defocusing case $\zeta = -1$, where ${\mathcal E}$ is the energy.
Moreover, if the initial data are also radial, we can prove the scattering of
the corresponding solutions by combining the Morawetz-type inequality, the
local theory and a pointwise estimate on radial $H^{0,1}({\mathbb H}^n)$
functions.
