  Up to now, the only known exact Foldy- Wouthuysen transformation (FWT) in
curved space is that concerning Dirac particles coupled to static spacetime
metrics. Here we construct the exact FWT related to a real spin-0 particle for
the aforementioned spacetimes. This exact transformation exists independently
of the value of the coupling between the scalar field and gravity. Moreover,
the gravitational Darwin term written for the conformal coupling is one third
of the relevant term in the fermionic case.
