<p>We present a systematic method for analytically computing time-dependent observables for a relativistic probe particle in Coulomb and Schwarzschild backgrounds. The method generates expressions valid both in the bound and unbound regimes, namely <i>bound-unbound universal</i> expressions. To demonstrate our method we compute the time-dependent radius and azimuthal angle for relativistic motion in a Coulomb background (relativistic Keplerian motion), as well as the electromagnetic field radiated by a relativistic Keplerian source. All of our calculations exhibit bound-unbound universality. Finally, we present an exact expression for the semi-classical wave function in Schwarzschild. The latter is crucial in applying our method to any time-dependent observable for probe-limit motion in Schwarzschild, to any desired order in velocity and the gravitational constant <i>G</i>.</p>