  We present a new method for calculation of the gravitational self-force (GSF)
in Kerr geometry, based on a time-domain reconstruction of the metric
perturbation from curvature scalars. In this approach, the GSF is computed
directly from a certain scalar-like self-potential that satisfies the
time-domain Teukolsky equation on the Kerr background. The approach is
computationally much cheaper than existing time-domain methods, which rely on a
direct integration of the linearized Einstein's equations and are impaired by
mode instabilities. At the same time, it retains the utility and flexibility of
a time-domain treatment, allowing calculations for any type of orbit (including
highly eccentric or unbound ones) and the possibility of self-consistently
evolving the orbit under the effect of the GSF. Here we formulate our method,
and present a first numerical application, for circular geodesic orbits in
Schwarzschild geometry. We discuss further applications.
