  It is well known that Einstein's equations assume a simple polynomial form in
the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the
gravitational dynamics in this framework and show that {\em time} evolution of
the gravitational field can be re-expressed as (a gauge covariant
generalization of) the Lie derivative along a novel shift vector field in {\em
spatial} directions. Thus, the canonical transformation generated by the
Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills
phase space, similar to that generated by the diffeomorphism constraint. In
classical general relativity this geometrical interpretation significantly
simplifies calculations and also illuminates the relation between dynamics in
the `integrable' (anti)self-dual sector and in the full theory. For quantum
gravity, it provides a point of departure to complete the Dirac quantization
program for general relativity in a more satisfactory fashion. This gauge
theory perspective may also be helpful in extending the `double copy' ideas
relating the Einstein and Yang-Mills dynamics to a non-perturbative regime.
Finally, the notion of generalized, gauge covariant Lie derivative may also be
of interest to the mathematical physics community as it hints at some
potentially rich structures that have not been explored.
