  We describe a new adaptive timestep criterion for integrating gravitational
motion, which uses the tidal tensor to estimate the local dynamical timescale
and scales the timestep proportionally. This provides a better candidate for a
truly general-purpose gravitational timestep criterion than the usual
prescription derived from the gravitational acceleration, which does not
respect the equivalence principle, breaks down when $\mathbf{a}=0$, and does
not obey the same dimensional scaling as the true timescale of orbital motion.
We implement the tidal timestep criterion in the simulation code GIZMO, and
examine controlled tests of collisionless galaxy and star cluster models, as
well as fully-dynamic galaxy merger and cosmological dark matter simulations.
The tidal criterion estimates the dynamical time faithfully, and generally
provides a more efficient timestepping scheme compared to an acceleration
criterion. Specifically, the tidal criterion achieves order-of-magnitude
smaller energy errors for the same number of force evaluations in potentials
with inner profiles shallower than $\rho \propto r^{-1}$ (ie. where
$\mathbf{a}\rightarrow 0$), such as star clusters and cored galaxies. For a
given problem these advantages must be weighed against the additional overhead
of computing the tidal tensor on-the-fly, but in many cases this overhead is
small.
