  While numerous numerical relativity simulations adopt a 1+log slicing
condition, shock-avoiding slicing conditions form a viable and sometimes
advantageous alternative. Despite both conditions satisfying similar equations,
recent numerical experiments point to a qualitative difference in the behavior
of the lapse in the vicinity of the black-hole puncture: for 1+log slicing, the
lapse appears to decay approximately exponentially, while for shock-avoiding
slices it performs approximately harmonic oscillation. Motivated by this
observation, we consider dynamical coordinate transformations of the
Schwarzschild spacetime to describe small perturbations of static trumpet
geometries analytically. We find that the character of the resulting equations
depends on the (unperturbed) mean curvature at the black-hole puncture: for
1+log slicing it is positive, predicting exponential decay in the lapse, while
for shock-avoiding slices it vanishes, leading to harmonic oscillation. In
addition to identifying the value of the mean curvature as the origin of these
qualitative differences, our analysis provides insight into the dynamical
behavior of black-hole punctures for different slicing conditions.
