  We study dynamics of interfaces in solutions of the equation $\epsilon \Box u
+ \frac 1 \epsilon f_\epsilon(u)=0$, for $f_\epsilon$ of the form
$f_\epsilon(u) = (u^2-1)(2u- \epsilon\kappa)$, for $\kappa\in {\mathbb R}$, as
well as more general, but qualitatively similar, nonlinearities. We consider
equations of this form both in $(1+n)$-dimensional Minkowski space, $n\ge 1$,
and on certain more general Lorentzian manifolds, and we prove that for
suitable initial data, solutions exhibit interfaces that sweep out timelike
hypersurfaces of mean curvature proportional to $\kappa$. In particular, in 1
dimension these interfaces behave like a relativistic point particle subject to
constant acceleration.
