  The effective potential for a dynamical Wick field (dynamical signature)
induced by the quantum effects of massive fields on a topologically non-trivial
$D$ dimensional background is considered. It is shown that when the radius of
the compactified dimension is very small compared with $\Lambda^{1/2}$ (where
$\Lambda$ is a proper-time cutoff), a flat metric with Lorentzian signature is
preferred on ${\bf R}^4 \times {\bf S}^1$. When the compactification radius
becomes larger a careful analysis of the 1-loop effective potential indicates
that a Lorentzian signature is preferred in both $D=6$ and $D=4$ and that these
results are relatively stable under metrical perturbations.
