  We discuss extra timelike dimensions and their effects on the gravitational
stability of spherical massive bodies. Here we specifically report our results
for the case of one extra timelike dimension where we have made analytically
rigorous investigations on the tachyonic graviton exchange due to the infinite
tower of the Kaluza-Klein mode. With the scale $L$ of the extra timelike
dimension we find that some spherical bodies of radius $R$ can be stable at
critical radii $R=2\pi Lp$ for some positive integer $p$. We also obtain the
generic property of massive bodies that for the short distance range $0<R\leq
\pi L$ the gravitational force due to the ordinary massless graviton exchange
is screened by the Kaluza-Klein mode exchange of tachyonic gravitons.
