Gravitational Stability and Screening Effect from Extra Timelike Dimensions

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.