Gravitational Stability and Extra Timelike Dimensions

Tachyonic Kaluza-Klein modes normally appear from the compactification of extra timelike dimensions with scale $L$ and lead to some instabilities of physical systems. We calculate the contribution of tachyonic gravitons to the self energies of spherical massive bodies of radius $R$ and discuss their possible gravitational stability. We find that some spherical bodies can be stable at critical radii $R=2\pi L p$ for some positive integers $p$. We also prove the generic property of massive bodies that for the 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.