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 Lp for some positive integers p. We also prove the generic property of massive bodies that for the range 0<R<less than or equal to>piL the gravitational force due to the ordinary massless graviton exchange is screened by the Kaluza-Klein mode exchange of tachyonic gravitons.