Gravitational Stability and Screening Effect from D Extra Timelike Dimensions

We study (3+1)+D dimensional spacetime, where D extra dimensions are timelike. Compactification of the D timelike dimensions leads to tachyonic Kaluza-Klein gravitons. We calculate the gravitational self-energies of massive spherical bodies due to the tachyonic exchange, discuss their stability, and find that the gravitational force is screened in a certain number of the extra dimensions. We also derive the exact relationship between the Newton constants in the full 4+D dimensional spacetime with the D extra times and the ordinary Newton constant of our 4 dimensional world.