  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.
