  A classical computation of gravitational bremsstrahlung in ultra-planckian
collisions of massive point particles is presented in an arbitrary number d of
toroidal or non-compact extra dimensions. Our method generalizes the
post-linear formalism of General Relativity to the multidimensional case. The
total emitted energy, as well as its angular and frequency distribution are
discussed in detail. In terms of the gravitational radius r_S of the collision
energy, the impact parameter b and the Lorentz factor in the CM frame, the
leading order radiation efficiency in the Lab frame is shown to be of order
(r_S/b)^{3(d+1)} gamma_{cm} for d=0, 1 and of order (r_S/b)^{3(d+1)}
gamma_{cm}^{2d-3} for d>1, up to a known d-dependent coefficient and a ln
gamma_{cm} factor for d=2, while the characteristic frequency of the radiation
is gamma/b. The contribution of the low frequency part of the radiation (soft
gravitons) to the total radiated energy is shown to be negligible for all
values of d. The domain of validity of the classical result is discussed.
Finally, it is shown that within the region of validity of our approach the
efficiency can obtain unnatural values greater than one, which is interpreted
to mean that the peripheral ultra-planckian collisions should be strongly
radiation damped.
