Eikonal Scattering in Kaluza-Klein Gravity

We study eikonal scattering in the context of Kaluza-Klein theory by considering a massless scalar field coupled to Einstein's gravity in 5D compactified to 4D on a manifold \(M_4\times S^1 \). We also examine various different kinematic limits of the resulting eikonal. In the ultra-relativistic scattering case we find correspondence with the time delay calculated for a massless particle moving in a compactified version of the Aichelburg-Sexl shock-wave geometry. In the case of a massless Kaluza-Klein scalar scattering off a heavy Kaluza-Klein mode a similar calculation yields the deflection angle of a massless particle in the geometry of an extremal, \(Q=2M\), Einstein-Maxwell-dilaton 4D black hole. We also discuss a related case in the scattering of dilatons off a large stack of \(D0\)-branes or \(D6\)-branes in dimensionally reduced \(D=10\) type IIA string theory.