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We discuss the total collision singularities of the gravitational N-bodyproblem on shape space. Shape space is the relational configuration space ofthe system obtained by quotienting ordinary configuration space with respect tothe similarity group of total translations, rotations, and scalings. For thezero-energy gravitating N-body system, the dynamics on shape space can beconstructed explicitly and the points of total collision, which are the pointsof central configuration and zero shape momenta, can be analyzed in detail. Itturns out that, even on shape space where scale is not part of the description,the equations of motion diverge at (and only at) the points of total collision.We construct and study the stratified total-collision manifold and show that,at the points of total collision on shape space, the singularity is essential.There is, thus, no way to evolve the solutions through these points. Thismirrors closely the big bang singularity of general relativity, where thehomogeneous-but-not-isotropic cosmological model of Bianchi IX shows anessential singularity at the big bang. A simple modification of thegeneral-relativistic model (the addition of a stiff matter field) changes thesystem into one whose shape-dynamical description allows for a deterministicevolution through the singularity. We suspect that, similarly, somemodification of the dynamics would be required in order to regularize the totalcollision singularity of the N-body model.

Total Collisions in the N-Body Shape Space