We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns 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 solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.

中文翻译：

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N 体形状空间中的总碰撞

我们讨论了形状空间上引力N体问题的总碰撞奇点。形状空间是系统的关系配置空间，通过将普通配置空间与总平移、旋转和缩放的相似群相商而获得。对于零能引力N体系统，可以显式构建形状空间的动力学，可以详细分析完全碰撞的点，即中心配置点和零形状动量。事实证明，即使在尺度不是描述一部分的形状空间上，运动方程也会在（并且仅在）完全碰撞的点处发散。我们构建并研究了分层的全碰撞流形，并表明，在形状空间上的全碰撞点处，奇点是必不可少的。因此，没有办法通过这些点来发展解决方案。这密切反映了广义相对论的大爆炸奇点，其中比安奇 IX 的同质但非各向同性的宇宙学模型显示了大爆炸的本质奇点。广义相对论模型的一个简单修改（添加一个刚性物质场）将系统改变为一个形状动力学描述允许通过奇点进行确定性演化的系统。我们怀疑，类似地，需要对动力学进行一些修改，以规范总碰撞奇异性N体模型。