Abstract
We analyze the points of total collision of the Newtonian gravitational system on shape space (the relational configuration space of the system). While the Newtonian equations of motion, formulated with respect to absolute space and time, are singular at the point of total collision due to the singularity of the Newton potential at that point, this need not be the case on shape space where absolute scale doesn’t exist. We investigate whether, adopting a relational description of the system, the shape degrees of freedom, which are merely angles and their conjugate momenta, can be evolved through the points of total collision. Unfortunately, this is not the case. Even without scale, the equations of motion are singular at the points of total collision (and only there). This follows from the special behavior of the shape momenta. While this behavior induces the singularity, it at the same time provides a purely shape-dynamical description of total collisions. By help of this, we are able to discern total-collision solutions from non-collision solutions on shape space, that is, without reference to (external) scale. We can further use the shape-dynamical description to show that total-collision solutions form a set of measure zero among all solutions.
Highlights
Is the Big Bang the beginning of Everything? Or did the universe evolve through the Big Bang which would provide a single instant in the eternal evolution of the world? In what follows, we want to analyze the points of total collision of the
Due to the special behavior of the shape momenta, the shape degrees of freedom cannot be evolved through the points of total collision
If there is a total collision at the mid-point of the evolution, we say there is a Big Bang in the common past of the Newtonian universe
Summary
Newtonian N-body system—the Big Bang of the Newtonian universe—on shape space, the relational configuration space of the system It is well-known that, in the usual description with respect to absolute space and time, the Newtonian solutions end at or begin at the point of total collision. There exists a unique description of the Newtonian system on shape space, a description, that is, which is free of scale, there is still a singularity at the points of total collision This has to do with the special behavior of the shape momenta at that point. We first analyze the E = = = 0 Newtonian gravitational system in its formulation with respect to absolute space and time We will study both the long-time behavior and the behavior at the points of total collision.
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