2026/07/29 by Julian Barbour, Francisco S. N. Lobo, Maria I. R. Lourenço
Physics and Astronomy · #gr-qc #astro-ph.HE #hep-th
9 pages, 3 figures
arxiv created 2026/07/29 · arxiv updated 2026/07/31
The Newtonian N-body problem in the zero-energy, zero-linear-momentum, and zero-angular-momentum sector provides a time-reversal-invariant setting in which generic complete solutions possess a Janus point and exhibit a gravitational arrow on both branches away from it. The dimensionless variety V, a global scale-invariant measure of clustering contrast, is not pointwise monotonic but fluctuates while growing between rising bounds away from the Janus region. Central configurations are critical shapes of the same scale-invariant landscape and therefore provide controlled probes of the structural information encoded by V. We investigate this question for two planar N=5000 central configurations using the numerical particle-coordinate data. Local morphology is quantified by the six-neighbour anisotropy A6, which ranges from 0 for an isotropic local environment to 1 for an effectively one-dimensional one. Although the varieties of the two configurations differ by only 1.686%, their mean anisotropies differ by 156.8%, from 0.1331 to 0.3419. Moreover, 18.1% of the particles in the higher-variety configuration satisfy A6>0.5, whereas none do so in the lower-variety configuration. The anisotropy ordering persists for all tested neighbourhood sizes 4≤ k≤12, and the principal contrast survives a dimensionless close-pair robustness test. For this pair of critical shapes, nearby values of the global variety therefore coexist with markedly different local geometrical organization. Thus the scalar quantity whose long-term behavior characterizes the BKM gravitational arrow does not, by itself, uniquely specify morphology. This identifies local relational observables as a complementary level of description and provides a quantitative bridge between the static shape-space landscape and morphology along genuine Janus-point histories.