In systems with time-reversible laws, the emergence of an arrow of time is usually attributed to special boundary conditions, such as the low-entropy past hypothesis. But gravity is long-range and promotes clustering, so self-gravitating systems may develop an intrinsic arrow without external conditions. This work builds on the model by Barbour, Koslowski, and Mercati (BKM), where for the Newtonian N-body problem with zero energy, momentum, and angular momentum, a Janus point is identified—a minimum of the moment of inertia, from which a dimensionless clustering measure, variety, grows in both directions. However, an open question remained: how fully does this global scalar quantity describe the actual geometric structure of the system?
Two planar central configurations (critical points of variety in shape space), each with 5000 equal-mass particles, were analyzed; these were obtained numerically in previous work. Their global variety differed by only 1.686% (0.5857 and 0.5956). To quantify local morphology, a scale-invariant measure—the anisotropy of the six nearest neighbors, A6—was introduced. For each particle, the tensor of second moments of neighbor displacements was constructed, and A6 was defined as the ratio of the difference to the sum of its eigenvalues: 0 for isotropic surroundings, 1 for collinear. Configuration averages and distributions of A6 were computed, and robustness was checked against neighborhood size (k=4..12) and merging of very close pairs (threshold 10⁻⁴ of the root-mean-square size).
The mean A6 for the configuration with lower variety was 0.1331, while for the higher-variety one it was 0.3419—a difference of 156.8%. The medians of the distributions differed even more: 0.1133 versus 0.3226. In the lower-variety configuration, no particle had A6 exceeding 0.5, whereas in the second, 18.1% did. The probability distributions of A6 have a Kolmogorov–Smirnov distance of 0.557, confirming their strong dissimilarity. Spatially, high A6 values concentrate along filaments and curved structures in the second configuration. The sign of the difference persisted for all tested neighborhood sizes (k from 4 to 12), though the magnitude varied from 8.8% to 161.8%. A robustness test for very close pairs (replacing connected components with centroids) reduced the mean A6 to 0.3169, but the difference with the first configuration remained at 138.1%.
The result shows that the global scalar variety, which characterizes the gravitational arrow of time in the BKM approach, does not uniquely determine the local geometric organization of the system. Two configurations with nearly identical variety can have radically different internal structure—from nearly isotropic to strongly filamentary. This does not diminish the role of variety as an indicator of the arrow, but points to the need for additional, local relational observables for a more complete description of structural evolution. Thus, the gravitational arrow may be a multidimensional phenomenon, where global clustering is only the first level in a hierarchy of descriptors.
This work opens two directions. Static: expand the sample of central configurations and construct the joint distribution P(Ak, V) to identify possible morphological branches or degeneracy at fixed variety. Dynamic: on ensembles of trajectories with a Janus point, measure how the distribution of Ak changes along the branches and whether it is statistically related to variety or to the distance from the Janus point. This will help understand which aspects of the static landscape of critical shapes are actually selected by dynamics.
The results are important for cosmology and astrophysics, where the large-scale structure of the Universe is described by statistical clustering measures, but local geometry (filaments, voids) carries additional information. The method can be applied to observational data and numerical simulations.
Perform a statistical analysis of a large set of central configurations with different variety values. Simulate an ensemble of N-body trajectories with zero energy and compute correlations between variety and local anisotropy along dynamical paths.
This paper addresses the fundamental problem of the arrow of time in classical and quantum systems, as well as the question of the completeness of describing complex systems via global order parameters. It also relates to the problem of structure formation in self-gravitating systems without dissipation.
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What the work does
The authors used numerical central configurations of 5000 particles to compare the global clustering measure variety with the local anisotropy A6. They found that configurations with nearly identical variety can differ in mean anisotropy by more than 150%, exhibiting different distributions and spatial organization.
Where its strength is
Introducing the scale-invariant local measure A6 and demonstrating its strong variability at nearly fixed global measure provides a concrete tool for distinguishing structures invisible to integral characteristics.
Where it could go next
Worth trying
Why this may matter
If local measures like A6 prove sensitive to the fine structure of gravitational systems, this could offer a new language for describing the transition from microscopic reversible laws to the macroscopic arrow of time, where not only global but also local aspects of clustering matter.
Nearby works in our archive
The closest work in the archive sits at 0.62 by meaning — the usual level for our corpus.
The difference in global variety between the two configurations is comparable to the height difference of two people of 1 cm at an average of 170 cm, whereas the difference in local anisotropy corresponds to one being a weightlifter and the other a marathon runner.