Chaos in an exact relativistic three-body self-gravitating system
Bibliographic record
Abstract
We consider the problem of three-body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the three-body Hamiltonian, implicitly determined in terms of the four coordinates and momentum degrees of freedom in the system. Nonrelativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal-mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtain orbits in both the hexagonal and three-body representations of the system, and plot the Poincaré sections as a function of the relativistic energy parameter eta. We find two broad categories of periodic and quasiperiodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase space between these two types. Despite the high degree of nonlinearity in the relativistic system, we find that the global structure of its phase space remains qualitatively the same as its nonrelativistic counterpart for all values of eta that we could study. However, the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing eta. For the post-Newtonian system we find that it experiences a chaotic transition in the interval 0.21<theta<0.26, above which some of the near-integrable regions degenerate into chaos.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".