Integrals of motion, turning points, and inertial points for orbits around spherical black-holes.
Bibliographic record
Abstract
A locally conserved angular quantity is obtained for non-circular orbits of a massive particle around a spherical black hole. This quantity yields the angle in the spatial plane of motion at which an orbit has either a turning point or an inertial point. It is an analogue of the angle of the globally conserved Laplace-Runge-Lenz vector and Hamilton's binormal vector which exist in Newtonian gravity, with the important difference that the angle jumps at every turning point in the case of elliptic-like orbits that precess. This feature is shared by the angle of the locally conserved Laplace-Runge-Lenz vector and Hamilton's binormal vector in Newtonian gravity with a cubic correction. The angular quantity is derived as part of obtaining a complete set of integrals of motion for the timelike geodesic equations in Schwarzschild spacetime. In addition, a locally conserved temporal quantity and a locally conserved proper-time quantity are obtained from the derivation. The properties of these (new) locally conserved quantities are explained and illustrated for each type of non-circular orbit. Their existence is shown to be related to a group of hidden dynamical symmetries of the timelike geodesic equations.
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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) | 0.000 | 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".