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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".