Geodesic motion around a distorted static black hole
Bibliographic record
Abstract
In this paper we study geodesic motion around a distorted Schwarzschild black hole. We consider both timelike and null geodesics which are confined to the black hole's equatorial plane. Such geodesics generically exist if the distortion field has only even interior multipole moments, and so the field is symmetric with respect to the equatorial plane. We specialize to the case of distortions defined by a quadrupole Weyl moment. An analysis of the effective potential for equatorial timelike geodesics shows that finite stable orbits outside the black hole are possible only for $q\ensuremath{\in}({q}_{\mathrm{min}},{q}_{\mathrm{max}}]$, where ${q}_{\mathrm{min}}\ensuremath{\approx}\ensuremath{-}0.0210$ and ${q}_{\mathrm{max}}\ensuremath{\approx}2.7086\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, while for null equatorial geodesics a finite stable orbit outside the black hole is possible only for $q\ensuremath{\in}[{q}_{\mathrm{min}},0)$. Moreover, the innermost stable circular orbits are closer to the distorted black hole horizon than those of an undistorted Schwarzschild black hole for $q\ensuremath{\in}({q}_{\mathrm{min}},0)$, and a null innermost stable circular orbit exists for $q={q}_{\mathrm{min}}$. These results show that an external distortion of a negative and sufficiently small quadrupole moment tends to stabilize the motion of massive particles and light.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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".