Gravitational Bending of Light Near Compact Objects
Bibliographic record
Abstract
A photon emitted near a compact object at an angle α with respect to the radial direction escapes to infinity at a different angle ψ > α. This bending of light is caused by a strong gravitational field. We show that, in a Schwarzschild metric, the effect is described by 1 - cos α = (1 - cos ψ)(1 - r g / R ), where R / r g is the emission radius in Schwarzschild units. The formula is approximate, and it applies at R ≥ 2 r g only; however, at these radii it has amazing accuracy, fully sufficient in many applications. As one application, we develop a new formulation for the light-bending effects in pulsars. It reveals the simple character of these effects and gives their quantitative description with practically no loss of accuracy (for the typical radius of a neutron star R = 3 r g the error is 1%). The visible fraction of a star surface is shown to be S v /4π R 2 = [2(1 - r g / R )] -1 , which is for R = 3 r g . The instantaneous flux of a pulsar comes from one or two antipodal polar caps that rotate in the visible zone. The pulse produced by one blackbody cap is found to be sinusoidal (light bending impacts the pulse amplitude but not its shape). When both caps are visible, the pulse shows a plateau: the variable parts of the antipodal emissions precisely cancel each other. The pulsed fraction of blackbody emission with antipodal symmetry has an upper limit A max = ( R - 2 r g )/( R + 2 r g ). Pulsars with A > A max must be asymmetric.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 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".