Numerical thermodynamic studies of classical gravitational collapse in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>3</mml:mn><mml:mo mathvariant="bold">+</mml:mo><mml:mn>1</mml:mn></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>4</mml:mn><mml:mo mathvariant="bold">+</mml:mo><mml:mn>1</mml:mn></mml:math>dimensions
Bibliographic record
Abstract
Recent numerical work on the gravitational collapse of a five-dimemsional (5D) [$4+1$] Yang-Mills instanton has provided numerical evidence that the free energy $F=E\ensuremath{-}TS=E/3$ of a 5D Schwarzschild black hole of mass $E$ can be obtained classically via the Lagrangian. Although there is no Hawking radiation, these numerical results suggest that the quantity $TS$ has a classical meaning. We investigate this association for the physically relevant case of $3+1$ dimensional collapse. We track numerically the negative of the total Lagrangian $\ensuremath{-}L$ during the gravitational collapse of a massless scalar field to a Schwarzschild black hole in isotropic coordinates. We show that $\ensuremath{-}L$ approaches the free energy $F=E\ensuremath{-}TS=E/2$ of a four-dimensional Schwarzschild black hole to within 5%. We also show that the matter contribution to the free energy tends towards zero so that the entropy at late stages of the collapse is gravitational in origin. The entropy $S$ makes a negative contribution to the free energy and this feature is observed in our numerical simulation. There is a pronounced dip (negative contribution) in a thin slice just inside the event horizon precisely where the metric field is nonstationary. This is in accord with recent work suggesting black hole entropy is connected with the nonstationary phase space hidden behind the event horizon. We also obtain thermodynamic results for the 5D collapse of a massless scalar field which confirms that previous 5D results are universal and independent of the type of matter undergoing the collapse.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".