Black hole entropy: Off-shell versus on-shell
Bibliographic record
Abstract
Different methods of calculation of quantum corrections to the thermodynamical characteristics of a black hole are discussed and compared. The relation between on-shell and off-shell approaches is established. The off-shell methods are used to explicitly demonstrate that the thermodynamical entropy $S~{TD}$ of a black hole, defined by the first thermodynamical law, differs from the statistical-mechanical entropy $S~{SM}$, determined as $S~{SM}=-\\mbox{Tr}(\\hat{\\rho}~H\\ln\\hat{\\rho}~H)$ for the density matrix $\\hat{\\rho}~H$ of a black hole. It is shown that the observable thermodynamical black hole entropy can be presented in the form $S~{TD}=\\pi {\\bar r}_+~2+S~{SM}-S~{SM}_{Rindler}$. Here ${\\bar r}_+$ is the radius of the horizon shifted because of the quantum backreaction effect, and $S~{SM}_{Rindler}$ is the statistical-mechanical entropy calculated in the Rindler space.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 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".