A CANONICAL ANALYSIS OF THE EINSTEIN–HILBERT IN FIRST ORDER FORM
Bibliographic record
Abstract
Using the Dirac constraint formalism, we examine the canonical structure of the Einstein–Hilbert action [Formula: see text], treating the metric g αβ and the symmetric affine connection [Formula: see text] as independent variables. For d>2 tertiary constraints naturally arise; if these are all first class, there are d(d-3) independent variables in phase space, the same number that a symmetric tensor gauge field ϕ μν possesses. If d = 2, the Hamiltonian becomes a linear combination of first class constraints obeying an SO (2, 1) algebra. These constraints ensure that there are no independent degrees of freedom. The transformation associated with the first class constraints is not a diffeomorphism when d = 2; it is characterized by a symmetric matrix ξ μν . We also show that the canonical analysis is different if [Formula: see text] is used in place of g αβ as a dynamical variable when d = 2, as in d dimensions, [Formula: see text]. A comparison with the formalism used in the ADM analysis of the Einstein–Hilbert action in first order form is made by applying this approach in the two-dimensional case with h αβ and [Formula: see text] taken to be independent variables.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".