Properties of the symplectic structure of general relativity for spatially bounded space–time regions
Bibliographic record
Abstract
We continue a previous analysis of the covariant Hamiltonian symplectic structure of general relativity for spatially bounded regions of space–time. To allow for wide generality, the Hamiltonian is formulated using any fixed hypersurface, with a boundary given by a closed spacelike two-surface. A main result is that we obtain Hamiltonians associated with Dirichlet and Neumann boundary conditions on the gravitational field coupled to matter sources, in particular a Klein–Gordon field, an electromagnetic field, and a set of Yang–Mills–Higgs fields. The Hamiltonians are given by a covariant form of the Arnowitt–Deser–Misner (ADM) Hamiltonian modified by a surface integral term that depends on the particular boundary conditions. The general form of this surface integral involves an underlying “energy-momentum” vector in the space–time tangent space at the spatial boundary two-surface. We give examples of the resulting Dirichlet and Neumann vectors for topologically spherical two-surfaces in Minkowski space–time, spherically symmetric space–times, and stationary axisymmetric space–times. Moreover, we establish the relation between these vectors and the ADM energy-momentum vector for a two-surface taken in a limit to be spatial infinity in asymptotically flat space–times. We also discuss the geometrical properties of the Dirichlet and Neumann vectors and obtain several striking results relating these vectors to the mean curvature and normal curvature connection of the two-surface. Most significantly, the part of the Dirichlet vector normal to the two-surface depends only on the space–time metric at this surface and thereby defines a geometrical normal vector field on the two-surface. We show that this normal vector is orthogonal to the mean curvature vector, and its norm is the mean null extrinsic curvature, while its direction is such that there is zero expansion of the two-surface, i.e., the Lie derivative of the surface volume form in this direction vanishes. This leads to a direct relation between the Dirichlet vector and the condition for a spacelike two-surface to be (marginally) trapped.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".