Decay of linear waves on higher-dimensional Schwarzschild black holes
Bibliographic record
Abstract
In this paper we consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation \Box_g\phi=0 on the domain of outer communications of the Schwarzschild spacetime manifold (M^n_m, g) (where n >= 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E[\phi](\Sigma_\tau) through a foliation of hypersurfaces (\Sigma_\tau) (terminating at future null infinity and to the future of the bifurcation sphere) decays, E[\phi](\Sigma_\tau) <= CD/\tau^2, where C is a constant only depending on n and m, and D < \infty is a suitable higher order initial energy on \Sigma_0; moreover we improve the decay rate for the first order energy to E[\partial_t\phi](\Sigma_\tau^R) <= CD/\tau^(4-2\delta) for any \delta > 0 where \Sigma_\tau^R denotes the hypersurface (\Sigma_\tau) truncated at an arbitrarily large fixed radius R < \infty provided the higher order energy D_\delta on \Sigma_0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate |\phi|_{\Sigma_\tau^R} <= (C D'_\delta) / \tau^(3/2-\delta). In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".