Beyond Buchdahl’s limit: Bilayered stars and thin-shell configurations
Bibliographic record
Abstract
One of the theoretical motivations behind the belief that black holes as described by general relativity exist in nature is that it is hard to find matter configurations that mimic their properties, especially their compactness. One of the classic results that go in this direction is the so-called Buchdahl limit: a bound for the maximum compactness that spherically symmetric isotropic fluid spheres in hydrostatic equilibrium can possibly achieve with an outward-decreasing energy density. However, physically realistic situations could violate both isotropy and the monotonicity of the density profile. Notably, Bondi already showed that if the density profile is allowed to be arbitrary (but remains non-negative), a less restrictive compactness bound emerges. Furthermore, if negative energy densities are permitted, configurations can approach the black hole compactness limit arbitrarily closely. In this work we introduce a set of simple bilayered and thin-shell toy models designed to illustrate the effect of relaxing separately the assumptions of Buchdahl’s theorem. Within these models we highlight the existence of two special examples that we have called anti–de Sitter stars and Einstein static stars. We also discuss how these toy models may represent some of the main features of realistic systems and how they could be extended to find more refined models. © 2025 American Physical Society.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".