Notes on nonsingular models of black holes
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Bibliographic record
Abstract
We discuss static spherically symmetric metrics which represent nonsingular black holes in four- and higher-dimensional spacetime. We impose a set of restrictions, such as a regularity of the metric at the center $r=0$ and Schwarzschild asymptotic behavior at large $r$. We assume that the metric besides mass $M$ contains an additional parameter $\ensuremath{\ell}$, which determines the scale where modification of the solution of the Einstein equations becomes significant. We require that the modified metric obeys the limiting curvature condition; that is, its curvature is uniformly restricted by the value $\ensuremath{\sim}{\ensuremath{\ell}}^{\ensuremath{-}2}$. We also make a ``more technical'' assumption that the metric coefficients are rational functions of $r$. In particular, the invariant $(\ensuremath{\nabla}r{)}^{2}$ has the form ${P}_{n}(r)/{\stackrel{\texttildelow{}}{P}}_{n}(r)$, where ${P}_{n}$ and ${\stackrel{\texttildelow{}}{P}}_{n}$ are polynomials of the order of $n$. We discuss first the case of four dimensions. We show that when $n\ensuremath{\le}2$ such a metric cannot describe a nonsingular black hole. For $n=3$ we find a suitable metric, which besides $M$ and $\ensuremath{\ell}$ contains a dimensionless numerical parameter. When this parameter vanishes, the obtained metric coincides with Hayward's one. The characteristic property of such spacetimes is $\ensuremath{-}{\ensuremath{\xi}}^{2}=(\ensuremath{\nabla}r{)}^{2}$, where ${\ensuremath{\xi}}^{2}$ is a timelike at infinity Killing vector. We describe a possible generalization of a nonsingular black-hole metric to the case when this equality is violated. We also obtain a metric for a charged nonsingular black hole obeying similar restrictions as the neutral one and construct higher dimensional models of neutral and charged black holes.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it