Formalizing Classical Gravity in Lean

Thu, 05 Nov 2026 00:00:00 +0800

Overview

This project develops machine-checked mathematics for classical gravity and general relativity using Lean 4 and Mathlib. The goal is to turn definitions, calculations, and derivations from the physics literature into explicit mathematical statements and reusable proofs checked by Lean’s kernel.

Each result makes its assumptions visible: parameter restrictions, coordinate domains, nonvanishing denominators, and smoothness of the fields.

Research directions

Exact spacetimes and coordinate transformations: verify metrics, curvature, vacuum equations, and relations between coordinate descriptions of classical solutions such as Schwarzschild and Kerr.
Einstein–Maxwell systems: certify charged backgrounds and first variations, and connect linearized equations to Kerr–Newman radiation systems.
Black-hole perturbations: check explicit changes of variables, matrix integrating factors, and equivalences between perturbation systems, building foundations for further work on separation and stability.

The Prove2Me profile collects the Exact Vacuum Atlas and Exact Spacetime Atlas missions. Individual mission pages record their precise targets, proof dependencies, and progress.

Standalone formalization

Hintz2026: Local Equivalence of Kerr–Newman Perturbation Systems is a separately published result within this research direction, with its own Lean source, pinned dependencies, and reproduction instructions.

中文简介

本项目使用 Lean 4 与 Mathlib，逐步形式化经典引力中的精确时空、坐标变换、Einstein–Maxwell 系统和黑洞扰动方程，明确每项结论的假设与适用区域。任务与进展汇总于 Prove2Me；Hintz2026 是其中独立发布的局部等价证明成果。

Hintz2026: Local Equivalence of Kerr–Newman Perturbation Systems

Tue, 27 Oct 2026 00:00:00 +0800

Overview

Hintz2026-formalization verifies a specific step in Hintz’s study of perturbations of rotating, charged Kerr–Newman black holes: the local equivalence of first-order systems (1) and (7) through chiral changes of variables and a matrix integrating factor.

Equation numbers follow Hintz’s paper, arXiv:2609.33661v1. The main theorem, Hintz2026.main, combines proofs for both spin signs.

Scope

Setting: subextremal parameters with positive mass and nonzero charge, outside the outer horizon and away from the polar axis.
Result: pointwise equivalence for arbitrary smooth complex fields on this region, with arbitrary real frequency and azimuthal parameters.
Boundary: deriving the original system from Einstein–Maxwell, the full separation and spectral argument, and mode stability are outside this repository’s scope.

Source and reproducibility

The standalone repository contains the 27 Lean modules in the theorem’s dependency closure, pinned Lean / Mathlib versions, build scripts, and an axiom audit. Reproduction instructions are available in English and Chinese. Version 0.1.0 was archived on Zenodo on 27 October 2026 under Apache-2.0.

Part of Formalizing Classical Gravity in Lean. See Exact Vacuum Atlas III on Prove2Me for the task and its mathematical context.

中文简介

本项目用 Lean 4 核验 Kerr–Newman 扰动方程组 (1) 与 (7) 的局部等价，覆盖正、负两种自旋。独立仓库提供证明源码和复现说明；证明范围限于该段等价推导，不包含完整的黑洞模稳定性证明。

Lean | A Quest After Perspectives

Formalizing Classical Gravity in Lean

Overview

Research directions

Standalone formalization

中文简介

Hintz2026: Local Equivalence of Kerr–Newman Perturbation Systems

Overview

Scope

Source and reproducibility

中文简介