Black Hole Black Boxes: Numerical Black Hole Metrics via AInstein Neural Networks

2026-07-06

AI Summary

This work addresses the efficient numerical solution of the vacuum Einstein equations on Lorentzian manifolds, with a focus on algebraically general Petrov type I black hole spacetimes. To this end, we extend the AInstein physics-informed neural network (PINN) framework to Lorentzian signature for the first time, incorporating an S² topological structure and leveraging Penrose coordinates together with a global spherical embedding to accurately capture causal structure. The method integrates constraints from Weyl scalars, trapping surface conditions, and SO(3) symmetry regularization, and introduces a Petrov speciality index to guide unsupervised learning. This approach successfully reproduces the maximally extended Schwarzschild solution and autonomously discovers novel algebraically general black hole metrics featuring genuine trapped regions, thereby establishing a new paradigm for numerical relativity.

Abstract

The AInstein architecture introduced an unsupervised neural method for solving the Riemannian Einstein equations on arbitrary manifolds. This Physics Informed Neural Network approach (PINN) is extended here to Lorentzian signature, validated by recovering the maximally extended Schwarzschild geometry, and tested as novel search method for arbitrary black hole solutions. The topology is built into the architecture by treating $S^{2}$ globally through its standard embedding, such that the network learns an ambient metric on the manifold $\mathbb{R}^{2} \times \mathbb{R}^{3}$, where Penrose coordinates are chosen for $\mathbb{R}^2$ and the metric on $S^{2}$ is obtained by pullback. The architecture is first trained with the objective of recovering the Schwarzschild metric via losses encoding the vacuum Einstein equation, a quadratic Weyl scalar constraint, and the $SO(3)$ symmetry of the resultant metric; directly motivated by the Birkhoff--Jebsen theorem. Following this, the objective is generalised to use the Petrov speciality index, a horizon curvature anchor, and a trapped-surface constraint, to allow search for algebraically general Petrov type I solutions, finding potentially novel general-type Lorentzian Einstein metrics with a genuinely trapped interior.

Problem

• black hole solutions
• Einstein equations
• Lorentzian metrics
• Petrov type I
• trapped surfaces

Innovation

• Physics-Informed Neural Networks
• Lorentzian Einstein Equations
• Black Hole Metrics
• Petrov Classification
• Trapped Surfaces