Quasi-isometric embedding of Kerr poloidal sub-manifolds

TL;DR

This paper introduces two methods for isometric embedding of Kerr poloidal sub-manifolds, one analytical using convex integration and the other numerical, to explore geometric and physical properties near Kerr black holes.

Contribution

It presents novel approaches combining convex integration and numerical solutions to embed Kerr sub-manifolds isometrically, enhancing understanding of their geometry and physics.

Findings

Convex integration yields a family of embeddings approaching isometry.

Numerical solutions reveal the ergoregion expansion with angular momentum.

Methods improve geometric modeling of Kerr black hole regions.

Abstract

We propose two approaches to obtain an isometric embedding of the poloidal Kerr sub-manifold. The first one relies on the convex integration process using the corrugation from a primitive embedding. This allows us to obtain one parameter family of embeddings reaching the limits of an isometric embedding. The second one consists in consecutive numerical resolutions of the Gauss-Codazzi-Mainardi and frame equations. This method requires geometric assumptions near the equatorial axis of the poloidal sub-manifold to get initial and boundary conditions. The second approach allows to understand some physical properties in the vicinity of a Kerr black hole, in particular the fast increasing ergoregion extent with angular momentum.