Embedding diagrams in stationary spacetimes

TL;DR

This paper derives spatial and dynamic embedding diagrams for various stationary spacetimes, including NUT and Kerr, using analytical and numerical methods, and compares their geometric properties such as curvature.

Contribution

It provides explicit embedding diagrams for NUT, pure NUT, and Kerr spacetimes, including solutions in elliptic integrals and numerical integrations, enhancing understanding of their geometry.

Findings

Spatial embeddings for NUT, pure NUT, and Kerr spacetimes obtained.

Pure NUT embeddings solved with elliptic integrals.

Comparison of embeddings via Gaussian and mean curvatures.

Abstract

We find the spatial and dynamic embedding diagrams in some stationary spacetimes. The spatial embeddings include the NUT, pure NUT and Kerr spacetimes. In the case of pure NUT spacetime, the spatial embedding equations are solved in terms of the elliptic integrals. In other cases we obtain the spatial embedding diagrams by numerical integration of the corresponding embedding equations. These embedding diagrams are then compared by calculating their Gaussian and mean curvatures. We also find the dynamic embedding diagrams of NUT and pure NUT spacetimes.