Extrinsic Curvature Embedding Diagrams

TL;DR

This paper introduces extrinsic curvature embedding diagrams as a new visualization tool in relativity, complementing intrinsic curvature diagrams to better understand how surfaces embed in curved spaces.

Contribution

It presents a novel type of embedding diagram based on extrinsic curvature, enhancing the visualization of surface embeddings in curved spacetime.

Findings

Extrinsic curvature embedding diagrams provide additional embedding information.

Combined use with intrinsic diagrams offers a fuller understanding of surface embedding.

Simple examples illustrate the effectiveness of the new diagrams.

Abstract

Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the {\it extrinsic} curvature (instead of the intrinsic curvature). Such an extrinsic curvature embedding diagram, when used together with the usual kind of intrinsic curvature embedding diagram, carries the information of how a surface is {\it embedded} in the higher dimensional curved space. Simple examples are given to illustrate the idea.