Embedding Versus Immersion in General Relativity

TL;DR

This paper examines the differences between embedding and immersion of space-times in higher dimensions, critiques Kasner's classical Schwarzschild immersion for not preserving causal structure, and discusses the physical implications.

Contribution

It clarifies the distinction between immersion and embedding in general relativity and highlights limitations of Kasner's Schwarzschild model regarding causal structure preservation.

Findings

Kasner's Schwarzschild immersion does not preserve causal structure

Embedding and immersion are conceptually distinct in space-time models

Physical adequacy of space-time immersions is questioned

Abstract

We briefly discuss the concepts of immersion and embedding of space-times in higher-dimensional spaces. We revisit the classical work by Kasner in which he constructs a model of immersion of the Schwarzschild exterior solution into a six-dimensional pseudo-Euclidean manifold. We show that, from a physical point of view, this model is not entirely satisfactory since the causal structure of the immersed space-time is not preserved by the immersion.