On non-equatorial embeddings into $\mathbb{R}^3$ of spherically symmetric wormholes with topological defects

TL;DR

This paper extends the embedding method for spherically symmetric spacetimes beyond the equatorial plane, enabling visualization of geometries with topological defects that cannot be embedded traditionally.

Contribution

It generalizes the embedding procedure to arbitrary polar angles, allowing analysis of spacetimes with topological defects where equatorial embeddings fail.

Findings

Derived explicit embedding constraints for non-equatorial slices.

Applied formalism to Schwarzschild-like wormholes and spacetimes with angular deficits.

Identified conditions for consistent embeddings in Euclidean space.

Abstract

Traditionally, the embedding procedure for spherically symmetric spacetimes has been restricted to the equatorial plane $\theta = \pi/2$. This conventional approach, however, encounters a fundamental limitation: not every spherically symmetric geometry admits an isometric embedding of its equatorial slice into three-dimensional Euclidean space. When such embeddings are not possible, the standard geometric intuition becomes inapplicable. In this work, we generalize the embedding procedure to slices with arbitrary polar angles $\theta \neq \pi/2$, thereby extending the visualization and analysis of spacetimes beyond the reach of traditional methods. The formalism is applied to Schwarzschild-like wormholes and to a generalized Minkowski spacetime with angular deficit or excess, which are particularly relevant since their equatorial slices cannot be consistently embedded in $\mathbb{R}^3$. In these cases, we identify the explicit constraints on the radial coordinate, polar angle, and geometric parameters required to guarantee consistent embeddings into three-dimensional Euclidean space.