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

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.