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

TL;DR

This paper addresses the limitation that equatorial slices of spherically symmetric spacetimes do not always admit isometric embeddings into $\mathbb{R}^3$. It generalizes the embedding procedure to slices at fixed polar angle $\theta\neq\pi/2$ and applies it to Schwarzschild-like wormholes with a defect parameter $\beta$ and to conical Minkowski spacetime, deriving explicit embedding conditions and formulas. The study shows that equatorial embeddings are restricted by $\frac{1}{1-\beta} - \sin^2\theta \ge 0$, while non-equatorial slices can embed even when the equator cannot, with embeddings existing over specific ranges of $\theta$ and $r$, depending on $\beta$. It also reveals that the Gaussian curvature of certain slices is negative, implying not all slices admit a complete embedding, and provides practical criteria for constructing embedding diagrams across topological defect parameters, thereby enhancing visualization and analysis of such spacetimes.

Abstract

Traditionally, the embedding procedure for spherically symmetric spacetimes has been restricted to the equatorial plane $θ= π/2$. 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 $θ\neq π/2$, 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.

Figures (8)

• Figure 1: The figure shows embedding diagrams of the metric (\ref{['sssss']}) derived from Eq. (\ref{['zderho']}), with $\theta = \pi/2$. Equatorial slices are plotted for $\beta = 9/10$ (dash-dotted line), $\beta = 1/2$ (solid line), $\beta = 1/10$ (dashed line), and $\beta = 1/100$ (dotted line). The plotted slices exhibit a conical singularity.
• Figure 2: The figure shows embedding diagrams of the metric (\ref{['sssss']}) derived from Eq. (\ref{['zderho']}), with $\theta = \pi/4$. The diagrams are plotted for $\beta = 96/100$ (dash-dotted line), $\beta = 1/2$ (solid line), $\beta = -7/10$ (dashed line), and $\beta = -96/100$ (dotted line). All slices exhibit a conical singularity.
• Figure 3: The figure shows embedding diagrams of the metric (\ref{['sssss']}) for $\beta =-1$. The $\theta$-slices are plotted by ussing Eq. (\ref{['zderho']}) for $\theta =0.1$ (dash-dotted line), $\theta =0.6$ (solid line), $\theta = 0.78$ (dashed line).
• Figure 4: The figure displays embedding diagrams of the slice $\theta = \pi/4$ for the metric (\ref{['sssss']}) across various values of the parameter $\beta$. The embeddings are shown for $\beta = -1/2$ (dashed line), $\beta = 0$ (solid line), and $\beta = 1/2$ (dash-dotted line). Comparing the values of $z_0(\beta)$ at $\rho = 1$ using Eq. (\ref{['zetadecero']}), we find that $z_0(-1/2) < z_0(0) < z_0(1/2)$.
• Figure 5: The figure displays embedding diagrams of equatorial slice $\theta = \pi/2$ for the metric (\ref{['schwarzschild_like_wormholes']}). The diagrams are plotted for $\rho_0=5$, $\beta = 1/2$ (dashed line), $\beta = 0$ (dotted line), and $\beta = -1/2$ (solid line).