Catenoid Inspired Hyperbolic Wormhole Geometry

TL;DR

This work introduces a novel catenoid-inspired, static, spherically symmetric wormhole geometry with a finite interior, described by the metric ds^2 = -$\alpha^2 \cosh^2\left(\frac{r}{\alpha}\right) dt^2 + \frac{dr^2}{\cosh^2\left(\frac{r}{\alpha}\right) - \beta} + r^2 d\Omega^2$. By computing the full curvature and Einstein tensors, the authors model the matter content as an anisotropic fluid, finding NEC violation ($\rho + p_r < 0$ in some regions) and hence exotic matter, while satisfying flare-out and horizon-free conditions. The solution is finite and joined to an exterior Schwarzschild spacetime at a junction, with throat radius determined by $\cosh^2(r_0/\alpha) = \beta$ and the junction conditions $b(a)=2M$, $\Phi(a)=\tfrac{1}{2}\ln(1 - 2M/a)$. Traversability is analyzed via tidal constraints and particle deflection using the Rindler–Ishak method, with stability assessed through a Darmois–Israel junction analysis showing a local minimum in the throat potential near the junction; these results suggest a physically viable, experimentally distinguishable finite wormhole geometry with rich observational implications such as lensing signatures.

Abstract

We unveil a novel class of traversable wormholes exhibiting exact spherical symmetry, geometrically inspired by the minimal surface structure of a catenoid. Introducing the spacetime metric, we rigorously derive its fundamental curvature properties, including the Riemann curvature tensor, and consequently compute the Einstein tensor and stress-energy tensor. This framework reveals that the wormhole is sustained by an anisotropic fluid. A detailed analysis of the energy conditions demonstrates the requisite presence of exotic matter, establishing the physical viability and constraints of this configuration. Subsequent investigations address the wormhole's traversability characteristics, gravitational lensing signatures, and dynamic stability. Crucially, we establish that this catenoid-inspired spacetime represents a finite wormhole, possessing bounded spatial extent.

Figures (10)

• Figure 1: Figure of a catenoid with $\alpha=1$
• Figure 2: Energy conditions with $\alpha=5$ and $\beta=2$
• Figure 3: The above diagrams are the graphical representation of embedding surface $z(r)$ (left panel) and visualization of the wormhole (right panel).We assume $\beta=2$ and red for $\alpha=0.8$, blue for $\alpha=1$, and green for $\alpha=1.2$.
• Figure 4: Figure of $b(r)-r$ vs $r$ taking the parameter values as $\alpha=5$, $\beta=2$
• Figure 5: Figure of $b'(r)$ vs $r$ taking the parameter values as $\alpha=5$