Traversable Wormholes with Spontaneous Symmetry Breaking

TL;DR

This paper demonstrates that a minimally coupled scalar field with a Higgs-type self-interaction can support spherically symmetric traversable wormholes in GR, with spontaneous symmetry breaking localized near the wormhole throat acting as a throat-formation threshold. It provides exact phantom-wwormhole and generalized Kiselev-wormhole solutions, showing curvature invariants remain finite and the NEC is violated in the throat region, while horizon structure depends on detailed parameter choices. The authors derive and analyze radial null geodesics, photon-sphere radii, Lyapunov exponents, shadow radii, and ISCOs to characterize the measurable lensing and orbital signatures of these geometries. The results reveal that the throat radius $a$ and the functional form of $M(\phi)$ govern whether the spacetime describes a one-way wormhole, a two-way traversable wormhole, or a regular black hole, highlighting a deep link between scalar-field phase structure and spacetime topology. The work suggests avenues for further study, including perturbative (Regge-Wheeler) analyses and extensions to other gravity theories and matter contents.

Abstract

We argue that a spherically symmetric traversable wormhole solution of the Einstein field equations can be supported by minimally coupled self-interacting scalar field which allows a spontaneous symmetry breaking of the field around the wormhole throat. We study two cases : (i) the phantom wormhole solution of Bronnikov and (ii) a generalized Kiselev wormhole. We study the property of radial null geodesics and show that the metric can describe either a two-way or a one-way traversable wormhole depending on certain parameter ranges. The scalar field exhibits spontaneous symmetry breaking within the coordinate range where a wormhole throat forms and helps one suggest that spontaneous symmetry breaking may act as a threshold for wormhole throat formation. We also compute the radius of the photon sphere, the Lyapunov exponent, the shadow radius, and the innermost stable circular orbits for the geometries.

Figures (19)

• Figure 1: Ricci scalar $R(r)$ and Kretschmann scalar $K(r)$ as a function of $r$ for different values of $a$. The parameter choices made in this graph are $C_{1} = 1$ and $C_{2} = 0.01$, while the parameter $a$ is varied.
• Figure 2: Null Energy Condition as a function of $r$ for different values of $a$, $C_1$ and $C_2$. Top left : $C_{1} = C_{2} = 1$ while $a$ is being varied. Top right : $a = 0.5$ and $C_{2} = 1$, $C_1$ is being varied from $0.01$ to $1$. Below : $a = 0.5$ and $C_{1} = 1$ while $C_2$ is being varied from $0.1$ to $10$.
• Figure 3: $\phi(r)$ as a function of $r$. The blue curve shows the profile for a minus sign in $C_{3} \pm 2 \tan^{-1}(r/a)$, while the orange curve is for the plus sign. $C_3 = 1$.
• Figure 4: Plot of $M(\phi)$ as a function of $r$ for $\phi(r) = C_{3} + 2 \tan^{-1}(r/a)$. Top left : different values of $a$ is considered while $C_{1}$, $C_{2}$ and $C_{3}$ are fixed. Top right : different values of $C_3$ is considered while $a$, $C_{1}$ and $C_{2}$ are fixed. Bottom left : different values of $C_1$ is considered while $a$, $C_{2}$ and $C_{3}$ are fixed. Bottom right : different values of $C_2$ is considered while $a$, $C_{1}$ and $C_{3}$ are fixed.
• Figure 5: Plot of $M(\phi)$ as a function of $r$ for $\phi(r) = C_{3} - 2 \tan^{-1}(r/a)$. Top left : different values of $a$ is considered while $C_{1}$, $C_{2}$ and $C_{3}$ are fixed. Top right : different values of $C_3$ is considered while $a$, $C_{1}$ and $C_{2}$ are fixed. Bottom left : different values of $C_1$ is considered while $a$, $C_{2}$ and $C_{3}$ are fixed. Bottom right : different values of $C_2$ is considered while $a$, $C_{1}$ and $C_{3}$ are fixed.