Characterization of exotic matter in PT-symmetric wormholes

TL;DR

This work analyzes a PT-symmetric, bimetric traversable wormhole by focusing on the null hypersurface at the throat and employing the Barrabès–Israel formalism to extract the surface stress-energy of the lightlike junction. The calculated jump in transverse curvature yields a surface energy density $ extmu<0$ and a positive tangential pressure $p>0$, signaling NEC violation and the presence of exotic matter that stabilizes the throat. The study demonstrates conservation laws on the null shell and discusses classical stability alongside quantum backreaction, outlining multiple observational signatures, including gravitational-wave echoes and through-throat imaging, as well as cosmological implications from a relic wormhole population. It also situates these results within a broader PT-symmetric and possibly bimetric cosmological context, offering a pathway to falsifiable tests with current and future multi-messenger observations.

Abstract

In our previous work [H. Zejli, Int. J. Mod. Phys. D 34, 2550052 (2025), arXiv:2508.00035], we introduced a PT-symmetric wormhole model based on a bimetric geometry, capable of generating closed timelike curves (CTCs). In this paper, we extend the analysis to the null hypersurface at the throat of this modified Einstein-Rosen bridge, where two regular Eddington-Finkelstein metrics render the geometry traversable. Using the Barrabes--Israel formalism in Poisson's reformulation, we evaluate the null shell's surface stress-energy tensor $S^{αβ}$ from the jump of the transverse curvature, revealing a violation of the null energy condition: a lightlike membrane of exotic matter with negative surface energy density and positive tangential pressure. This exotic fluid acts as a repulsive source stabilizing the throat, ensuring consistency with the Einstein field equations, including conservation laws on the shell. Beyond the local characterization, we outline potential observational signatures: (i) gravitational-wave echoes from the photon-sphere cavity; (ii) horizon-scale imaging with duplicated and through-throat photon rings, and non-Kerr asymmetries; (iii) quantum effects such as PT-induced frequency pairing with possible QNM doublets and partial suppression of vacuum flux at the throat; and (iv) a relic cosmological population yielding an effective $Λ_{\mathrm{eff}}$ and seeding voids. Compared with timelike thin-shell constructions, our approach is based on a null junction interpreted as a lightlike membrane, combined with PT symmetry, providing a distinct route to traversability and clarifying the conditions under which CTCs can arise in a self-consistent framework.

Figures (4)

• Figure 1: Schematic $3D$ representation of the vector basis on the null hypersurface $\Sigma$ at the wormhole throat ($r = \alpha$), with $\alpha = 1$ for illustrative purposes. The wireframe sphere depicts the $2D$ intrinsic geometry of $\Sigma$, parameterized by angular coordinates $\theta$ and $\phi$. Vectors are shown originating from a reference point on the sphere (corresponding to Cartesian coordinates $X=1$, $Y=0$, $Z=0$). The red arrow represents the spacelike tangent vector $e^\mu_\theta$ (pointing in the negative $Z$-direction). The green arrow denotes the spacelike tangent vector $e^\mu_\phi$ (pointing in the positive $Y$-direction). The magenta arrow illustrates the null transverse vector $N^\mu$ for the incoming side (associated with metric $g^{(+)}$, directed along the negative radial direction). The cyan arrow shows $N^\mu$ for the outgoing side (associated with metric $g^{(-)}$, directed along the positive radial direction). The yellow dashed arrow symbolizes the null tangent vector $k^\mu$ (or $\ell^\mu$ in alternative notation), aligned with the temporal direction and pointing in the positive $Z$-direction (represented as dashed to emphasize its projected, non-spatial nature in this Euclidean visualization). This basis, as detailed in Section \ref{['sec:section3']}, satisfies conditions $k^\mu k_\mu = 0$, $N^\mu N_\mu = 0$, $k^\mu N_\mu = 1$, $k^\mu e^\nu_A = 0$, and $N^\mu e^\nu_A = 0$ (where $A = \{\theta, \phi\}$), facilitating the computation of the transverse curvature tensor and the invariant null–shell decomposition of the surface stress-energy tensor $S^{\alpha\beta}$ using the Barrabès-Israël formalism reformulated by Poisson Poisson2002Reformulation. Note that this is a simplified Euclidean projection for visualization. The actual geometry is embedded in the $4D$ spacetime with $\mathcal{PT}$ symmetry.
• Figure 2: Echo delays as a function of the throat proximity parameter $\epsilon=\ell/M$ with $r_0=2M+\ell$, for a mass $M=30\,M_{\odot}$. The solid curve shows the arrival time of the first late-time response ($\approx L_{\rm cav}/c$) and the dashed curve shows the spacing between successive echoes ($\approx 2L_{\rm cav}/c$). The logarithmic behaviour in the near–black-hole limit ($\epsilon\ll 1$) is recovered, consistent with Cardoso2016.
• Figure 3: Conceptual schematic of the cavity in the tortoise coordinate $r_*$ between the outer barrier near the light ring ($r=3M$) and the throat located at $r_0=\alpha$. The two peaks depict the effective potential in $r_*$. The optical length $L_{\rm cav}=|r_*(3M)-r_*(r_0)|$ sets the characteristic spacing between successive echoes, $\Delta t_{\rm echo}\simeq 2L_{\rm cav}/c$. Arrows indicate multiple partial reflections that generate the late-time echo train.
• Figure 4: Schematic view of the tortoise coordinate $r_*$ showing the location of the throat $r_0=2M+\ell$, the photon sphere $r=3M$, and the effective cavity of length $L_{\rm cav}$ trapping gravitational waves.

Theorems & Definitions (2)

• proof
• proof : Frequency–parity argument at the throat