⚛️ general relativity

Lorentz Symmetry Breaking Traversable Wormhole Models Supported by Einasto Dark Matter

This paper proposes and analyzes static, spherically symmetric traversable wormhole solutions in Kalb-Ramond gravity supported by Einasto dark matter, demonstrating that the required Lorentz symmetry breaking and exotic matter can be localized near the throat while satisfying geometric, energetic, and observational constraints.

Original authors: Rana Muhammad Zulqarnain, M. Yousaf, Allah Ditta, Asifa Ashraf, Ahmadjon Abdujabbarov, Farruh Atamurotov

Published 2026-08-13

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CC BY 4.0

Original authors: Rana Muhammad Zulqarnain, M. Yousaf, Allah Ditta, Asifa Ashraf, Ahmadjon Abdujabbarov, Farruh Atamurotov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, flexible trampoline. In the old days, scientists thought gravity was just an invisible rope pulling things together. But then, Einstein came along and said, "Nope, it's actually the trampoline itself curving." Massive objects like stars and planets sit on this cosmic trampoline, bending it so deeply that other things roll toward them. This bending is what we feel as gravity. But what if you could fold that trampoline over and poke a hole through it, connecting two distant points instantly? That's the wild idea of a "wormhole"—a theoretical tunnel through space and time.

However, there's a catch. For a wormhole to stay open and not collapse instantly, you need something strange to hold the walls apart. Think of it like trying to keep a tunnel in a sandcastle from caving in; you'd need a magical, repulsive force pushing the sand outward. In physics, this requires "exotic matter," a substance with negative energy that we haven't found in the real world yet. Most scientists think this is just a cool math problem, but some are trying to figure out if the universe might have a natural way to create these tunnels, perhaps using the invisible "dark matter" that holds galaxies together or by tweaking the rules of gravity itself.

This paper dives into that exact puzzle. The authors, a team of researchers from universities in China, Pakistan, Thailand, and beyond, ask: "What if we combine two specific ideas to build a stable wormhole?" First, they look at a modified version of gravity called "Kalb-Ramond gravity," which suggests that the fabric of space might have a hidden, broken symmetry (like a spinning top that suddenly decides to tilt). Second, they fill this tunnel with a specific recipe for dark matter called the "Einasto profile," which is a mathematical way to describe how dark matter is spread out in galaxies.

The team didn't build a physical wormhole in a lab (that would be impossible right now). Instead, they used complex math to simulate what would happen if these two ingredients were mixed. They found that, mathematically, it is possible to create a traversable wormhole—a tunnel you could theoretically fly through without hitting a wall. Their calculations show that the "exotic matter" needed to keep the tunnel open doesn't have to fill the entire universe; it can be concentrated in a small, specific zone right at the throat of the wormhole. They also checked how light would bend around this tunnel and how long a signal would take to bounce back, finding that the shape of the tunnel depends heavily on the specific "recipe" of dark matter used. While this is still just a theoretical model and not a blueprint for building a time machine, it suggests that if the universe does play by these specific modified rules of gravity and dark matter, stable wormholes might be a natural possibility rather than just a science fiction dream.

Technical Summary: Lorentz Symmetry Breaking Traversable Wormhole Models Supported by Einasto Dark Matter

Problem Statement
The construction of traversable wormholes (WHs) within General Relativity (GR) typically requires "exotic matter" that violates the Null Energy Condition (NEC), specifically near the throat. While quantum field theory allows for localized negative energy densities, macroscopic astrophysical WHs remain elusive due to the lack of known matter sources satisfying the necessary tension-to-density ratios. Furthermore, standard GR models often rely on idealized matter distributions. This paper addresses the challenge of constructing physically viable, static, and spherically symmetric traversable WHs by combining two distinct theoretical frameworks: a modified gravity theory involving spontaneous Lorentz symmetry breaking (Kalb-Ramond gravity) and a realistic dark matter density profile (the Einasto profile). The goal is to determine if the interplay between these mechanisms can support a traversable geometry where exotic matter is localized and the solution remains geometrically and physically consistent.

Methodology
The authors employ the Kalb-Ramond (KR) gravity framework, where a non-vanishing vacuum expectation value (VEV) of an antisymmetric tensor field induces spontaneous Lorentz symmetry breaking. The gravitational action includes a non-minimally coupled KR field with parameters $\xi_2$ and $\xi_3$ , leading to modified Einstein Field Equations (EFEs).

Metric and Field Equations: The study utilizes the Morris-Thorne metric for a static, spherically symmetric spacetime. The authors assume a constant redshift function ( $\Phi(r) = \text{constant}$ ) to eliminate tidal forces. The modified field equations are derived, incorporating the energy-momentum tensor (EMT) contributions from both the Lorentz-violating KR vacuum and the matter sector.
Matter Source: The matter sector is modeled using the Einasto dark matter density profile, $\rho(r) = \rho_s e^{-(r/r_s)^{1/n}}$ , where $\rho_s$ is the characteristic density, $r_s$ is the scale radius, and $n$ is the Einasto index. This profile is chosen for its empirical success in describing galactic halos.
Shape Function Derivation: By equating the modified EFEs with the Einasto density profile, the authors derive an analytical expression for the wormhole shape function, $Y(r)$ . This solution involves the incomplete Gamma function and is constrained by the throat condition $Y(r_0) = r_0$ .
Analysis Framework: The resulting geometry is subjected to a comprehensive analysis including:
- Geometric Viability: Checking the throat condition, flare-out condition ( $Y'(r_0) < 1$ ), and asymptotic flatness ( $Y(r)/r < 1$ ).
- Embedding Diagrams: Visualizing the spatial curvature via 2D and 3D embedding surfaces.
- Energy Conditions: Evaluating the Weak (WEC), Strong (SEC), Null (NEC), and Dominant (DEC) energy conditions to identify regions of exotic matter.
- Structural Complexity: Calculating the complexity factor ( $Y_{TF}$ ) to assess the internal structural organization of the fluid.
- Equilibrium: Analyzing the balance of forces using the generalized Tolman-Oppenheimer-Volkoff (TOV) equation.
- Optical Observables: Investigating the photon sphere, critical impact parameter, light deflection angle, and echo time delay.
- Exotic Matter Quantification: Using the Volume Integral Quantifier (VIQ) to estimate the total amount of NEC-violating matter.

Key Contributions and Results

Analytical Shape Function: The paper successfully derives an analytical shape function $Y(r)$ supported by the Einasto profile within Kalb-Ramond gravity. The solution depends on the KR gravitational parameter $\lambda$ (related to the VEV and coupling constants) and the Einasto parameters ( $\rho_s, r_s, n$ ).
Geometric Viability: The derived shape function satisfies all necessary geometric criteria for a traversable WH. The flare-out condition is met at the throat, and the geometry is asymptotically flat. Embedding diagrams confirm the characteristic two-sheeted structure with a smooth throat.
Localization of Exotic Matter: The analysis of energy conditions reveals that while the energy density $\rho$ remains positive, the radial null energy condition ( $\rho + P_{rad} \geq 0$ ) is violated in the vicinity of the throat. Crucially, this violation is localized; as the radial distance increases, the exotic contribution diminishes, and standard energy conditions are satisfied in the outer regions.
Complexity Factor: The complexity factor $Y_{TF}$ is found to be most pronounced near the throat, driven by anisotropic pressure and density inhomogeneity. It gradually approaches zero at larger distances, indicating that the spacetime becomes structurally simpler away from the strong-field region.
Gravitational Energy and Equilibrium: The total gravitational energy is calculated to be positive, suggesting a repulsive gravitational character that helps counteract collapse. Equilibrium analysis via the TOV equation shows that, under the constant redshift assumption, the gravitational force vanishes, and stability is maintained by the counterbalancing of hydrostatic and anisotropic forces. The pressure anisotropy is positive, providing an outward-directed force.
Optical Properties: The study computes the photon sphere radius, critical impact parameter, light deflection angle, and echo time. These observables are shown to be sensitive to the Einasto scale radius $r_s$ and the KR parameter $\lambda$ , offering potential observational signatures for distinguishing such WHs from black holes.
Quantification of Exotic Matter: The Volume Integral Quantifier (VIQ) confirms that the total amount of NEC-violating matter is finite. The analysis demonstrates that the magnitude of this exotic matter can be minimized and controlled by adjusting the Einasto scale radius and the KR parameter.

Significance
The paper claims that the combination of Lorentz symmetry breaking (via the Kalb-Ramond field) and a realistic dark matter distribution (Einasto profile) provides a unified setting for constructing traversable wormholes that are geometrically consistent and physically plausible. The primary significance lies in demonstrating that the required exotic matter does not need to permeate the entire spacetime but can be confined to a restricted region near the throat. Furthermore, the study highlights how the Kalb-Ramond gravitational parameter and the characteristics of the dark matter halo jointly determine the geometric, energetic, and optical properties of the wormhole. This work contributes to the broader effort of identifying viable astrophysical candidates for traversable wormholes by moving beyond idealized matter models and incorporating modified gravity effects.

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