⚛️ general relativity

Holographic Dark Energy as a Source for Wormholes in Modified Gravity

This paper explores traversable wormhole solutions in $f(\mathcal{R},\mathbb{T})$ gravity supported by Rényi, Moradpour, and Bekenstein–Hawking holographic dark energy models, demonstrating that while the resulting shape functions satisfy geometric traversability criteria, the null energy condition is inevitably violated, necessitating exotic matter or an effective exotic sector.

Original authors: G. G. L. Nashed, A. Eid

Published 2026-02-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: G. G. L. Nashed, A. Eid

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, stretchy trampoline. Usually, we think of gravity as something that pulls things down, creating dips in the fabric. But what if you could fold that trampoline over and stitch two distant points together, creating a shortcut? That's the basic idea of a wormhole: a tunnel through space that connects two faraway places, potentially allowing for travel that would otherwise take eons.

For decades, scientists have said, "You can't build a wormhole." Why? Because to keep the tunnel open, you need a special kind of "anti-gravity" material (called exotic matter) that pushes outward instead of pulling inward. In standard physics, this stuff doesn't exist, or at least, we haven't found enough of it.

This paper by Nashed and Eid asks a new question: What if we change the rules of gravity itself?

Here is a simple breakdown of their findings:

1. The New Rulebook: f(R, T) Gravity

Standard gravity (Einstein's General Relativity) is like a very strict recipe book. The authors decided to tweak the recipe. They introduced a new theory called f(R, T) gravity.

The Analogy: Think of standard gravity as a car that runs only on gasoline. This new theory is like a hybrid car that can run on gasoline and electricity. The "electricity" part comes from a new interaction between the shape of space (curvature) and the stuff inside it (matter).
The Result: By adding this extra "electricity" to the engine, the theory creates a natural repulsive force. This means you might not need as much of that impossible "exotic matter" to keep the wormhole open. The modified gravity itself does some of the heavy lifting.

2. The Fuel: Holographic Dark Energy

To build their wormhole, the authors needed a specific type of fuel. They didn't just pick random fuel; they used three different recipes based on Holographic Dark Energy.

The Analogy: Imagine you are trying to keep a balloon inflated. You have three different types of air pumps:
1. The Rényi Pump: A complex pump that adjusts its pressure based on how "fractal" or jagged the balloon's surface is.
2. The Moradpour Pump: A pump that changes its behavior based on how the air molecules interact with each other in a non-standard way.
3. The Bekenstein-Hawking Pump: The classic pump, based on the surface area of the balloon.
The authors tested all three. They found that all three could successfully inflate the wormhole tunnel, but they behaved slightly differently depending on how far you were from the center.

3. The Shape of the Tunnel

The paper calculates the exact shape of these tunnels (called shape functions).

The Flare-Out: For a wormhole to work, the throat (the narrowest part) must be wide enough to pass through, and it must flare out like a trumpet bell as you move away from the center. The authors showed that with their new gravity rules and the holographic fuel, the tunnel naturally flares out correctly.
The Flatness: For the Rényi and Moradpour models, the tunnel eventually smooths out and becomes flat again, just like normal space. However, the Bekenstein-Hawking model creates a tunnel that never quite flattens out completely; it stays slightly curved forever. The authors suggest this is fine if you only care about the "inside" of the tunnel and plan to connect it to normal space at a specific point.

4. The "Exotic Matter" Problem

The biggest hurdle for wormholes is the Null Energy Condition (NEC). In simple terms, this is a rule that says "energy density plus pressure must be positive." To keep a wormhole open, you usually have to break this rule (which requires exotic matter).

The Finding: The authors found that in their new theory, the rule is still broken near the center of the wormhole (you still need some "weird" physics there). However, the modified gravity helps significantly. It reduces the amount of "weirdness" needed compared to standard gravity. It's like needing a smaller amount of a rare, expensive ingredient to bake a cake because your oven (the gravity theory) is more efficient.

5. Keeping the Balance (Stability)

A wormhole is useless if it collapses the moment you step inside. The authors checked if these tunnels are stable using a concept called the TOV equation (which is like checking if a bridge is in balance).

The Forces: They looked at three forces:
1. Gravity: Trying to crush the tunnel.
2. Pressure: Trying to push it apart.
3. The New Force: A unique force created by the interaction between matter and the new gravity rules.
The Result: They found that these forces cancel each other out perfectly. The tunnel is in hydrostatic equilibrium. It's like a tightrope walker who is perfectly balanced; the forces pulling left and right are equal, so the walker (the wormhole) stays stable.

6. What Would You See? (Gravitational Lensing)

If you looked at one of these wormholes from far away, what would you see?

The Analogy: Just like a black hole bends light around it, a wormhole would act like a cosmic magnifying glass. Light from stars behind it would bend around the tunnel.
The Finding: The authors calculated exactly how much the light would bend. They found that the wormhole acts like an unstable "photon sphere"—a ring where light can orbit the tunnel. If you were close enough, you might see multiple images of the same star, distorted and wrapped around the tunnel's mouth.

Summary

In plain English, this paper says:
"If we update our understanding of gravity to include a specific new interaction, and we use a specific type of 'holographic' energy to fuel it, we can mathematically build stable wormholes. These wormholes don't require as much impossible exotic matter as before, they stay balanced, and they would bend light in predictable ways. While we can't build one yet, the math says it's possible within this new framework."

Technical Summary: Holographic Dark Energy as a Source for Wormholes in Modified Gravity

Problem Statement
Traversable wormholes are theoretical spacetime geometries connecting distant regions, yet their existence within General Relativity (GR) is hindered by the requirement of "exotic matter"—a fluid that violates the Null Energy Condition (NEC). This violation is necessary to satisfy the "flare-out" condition at the wormhole throat. To address this, researchers have turned to modified gravity theories, where the effective energy-momentum tensor can mimic exotic behavior without necessarily relying on hypothetical fluids. This paper investigates whether traversable wormhole solutions can be constructed within the framework of $f(R, T)$ gravity (a curvature-matter coupling theory) when supported by specific Holographic Dark Energy (HDE) density profiles derived from entropy formalisms (Rényi, Moradpour, and Bekenstein-Hawking).

Methodology
The authors employ a linear realization of $f(R, T)$ gravity, defined by the functional form $f(R, T) = R + \beta T$ , where $R$ is the Ricci scalar, $T$ is the trace of the energy-momentum tensor, and $\beta$ is a constant parameter. The study utilizes a static, spherically symmetric metric characterized by a shape function $s(r)$ and a constant redshift function $\zeta(r)$ .

The methodology proceeds as follows:

Field Equations: The modified Einstein field equations are derived for the linear $f(R, T)$ model. These equations relate the geometry (shape function) to the matter sector (energy density $\rho$ , radial pressure $p_r$ , and tangential pressure $p_t$ ).
HDE Models: Three distinct holographic dark energy density profiles are prescribed based on different entropy formalisms:
- Rényi HDE: Derived from Rényi entropy, incorporating a parameter $\alpha_1$ to account for non-local interactions.
- Moradpour HDE: A modification of the Bekenstein-Hawking entropy based on Tsallis statistics, introducing a specific correction to the area law.
- Bekenstein-Hawking HDE: The standard holographic model where energy density scales inversely with the horizon area.
Solution Construction: By equating the prescribed HDE densities with the field equations, the authors analytically derive the corresponding shape functions $s(r)$ for each model.
Analysis: The solutions are tested against:
- Geometric Constraints: Asymptotic flatness ( $s(r)/r \to 0$ as $r \to \infty$ ) and the flare-out condition ( $s'(r_0) < 1$ at the throat $r_0$ ).
- Energy Conditions: Evaluation of the Null (NEC), Weak (WEC), Dominant (DEC), and Strong (SEC) energy conditions.
- Stability: Application of the generalized Tolman-Oppenheimer-Volkoff (TOV) equation to verify hydrostatic equilibrium by balancing gravitational, hydrostatic, anisotropic, and matter-geometry coupling forces.
- Observables: Calculation of gravitational lensing effects, specifically the deflection angle and critical impact parameter, using the strong-deflection limit.

Key Contributions and Results

Analytical Solutions: The paper successfully derives explicit analytical expressions for the shape functions $s(r)$ and the associated pressure components for all three HDE models within the linear $f(R, T)$ framework.
Geometric Viability:
- The Rényi and Moradpour models yield shape functions that satisfy asymptotic flatness and the flare-out condition, representing valid traversable wormhole geometries.
- The Bekenstein-Hawking model yields a shape function that is not asymptotically flat ( $s(r)/r \to A_\infty \neq 0$ ) for non-trivial parameters. The authors interpret this solution as a local wormhole interior that would require matching to an exterior vacuum at a finite radius.
Energy Condition Violation: In all three models, the radial Null Energy Condition (NEC), defined as $\rho + p_r \geq 0$ , is violated near the throat. This confirms that exotic matter (or an effective exotic sector) is still required to sustain the wormhole geometry. However, the tangential NEC ( $\rho + p_t \geq 0$ ) is satisfied. The study notes that the modified gravity framework helps regulate the distribution of this exotic matter.
Stability Analysis: The generalized TOV equation is satisfied for all models. The analysis demonstrates that the hydrostatic force, anisotropic force, and the new force arising from the matter-geometry coupling ( $F_c$ ) balance each other perfectly ( $\sum F = 0$ ), indicating that the wormhole configurations are in dynamic equilibrium.
Gravitational Lensing: The authors calculate the deflection angle for photons passing near the throat. They find that the throat acts as an unstable photon sphere. The deflection angle follows a universal form dependent on the second derivative of the shape function at the throat, providing a potential observational signature to distinguish these wormholes from black holes.
Exotic Matter Reduction: A volume integral quantifier for the Averaged Null Energy Condition (ANEC) is computed. The results indicate that the amount of exotic matter required in the $f(R, T)$ framework is reduced compared to standard GR (where $\beta \to 0$ ), with reductions ranging from approximately 26% to 32% depending on the specific HDE model.

Significance and Claims
The paper claims to demonstrate that $f(R, T)$ gravity, when coupled with holographic dark energy formalisms, provides a viable and physically consistent framework for constructing traversable wormholes. The primary significance lies in showing that while the NEC violation (exotic matter requirement) is intrinsic to wormhole physics, the specific entropy-based HDE models combined with curvature-matter coupling can:

Satisfy the geometric requirements for traversability (flare-out and asymptotic flatness for two of the models).
Ensure dynamical stability through force balance.
Reduce the total volume of exotic matter required compared to standard GR solutions.

The authors conclude that these findings suggest the "exotic matter" problem may be less restrictive than previously believed when viewed through the lens of modified gravity and holographic principles, offering a theoretical basis for stable wormhole configurations without relying solely on unphysical fluid assumptions.