Analytical solution of traversable wormholes in the presence of positive cosmological constant

This paper presents an analytical solution for traversable wormholes with a positive cosmological constant using the gravitational decoupling method, revealing a configuration with both a standard and a cosmological throat that satisfies the flare-out condition and allows for safe human passage despite violating the null energy condition.

The Gist

Imagine the universe as a giant, stretchy fabric. For a long time, physicists have been fascinated by the idea of "wormholes"—tunnels through this fabric that could connect two distant points, like a shortcut through a mountain instead of driving around it.

However, building a stable tunnel is tricky. In the real world, gravity usually tries to crush things together. To keep a wormhole open, you need something that pushes back, a kind of "anti-gravity" force. Usually, this requires exotic, imaginary matter that doesn't behave like normal stuff.

This paper is about a team of physicists who asked a specific question: What happens to these wormhole tunnels if the universe itself is pushing outward?

The Setting: An Expanding Universe

We know the universe isn't just sitting still; it's expanding, and that expansion is speeding up. Scientists describe this push as a "positive cosmological constant" (think of it as a gentle, universal wind blowing outward).

The researchers wanted to see how this "universal wind" affects the shape of a wormhole. They didn't want to start from scratch; instead, they took a known, stable wormhole design (called the Ellis–Bronnikov wormhole) and asked, "How does this design change if we add the universe's expansion wind to the mix?"

The Method: The "Deformation" Trick

To solve this, they used a mathematical tool called Gravitational Decoupling.

Think of it like this: Imagine you have a perfectly round, flat balloon (the original wormhole). Now, you want to see what happens if you blow a steady stream of air at it. Instead of trying to calculate the complex physics of the air hitting the rubber all at once, you treat the air as a separate "layer" that gently stretches and reshapes the balloon.

The researchers did exactly this. They took their flat balloon (the wormhole) and applied the "wind" (the cosmological constant) as a separate force. They calculated exactly how the balloon's shape would stretch and warp under this pressure.

The Big Discovery: Two Doors, Not One

The most surprising result was that the wormhole didn't just get bigger or smaller; it changed its structure entirely.

1. The Original Door (The Inner Throat): The wormhole still has its original tunnel entrance, but it's slightly different than before.
2. The New Door (The Cosmological Throat): Because of the outward "wind" of the universe, a second tunnel entrance appeared much further out.

Imagine a tunnel that starts in a cave, goes through a mountain, and then suddenly opens up into a vast, open field. In this new model, the wormhole has an "inner throat" (the cave) and an "outer throat" (the edge of the field). You can pass through both, but they are separated by a region of space that is being stretched by the universe's expansion.

Is It Safe to Travel?

A wormhole is useless if it collapses on you or rips you apart. The researchers checked two main safety factors:

• The "Flare-Out" Condition: This is a fancy way of saying, "Does the tunnel stay open?" They confirmed that at both the inner and outer doors, the tunnel flares open wide enough to let things through. It doesn't pinch shut.
• Tidal Forces (The "Spaghettification" Test): When you go through a wormhole, gravity can stretch you like spaghetti. The team calculated the forces a human traveler would feel. They found that if you travel at a reasonable speed (not too fast, not too slow), the stretching forces are manageable—similar to the gravity we feel on Earth. You wouldn't be ripped apart.

The Catch: It's Not a Perfect "De Sitter" Universe

There is one small twist. Usually, when you add this "universal wind," you expect the space to look like a standard "De Sitter" universe (a specific mathematical model of an expanding universe).

However, this wormhole solution is a bit unique. It behaves like a De Sitter universe in the middle, but as you get very far away, it doesn't quite match the standard textbook definition of that universe. It's a "modified" version. The researchers note that while it's not a perfect textbook match, it is a valid, stable, and traversable tunnel.

Summary

In simple terms, this paper shows that if you take a theoretical wormhole and place it in our real, expanding universe, it doesn't break. Instead, it evolves. It gains a second "exit" far away, creating a double-door structure. As long as you travel at a safe speed, you could theoretically walk through this tunnel without being crushed, proving that such cosmic shortcuts might be mathematically possible even in our expanding universe.

Technical Summary

Technical Summary: Analytical Solution of Traversable Wormholes in the Presence of Positive Cosmological Constant

Problem Statement
The construction of traversable wormhole (WH) solutions faces a central challenge: sustaining the throat against collapse requires matter sources that violate the Null Energy Condition (NEC). While analytical solutions exist for asymptotically flat WHs (e.g., the Ellis–Bronnikov solution supported by phantom scalar fields), the introduction of a non-vanishing, positive cosmological constant ($\Lambda > 0$) significantly modifies spacetime geometry and asymptotic behavior. Previous attempts to incorporate $\Lambda$ often relied on matching asymptotically flat regions to Schwarzschild–de Sitter (dS) black hole spacetimes or introducing matter shells. This work addresses the need for a systematic analytical approach to derive a traversable WH solution deformed by a positive cosmological constant without resorting to numerical methods or ad-hoc matching procedures.

Methodology: Gravitational Decoupling (GD)
The authors employ the Gravitational Decoupling (GD) method to generate the solution. The procedure involves:

1. Seed Solution: Selecting the static, spherically symmetric, asymptotically flat Ellis–Bronnikov WH geometry as the seed. This solution is supported by a phantom-like scalar field with negative kinetic energy.
2. Decoupling Framework: Treating the cosmological constant $\Lambda$ as an effective source term ($S_{\mu\nu}$) added to the Einstein equations. The total energy-momentum tensor is split into the seed matter sector ($T_{\mu\nu}$) and the $\Lambda$-induced sector ($\alpha S_{\mu\nu}$), where $\alpha$ is a counting parameter.
3. Metric Deformation: Assuming a minimal modification to the seed metric functions (redshift function $\xi(r)$ and shape function $\mu(r)$) via deformation functions $\eta_1(r)$ and $\eta_2(r)$. The deformed metric takes the form:
   $$\nu(r) = \xi(r) + \alpha\eta_1(r), \quad e^{-\sigma(r)} = e^{-\mu(r)} + \alpha\eta_2(r)$$
4. Solving the Quasi-Einstein Equations: By linearizing the Einstein equations with respect to $\alpha$, the authors derive a set of differential equations governing the deformation functions. These equations are solved analytically to determine the specific form of the deformed metric in the presence of $\Lambda$.

Key Results
The application of the GD method to the Ellis–Bronnikov seed yields a new analytical metric describing a traversable WH in a de Sitter-like background:
$$ds^2 = -e^{-\frac{\Lambda}{3}r^2 - \frac{\Lambda r_0^2}{3}\ln|r^2-r_0^2|}dt^2 + \frac{dr^2}{1 - \frac{r_0^2}{r^2} - \frac{r^2\Lambda}{3}} + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$
where $r_0$ is the throat radius of the original flat seed.

• Dual Throat Structure: The presence of $\Lambda$ modifies the shape function, leading to the emergence of two distinct throat positions, $r_{th}^{\pm}$, determined by the roots of the shape function.
  • $r_{th}^-$: A smaller-radius throat corresponding to the standard WH geometry.
  • $r_{th}^+$: A larger-radius "cosmological throat."
  • Both throats exist provided $\Lambda \leq \Lambda_{max} = 3/(4r_0^2)$.
• Flare-out Condition: The authors verify that the flare-out condition (minimality of the throat) is satisfied in the vicinity of both $r_{th}^-$ and $r_{th}^+$.
• Energy Conditions: The analysis confirms that the Null Energy Condition (NEC) is violated in the vicinity of both throats. This violation is attributed to the phantom scalar field from the seed solution, as the $\Lambda$ term vanishes when contracted with null vectors.
• Asymptotic Behavior: The resulting spacetime is not asymptotically de Sitter in the standard sense. While the radial metric component diverges at $r = \sqrt{3/\Lambda}$ (similar to the static patch of dS space), the lapse function ($g_{tt}$) vanishes exponentially rather than linearly, indicating this surface is not a standard cosmological Killing horizon.
• Traversability Analysis:
  • Tidal Forces: The tidal accelerations (radial and lateral) acting on a traveler are calculated and found to be finite throughout the region between the two throats.
  • Velocity Constraints: By imposing the condition that tidal forces must not exceed Earth's gravitational acceleration ($g_\oplus$), the authors derive constraints on the traveler's velocity. Interestingly, the presence of $\Lambda$ relaxes the lateral tidal constraints near the throat compared to the asymptotically flat case, allowing for higher traversal velocities.
• Geodesic Motion: The affine parameter for massless particles is shown to be finite and integrable at the throat, confirming that null geodesics can smoothly cross the throat.

Significance and Claims
The paper claims to provide a systematic analytical framework for studying WH deformations induced by a positive cosmological constant. By utilizing the GD method, the authors successfully derive a solution that preserves spherical symmetry and avoids the need for numerical integration.

The primary significance lies in the demonstration that a positive $\Lambda$ naturally induces a dual-throat structure (a standard WH throat and a cosmological throat) while maintaining traversability. The authors note that while the solution is traversable and satisfies necessary physical conditions (flare-out, NEC violation, finite tidal forces), the deviation from standard asymptotic dS behavior suggests that the direct application of GD to obtain a fully consistent dS WH solution may require further refinement, such as matching the exterior to a consistent dS region.

The work concludes that the developed methodology can be extended to other asymptotically flat, static, spherically symmetric WH solutions (e.g., generalized Ellis–Bronnikov configurations), though solutions with non-vanishing redshift functions (like Damour–Solodukhin WHs) may present additional integration challenges.