Short-range approximation to Casimir wormholes inspired by scalar and electric fields

This paper investigates static traversable wormholes sustained by a combination of minimally coupled scalar and electric fields with Casimir energy as exotic matter, deriving analytical near-throat solutions for both variable and fixed plate separations that yield well-behaved manifolds with throat sizes scaling proportionally to the supported electric charge.

The Gist

Imagine the universe as a giant, stretchy fabric. Usually, this fabric is flat or curves gently around heavy objects like stars. But what if you could fold this fabric so that two distant points touch, creating a shortcut? That's a wormhole.

The problem is that to keep this shortcut open, you need something weird. In physics, we call this "exotic matter." It's a substance that pushes outward (negative pressure) instead of pulling inward like normal gravity. Since we haven't found this "exotic matter" in nature yet, scientists have to get creative.

This paper by Garattini and Tzikas is like a recipe book for building a wormhole using ingredients we do know exist, but mixing them in a very specific, theoretical way.

The Three Ingredients

The authors try to build a wormhole using a "smoothie" of three distinct physical effects:

1. The Casimir Effect (The Negative Energy): Imagine two very smooth, flat metal plates floating in a vacuum. Quantum physics tells us that even in empty space, there is a "buzz" of energy. When you put these plates very close together, they squeeze out some of that energy, creating a pressure that pushes the plates together. This creates a region of negative energy. The authors use this as their primary "exotic" ingredient to hold the wormhole open.
2. The Electric Field (The Charge): Think of this as the static electricity on a balloon. The wormhole isn't just empty space; it's charged. The more electric charge (specifically, elementary charges like electrons) the wormhole holds, the bigger the shortcut becomes.
3. The Scalar Field (The Invisible Glue): This is a bit like an invisible field that permeates space. The authors test two versions: one that is "massless" (like a wave with no weight) and one that is "massive" (like a wave with a heavy payload). This field helps balance the equations so the math works out.

The "Thermal" Safety Net

When the authors mixed these three ingredients, the math didn't quite balance perfectly. The pressures were getting out of whack. To fix this, they added a fourth, invisible ingredient called a thermal tensor.

Think of this like a shock absorber in a car. It doesn't add weight to the car (it doesn't add energy density), but it adjusts the pressure to make the ride smooth. In their model, this "thermal" part acts as a pressure adjuster that vanishes right at the center of the wormhole (the throat), ensuring the structure doesn't collapse.

Two Ways to Build the Plates

The authors tested two different construction methods for the "Casimir" part (the metal plates):

• Variable Plates: Imagine the distance between the plates changes as you move along the wormhole.
• Fixed Plates: Imagine the plates are stuck at a specific, unchanging distance.

The Big Discovery: Size Matters

The most exciting result of their recipe is the relationship between charge and size.

They found that the size of the wormhole's opening (the throat) is directly tied to how much electric charge it can hold.

• The Analogy: Think of the wormhole like a balloon. The more air (electric charge) you blow into it, the bigger it gets.
• The Result: If the wormhole holds a few elementary charges, the opening is tiny (sub-atomic). But if it holds a lot of charges, the opening gets larger. They calculated that the size scales proportionally with the number of charges.

Does it Work?

In physics, to keep a wormhole open, you have to break a rule called the Null Energy Condition (NEC). This rule basically says "energy density plus pressure must be positive." To keep a wormhole open, you need it to be negative.

The authors found that in all their scenarios, the radial pressure (the pressure pushing along the length of the tunnel) successfully broke this rule, keeping the tunnel open. However, the "sideways" pressure (tangential) sometimes behaved normally. This means the wormhole is stable, but only because of that specific push-and-pull along the tunnel's length.

The Bottom Line

This paper doesn't say we can build a wormhole tomorrow. Instead, it says: "If we combine Casimir energy, electric charge, and a scalar field in a very specific mathematical way, we can create a stable, theoretical wormhole."

It's a proof of concept that shows how the universe's known forces could theoretically conspire to create a shortcut through space, with the size of that shortcut being controlled by how much electric charge you pack into it. The "thermal" part is just the mathematical glue needed to make the whole structure hold together without falling apart.

Technical Summary

Technical Summary: Short-range approximation to Casimir wormholes inspired by scalar and electric fields

Problem Statement
Traversable wormholes (TWs) generally require exotic matter to satisfy the geometric "flare-out" condition, a requirement that violates the Null Energy Condition (NEC). While the Casimir effect is a known candidate for providing the necessary negative energy density, previous studies have largely examined Casimir wormholes in isolation or combined with only one other field (e.g., scalar or electric). This paper addresses the gap in understanding how a traversable wormhole can be sustained by the simultaneous interaction of three distinct physical sources: the Casimir effect, a static electric field, and a scalar field (both massless and massive). The authors aim to determine if these sources can coexist to form a consistent solution to the Einstein Field Equations (EFE) without introducing ad hoc exotic matter components.

Methodology
The authors analyze a static, spherically symmetric spacetime described by the standard wormhole metric with a redshift function $\Phi(r)$ and a shape function $b(r)$. The investigation focuses on a "near-throat" approximation, where the shape function is linearized as $b(r) = r_0 + B(r - r_0)$ with $B < 1$ to satisfy the flare-out condition.

The total Stress-Energy Tensor (SET) is decomposed into four components:

1. Casimir Source: Modeled either with radially variable plate separation or with plates fixed at a parametric distance $d$.
2. Scalar Field: Analyzed in two regimes: massless (kinetic energy only) and massive (including a potential $V(\psi)$).
3. Electric Field: Represented by a charge $Q = Ne$, where $N$ is the number of elementary charges.
4. Thermal Tensor: A relativistic thermodynamic contribution consisting solely of pressure terms ($\tau_r, \tau_t$) with zero energy density, introduced to ensure the consistency of the field equations.

The authors solve the EFEs and the conservation of the SET for both massless and massive scalar fields under the two Casimir configurations. They impose the condition that the thermal pressures vanish at the throat ($r=r_0$) to minimize the thermal contribution and derive analytical expressions for the throat radius $r_0$, the redshift function, and the constraints on the parameters.

Key Contributions and Results

• Analytical Near-Throat Solutions: The paper derives explicit analytical solutions for the wormhole geometry in all four scenarios (massless/massive $\times$ variable/fixed plates).
• Role of the Thermal Tensor: The study demonstrates that a thermal tensor with vanishing energy density but non-zero pressure is necessary to balance the field equations. Crucially, the authors show that this tensor vanishes at the throat, acting as a backreaction of the geometry rather than an independent energy source.
• Throat Size and Charge Dependence: A primary result is the derivation of the throat radius $r_0$ as a function of the number of elementary charges $N$.
  • For the massless scalar field with variable plates, the throat size scales proportionally with $N$ (specifically $r_0 \propto N$ for $N \ge 7$).
  • For the massless scalar field with fixed plates, a specific relation between the plate separation $d$ and the charges is required ($d = 2\sqrt{r_1 r_2}$), yielding $r_0 \approx 0.27 N \ell_P$.
  • For the massive scalar field, the throat size is also found to be proportional to $N$ (e.g., $r_0 = \ell_P \sqrt{\frac{\pi^3}{45} + \frac{2N^2}{137}}$ for variable plates).
• NEC Violation: In all cases, the Null Energy Condition is violated exclusively through the radial pressure component ($\rho + p_r < 0$), which is a requisite for maintaining the wormhole. The tangential pressure generally satisfies the NEC, except in specific limited ranges of $N$ for the massive field cases.
• Exotic Matter Quantification: The total amount of exotic matter required is calculated via a volume integral of the NEC-violating term. The authors find this quantity depends logarithmically on the external boundary $\bar{r}$ and is determined by the geometric parameter $B$. As $B \to 1$, the amount of exotic matter decreases.

Significance and Claims
The paper claims to provide the first investigation into the combined effects of Casimir energy, electric fields, and scalar fields in sustaining a traversable wormhole. The authors emphasize that their model does not rely on phantom fields or ad hoc exotic matter sources; instead, the wormhole is supported by a well-defined physical matter sector comprising three distinct contributions.

Key claims regarding the model's validity include:

• Physical Consistency: The solutions satisfy the Einstein Field Equations entirely within the defined physical matter content, with the thermal tensor serving only to balance pressures.
• Scalability: The results establish a clear, proportional relationship between the wormhole's throat size and the accumulated electric charge, suggesting that the geometry can be "tuned" by the number of charges.
• Limitations: The authors modestly note that their analysis is restricted to a short-range approximation near the throat. They acknowledge that while the thermal tensor vanishes at the throat, it does not necessarily vanish at the external boundary, and the validity of the approximation is limited to the region where the linearized shape function holds. They explicitly state that a full treatment without the near-throat approximation and the inclusion of rotation effects are left for future work.

The paper concludes that while Casimir-type effects have been studied extensively, this multi-component approach offers a more structured physical interpretation of the sources supporting traversable wormholes, reducing the reliance on purely theoretical exotic matter constructs.