The scalar--Maxwell--$\Lambda(x)$ system: Wormhole spacetimes without nonlinear electrodynamics in unimodular gravity

This paper demonstrates that Unimodular Gravity enables the construction of exact traversable wormholes supported by a phantom scalar field and standard linear Maxwell electrodynamics, thereby eliminating the need for Non-Linear Electrodynamics typically required in General Relativity.

The Gist

Imagine you are an architect trying to build a wormhole—a magical tunnel through space that connects two distant points in the universe. In the standard rules of physics (General Relativity), building this tunnel is incredibly difficult.

Usually, to keep the tunnel open and prevent it from collapsing, you need "exotic" building materials. Think of these materials as anti-gravity bricks that push outward instead of pulling inward. In the real world, we don't have these bricks. To get around this, physicists have historically used complicated, made-up theories (like Non-Linear Electrodynamics) to invent these exotic materials mathematically. It's like trying to build a house by inventing a new type of wood that doesn't exist in nature.

This paper proposes a brilliant shortcut.

The authors, working with a theory called Unimodular Gravity, show that you don't need exotic, imaginary materials at all. You can build a stable wormhole using only standard, everyday physics: a simple scalar field (a type of energy field) and standard electricity (the kind that powers your phone).

Here is how they did it, explained through a few simple analogies:

1. The "Leaky Bucket" Analogy (Breaking the Rules)

In standard physics, energy is like water in a perfectly sealed bucket. If you have water in one spot, it stays there unless you move it. The total amount of water is always conserved. This rule makes building a wormhole very hard because the "water" (energy) needed to hold the tunnel open just won't stay in the right place without exotic help.

Unimodular Gravity changes the rules. It's like having a bucket with a tiny, controlled leak.

• In this new framework, energy doesn't have to stay perfectly conserved in one spot.
• Instead, energy can "leak" between the matter (the wormhole) and the vacuum of space itself.
• This leak is managed by a variable term called $\Lambda(x)$ (a dynamic cosmological constant). Think of this as a smart pump that can inject energy into the tunnel when it needs to stay open, or suck energy out when it doesn't.

2. The "Tug-of-War" Analogy

To keep a wormhole open, you need a force pushing outward (to stop it from collapsing) and a force holding it together.

• The Phantom Scalar Field: Imagine a ghostly rope pulling the walls of the tunnel outward. This is the "phantom" field. It's a bit weird, but it's a known type of field in physics.
• The Electric Field: Usually, electricity (Maxwell's equations) is too "tame" to help hold a wormhole open. It follows strict rules and refuses to bend.
• The Solution: Because of the "leaky bucket" (the non-conservation of energy), the electric field gets a boost. The smart pump ($\Lambda$) injects just the right amount of energy into the electric field, allowing it to act like a super-strong cable that helps hold the tunnel open.

The Result: You get a stable wormhole held up by a ghostly rope and a standard electric wire, powered by a vacuum pump. No exotic, imaginary materials required!

3. The "Blueprint" Test

The authors didn't just say "it works"; they drew up blueprints to prove it.

• The Failed Blueprint: They tried to use a famous, complex wormhole shape (the Generalized Ellis-Bronnikov). They found that for this specific shape, the "pump" would have to work in reverse, breaking the laws of physics. This proved that not every wormhole shape works with this method; the shape of the tunnel matters.
• The Success Stories: They then tested simpler shapes (like a constant width tunnel and a specific "power-law" curve). For these shapes, the math worked perfectly. They showed that with the right geometry, the standard electric field and the phantom field could dance together perfectly, supported by the energy exchange with the vacuum.

Why This Matters

This paper is a game-changer because it simplifies the universe.

• Before: To build a wormhole, you needed to invent complex, weird physics (Non-Linear Electrodynamics) that we don't understand well.
• Now: We can build wormholes using standard electricity and known fields, provided we accept that the universe allows for a little bit of energy exchange with the vacuum (a feature of Unimodular Gravity).

In a nutshell:
The authors found that if you relax the rule that "energy must always stay put," you can use simple, well-understood tools (like electricity) to build complex structures (wormholes) that were previously thought to require magic. It's like realizing you don't need a jet engine to fly a plane; you just need to understand the wind currents a little better.

Technical Summary

1. Problem Statement

In standard General Relativity (GR), constructing exact traversable wormhole solutions coupled to electromagnetic fields presents a significant "geometric frustration."

• The Constraint: To maintain a traversable throat (satisfying the flare-out condition) without event horizons, the stress-energy tensor must violate the Null Energy Condition (NEC).
• The Limitation: Standard linear Maxwell electrodynamics cannot support these geometries on its own. Historically, researchers have been forced to introduce Non-Linear Electrodynamics (NED) or exotic, ad-hoc anisotropic fluids to close the system of equations and support the wormhole structure.
• The Goal: The authors aim to demonstrate that exact traversable wormholes can be supported by standard linear Maxwell electrodynamics and a phantom scalar field, provided the framework is shifted from standard GR to Unimodular Gravity (UG).

2. Methodology and Theoretical Framework

The paper utilizes Unimodular Gravity (UG), a theory where the metric determinant is fixed ($\sqrt{-g} = \text{const}$), breaking full diffeomorphism invariance down to volume-preserving transformations.

Key Theoretical Mechanisms:

• Relaxed Conservation Laws: In UG, the standard conservation law $\nabla_\mu T^{\mu\nu} = 0$ is relaxed. Instead, the divergence of the stress-energy tensor is non-zero and related to a dynamical cosmological term:
  $$\nabla_\mu T^{\mu\nu} = \nabla^\nu \Lambda(x)$$
  Here, $\Lambda(x)$ is not a fundamental constant but an integration function arising from the field equations, representing a semi-classical energy exchange between matter and the vacuum.
• Breaking the $V + \Lambda$ Degeneracy:
  • In a pure scalar field model within UG, the scalar potential $V(\phi)$ and the cosmological term $\Lambda(x)$ appear only as an effective sum ($V + \Lambda/\kappa^2$), making them mathematically indistinguishable.
  • Solution: The authors introduce an electromagnetic sector (Maxwell field). Since the Maxwell energy-momentum tensor is traceless ($T^{(M)} = 0$) in 4D, the trace of the Einstein equations allows $\Lambda(x)$ to be determined independently of the scalar potential $V(\phi)$. This breaks the degeneracy and allows for a unique reconstruction of the fields.
• Sector-wise Conservation: The authors assume the scalar field is individually conserved ($\nabla_\mu T^{(\phi)\mu\nu} = 0$), while the non-conservation is entirely carried by the electromagnetic sector. This implies the cosmological gradient $\nabla_\nu \Lambda$ acts as an effective current source for the electromagnetic field.

Construction Algorithm:

The authors develop a systematic algorithm to construct exact solutions for static, spherically symmetric wormholes (Morris-Thorne metric with zero redshift function $\Phi=0$):

1. Fix Geometry: Choose a shape function $b(r)$ satisfying geometric constraints (no horizons, $b(r) + rb'(r) > 0$).
2. Scalar Field: Derive the scalar field profile $\phi(r)$ directly from the geometry using the difference of Einstein equations.
3. Electromagnetic Sector: Use the master equation derived from the angular components of the Einstein equations to determine the electric field $E(r)$, assuming the standard Maxwell Lagrangian $L(F) = -F$.
4. Cosmological Term: Calculate $\Lambda(r)$ using the trace equation, which acts as the energy reservoir.
5. Consistency Check: Verify that the non-conservation equation (Eq. 73) is satisfied, ensuring the effective current $J^\mu$ generated by $\Lambda'(r)$ sustains the electric field.

3. Key Contributions and Results

A. The Generalized Ellis-Bronnikov (GEB) Counter-Example

The authors tested the GEB geometry (a popular wormhole model).

• Result: They found that for the GEB shape function, the geometric master function $H(r)$ becomes negative at spatial infinity.
• Implication: This leads to an imaginary electric field ($E^2 < 0$) in the linear Maxwell limit.
• Conclusion: Not all wormhole geometries are compatible with this framework. The shape function $b(r)$ must satisfy strict geometric conditions (specifically regarding the sign of $b(r) + rb'(r)$) to allow for a real, physical linear electromagnetic field.

B. Exact Solution 1: Constant Shape Function ($b(r) = b_0$)

The authors constructed an exact solution for a toy model with a constant shape function.

• Fields:
  • Scalar: A phantom scalar field with a kink-like profile ($\phi \sim \arctan$).
  • Potential: A strictly positive potential $V(r) \propto r^{-3}$.
  • Electric Field: A real, finite radial electric field $E(r) \propto (r-b_0)^{-1/2}$.
  • Cosmological Term: $\Lambda(r) \propto r^{-3}$.
• Significance: This proves that standard linear electrodynamics can support a wormhole if the energy exchange mechanism ($\Lambda'$) is active. The electric field does not need to be source-free; it is sustained by the vacuum energy gradient.

C. Exact Solution 2: Power-Law Shape Functions

The framework was extended to a family of power-law geometries: $b(r) = b_0(b_0/r)^\gamma$ with $0 < \gamma < 1$.

• Results: Exact analytical solutions were derived for the scalar field, potential, electric field, and $\Lambda(r)$ for the entire family.
• Limits:
  • As $\gamma \to 0$, it recovers the constant shape function model.
  • As $\gamma \to 1$, the solution smoothly reduces to the classic Ellis-Bronnikov wormhole (pure phantom scalar, no electromagnetic field, $\Lambda=0$).
• Significance: This demonstrates a continuous family of solutions where the "missing" energy to support the wormhole with linear electrodynamics is provided dynamically by the non-conservative term.

4. Significance and Implications

1. Bypassing Non-Linear Electrodynamics (NED): The most significant finding is that NED is not required to support traversable wormholes. Standard, well-understood linear Maxwell electrodynamics is sufficient when embedded in Unimodular Gravity.
2. Physical Interpretation of $\Lambda(x)$: The dynamical cosmological term $\Lambda(x)$ is reinterpreted not just as a vacuum energy, but as an active energy exchange channel. It injects or absorbs energy from the electromagnetic sector via an effective current $J^\mu$, allowing the system to bypass the restrictive "no-go" theorems of conservative GR.
3. Simplification of Matter Sources: The approach replaces complex, phenomenological NED Lagrangians with a simpler matter sector (Phantom Scalar + Linear Maxwell + Geometric $\Lambda$).
4. Broader Applicability: The authors suggest this mechanism could apply to other compact objects, such as Regular Black Holes, potentially resolving singularities or horizon issues using standard fields within a non-conservative gravitational framework.

Conclusion

The paper establishes that Unimodular Gravity provides a robust and elegant framework for modeling non-trivial spacetime topologies. By relaxing energy-momentum conservation, UG allows for a semi-classical energy exchange that enables standard linear electrodynamics to support exact traversable wormholes, a feat previously thought to require exotic or non-linear matter sources. The work highlights the power of geometric constraints and the dynamical nature of the cosmological term in modern gravitational physics.