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

Abstract: In General Relativity, constructing exact traversable wormholes coupled to electromagnetic fields typically requires complex Non-Linear Electrodynamics (NED). We demonstrate that Unimodular Gravity (UG) elegantly resolves this limitation. By relaxing energy-momentum conservation, UG introduces a dynamical cosmological term, $Λ(x)$, enabling a semi-classical energy exchange between matter and the vacuum. Exploiting this mechanism, we construct exact Scalar-Maxwell-$Λ(x)$ wormholes. We show that, provided the shape function $b(r)$ satisfies specific geometric conditions, these exact spacetimes can be fully supported by a phantom scalar field and standard linear Maxwell electrodynamics. This approach entirely bypasses NED, highlighting UG as a powerful framework for modeling non-trivial topologies with simplified, well-understood classical fields.

• The paper introduces a framework for constructing traversable wormholes in unimodular gravity using phantom scalar fields, standard Maxwell electrodynamics, and a dynamic cosmological term Λ(x).
• It decouples the scalar potential from Λ(x), enabling distinct sector-wise energy exchange without relying on nonlinear electrodynamics.
• Numerical results with constant and power-law shape functions demonstrate regular physical behavior and enforce strict geometric selection rules for viable wormhole solutions.

Wormhole Spacetimes in the Scalar–Maxwell–$\Lambda(x)$ System under Unimodular Gravity

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Introduction and Motivation

The paper introduces a systematic framework for constructing traversable wormhole geometries within unimodular gravity (UG), specifically emphasizing the scalar–Maxwell–$\Lambda(x)$ system. Unimodular gravity distinguishes itself from general relativity (GR) through the preservation of a fixed metric determinant, yielding field equations where the cosmological term is a dynamic integration function $\Lambda(x)$ rather than a static Lagrangian parameter. This variance facilitates novel semi-classical energy exchanges between matter and the vacuum, breaking the standard energy-momentum conservation paradigm.

While GR requires complex sources—such as Nonlinear Electrodynamics (NED)—to support wormhole geometries, UG circumvents this necessity. By relaxing conservation laws, UG allows standard linear Maxwell electrodynamics, supplemented by phantom scalar fields and a dynamical cosmological term, to sustain exact wormhole solutions. The framework critically breaks the degeneracy between scalar potential $V(\phi)$ and the cosmological contribution, which arises in purely scalar sectors, making the introduction of an electromagnetic sector essential for unambiguous modeling.

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Non-Conservative Dynamics and Sector-Wise Field Separation

The UG field equations in the non-conservative form are: $$G_{\mu\nu} + g_{\mu\nu}\Lambda(x) = \kappa^2 T_{\mu\nu}, \qquad \nabla_\mu T^{\mu\nu} = \nabla^\nu \Lambda$$ In the purely scalar scenario, $V(\phi)$ and $\Lambda(x)$ enter gravitational and scalar equations only via an inseparable combination—rendering individual identification impossible. This motivates the inclusion of the Maxwell sector, since the energy-momentum trace of Maxwell electrodynamics vanishes, thus enabling sector-wise conservation laws. Specifically:

• The scalar field remains conserved: $\nabla^\mu T_{\mu\nu}^{(\phi)} = 0$.
• Non-conservation is carried by the electromagnetic sector: $$\nabla^\mu T_{\mu\nu}^{(M)} = \frac{1}{\kappa^2}\nabla_\nu\Lambda$$.

Introducing the Maxwell field allows the separation of $V(\phi)$ and $\Lambda(x)$0 via the trace relation: $\Lambda(x)$1 The dynamical cosmological term now acts as a local source for the electromagnetic sector, linking its spatial gradient directly to an effective current: $\Lambda(x)$2. This is instrumental for modeling the energy exchange necessary for wormhole configurations outside of standard conservative frameworks.

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Wormhole Construction: Algorithm and Consistency

The authors detail a construction algorithm for wormholes in the UG framework with vanishing redshift function (Morris–Thorne metric). The matter sector consists of:

• Phantom scalar field ($\Lambda(x)$3),
• Radial Maxwell field.

The algorithm proceeds as follows:

1. Shape function selection: $\Lambda(x)$4 is chosen to satisfy $\Lambda(x)$5 and $\Lambda(x)$6, ensuring throat regularity and electromagnetic sector viability.
2. Scalar field profile: $\Lambda(x)$7 is fixed by geometry: $\Lambda(x)$8.
3. Scalar potential: $\Lambda(x)$9 is obtained via integration, yielding explicit functional dependence on the geometry.
4. Electromagnetic master relation: For the Maxwell limit, $\Lambda(x)$0, with $\Lambda(x)$1.
5. Dynamical cosmological term: $\Lambda(x)$2, with $\Lambda(x)$3 determined from electromagnetic sector.
6. Consistency and energy exchange: Non-conservative current $\Lambda(x)$4 ensures the modified Maxwell equations are satisfied.

This construction avoids the pitfalls of NED, facilitating exact wormhole solutions supported by linear Maxwell electrodynamics—conditioned solely on geometric constraints.

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Numerical Results and Explicit Solutions

The approach yields a family of exact solutions:

• Constant shape function ($\Lambda(x)$5): All physical quantities (scalar field, electric field, potential, cosmological term) are real, regular, and decay properly at infinity. Closure and consistency are ensured, with energy exchange governed by $\Lambda(x)$6.
• Power-law shape functions ($\Lambda(x)$7, $\Lambda(x)$8): A continuous spectrum of solutions exist, interpolating from the constant case ($\Lambda(x)$9) to the canonical Ellis–Bronnikov geometry ($V(\phi)$0). Scalar potential, electric field, cosmological term, and source current decay as explicit power laws, ensuring regularity and physical acceptability.
• GEB Geometry Counter-Example: Not all wormhole shapes are permitted; the GEB case, with $V(\phi)$1, fails the positivity test for $V(\phi)$2, leading to unphysical (imaginary) electric fields at spatial infinity. This enforces strict geometric selection rules.

These results demonstrate that UG supports traversable wormholes with standard electromagnetic fields under precise geometric criteria, bypassing the intricacies of NED and exotic matter.

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Theoretical and Practical Implications

The work fundamentally alters the landscape of wormhole modeling. In GR, exact traversable wormhole solutions require contrived sources—often nonlinear or phenomenological. The unimodular framework, via a relaxable conservation law and dynamical $V(\phi)$3, reveals that semi-classical energy exchange mechanisms are sufficient. Phantom scalar fields and standard Maxwell electrodynamics suffice, provided suitable geometric constraints are met.

• Theoretical Impact: The framework clarifies the interplay between geometry, matter, and cosmological sector in UG, and demonstrates sector-wise conservation as a tool for unambiguous field separation.
• Practical Impact: Exact construction algorithms are provided, enabling systematic exploration of wormhole geometries with well-understood classical fields. Models can be generalized to various compact objects—regular black holes, bounces—where similar mechanisms apply.
• Future Directions: The paper suggests perturbative stability analyses of these structures, potential extensions to axisymmetric and rotating wormholes, and broader applications in astrophysical modeling within UG.

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Conclusion

The scalar–Maxwell–$V(\phi)$4 system under unimodular gravity yields a concise, rigorous prescription for constructing exact traversable wormhole geometries without reliance on nonlinear electrodynamics or exotic matter. By capitalizing on UG's relaxed diffeomorphism symmetry and dynamic cosmological term, the authors demonstrate sector-wise field separation, geometric selection rules, and robust numerical results for a broad class of wormhole spacetimes.

This framework not only streamlines the matter sector but also elucidates semi-classical energy exchange mechanisms as central to supporting non-trivial topologies. It positions unimodular gravity as a foundational platform for exploring compact objects and intricate geometries with standard, analytically tractable fields, inviting further theoretical and practical investigation in gravitational modeling and quantum cosmology.

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Reference:

"The scalar–Maxwell–$V(\phi)$5 system: Wormhole spacetimes without nonlinear electrodynamics in unimodular gravity" (2603.30003)