⚛️ general relativity

Neutral scalar particle in a charged wormhole background

This paper investigates the dynamics of a neutral scalar particle in a charged traversable wormhole background by solving the Klein-Gordon equation with harmonic and modified oscillator couplings, revealing that the wormhole's charge indirectly influences the bound-state energy spectrum and quantized frequencies through spacetime geometric deformation.

Original authors: Matheus D. de Oliveira, Alexandre G. M. Schmidt

Published 2026-09-09

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CC BY 4.0

Original authors: Matheus D. de Oliveira, Alexandre G. M. Schmidt

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

Deep within the fabric of the universe, general relativity allows for the existence of wormholes, theoretical tunnels that connect distant regions of space and time. While these structures are often imagined as gateways for travel, physicists are equally interested in how they behave as cosmic laboratories. A wormhole is defined by its "throat," the narrowest point of the tunnel, and its shape is determined by the geometry of the space around it. In many theoretical models, this geometry can be altered by adding electric charge, much like how a heavy object bends a rubber sheet. When scientists study how particles move through these warped spaces, they usually look at how the path of the particle curves. However, quantum particles, which are far smaller than anything we can see, do not just follow a path; they exist as waves that feel the shape of space itself. By studying how these waves behave, researchers can learn about the hidden structure of the wormhole, even if the particle itself carries no electric charge.

In a recent study, researchers investigated what happens to a neutral, wave-like particle trapped inside a charged wormhole. They focused on a specific type of interaction called a harmonic oscillator, which is a mathematical way of describing a particle that is tethered to a central point, similar to a weight on a spring that bounces back and forth. The scientists wanted to see how the electric charge of the wormhole would change the energy levels of this bouncing particle. Even though the particle has no electric charge and should not feel the wormhole's electric field directly, the charge still changes the shape of the space the particle moves through. The team solved the complex equations that govern this motion and found that the charge of the wormhole leaves a distinct fingerprint on the particle's energy, even without a direct electrical connection.

The researchers discovered that the energy levels of the particle are not random; they are locked into specific, discrete values, a phenomenon known as quantization. This means the particle can only exist at certain energy states, much like a ladder where you can stand on a rung but not in the space between them. The study showed that the frequency at which the particle oscillates, or bounces, is not a fixed number but depends on several factors, including the particle's angular momentum, which describes how it swirls around the center. Crucially, the electric charge of the wormhole changes the geometry of the throat, and this geometric shift forces the oscillation frequency to adjust. The team found that as the charge increases, the frequency of the bounce changes in a predictable way, proving that the wormhole's electric charge influences the quantum world indirectly by reshaping the stage upon which the particle performs.

The team also explored a more complex version of this system where an additional force was introduced, acting like a second tether that pulls the particle differently depending on its distance from the center. In this modified scenario, the relationship between the charge, the shape of the wormhole, and the particle's energy became even more intricate. The calculations revealed that the allowed energy levels and the required oscillation frequencies shift in response to this new force. The researchers confirmed that their results make sense by checking what happens when the electric charge is removed; in that case, their equations matched previous studies of uncharged wormholes perfectly. They also looked at what happens if the wormhole's throat becomes extremely large, finding that the particle eventually loses its ability to oscillate in a stable way, effectively stopping its quantum dance.

Ultimately, this work demonstrates that the geometry of space is a powerful force in quantum mechanics. The study proves that a neutral particle can "feel" the electric charge of a wormhole not because it is electrically attracted to it, but because the charge warps the space the particle inhabits. This subtle influence changes the rules of the game, dictating exactly how the particle can move and what energy it can possess. By solving these equations, the researchers have provided a clearer picture of how charged wormholes might interact with the quantum world, offering a new way to probe the nature of these exotic cosmic structures through the behavior of the smallest particles.

Technical Summary: Neutral Scalar Particle in a Charged Wormhole Background

Problem Statement
This work investigates the quantum dynamics of a neutral scalar particle propagating in the background of a charged traversable wormhole. While previous studies have examined relativistic oscillators in uncharged wormhole geometries, wormholes with topological defects, or in the presence of external scalar/vector potentials, the specific case of a neutral particle in a charged wormhole background (specifically the Kim-Lee solution) had not been fully explored. The central physical question is how the electric charge of the wormhole, despite the particle being electrically neutral and lacking direct electromagnetic coupling, influences the quantum bound states and energy spectrum through the geometric deformation of spacetime.

Methodology
The authors formulate the problem using the Klein-Gordon equation for a neutral scalar particle of mass $m_0$ in a curved spacetime defined by the charged traversable wormhole metric. The metric is characterized by a charge $Q$ and a shape function $b(r) = b^2/r$ , where $b$ relates to the throat radius.

Two interaction models are considered:

Harmonic Oscillator: Introduced via a four-potential $X_\mu = (0, m_0\omega r, 0, 0)$ .
Modified Harmonic Oscillator: Introduced via $X_\mu = (0, m_0\omega r + V_0/r, 0, 0)$ , adding an inverse-radial interaction parameter $V_0$ .

The methodology proceeds as follows:

Coordinate Transformation: The radial coordinate is transformed to $x = \sqrt{r^2 - \beta^2}$ , where $\beta = \sqrt{b^2 - Q^2}$ , to map the domain to $-\infty < x < \infty$ .
Reduction to Heun Equation: By applying specific ansatzes for the radial wave function (involving exponential and power-law factors), the radial differential equations are reduced to the form of a generalized Heun equation.
Quantization via Polynomial Truncation: To obtain physically acceptable bound-state solutions (normalizable wave functions), the power series solution of the Heun equation is truncated. This requires satisfying a set of conditions derived from a four-term recurrence relation, specifically forcing the series to terminate at a finite order $n$ .
Numerical Analysis: The resulting quantization conditions yield a system of homogeneous linear equations. The authors solve these numerically to determine the oscillator frequency $\omega$ as a function of angular momentum $l$ , throat parameter $b$ , charge $Q$ , and interaction parameter $V_0$ .

Key Contributions and Results

Exact Solutions and Energy Spectrum:
- For both the harmonic and modified oscillators, the radial wave functions are expressed in terms of generalized Heun functions.
- The energy spectrum for the harmonic oscillator is derived as $E_n = \pm \sqrt{m_0^2 + 4m_0\omega(\frac{n-3}{2})}$ , and for the modified oscillator as $E_n = \pm \sqrt{m_0^2 + 4m_0\omega(\frac{n+V_0-3}{2})}$ , where $n \geq 3$ .
- Crucially, the functional form of the energy eigenvalues is identical to that found in uncharged wormhole backgrounds. This confirms that for a neutral particle, there are no direct electrostatic contributions to the energy levels.
Geometric Dependence of Oscillator Frequency:
- While the energy formula appears independent of charge $Q$ , the oscillator frequency $\omega$ is not a free parameter; it is quantized and determined by the boundary conditions.
- Numerical results demonstrate that $\omega$ depends implicitly on the wormhole charge $Q$ and the throat parameter $b$ through the geometric parameter $\beta = \sqrt{b^2 - Q^2}$ .
- Behavior of $\omega$ : The frequency increases with angular momentum $l$ . It diverges ( $\omega \to \infty$ ) as $\beta \to 0$ (the limit where the throat radius approaches the charge scale) and vanishes ( $\omega \to 0$ ) as $\beta \to \infty$ (where the throat becomes arbitrarily large, suppressing oscillatory behavior).
- In the modified oscillator, the parameter $V_0$ further modulates the frequency, showing a dependence qualitatively similar to that of the electric charge.
Lorentz Symmetry Violation:
- The study identifies that the oscillator frequency becomes a function of the angular momentum $l$ . This dependence implies a violation of Lorentz symmetry, a feature noted in previous works involving Heun polynomials in curved spacetimes and topological defect backgrounds.
Asymptotic Limits:
- Uncharged Limit ( $Q \to 0$ ): The results consistently recover the known solutions for the uncharged Ellis-Bronnikov wormhole.
- Black Hole Limit ( $b \to 0$ ): Taking the limit where the throat vanishes recovers the massless Reissner-Nordström black hole background. In this case, the frequency and energy spectrum are no longer quantized with respect to angular momentum and depend solely on the charge $Q$ . The authors note that the resulting values are complex, referring to them as "quasi-frequency" and "quasispectrum," consistent with quantum systems in black hole backgrounds.

Significance and Claims
The paper claims that the dynamics of a neutral scalar particle in a charged wormhole background provide a sensitive probe of the spacetime geometry. The primary significance lies in demonstrating that the electric charge of the wormhole modifies the quantum bound states indirectly through the deformation of the metric, even in the absence of direct electromagnetic coupling.

The authors emphasize that the interplay between the wormhole geometry (specifically the charge and throat radius) and the oscillator coupling creates a rich structure of relativistic bound states. The quantization of the oscillator frequency in terms of angular momentum serves as a distinct signature of the underlying geometry. Furthermore, the work extends the applicability of generalized Heun equations to charged traversable wormholes, providing exact analytical frameworks for studying quantum systems in these non-trivial topological settings. The results are presented as a theoretical extension of previous work on uncharged wormholes and topological defects, offering a more complete picture of how gravitational and electromagnetic parameters influence quantum spectra in relativistic systems.

Technical Summary: Neutral Scalar Particle in a Charged Wormhole Background

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