Realistic classical charge from an asymmetric wormhole

This paper presents a solution within Einstein-Dirac-Maxwell theory where an asymmetric wormhole supported by a complex spinor field and electromagnetic fields connects two universes with distinct physical parameters, effectively modeling a classical particle with spin by exhibiting Standard Model mass and charge at one end while displaying Planck-scale values at the other.

The Gist

The Big Idea: A Cosmic Funnel That Hides the Truth

Imagine you have a giant, magical cosmic funnel (a wormhole) that connects two different universes. On one side of the funnel, the laws of physics look "normal" to us—particles have tiny masses and small electric charges, just like the electrons in your phone or the atoms in your body.

On the other side of the funnel, however, the physics is wild and extreme. Everything there is heavy as a mountain and charged like a lightning storm.

This paper proposes a new way to understand the building blocks of our universe. The authors suggest that the tiny particles we see (like electrons) aren't just "dots" of matter. Instead, they might be wormholes themselves. The "mass" and "charge" we measure are just the view from one side of the funnel, while the "real" underlying stuff is actually much, much heavier and more energetic.

The Problem: The "Hierarchy" Mystery

To understand why this is cool, we have to look at a big problem in physics called the Hierarchy Problem.

• The Puzzle: In our universe, particles like electrons are incredibly light. But the "fundamental" scale of gravity (the Planck scale) is unimaginably heavy—about $10^{19}$ times heavier than an electron.
• The Question: Why is there such a huge gap? Why is the electron so light when the "raw material" of the universe seems so heavy? Usually, physicists have to invent complicated new rules to explain why the numbers cancel out so perfectly.

The Solution: "Mass Without Mass"

The authors, Vladimir and Vladimir, suggest a different approach. They say: "What if the electron is heavy, but it looks light because of how it's shaped?"

They use a concept from the 1960s by physicist John Wheeler called "Mass without mass" and "Charge without charge."

• The Analogy: Imagine a whirlpool in a bathtub. The whirlpool has energy and can push things around (like it has "mass"), but if you look closely, there is no actual "stuff" (like a rock) inside the water. The whirlpool is the water moving in a specific shape.
• The Paper's Twist: They propose that an electron is a whirlpool in the fabric of space-time. The "stuff" making up the whirlpool is actually super-heavy (Planck mass), but because it is twisted into a specific wormhole shape, the "weight" we feel on our side of the funnel is tiny.

How It Works: The Asymmetric Wormhole

The paper describes a specific type of wormhole that is asymmetric. Think of it like a tunnel connecting two rooms, but the tunnel is lopsided.

1. The Ingredients: They built this model using three main ingredients:

   • Spinor Fields: Think of these as "twisting energy waves" that carry a tiny spin (like a spinning top).
   • Electric & Magnetic Fields: These act like the glue holding the shape together.
   • The Wormhole: The tunnel connecting two different versions of space.

2. The Two Sides:

   • Side A (The "Heavy" Side): Here, the raw energy is huge. The mass is at the "Planck scale" (the heaviest possible scale in physics).
   • Side B (The "Light" Side - Our Universe): Here, the geometry of the wormhole stretches and dilutes the energy. To an observer on this side, the object looks like a standard electron or positron with a tiny mass and a tiny electric charge.

3. The "Magic" Adjustment: The authors found that by tweaking a few numbers (like the size of the wormhole's throat), they could make the "Light Side" look exactly like an electron, even though the "Heavy Side" is still Planck-scale heavy.

The "Spin" and the "Charge"

• Spin: The model naturally creates a spinning object. Just like a real electron spins, this wormhole configuration has an intrinsic spin of 1/2. This happens because the "twisting energy" (the spinor field) is built into the shape of the wormhole.
• Charge: The electric field lines enter the wormhole from one universe and exit into the other. To an observer on one side, it looks like a charged particle. To an observer on the other side, it looks like a particle with the opposite charge. It's like a hose: water flows in one end and out the other, but the hose itself is just a connection.

Why This Matters

This is a classical model, meaning they didn't use quantum mechanics (the weird rules of tiny particles) to build it. They used the standard rules of gravity and fields.

• The Takeaway: They showed that you don't need to invent new, mysterious particles to explain why electrons are light. You just need a specific shape of space-time (a wormhole) that hides the true heaviness of the object from our perspective.
• The "Classical Charge": They are essentially saying, "We can build a model of a charged particle using only classical fields and gravity, without needing to invoke the full complexity of quantum theory yet."

Summary in One Sentence

The paper suggests that the tiny particles we see in our universe might actually be heavy, spinning wormholes where the "heaviness" is hidden on the other side of the tunnel, making our universe look light and simple by comparison.

It's a bit like looking at a shadow on the wall: the shadow is small and simple, but the object casting it (the wormhole) is massive and complex.

Technical Summary

1. Problem Statement

The paper addresses the hierarchy problem in particle physics, specifically the vast discrepancy between the Planck mass ($M_P \sim 10^{19}$ GeV), which characterizes the strength of gravity, and the masses of Standard Model (SM) particles (e.g., the electron mass $\sim 0.5$ MeV).

• The Challenge: In the Standard Model, fermion masses arise from Yukawa couplings to the Higgs field, but the origin of the specific mass hierarchy (ranging over 5 orders of magnitude) and the gap between the electroweak scale and the Planck scale remains unexplained.
• The Goal: The authors propose a mechanism to generate localized field configurations with effective masses and charges typical of SM particles (like an electron or positron) while starting from a fundamental Lagrangian where the bare mass of the spinor field is of the order of the Planck mass. This aims to realize Wheeler's concept of "mass without mass" and "charge without charge."

2. Methodology

The authors employ the Einstein-Dirac-Maxwell (EDM) theory within the framework of General Relativity.

• Theoretical Framework:

  • Action: The system is defined by the total action $S_{tot} = S_{grav} + S_{sp} + S_{EM}$, comprising the Einstein-Hilbert term, the Dirac spinor field action, and the Maxwell electromagnetic action.
  • Fields:
    • A complex, non-phantom spinor field ($\psi$) with a bare mass $m_s \sim M_P$. This field provides the nontrivial spacetime topology (wormhole) and intrinsic angular momentum.
    • Electromagnetic fields (electric and magnetic) generated by the spinor current, providing the system's charge.
  • Metric: An axially symmetric, stationary wormhole metric is used, connecting two asymptotically flat spacetimes (two "universes") with potentially different physical parameters.
  • Classical Spinor Formalism: To handle the mathematical difficulties of classical spinors (which are typically Grassmann numbers), the authors define a rigorous Lagrangian by separating variables into ordinary complex functions and Grassmann numbers. They integrate over the Grassmann variables to obtain a well-defined energy-momentum tensor and current, treating the system as a quasiclassical approximation (limit $\hbar \to 0$).

• Numerical Approach:

  • The coupled system of Einstein, Dirac, and Maxwell partial differential equations (PDEs) is solved numerically.
  • The infinite radial domain is mapped to a finite interval using a compactified coordinate.
  • Boundary conditions are imposed at spatial infinity (asymptotic flatness) and on the symmetry axis to ensure regularity and the absence of conical singularities.
  • Solutions are sought for a small throat parameter ($x_0$) and specific spinor frequencies ($\bar{\Omega}$).

3. Key Contributions

1. Rigorous Classical Spinor Definition: The authors provide a mathematically consistent definition of a classical spinor field Lagrangian by explicitly separating Grassmann numbers and integrating them out. This resolves issues in defining the energy-momentum tensor and current for classical spinors.
2. Asymmetric Wormhole Solutions: They extend previous work to find asymmetric wormhole solutions. Unlike symmetric wormholes, these connect two universes where the observed physical quantities (mass, charge, angular momentum) differ significantly at the two asymptotic ends.
3. Mechanism for Mass/Charge Hierarchy: The paper demonstrates that by tuning free parameters (throat size and spinor frequency), the system can exhibit Planck-scale physics at one end of the wormhole and Standard Model-scale physics at the other, despite the bare mass being Planckian everywhere.
4. Electron/Positron Model: The authors construct a specific solution that mimics the physical characteristics (mass, charge, spin) of an electron or positron.

4. Key Results

• Asymmetry of Observables:
  • At one end of the wormhole ($x > 0$), the observed mass is of the order of the Planck mass ($M_P$).
  • At the other end ($x < 0$), the observed mass can be tuned to be of the order of SM particles. For a specific configuration (bold dot in Fig. 2), the mass corresponds to that of a positron ($m_e \approx 0.511$ MeV), even though the bare spinor mass $m_s \sim M_P$.
  • Similarly, the electric charge observed at the $x < 0$ end can be tuned to match the elementary charge ($q_e$), while the charge at the $x > 0$ end differs.
• Angular Momentum and Spin:
  • The system possesses a total intrinsic angular momentum (spin) of $J = 1/2$ (in natural units), consistent with fermions.
  • The ratio of angular momentum to Noether charge ($J/Q_\psi$) is exactly $1/2$ for the total system.
  • For the specific "positron-like" configuration, the ratio observed at the $x < 0$ end is approximately $1/2$.
• Gyromagnetic Ratio:
  • The calculated gyromagnetic ratio $g$ for these configurations is generally not equal to 2 (the Dirac value for electrons). For the specific positron-like solution, $g \to 0$. The authors note this is a limitation of the current model, as the classical nature of the solution does not fully reproduce the quantum magnetic moment.
• Field Topology:
  • The electric field lines enter the wormhole from one asymptotic region and exit to the other. The configuration is a source-free solution to Maxwell's equations coupled to gravity, where the "charge" is a topological feature of the wormhole rather than a point source.

5. Significance

• Geometric Origin of Particle Properties: The paper supports the idea that fundamental particle properties (mass, charge, spin) could be emergent phenomena arising from the topology of spacetime (wormholes) rather than intrinsic properties of point particles.
• Solving the Hierarchy Problem: It offers a novel, geometric mechanism to explain why observed masses are so small compared to the Planck scale without invoking supersymmetry or extra dimensions in the traditional sense. The "smallness" is a result of the observer's location relative to the wormhole throat.
• Classical Approximation of Quantum Objects: By treating the spinor field classically (with Grassmann integration), the authors provide a bridge between classical general relativity and quantum particle physics, suggesting that "Dirac stars" or wormholes could be classical analogs of elementary particles.
• Wheeler's Geometrodynamics: The work revitalizes John Wheeler's "mass without mass" and "charge without charge" hypothesis, demonstrating that a localized, rotating, charged object with fermionic properties can exist as a solution to classical field equations.

Conclusion: The paper successfully constructs a theoretical model of an asymmetric wormhole that, under specific parameter tuning, appears to a distant observer in one universe as a particle with the mass and charge of an electron/positron, while the underlying fundamental parameters remain at the Planck scale. This provides a compelling classical geometric explanation for the hierarchy of mass scales in nature.