Rotating wormholes in Einstein-Dirac-Maxwell theory

This paper presents asymmetric, regular, and asymptotically flat rotating wormhole solutions in Einstein-Dirac-Maxwell theory, supported by a complex non-phantom spinor field and electromagnetic fields, which connect two identical Minkowski spacetimes with potentially different masses and global charges while being fully determined by the throat parameter, spinor frequency, and electromagnetic coupling constant.

The Gist

Imagine the universe as a vast, flat sheet of fabric. Usually, if you want to travel from point A to point B, you have to walk across the fabric. But what if you could fold the fabric over and poke a tunnel through it, creating a shortcut? That tunnel is a wormhole.

For decades, physicists have struggled to build a stable wormhole. The problem? To keep the tunnel open, you usually need "exotic matter"—a weird, ghost-like substance that pushes outward instead of pulling inward (like anti-gravity). This exotic matter is theoretical and has never been found.

This paper proposes a new way to build a wormhole without needing that ghostly exotic matter. Instead, the authors use a "quantum spin" and electricity to hold the tunnel open.

Here is the breakdown of their discovery using simple analogies:

1. The Ingredients: A Spinning Quantum Top

Instead of exotic matter, the authors use a spinor field.

• The Analogy: Think of a spinning top. In quantum physics, particles like electrons have an intrinsic "spin" (they aren't literally spinning like a top, but they act like it). The authors use a cloud of these spinning particles.
• The Twist: They make this cloud rotate and give it an electric charge.
• The Result: The energy and pressure from this spinning, charged cloud create a "push" that keeps the wormhole throat from collapsing, replacing the need for exotic matter.

2. The Shape: A One-Way Street with a Twist

The wormholes they found are asymmetric.

• The Analogy: Imagine a tunnel connecting two identical cities. Usually, you'd expect the tunnel to look the same from both sides. But in this model, the tunnel is lopsided.
• What happens: If you enter from the "left" side, you might see a massive city. If you exit on the "right" side, you might see a tiny village. The wormhole connects two identical universes (flat, empty space), but the mass and charge of the wormhole itself look different depending on which side you are standing on.
• The Rotation: Because the "spinor cloud" is spinning, it drags the space around it. This is like a spinning ball in a pool of water; the water swirls around it. This dragging effect allows the wormhole to have angular momentum (it spins), which is a first for this type of solution.

3. The "Mass Without Mass" and "Charge Without Charge"

This is the most mind-bending part, inspired by the physicist John Wheeler.

• The Analogy: Imagine a whirlpool in a bathtub. The whirlpool has a shape and a center, but there is no actual "object" sitting in the middle of the water. The whirlpool is the water moving.
• The Physics: In this wormhole, the "mass" and "electric charge" aren't sitting inside a solid object. They are just the shape of the space and the flow of the fields.
  • Mass without mass: The wormhole has weight (gravity), but there is no "rock" or "star" causing it. The geometry of the tunnel itself creates the gravity.
  • Charge without charge: The wormhole has an electric charge, but there is no "battery" or "electron" sitting inside. The charge is a property of the tunnel's shape and the spinning fields.

4. The "Electron" Connection

The authors suggest this could be a model for an electron.

• The Idea: An electron has a tiny mass, a specific electric charge, and a "spin" of 1/2.
• The Match: Their wormhole solutions naturally have a spin of 1/2 and carry charge.
• The Catch: While it looks like an electron, it's not a perfect match. An electron has the same charge no matter which side you look at it from. This wormhole has different charges on the left and right sides. So, while it's a cool "classical" model of a spinning charged particle, it's not quite the electron we know in the lab yet.

Summary of the Discovery

The authors solved a complex set of equations (Einstein's gravity + Quantum spin + Maxwell's electricity) to show that:

1. You don't need exotic matter to make a wormhole; a spinning, charged quantum field can do the job.
2. The wormhole can spin, dragging space around it.
3. The wormhole is asymmetric, looking different from the two ends it connects.
4. It connects two empty universes, but the wormhole itself acts like a heavy, charged object.

In a nutshell: They found a way to build a spinning, charged tunnel through space using only the "twist" of quantum particles and electricity, proving that you don't need magic "ghost matter" to keep a wormhole open. It's a theoretical "proof of concept" that the universe might be able to fold itself into shortcuts using only the laws of physics we already know.

Technical Summary

1. Problem Statement

The paper addresses the challenge of constructing rotating, asymptotically flat wormhole solutions within the framework of General Relativity (GR) without invoking "exotic" or phantom matter (matter that violates the Null Energy Condition).

• Context: Traditional wormhole models in GR typically require exotic matter to sustain the throat. While some models avoid this by modifying gravity or using cylindrical symmetry (losing asymptotic flatness), finding spherically or axially symmetric, asymptotically flat solutions using standard GR and non-exotic matter remains difficult.
• Specific Gap: Previous work by the authors (Ref. [28]) established static asymmetric wormholes supported by a complex, non-phantom spinor field and electromagnetic fields. However, the rotating case had not been explored.
• Goal: To generalize the static Einstein-Dirac-Maxwell (EDM) wormhole model to include rotation, determining if stable, regular, rotating configurations exist and characterizing their physical properties (mass, angular momentum, charge).

2. Methodology

Theoretical Framework

The authors utilize the Einstein-Dirac-Maxwell (EDM) theory, combining:

1. Einstein Gravity: Standard GR with the metric signature $(+, -, -, -)$.
2. Dirac Field: A classical, complex spinor field $\psi$ with mass $m_s$. Crucially, this is treated as a non-phantom field (standard mass term), yet it supports the wormhole topology.
3. Maxwell Field: Electromagnetic fields ($A_\mu$) coupled to the spinor field via a gauge coupling constant $e$.

Ansatz and Symmetries

• Metric: A stationary, axially symmetric metric is used, characterized by functions $f, q, b, \omega$ depending on radial coordinate $r$ and polar angle $\theta$. The radial coordinate $r$ spans $(-\infty, +\infty)$, connecting two asymptotically flat Minkowski regions.
• Spinor Field: The spinor field is parameterized with explicit dependence on time $t$ and azimuthal angle $\phi$:
  $$\psi^T = e^{i(M_\psi \phi - \Omega t)} (\psi_1, \psi_2, \psi_2^*, \psi_1^*)$$
  where $\Omega$ is the spinor frequency and $M_\psi = 1/2$ is the azimuthal number. This specific phase dependence induces rotation.
• Electromagnetic Field: Defined by electric potential $\phi(r, \theta)$ and magnetic potential $\sigma(r, \theta)$.

Numerical Approach

• Equations: The variation of the action yields coupled non-linear partial differential equations (Einstein, Dirac, and Maxwell equations).
• Boundary Conditions:
  • Asymptotic Flatness: Metric functions vanish at $r \to \pm \infty$.
  • Regularity: Conditions imposed on the symmetry axis ($\theta = 0, \pi$) to avoid conical singularities.
  • Constraints: The system includes constraint equations ($E^x_\theta = 0$ and $d \equiv E^x_x - E^\theta_\theta = 0$). The authors adjust the asymptotic electric potential $\phi_{+\infty}$ numerically to satisfy the second constraint.
• Discretization: The infinite radial domain is mapped to a finite interval $[-1, 1]$ using a compactified coordinate. The equations are solved using a modified Newton method with the Intel MKL PARDISO solver.
• Normalization: To ensure physical relevance as "one-particle" solutions, the Noether charge $Q_\psi$ is fixed to 1. This implies the curves in the results represent sequences of models with varying spinor mass $m_s$ for fixed dimensionless parameters.

3. Key Contributions

1. First Rotating Non-Phantom Wormholes: This work presents the first known example of rotating, asymptotically flat wormholes supported solely by a non-phantom spinor field and electromagnetic fields within standard GR.
2. Asymmetry: The solutions are asymmetric with respect to the throat ($x \to -x$). The two asymptotic regions possess different masses ($M_+$ vs $M_-$) and global charges ($Q_+$ vs $Q_-$), despite connecting identical Minkowski spacetimes.
3. Angular Momentum Relation: The total angular momentum $J$ is found to be proportional to the Noether charge $Q_\psi$ via the azimuthal number:
   $$J = M_\psi Q_\psi$$
   With $M_\psi = 1/2$, this mirrors the relation found in spinning Dirac stars, confirming the spinor field carries the intrinsic angular momentum.
4. Topology and "Charge without Charge": The solution realizes Wheeler's concept of "charge without charge." The electric field lines enter the wormhole from one asymptotic region and exit the other, creating a net charge in both regions without a central source singularity.

4. Results

The numerical analysis reveals the following physical characteristics:

• Mass Behavior:

  • The ADM masses ($M_+$ and $M_-$) depend on the spinor frequency $\Omega$ and coupling constant $e$.
  • For uncharged fields ($e=0$), masses approach zero as $\Omega \to -1$ and increase as $\Omega \to 0$.
  • For charged fields ($e \neq 0$), a critical frequency $\Omega_{crit} \approx e$ exists where mass diverges. Unlike static cases, rotating charged wormholes can have non-zero mass even as $\Omega \to -1$.
  • The mass curves are monotonic, unlike the spiral-like behavior seen in Dirac stars.

• Angular Momentum and Rotation:

  • The dimensionless spin parameter $a^* = J/M^2$ can exceed 1 (unlike Kerr-Newman black holes where $a^* \leq 1$), reaching values $a^* \gg 1$ for certain configurations.
  • The rotation of the spinor field "drags" the spacetime and the throat, allowing for the existence of these rotating solutions.

• Geometry:

  • Throat Shape: The wormhole throat is slightly oblate (flattened along the rotation axis). The ratio of equatorial to polar radius $R_e/R_p \approx 1.02$ at maximum rotation.
  • Embedding: Embedding diagrams show that to maintain a fixed equatorial throat radius, increasing the throat parameter $x_0$ requires significantly higher rotational velocities, resulting in a "squeezed" geometry along the rotation axis.

• Field Configurations:

  • Electric Field: Lines of force pass through the throat from the $x>0$ region to the $x<0$ region.
  • Magnetic Field: Behaves as an axially symmetric dipole sourced by the spinor current.
  • Asymmetry: The energy density and field distributions are asymmetric across the throat but symmetric with respect to the equatorial plane.

5. Significance

• Theoretical Viability: The study demonstrates that exotic matter is not strictly necessary for rotating wormholes if one utilizes the non-linear self-interaction of a classical spinor field within the EDM framework.
• Particle Models: The solutions possess mass, spin ($1/2$), and electric/magnetic charges, making them potential classical models for elementary particles (specifically electrons). However, the authors note a limitation: the charges on either side of the throat are generally unequal and depend on model parameters, preventing a direct 1:1 identification with a single electron.
• New Class of Solitons: These configurations represent a new family of regular, localized solitonic solutions in GR that bridge the gap between black holes (singularities) and Dirac stars (particle-like objects), offering a unique topology where two distinct universes (or regions) are connected with different global charges.
• Constraints on Parameters: The study identifies the viable parameter space, specifically noting that solutions likely only exist for the coupling constant range $-1 \leq \bar{e} \leq 0$.

In conclusion, the paper successfully extends the theory of spinor-supported wormholes to the rotating regime, providing a robust, regular, and asymptotically flat solution that challenges the necessity of phantom matter for such topological structures.