Wormholes in Einstein-Dirac-Maxwell theory with identical spacetime asymptotics

This paper constructs complete families of regular, asymmetric wormhole solutions in Einstein-Dirac-Maxwell theory that connect two identical Minkowski spacetimes, demonstrating that their physical properties, including the possibility of negative ADM mass, are fully determined by the throat parameter, spinor frequency, and coupling constant.

The Gist

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

For decades, physicists have struggled to build these shortcuts using the rules of Einstein's General Relativity. The problem? To keep the hole open, you usually need "exotic matter"—stuff that acts like anti-gravity, pushing space apart instead of pulling it together. This stuff is hypothetical and hard to find.

This paper, by a team of researchers from Kazakhstan and Kyrgyzstan, asks a bold question: Can we build a stable wormhole using only "normal" ingredients? Specifically, they tried using two things we know exist:

1. Electrons (Spinor fields): Tiny particles with a "spin" (like a tiny spinning top).
2. Electric fields: The force that makes magnets stick or lightning strike.

Here is the simple breakdown of their discovery, using some creative analogies.

1. The "Tug-of-War" Construction

Think of the wormhole as a tunnel connecting two identical, empty rooms (Minkowski spacetimes). Usually, gravity tries to crush this tunnel shut, like a heavy blanket falling over a hole. To keep it open, you need a force pushing back.

In this paper, the authors found a way to use the spin of the electrons and their electric charge to create a "tug-of-war."

• The electrons act like a fluid swirling inside the tunnel.
• The electric field acts like a tensioned rubber band.
• When these two forces interact just right, they create a perfect balance that holds the tunnel open without needing any "magic" anti-gravity matter.

2. The "Lopsided" Tunnel

Most people imagine a wormhole as a perfectly symmetrical tube, like a donut. But the universe doesn't always play fair.

The researchers discovered that these wormholes are asymmetrical. Imagine a tunnel that looks like a funnel.

• One side (let's call it the "Left Room") might look small and cozy.
• The other side (the "Right Room") might look huge and expansive.

Even though both rooms are made of the same "fabric" (identical empty space), if you stand in the Left Room, the wormhole looks tiny. If you stand in the Right Room, it looks massive. Furthermore, the "weight" of the wormhole looks different depending on which side you measure it from. It's like a scale that is broken: one side says the object weighs 5 pounds, and the other says it weighs 50 pounds.

3. The "Dial" of the Universe

The team found that they could control the shape and weight of these wormholes by turning three "dials":

1. The Throat Size: How wide the narrowest part of the tunnel is.
2. The Spin Frequency: How fast the electrons are "vibrating" or oscillating inside the tunnel.
3. The Coupling Constant: How strongly the electrons and the electric field talk to each other.

By tweaking these dials, they found something surprising: Some of these wormholes have "negative mass."

• Positive Mass: Pulls things in (like a planet).
• Negative Mass: Pushes things away (like a repulsive ghost).
• The paper shows that depending on how you tune the dials, the wormhole can act like a cosmic vacuum cleaner that pushes everything away, or a heavy anchor that pulls everything in.

4. The "Extreme" Limit

The researchers also looked at what happens when the electric charge gets very strong. They found that in this extreme state, the wormhole starts to look less like a tunnel and more like a specific type of black hole known as a Reissner-Nordström black hole.

Think of it like this: If you squeeze the tunnel too hard with electric force, the walls of the tunnel flatten out, and the shortcut disappears, turning into a one-way trap (a black hole). However, right before it collapses, the math shows a beautiful, stable state where the tunnel is perfectly balanced.

5. Is it Stable? (The "Wobbly" Factor)

The big question is: If you built one of these, would it stay open, or would it collapse instantly?

• The authors admit they haven't done the full "stress test" yet. Previous studies suggest that similar wormholes might be unstable and could collapse into black holes over time.
• However, they found that for certain settings, the "binding energy" (the glue holding the system together) is positive, which is a good sign. It's like finding a bridge that might hold a car, even if we haven't driven a truck across it yet.

The Bottom Line

This paper is a theoretical "blueprint." It proves that you don't need magic anti-gravity to make a wormhole. You just need a very specific, complex dance between spinning electrons and electric fields.

While we can't build these in a lab today (they require energies far beyond our reach), this discovery changes the rules of the game. It tells us that the universe might be full of these "lopsided" shortcuts, hidden in the fabric of space, held open by the very particles that make up our own bodies.

In short: They found a way to build a cosmic shortcut using only standard physics, but the shortcut is lopsided, might weigh "negative" pounds, and requires a very precise recipe to keep from collapsing.

Technical Summary

Here is a detailed technical summary of the paper "Wormholes in Einstein-Dirac-Maxwell theory with identical spacetime asymptotics."

1. Problem Statement and Motivation

The paper addresses the construction of traversable wormhole solutions within General Relativity (GR) without resorting to "exotic matter" (such as phantom scalar fields) that violates energy conditions.

• Context: Traditional wormhole solutions in GR typically require matter violating the Null Energy Condition (NEC). While dark energy is a candidate, the authors aim to find solutions using standard classical fields: spinor (Dirac) fields and electromagnetic (Maxwell) fields.
• Previous Work: Earlier studies within Einstein-Dirac-Maxwell (EDM) theory (e.g., Refs. [9, 12, 13]) produced wormhole solutions, but they suffered from limitations:
  • Some required thin shells (discontinuities).
  • Others connected two non-identical Minkowski spacetimes (asymmetric asymptotics).
• Goal: The authors seek regular, asymmetric solutions that connect two identical Minkowski spacetimes, supported solely by smooth metric, spinor, and electric fields, without thin shells.

2. Methodology

The study employs a spherically symmetric ansatz within the Einstein-Dirac-Maxwell framework.

A. Theoretical Framework

• Action: The total action includes the Einstein-Hilbert term, the Dirac spinor field action ($S_{sp}$), and the Maxwell electromagnetic field action ($S_{EM}$).
• Fields:
  • Metric: A spherically symmetric metric with a throat parameter $r_0$, defined by functions $A(r)$ and $B(r)$.
  • Spinor Field: A stationary ansatz for two fermions with opposite spins and equal mass $\mu$, oscillating with frequency $\Omega$. This ensures a spherically symmetric energy-momentum tensor despite individual fermion asymmetry.
  • Electromagnetic Field: A static electric potential $A_\mu = \{\phi(r), 0, 0, 0\}$.
• Equations: The system yields a set of coupled non-linear ordinary differential equations (ODEs) for the metric functions ($A, B$), spinor components ($u, v$), and electric potential ($\phi$).

B. Boundary Conditions and Numerical Approach

• Asymptotics: Unlike previous works, boundary conditions are imposed at both $x \to +\infty$ and $x \to -\infty$ (where $x$ is a dimensionless radial coordinate).
  • The metric approaches Minkowski space ($A \to 0, B \to 1$) on both sides.
  • Spinor fields decay exponentially to zero.
  • This setup forces the solution to describe a wormhole connecting two identical asymptotic regions.
• Numerical Method:
  • The infinite domain $(-\infty, \infty)$ is mapped to a finite interval $[-1, 1]$ using a compactified coordinate.
  • The resulting boundary value problem is solved using a modified Newton method with sparse direct solvers (Intel MKL PARDISO).
  • The constraint equation (the $(r,r)$-component of Einstein's equations) is used to verify consistency and determine specific asymptotic values.

3. Key Contributions

1. Identical Asymptotics: The paper successfully constructs a new family of wormhole solutions where the two universes connected are identical (same ADM mass and geometry at infinity), a feature not achieved in previous smooth EDM wormhole models.
2. Asymmetry in the Interior: While the asymptotic ends are identical, the solutions are asymmetric with respect to the center ($x=0$). The throat, mass distribution, and field maxima are shifted away from the geometric center.
3. Parameter Space Exploration: The study systematically explores the parameter space defined by:
   • Throat parameter ($x_0$).
   • Spinor frequency ($\bar{\Omega}$).
   • Coupling constant ($\bar{e}$, representing the interaction between spinor and electric fields).

4. Key Results

A. Mass and Geometry

• Negative ADM Mass: For uncharged spinor fields ($\bar{e}=0$), the configurations can possess negative ADM masses depending on the spinor frequency $\bar{\Omega}$ and throat parameter $x_0$. This is attributed to the negative energy density of the spinor field violating the weak energy condition.
• Mass Divergence: As $\bar{\Omega} \to 0$ (or $\bar{\Omega} \to \bar{\Omega}_{crit} \approx -\bar{e}$ for charged cases), the ADM mass diverges ($M \sim |\bar{\Omega}|^{-1}$).
• Throat Location: Due to asymmetry, the throat (minimum circumferential radius) is always located to the left of the center ($x < 0$).
• Single Throat: All numerical solutions found possess a single throat. While double-throat solutions exist in other theories (e.g., ghost scalar fields), none were found here within the explored parameter range.

B. Charged vs. Uncharged Systems

• Uncharged ($\bar{e}=0$): Masses approach zero (or become negative) as $\bar{\Omega} \to -1$. The ratio of charge to mass is not applicable here as the system is neutral.
• Charged ($\bar{e} \neq 0$):
  • The mass curves shift leftward as the coupling constant increases.
  • A critical frequency $\bar{\Omega}_{crit} \approx -\bar{e}$ exists where mass diverges.
  • In the limit $\bar{\Omega} \to \bar{\Omega}_{crit}$, the spinor fields vanish, and the solution approaches the extremal Reissner-Nordström metric ($Q \approx M$).
  • The coupling constant is restricted to the range $0 \le \bar{e} < 1$ for these solutions to exist.

C. Stability Analysis

• Binding Energy (BE): The authors calculate the binding energy ($BE = E_{free} - E_{total}$). Configurations with negative BE are deemed energetically unstable.
• Mass-Radius Relations: The study constructs mass-radius curves. Unlike neutron or boson stars where a turning point often signals instability, these wormhole configurations show complex behavior:
  • Some curves have no turning points.
  • Regions where mass increases as radius decreases can still be unstable if the binding energy is negative.
  • The paper suggests that energetically stable configurations likely exist for all values of $\bar{e}$, though a full linear perturbation analysis (which is complicated by the asymmetric spinor field) is required for definitive proof.

D. Energy Conditions

• The solutions inherently violate the Null Energy Condition (NEC) and Weak Energy Condition (WEC), which is necessary for the non-trivial topology (wormhole). The violation is driven by the spinor field's negative energy density.

5. Significance

• Theoretical Advancement: This work demonstrates that wormholes connecting identical universes can be constructed using only standard classical fields (Dirac and Maxwell) within GR, provided one accepts the violation of energy conditions inherent to the spinor field.
• Refinement of Models: It resolves issues with previous models (thin shells, non-identical asymptotics) by providing a smooth, fully regular solution.
• Physical Insight: The discovery of negative ADM masses in these regular configurations highlights the unique gravitational properties of spinor fields in strong gravity regimes.
• Future Directions: The paper identifies the need for a rigorous stability analysis (linear and nonlinear perturbations) to confirm the physical viability of these configurations, noting that the asymmetry makes this analysis significantly more complex than for symmetric systems.

In summary, the paper provides a robust numerical and analytical framework for a new class of asymmetric wormholes in Einstein-Dirac-Maxwell theory, expanding the landscape of possible regular solutions in General Relativity.