Obstructions to Traversable Wormholes in Einstein-Dirac Theory

Technical Summary: Obstructions to Traversable Wormholes in Einstein-Dirac Theory

Problem Statement
Traversable wormholes are hypothetical solutions to Einstein's equations that require matter violating the Averaged Null Energy Condition (ANEC). While classical bosonic fields generally satisfy energy conditions, classical Dirac fields have been proposed as candidates for sourcing such geometries due to their ability to violate the null energy condition. Recent works (e.g., Blázquez-Salcedo et al., Konoplya and Zhidenko, Kain) claimed to find traversable wormhole solutions in the Einstein-Dirac-Maxwell system. However, these claims suffered from two critical flaws: they treated single-particle states as having negative frequencies (which lack a physical semiclassical interpretation in this context) and they incorrectly modeled the electromagnetic self-interaction of charged Dirac fields. This paper investigates whether a physically meaningful, normalized, positive-frequency, neutral classical Dirac field can source a static, spherically symmetric, asymptotically flat traversable wormhole within the Einstein-Dirac system.

Methodology
The author employs a rigorous semiclassical approach, first establishing the correct physical interpretation of classical Dirac fields in curved spacetime and then performing numerical evolutions of the fully backreacting Einstein-Dirac system.

1. Semiclassical Framework (Sec. II): The paper clarifies that single-particle states of a Dirac field, even when charged, evolve according to the free Dirac equation (without the self-generated vector potential) because interactions are absorbed into the definition of the mass and spin in the one-particle Hilbert space. Consequently, the stress-energy tensor must be derived from the free field expression to satisfy local conservation laws when coupled to gravity. The author restricts their analysis to neutral Dirac fields to avoid inconsistencies in the stress-energy conservation of the Einstein-Dirac-Maxwell system.
2. Fixed Background Analysis (Sec. III & IV): The author analyzes Dirac fields on fixed wormhole geometries. They demonstrate that while pointwise violations of the null energy condition are easy to construct, obtaining a static, spherically symmetric solution requires an incoherent superposition of stationary bound states with different angular momenta ($m$) but fixed frequency ($\omega$) and total angular momentum ($\ell$). They show that such stationary bound states can indeed violate the ANEC on fixed backgrounds.
3. Numerical Evolution of the Full System (Sec. V): The core of the investigation involves numerically solving the coupled Einstein-Dirac equations. The author parameterizes asymptotic initial data at one end of the wormhole ($x \to -\infty$) using four parameters: dimensionless frequency ($\Omega$), dimensionless mass ($\tilde{M}$), spinor magnitude ($I$), and a "parity angle" ($\rho$) determining the mix of sphere-parity even and odd components. They evolve these data inward using an adaptive Runge-Kutta method to search for solutions that form a regular throat and extend to a second asymptotically flat end ($x \to +\infty$).

Key Contributions and Results

• Positive Mass Theorem: The author proves that any static, spherically symmetric, asymptotically flat traversable wormhole solution to the Einstein-Dirac equations must have a positive ADM mass at each asymptotic end, bounded below by half the throat radius. This is derived from the positivity of the time-time component of the Dirac stress-energy tensor ($T_{\hat{t}\hat{t}} > 0$).
• Partial-Wormhole Solutions: The author successfully constructs "partial-wormhole solutions." These are regular solutions that possess a throat and correct asymptotic behavior at one end ($x \to -\infty$) but cannot be extended to a second asymptotic end. This confirms that the Einstein-Dirac equations admit local solutions near a throat without immediate singularities.
• Obstruction via Sphere-Parity (Sec. V D): For generic wormhole geometries (without reflection symmetry), the Dirac bound states at a given frequency possess a definite sphere-parity (either even or odd). The author's numerical scans reveal a strict obstruction:
  • For a fixed frequency $\omega$ and angular momentum $\ell$, only one sphere-parity (either even or odd) allows the formation of a throat when evolving from a specific asymptotic end.
  • Crucially, the parity required to form a throat from the left end ($x \to -\infty$) is the opposite of the parity required to form a throat from the right end ($x \to +\infty$).
  • Since sphere-parity is a global label for the solution, a single solution cannot satisfy the throat-forming conditions at both ends simultaneously. Thus, no global traversable wormhole solution exists for definite-parity states.
• Obstruction in Reflection-Symmetric Cases (Sec. V E): The author addresses the special case where the geometry is reflection-symmetric, which theoretically allows for linear combinations of both parities at the same frequency. They derive necessary conditions for the spinor components at the throat ($x=0$) to maintain reflection symmetry (specifically, that the spinor must be an eigenstate of the reflection operator). An extensive numerical scan of the parameter space shows that the condition $Q^2_t = 1$ (required for the throat conditions) is never approached; the maximum value found was $\approx 1.4 \times 10^{-4}$. This indicates that even in the symmetric case, the necessary conditions for a throat cannot be met.

Significance and Claims
The paper claims that the Einstein-Dirac system, when sourced by physically realistic, normalized, positive-frequency Dirac fields, does not admit static, spherically symmetric, asymptotically flat traversable wormhole solutions.

The significance of this work lies in its correction of previous literature's handling of single-particle states and self-interactions, and its provision of strong numerical evidence against the existence of such wormholes. The author argues that previous claims of traversable wormholes in this system were artifacts of unphysical assumptions (negative frequencies or incorrect self-interaction terms). By demonstrating that the parity requirements for throat formation are mutually exclusive at the two asymptotic ends, and that reflection-symmetric configurations fail to satisfy necessary throat constraints, the paper concludes that the obstructions to traversable wormholes in Einstein-Dirac theory are robust, arising from the self-consistency requirements between the Dirac field and the geometry it sources.