Obstructions to Traversable Wormholes in Einstein-Dirac Theory
Traversable wormholes are among the most striking hypothetical solutions of general relativity, but every such geometry requires matter that violates the averaged null energy condition (ANEC). A possible candidate for such matter is the Dirac field. Several recent works have claimed traversable wormhole solutions of the classical Einstein-Dirac-Maxwell system, but these works did not properly take into account that single-particle states do not self-interact electromagnetically and that the mode functions of single-particle states are of positive frequency. In this paper, we show that classical, positive-frequency Dirac fields on certain fixed wormhole geometries can violate ANEC, so they are indeed candidates for sourcing traversable wormholes. We then numerically search for static, spherically symmetric, asymptotically flat traversable wormhole geometries sourced by Dirac fields of a definite frequency $ω>0$, angular momentum quantum number $\ell$, and definite parity. We find "partial-wormhole solutions," describing a regular wormhole throat with correct asymptotics at one end of the wormhole, but we find that these solutions cannot be continued to a second asymptotically flat end. In the case of reflection-symmetric wormholes, we perform an additional search where we do not assume that the Dirac solution has a definite parity. In this case, after an extensive scan of throat-forming asymptotic data, we find that the conditions necessary for a reflection-symmetric wormhole throat cannot be obtained. Taken together, these results strongly support that the Einstein-Dirac system does not admit traversable wormhole solutions when sourced by a physically meaningful Dirac field.