Non-Existence of Traversable Wormholes in Einstein-Dirac Theory

Traversable wormholes are among the most striking hypothetical solutions of general relativity, but every such geometry necessarily violates the averaged null energy condition. To evade this obstruction, most constructions invoke hypothetical "exotic matter." A more realistic candidate is the classical Dirac field, which describes spin-1/2 particles such as electrons and can violate the averaged null energy condition even at the classical level. Several recent works have accordingly claimed traversable wormhole solutions of the classical Einstein-Dirac-Maxwell system. In this dissertation, we argue that these constructions break down once one restricts to classical configurations that admit a legitimate semiclassical interpretation.

We first analyze when a classical Dirac field is a faithful model of the underlying quantum field. We show that a free, positive-frequency, normalizable, classical Dirac solution captures the physical behavior of a single-particle state of the Dirac field, whose classical stress-energy matches the expected quantum stress-energy, when vacuum stress-energy is neglected. Consequently, earlier Einstein-Dirac-Maxwell constructions (which use negative frequencies and improperly treat the electromagnetic coupling as a self-interaction) do not meet the requirements for physical validity.

Working within this system, we present numerical evidence against the existence of static, spherically symmetric, asymptotically flat traversable wormhole solutions. We find that "partial-wormhole" solutions exist, which have correct asymptotics at one end and a regular wormhole throat, but our numerical results reveal a structural problem that arises when attempting to join such solutions into a global traversable wormhole. In the generic case, each Dirac solution carries a definite parity, and the parity permitting a throat to form when evolving inward from one asymptotic end is precisely the opposite of the parity permitting one from the other end; since parity is a global label, this would imply that no two-sided wormhole solutions exist. The exception, a reflection-symmetric wormhole in which both parities are accessible at a single frequency, is evaluated by numerical search: an extensive scan of throat-forming asymptotic data shows that the spinor condition required at a reflection-symmetric throat is never approached. Taken together, these results strongly indicate that the Einstein-Dirac system admits no traversable wormhole solutions when sourced by a physically meaningful classical Dirac field.