  The recent discovery of day-long gamma-ray flares in the Crab Nebula,
presumed to be synchrotron emission by PeV (10^{15} eV) electrons in milligauss
magnetic fields, presents a strong challenge to particle acceleration models.
The observed photon energies exceed the upper limit (~100 MeV) obtained by
balancing the acceleration rate and synchrotron radiation losses under standard
conditions where the electric field is smaller than the magnetic field. We
argue that a linear electric accelerator, operating at magnetic reconnection
sites, is able to circumvent this difficulty. Sufficiently energetic electrons
have gyroradii so large that their motion is insensitive to small-scale
turbulent structures in the reconnection layer and is controlled only by
large-scale fields. We show that such particles are guided into the
reconnection layer by the reversing magnetic field as they are accelerated by
the reconnection electric field. As these electrons become confined within the
current sheet, they experience a decreasing perpendicular magnetic field that
may drop below the accelerating electric field. This enables them to reach
higher energies before suffering radiation losses and hence to emit synchrotron
radiation in excess of the 100 MeV limit, providing a natural resolution to the
Crab gamma-ray flare paradox.
