  We simulate the magnetosphere of the nearby millisecond pulsar PSR
J0437-4715, which is expected to have an unscreened electric potential due to
the lack of magnetic pair production. We incorporate General Relativistic (GR)
effects and study curvature radiation (CR) by primary electrons, but neglect
inverse Compton (IC) scattering of thermal X-ray photons by these electrons. We
find that the CR spectrum cuts off at energies below ~ 17 GeV, well below the
threshold of the H.E.S.S. telescope <~ 100 GeV, while other models predict a
much higher cutoff of >~ 100 GeV. GR theory also predicts a relatively narrow
pulse of ~ 0.2 phase width centered on the magnetic axis. EGRET observations
above 100 MeV significantly constrain the application of the Muslimov & Harding
(1997) model for gamma-ray production as a result of GR frame dragging, and
ultimately its polar cap (PC) current and accelerating potential. Whereas the
standard prediction of this pulsar's gamma-ray luminosity due to GR frame
dragging is ~10% of the spindown power, a non-detection by forthcoming H.E.S.S.
observations will constrain it to <~ 0.3%, enforcing an even more severe
revision of the accelerating electric field and PC current.
