  We study the initial value problem of the Einstein-Dirac system, and show the
stability of the Minkowski solution in the massless case with the use of
generalized wave coordinates. This requires the understanding of the Dirac
equation in curved spacetime, for which we establish various estimates. The
proof is based on the vector-field method which is widely used in works on the
stability of Minkowski problems for other Einstein-coupled systems. Under a
specific choice of the tetrad, we show that components of the Dirac field
satisfy a quasilinear wave equation, by resolving a potential loss of
derivative problem. We also show that the semilinear nonlinearity of this
equation behaves like a null form. In this way, we obtain the sharp decay of
the field along the light cone. The structure of the energy-momentum tensor
causes worse behavior of some components of the metric than the vaccum case,
but an additional structure shows that there is no harm to the global existence
result. In addition, we develop an estimate of the Dirac equation itself
adpated to the decay of the metric, as this provides better estimates in the
interior compared with the estimates from the second order equation. The
combination of these estimates leads to a good control of the Dirac field that
helps close the proof. We shall also see how our argument here gives a proof of
the wellposedness of the system in the massive case.
