The metric and curvature properties of H -space

The space H of asymptotically (left-) shear-free cuts of the J+ (good cuts) of an asymptotically flat space-time M is defined. The connection between this space and the asymptotic projective twistor space PJ of M is discussed, and this relation is used to prove that H is four-complex-dimensional for sufficiently ‘calm’ gravitational radiation in M. The metric on H-space is defined by a simple contour integral expression and is found to be complex Riemannian. The good cut equation governing H-space is solved to three orders by a Taylor series and the solution is used to demonstrate that the curvature of H-space is always a self dual (left flat) solution of the Einstein vacuum equations.