  We obtain necessary and sufficient conditions for the existence of
"conservation laws" on null hypersurfaces for the wave equation on general
four-dimensional Lorentzian manifolds. Examples of null hypersurfaces
exhibiting such conservation laws include the standard null cones of Minkowski
spacetime and the degenerate horizons of extremal black holes. Another
(limiting) example of such a conservation law is that which gives rise to the
well-known Newman-Penrose constants along the null infinity of asymptotically
flat spacetimes. The existence of such conservation laws can be viewed as an
obstruction to a certain gluing construction for characteristic initial data
for the wave equation. We initiate the general study of the latter gluing
problem and show that the existence of conservation laws is in fact the only
obstruction. Our method relies on a novel elliptic structure associated to a
foliation with 2-spheres of a null hypersurface.
