We discuss the Hamiltonian dynamics of general relativity with real
connection variables on a null foliation, and use the Newman-Penrose formalism
to shed light on the geometric meaning of the various constraints. We identify
the equivalent of Sachs' constraint-free initial data as projections of
connection components related to null rotations, i.e. the translational part of
the ISO(2) group stabilising the internal null direction soldered to the
hypersurface. A pair of second-class constraints reduces these connection
components to the shear of a null geodesic congruence, thus establishing
equivalence with the second-order formalism, which we show in details at the
level of symplectic potentials. A special feature of the first-order
formulation is that Sachs' propagating equations for the shear, away from the
initial hypersurface, are turned into tertiary constraints; their role is to
preserve the relation between connection and shear under retarded time
evolution. The conversion of wave-like propagating equations into constraints
is possible thanks to an algebraic Bianchi identity; the same one that allows
one to describe the radiative data at future null infinity in terms of a shear
of a (non-geodesic) asymptotic null vector field in the physical spacetime.
Finally, we compute the modification to the spin coefficients and the null
congruence in the presence of torsion.Comment: 23 pages + Appendix, 2 figures. v2: Improved text and some amendments
throughout, added more details on the relation between 2+2 foliations and
null tetrads, updated references. Version submitted for peer reviewing. v3:
Few minor amendments, footnote added on a null congruence in the presence of
torsion; matches published versio
