Null-coordinate gravidynamics and the spin coefficients

By choosing a very natural tetrad we show that the null-energy density, found previously by two of us from a first-order action principle, is four times the modulus squared of the Newman-Penrose coefficient $\tau$. Moreover, we compute in terms of the usual tensorial field variables of gravity the more important three tetradic scalars of the conformal Weyl tensor. Consequently, we are able to write down the mass formulas in terms of the field variables. Also given is a null-energy formula in terms of line and surface integrals. All the results are given for fields which are asymptotically flat and asymptotically "uniformly smooth."