PHYSICAL REVIEW D VOLUME 13, NUMBER 2 15 JANUARY 1976 Null-coordinate gravidynamics and the spin coefficients C. Aragone* International Centre for Theoretical Physics, Trieste, Italy R. Gambini and A. Restuccia Departamento de Fisica, Universidad Simon Bolivar, Apartado 80659, Caracas 108, Venezuela (Received 2 September 1975) 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 T. 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." I. INTRODUCTION The dynamics of the Einstein field, either in the vacuum or in the presence of a reasonable matter field (for instance, an Abelian massive vector field), shows many interesting features when it is analyzed by starting from a first-order action (with the usual variables g,, , I?,, a) and a set of coordinate conditions is chosen which imply that one of them (u) (at least) is null, i.e., guu =O and that one (r) of the three remaining coordinates (r, xl, x2) is the affine parameter along the light rays contained in each of the null hypersurfaces. For instance, it has been shown by Aragone and Chela-Flores (ACh)' that after the reduction process is accomplished, taking a s the evolution coordinate the null one (u), one is left with an action depending upon the minimum number of physical variables allowed for the pure helicity- 2, massless, self-interacting field: 2. This is half the number of independent variables needed in the canonical 3 + 1 analysis of Arnowitt, Deser, and Misner,' even if in this frame one does not need to make an apriori choice of coordinates. Another relevant property of the 2 + 2 null dy - namics is the simplicity of the differential con- straints. Instead of being real, coupled, partial differential equations mixing democratically the derivatives with respect to the three internal co- ordinates of the null hypersurfaces, they are now mostly3 ordinary differential equations with the affine parameter r a s the independent variable. The third interesting property that Aragone and Restuccia (AR)4 have shown is that, after solving the differential vectorial constraint, a reduced action can always5 be reached which ex- plicitly shows a very simple structure -4q' - Ju: It has a dynamical germ of the null-canonical type -4q' minus an explicitly non-negative quan- tity JU 3 0 which was called the total null-energy density of the system. This non-negative quan- tity 5' = JE + Ji is the sum of the pure gravita- tional quantity J;, the null energy of gravity, and J f$ (a non-negative quantity also) which provides the contribution of the matter-independent dynam- ical variables to the null energy of the interacting sy~tem.~ It was also shown6 that for linearized gravity7 S J i d G d r goes to the flat-space generator P,. In spite of this, many other questions related to the physical significance of Ji could also be raised. For instance, we can ask ourselves whether the integration of the null energy over a null hypersurface, assuming some definite asymptotic behavior of the field, might give a constant. Or, also, one can look for the connection between the mass of the system (in the sense of the definition given by Newman and Untie) and J t. Or, even more generally: Is there some connection between the spin-coefficient approach developed by New- man and Penroseg (where the null coordinates play a substantial role too) and the simpler dy - namical approach developed in Refs. 1 and 4? In order to give an answer to some of these questions, we shall give in the next section the minimum nontrivial review of the results already obtained for pure gravity in the 2 + 2 tensorial approach. Therefore, we shall recall some of the definitions introduced by Newman and Penrose (NP)9 in their spin-coefficient approach, either for representing the dynamics of the local tetrad or for analyzing the behavior of the conformal Weyl tensor.'' After doing that, in Sec. 111, we evaluate the NP scalars and the Newman-Unti mass formula in terms of the tensorial variables, showing very simple results.