Radiation in Yang-Mills formulation of gravity and a generalized pp-wave metric

The variational methods implemented on a quadratic Yang-Mills type Lagrangian yield two sets of equations interpreted as the field equations and the energy-momentum tensor for the gravitational field. A covariant condition is imposed on the energymomentum tensor to represent the radiation field. A generalized pp-wave metric is found to simultaneously satisfy both the field equations and the radiation condition. The result is compared with that of Lichnérowicz. Based on its gauge structure which was first recognized by Utiyama [1], alternative formulations of gravitation especially the ones that are derivable from quadratic Lagrangians of Yang-Mills (YM) type have received substantial interest in the literature [2, 3, 4], not merely as an academic curiosity, but essentially because when the gravitational Lagrangian is coupled to matter the renormalization problems are much less severe [5]. For massless particles like photons and neutrinos radiation presents itself, if for all observers their respective energy flows in the same direction with that of light. As to the electromagnetic field this situation is well known and the neutrino radiation in curved space is also well established [6]. It has been shown that any solution to Einstein-Weyl equations represent neutrino radiation if the energy-momentum tensor of the Weyl field can be expressed in terms of a null four vector which is collinear to its four-momentum. Therefore, there is no reason not to expect that the gauge quanta of gravitation retains the same conditions to be in a radiative state. In this article, we begin with a YM type Lagrangian which is quadratic in the Riemann tensor and employ Palatini’s variational method to derive two sets of equations whose interpretations are given as the field equations and the gravitational energy-momentum tensor. According to Palatini’s method, which was first implemented by Stephenson[7] and then elaborated by Fairchild[2], the equations may be derived by independent variations of the Lagrangian for the connection and for the metric. We introduce covariant conditions on the energy-momentum tensor to represent gravitational radiation and for the solutions specialize on a generalized form of a metric which represents plane fronted waves with parallel rays (pp-wave). We compare the radiation criteria with that of Lichnérowicz.