Conformally covariant massless spin-two field equations

Abstract

An explicit proof is constructed to show that the field equations for a symmetric tensor fieldhab describing massless spin-2 particles in Minkowski space-time are not covariant under the 15-parameter groupSO4,2; this group is usually associated with conformal transformations on flat space, and here it will be considered as aglobal gauge group which acts upon matter fields defined on space-time. Notwithstanding the above noncovariance, the equations governing the rank-4 tensorSabcd constructed fromhab are shown to be covariant provided the contractionSab vanishes. Conformal covariance is proved by demonstrating the covariance of the equations for the equivalent 5-component complex field; in fact, covariance is proved for a general field equation applicable to massless particles of any spin >0. It is shown that the noncovariance of thehab equations may be ascribed to the fact that the transformation behaviour ofhab is not the same as that of a field consisting of a gauge only. Since this is in contradistinction to the situation for the electromagnetic-field equations, the vector form of the electromagnetic equations is cast into a form which can be duplicated for thehab-field. This procedure results in an alternative, covariant, field equation forhab.