A generalization of Penrose’s helicity theorem for space-times with nonzero dual mass

Abstract

An algebraic definition of the helicity operator ℋ is proposed for vacuum stationary and asymptotically flat wormholes (i.e., space-times where the manifold of orbits of the stationary Killing field has S2×R topology). The definition avoids the use of momentum space or Fourier decomposition of the gravitational degrees of freedom into positive and negative frequency parts, and is essentially geared to emphasize the role of nontrivial topology. It is obtained via the introduction of a total spin vector Sα derived from the dual Bondi four-momentum *Pα, both vectors originating in the presence of nontrivial homotopy groups. (Space-times with nonzero dual mass can be characterized by a conformal null boundary ℐ having the topology of an S1 fiber bundle over S2 with possible identifications along the fiber—lens space—or equivalently vanishing Bondi–News.) It is shown that Sα is a constant multiple of Pα, the total Bondi four-momentum, and if in addition the space-time admits a point at spacelike infinity, there is strong support for the past limit of Sα to be a null vector. This can be viewed as a generalization of Penrose’s result on the Pauli–Lubanski vector for classical zero rest-mass particles. The helicity operator at null infinity is rooted in the topology and turns out to be essentially the Hodge duality operator(*). The notion of duality appears as a global concept. Under such conditions, self- and anti-self-dual modes of the Weyl curvature could be viewed as states originating in the nontrivial topology. These results depend crucially on the presence of topological charges; it is tempting to speculate that such wormholes might be basic building blocks.