Duality Symmetry and Anomaly for Gravitational Waves in Curved Spacetime

TL;DR

This work reveals that linearized gravitational perturbations in flat spacetime exhibit a Maxwell-like duality between gravitoelectric and gravitomagnetic components, accompanied by a Noether helicity current that counts the difference between right- and left-handed graviton polarizations. By reformulating in self-dual variables, the authors cast the dynamics into a Dirac-like, spinorial framework where duality acts as a chiral rotation, and extend the construction to curved backgrounds using tetrads and a covariant spinor structure. A nonvanishing quantum anomaly arises in curved spacetimes, computed via DeWitt-Schwinger renormalization, yielding a curvature-squared term proportional to $R_{\alpha\beta\mu\nu}{}^{\star}R^{\alpha\beta\mu\nu}$ that signals a breakdown of the classical duality at the quantum level. The results generalize chiral anomalies to massless spin-2 fields and imply possible gravitonic polarization effects in curved backgrounds, while the classical theory remains valid in the geometric optics limit. Overall, the paper provides a coherent Maxwell-like description of linearized gravity, develops a spinor formulation, and identifies a curvature-induced quantum anomaly with potential physical consequences for gravitational radiation in curved spacetimes.

Abstract

The vacuum Einstein equations admit a formulation closely analogous to the source-free Maxwell theory. In particular, the linearized equations exhibit an electric-magnetic duality symmetry. We develop a framework that makes this analogy manifest by explicitly identifying the electric and magnetic components of perturbative gravitational waves. Within this formulation, we show that duality rotations between these gravitoelectric and gravitomagnetic fields constitute a Noether symmetry of the linearized theory, and we derive the associated conserved current. The corresponding conserved charge encodes the difference in intensity between the right- and left-handed circularly polarized components of the gravitational wave - that is, between its self-dual and anti-self-dual parts. Remarkably, this conservation law remains valid even when the gravitational perturbations propagate on generic curved backgrounds. We then investigate whether this symmetry survives quantization. While the duality symmetry is preserved at the quantum level in flat spacetime, we find that it is anomalously broken in curved backgrounds. As a result, an imbalance between right- and left-handed gravitons could be excited from the vacuum. This effect represents a chiral anomaly for massless spin-two fields, generalizing known results for fermions and spin-one photon fields.