Gravitational Electric-Magnetic Duality, Gauge Invariance and Twisted Self-Duality

Abstract

The concept of electric-magnetic duality can be extended to linearized gravity. It has indeed been established that in four dimensions, the Pauli-Fierz action (quadratic part of the Einstein-Hilbert action) can be cast in a form that is manifestly invariant under duality rotations in the internal 2-plane of the spacetime curvature and its dual. In order to achieve this manifestly duality-invariant form, it is necessary to introduce two "prepotentials", which form a duality multiplet. These prepotentials enjoy interesting gauge invariance symmetries, which are, for each, linearized diffeomorphisms and linearized Weyl rescalings. The purpose of this note is twofold: (i) To rewrite the manifestly-duality invariant action obtained in previous work in a way that makes its gauge invariances also manifest. (ii) To explicitly show that the equations of motion derived from that action can be interpreted as twisted self-duality conditions on the curvature tensors of the two metrics obtained from the two prepotentials.