  We revisit the question of whether classical general relativity obeys, beyond
the linearised order, an analogue of the global U(1) electric-magnetic duality
of Maxwell theory, with the Riemann tensor playing the role analogous to the
field strength. Following contradictory claims in the literature, we present a
simple gauge-invariant argument that the duality does not hold. The duality
condition is the conservation of the helicity charge. Scattering amplitudes of
gravitons in general relativity, and of gluons in Yang-Mills theory, violate
this selection rule already at tree level. Indeed, the
maximally-helicity-violating (MHV) amplitudes are famous for their simplicity.
The duality in the linearised theories is, therefore, broken by the
interactions. In contrast, the tree-level scattering amplitudes in
duality-invariant theories of non-linear electromagnetism are known to obey
helicity conservation. While the duality is not a symmetry of the full theory
of general relativity, it does hold within a sector of the solution space,
including vacuum type D solutions, where the duality is known to rotate between
mass and NUT charge.
