Non-existence of dynamical perturbations of Schwarzschild with vanishing self-dual part

A recent proposal of Ashtekar (1985) for the canonical quantisation of gravity involves a new choice of a (complex) polarisation on the (real) phase space of general relativity. In order for the 'four dimensional' version of Ashtekar's proposal to be viable, it is necessary that for an arbitrary (real) background solution there exist a sufficient number of (complex) solutions of the linearised Einstein equation with perturbed Weyl spinor having vanishing self-dual part. It is shown that this is not the case by explicitly demonstrating that for the Schwarzschild spacetime, aside from stationary perturbations, there are no such linearised solutions which can be expressed as superpositions of modes of real frequencies. The viability of the '3+1 version' of Ashtekar's proposal is not affected by these considerations.