  In the absence of gravity, one can prove that tunnelling instantons exhibit
exactly one negative mode in their spectrum of fluctuations. It is precisely
the existence of this tunnelling negative mode that warrants an interpretation
of these solutions as mediating the decay of a metastable vacuum. In the
presence of gravity the situation is much more subtle, not least because of
diffeomorphism invariance. New complications arise here: in particular, the
kinetic term of the fluctuations can change sign somewhere along the instanton.
We show that in this case the mode functions remain non-singular, and the
tunnelling negative mode continues to exist. Moreover, the eigenvalues vary
continuously when the potential is varied such that the kinetic term of the
fluctuations switches sign. However, the negative kinetic term implies the
additional existence of an infinite tower of negative modes, whose significance
and interpretation remain elusive.
