Positive-action conjecture

The Euclidean action of the gravitational field is not positive definite under conformal transformations. This poses difficulties for the convergence of path integrals in quantum gravity. Hawking has suggested that this difficulty may be overcome by dividing the space of all metrics into conformal equivalence classes. Then real conformal transformations within each class are to be replaced by complex conformal transformations of the member of the class having $R=0$. This gives a positive-definite action in nonsingular cases if Hawking's positive-action conjecture is true: All nonsingular asymptotically flat metrics with $R=0$ everywhere have non-negative action. This conjecture is shown to be true for static spacetimes without horizons if the positive-energy conjecture is true. However, it is false for spacetimes with horizons. Such spacetimes are asymptotically flat only in spatial but not in timelike directions. This suggests that the positive-action conjecture should be restricted to asymptotically Euclidean spacetimes. The failure of the conjecture when there is a horizon present seems to introduce a factor of $i$ into the partition function which neatly cancels another factor of $i$ to give a real density of states for the gravitational field.