Asymptotically Flat Self-dual Solutions to Euclidean Gravity

The discovery of pseudoparticle solutions to the euclidean SU(2) Yang-Mills theory [1 ] has suggested the possibility that analogous solutions might occur in Einstein's theory of gravitation. The existence of such solutions would have a profound effect on the quantum theory of gravitation [2,3]. Since fire Yang-Mills pseudoparticles possess self-dual field strengths, one likely possibility is that gravitational pseudoparticles are characterized by self-dual curvature. In fact it has been pointed out by Hawking [3] that the Taub-NUT metric [4], when appropriately continued to euclidean space-t ime, produces a selfdual curvature and hence is a possible candidate for a gravitational pseudoparticle. He has also given a generalized multi-Taub-NUT metric. However, these metrics do not approach a fiat metric at infinity [5]. To see this, let us write the euclidean Taub-NUT solution as