Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 975universe evolves in time. They would feel that the boundary conditionsfor the universe that determine how the universe began were a questionfor metaphysics or religion rather than science [Hawking and Israel (1979);Hawking and Penrose (1996)].Hawking–Penrose theorems showed that according to general relativitythere should be a singularity in our past. At this singularity the field equationscould not be defined. Thus classical general relativity brings about itsown downfall: it predicts that it cannot predict the universe. For Hawkingthis sounds rally disturbing: If the laws of physics could break down at thebeginning of the universe, why couldn’t they break down any where. Inquantum theory it is a principle that anything can happen if it is not absolutelyforbidden. Once one allows that singular histories could take partin the path integral they could occur any where and predictability woulddisappear completely. If the laws of physics break down at singularities,they could break down any where.The only way to have a scientific theory is if the laws of physics holdeverywhere including at the beginning of the universe. One can regard thisas a triumph for the Principle of Democracy: Why should the beginningof the universe be exempt from the laws that apply to other points. If allpoints are equal one cannot allow some to be more equal than others.To implement the idea that the laws of physics hold everywhere, oneshould take the path integral only over non–singular metrics. One knowsin the ordinary path integral case that the measure is concentrated on non–differentiable paths. But these are the completion in some suitable topologyof the set of smooth paths with well defined action. Similarly, one wouldexpect that the path integral for quantum gravity should be taken over thecompletion of the space of smooth metrics. What the path integral cannotinclude is metrics with singularities whose action is not defined.In the case of black holes we saw that the path integral should be takenover Euclidean, that is, positive definite metrics. This meant that thesingularities of black holes, like the Schwarzschild solution, did not appearon the Euclidean metrics which did not go inside the horizon. Insteadthe horizon was like the origin of polar coordinates. The action of theEuclidean metric was therefore well defined. One could regard this as aquantum version of Cosmic Censorship: the break down of the structure ata singularity should not affect any physical measurement.It seems, therefore, that the path integral for quantum gravity should betaken over non–singular Euclidean metrics. But what should the boundaryconditions be on these metrics. There are two, and only two, nat-