Ivancevic_Applied-Diff-Geom

980 Applied Differential Geometry: A Modern Introductiongood, and that the calculation of the wave function of the universe will notbe affected by our ignorance of what happens in very high curvatures.One can also solve the field equations for boundary metrics that aren’texactly the round three sphere metric. If the radius of the three sphere isless than 1 H, the solution is a real Euclidean metric. The action will be realand the wave function will be exponentially damped compared to the roundthree sphere of the same volume. If the radius of the three sphere is greaterthan this critical radius there will be two complex conjugate solutions andthe wave function will oscillate rapidly with small changes in h ij .Any measurement made in cosmology can be formulated in terms ofthe wave function. Thus the no boundary proposal makes cosmology intoa science because one can predict the result of any observation. The casewe have just been considering of no matter fields and just a cosmologicalconstant does not correspond to the universe we live in. Nevertheless, itis a useful example, both because it is a simple model that can be solvedfairly explicitly and because, as we shall see, it seems to correspond to theearly stages of the universe.Although it is not obvious from the wave function, a de Sitter universehas thermal properties rather like a black hole. One can see this by writingthe de Sitter metric in a static form (rather like the Schwarzschild solution)ds 2 = −(1 − H 2 r 2 )dt 2 + (1 − H 2 r 2 ) −1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ).There is an apparent singularity at r = 1 H. However, as in the Schwarzschildsolution, one can remove it by a coordinate transformation and it correspondsto an event horizon.If one returns to the static form of the de Sitter metric and put τ = itone gets a Euclidean metric. There is an apparent singularity on thehorizon. However, by defining a new radial coordinate and identifyingτ with period 2π H, one gets a regular Euclidean metric which is just thefour sphere. Because the imaginary time coordinate is periodic, de Sitterspace and all quantum fields in it will behave as if they were at atemperature H 2π . As we shall see, we can observe the consequences ofthis temperature in the fluctuations in the microwave background. Onecan also apply arguments similar to the black hole case to the actionof the Euclidean–de Sitter solution [Witten (1998b)]. One finds that itπhas an intrinsic entropy ofH, which is a quarter of the area of the2event horizon. Again this entropy arises for a topological reason: theEuler number of the four sphere is two. This means that there cannotbe a global time coordinate on Euclidean–de Sitter space. One can in-