New PDF release: Hamiltonian Cosmology

By Michael Ryan (auth.)

ISBN-10: 3540057412

ISBN-13: 9783540057413

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22 In the orthonormal frame (d~, 1 , 52, ~3~ d3x = 82 ^ 5 2 A 5 $ and if we integrate over the ~i for closed spaces and artificially close open spaces and integrate we get I J IX universes this constant is A 52 A @3 = (4~)2, type I universes which are open. ~escaling the forms and the COnSt. Misner 21 Because for type chooses this condition for We shall make this choice in general Cijk if necessary). 9) This action will be in canonical form if we can find a set of parameters BA for 8ij such that and a matrix Pij(parametrized by a set of parameters (2~) (eg~**e-B)ij ~gij = pij~Bij = pJdBA, k the constraints for ~ k in terms of constraint C° = 0 we Pij' ~ij and ~.

Sin(~) e 2~ e B . 28) where C ijk are the structure constants of the particular Bianchi type considered, and the D a are three constants subject to the constraint D a Ci • = O. ~a = 0. additional constraint e ~ij Bj C i lm -- 0. There is an These are used to show that 48 there can be no pure magnetic field in types Vlll, IX, IV, V, Vl (h # - 7) and VII (h @ 0). Types III, VI (h = -1) and VII (h = 0) allow only one independent component of the magnetic field. Type II allows two inde- pendent magnetic field components while type I puts no constraints on the B i.

1) is a sketch of the motion of the universe point for where ~m/n corresponds to the maximum of expansion m~n (turnaround). • m~n and 8(~ Choices of constants of motion, ~ • ) m~n o and c, merely place at various points on the 8~-plane. The eqality of H to zero allows the following potential condition to be given 2 P8 where ~ ~ _ ~2 d~/dh. 2) shows the effect of this condition and points up the fact that the problem is equivalent to that of a particle scattering off a one-dimensional exponential potential in a two-dimensional space.

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Hamiltonian Cosmology by Michael Ryan (auth.)


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