On a wave-particle in closed and open isotropic universes

Mathematics – Mathematical Physics

Scientific paper

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Cosmology, Einstein Field Equations, Nonlinear Differential Equations, Nonlinear Field Theory, Quantum Gravity, Space-Time Configurations, Wave Functions, Exact Solutions, Quantum Gravity, Nonlinear Or Nonlocal Theories And Models

Scientific paper

The Klein-Gordon equation satisfied by the wave function in general relativity is solved for the metric of the closed and open universe corresponding to Einstein-De Sitter-Friedmann isotropic cosmological model. The angular dependences are specified by spherical harmonics for the longitude and latitude, and for the hyperlatitude by modified spherical harmonics having as variable circular functions for the closed universe and hyperbolic functions for the open universes. The time dependence of the probabilistic wave function is similar for the closed and open universes and is obtained in the following three cases: (I) constant Hubble parameter, (II) constant decceleration parameter, and (III) uniform matter and energy distribution, which corresponds to the Hubble parameter a linear function of time. Thus six solutions are obtained, namely, the three cases I-III each for closed and open isotropic universes. For each of these six solutions is considered: (i) the existence of singularities in space-time including asymptotic time in the future or past, (ii) the square integrability of the wave function over the full extent of the four-dimensional space-time, and (iii) the existence or otherwise of a positive probability density associated with the wave function.

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