Studies of Non-Homogenous Cosmological Models

Citation

Omer, Guy Clifton, Jr. (1947) Studies of Non-Homogenous Cosmological Models. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/gjbh-1427. https://resolver.caltech.edu/CaltechTHESIS:06022025-214737280

Abstract

A family of spherically symmetric, non-static, non-homogeneous relativistic models with zero pressure is considered. In the first chapter the pioneer work of Lemaitre and Tolman is reviewed. The fundamental partial differential equation is transformed into a parametric pair which may be solved explicitly in terms of known functions.

In the second chapter the nature of the most general solutions and the conditions under which they will exist are determined. The Robertson-Tolman notation is extended to apply to non-homogeneous models. The fundamental equation is solved in terms of Weierstrassian elliptic functions. The solutions in terms of elliptic functions are then shown to behave in the expected manner.

Approximate solutions not involving elliptic functions which will apply when the cosmological constant is small as compared to another parameter are derived in the following chapter. Further information about the relations between the various solutions is obtained.

In the fourth chapter all of the special non-elliptic solutions are found for which the coefficients of the fundamental equation are finite. The well known static and non-static homogeneous models with zero pressure are shown to be special cases of this family of cosmological models. The Robertson two-dimensional graphical representation of the existence conditions for homogeneous models is extended to its equivalent three-dimensional representation for these models. A general expression for the local proper density within these models is derived and further physical restrictions upon the solutions are developed.

In the final chapter the usefulness of this family of models is illustrated by applying the special solutions for a zero cosmological constant to the cosmological problem. Suitable expressions are derived for the red-shift, for the number of nebulae which would be counted to a limiting magnitude, and for the observed magnitude of a source of known luminosity located at a stated coordinate. A model is then constructed which agrees with Hubble's observational data to a first approximation.

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