A Topological Perspective of Cosmology
IN 1948 Landau and Lifschitz published in their volume on statistical mechanics a study of the fluctuations of a real (almost ideal) gas about its critical point. It is known that near the critical point the gas exhibits extreme fluctuations in density (and entropy). These density fluctuations are correlated with a correlation function that implies a force of attractive interaction that falls off as 1/r^2. That is, the density flucations attract one another according to the law of Newtonian gravity!

In 1965 after rereading Landau's satistical derivations, I made the conjecture that if the universe was a very dilute ideal gas near its critical point, and the condensation fluctuations were the stars and galaxies, then the cosmological model of a very dilute gas near the critical point gave a simple explanation for the granularity of the night sky, and for the attractive gravitational force law. Such results are not inherent in the homogeneous metrical models of current general relativity theory.

When Cartan's methods of exterior differential forms are applied to the study of thermodynamics, it appears that on a space time of 4 geometric dimensions, a turbulent non-equilibrium open thermodynamic system can exist and will be of Pfaff topological dimension 4. Processes that are initialized in such a turbulent state can decay to closed thermodynamic states of Pfaff dimension 3, and thereby create long lived topological defects representing stationary states far from equilibrium. (Equilibrium requires a Pfaff dimension of 2 or less).

When the universe is represented by an exterior differential 1-form of Pfaff topological dimension 4, the Jacobian matrix has a Cayley-Hamilton characteristic polynomial of degree 4. At the system evolves, one of the eigenvalues of the characteristic equation can approach zero, which will reduce the Pfaff topological dimension to 3. The resulting cubic polynomial constructed from the similarity invariants of the Jacobian have a representation as a universal van der Waals gas. The remaining eigen values of the characteristic polynomial can be complex, which would describe a non-equilibrium van der Waals gas. The critical point can be determined in terms the similarity invariants (curvatures) of the Jacobian matrix.


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Last update 01/23/2009
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