Previous Topics on Cartan's Corner 2003-2006

Updated November 12, 2006

Topological Torsion, Negative Pressure and the Internal Energy of a Non-equilibrium Fluid in 4D

The thermodynamic Internal Energy of a closed, non-equilibrium, rotating fluid or gas can have a classical component related to potential energy and a non-classical component related to the spiral helicity density, (v,curl v), of the gas dynamics. The existence of non-zero helicity density is a topological property (of topological, not affine, torsion) not found in isolated equilibrium thermodynamic systems.
For a non-equilibrium gas, whose molecules are stars,
the spiral helicity density could play the role of negative pressure, thereby giving a palatable explanation for the observed rotational velocities of spiral galaxies.

Updated July 16, 2006

Experimental Evidence indicates that Physical 3D space is not necessarily Euclidean

Conventional physical dogma, justified by the local success of Newtonian dynamics for particles, assigns a Euclidean metric with signature (plus, plus, plus) to the three spatial dimensions. However, experimental evidence now indicates that the intransitive, non-affine, (rotational) dynamics of a fluid admits a better description in terms of a 3 dimensional space with a Lorentz metric of signature (plus, plus, minus), or a Majorana metric of signature (minus, minus, plus). Three dimensional spaces with such non-Euclidean metric signatures admit what mathematicians have described as maximal surfaces of zero mean curvature, with conical or isolated singularities. The zero mean curvature surfaces in Majorana space have negative Gauss curvature (similar to minimal surfaces in Euclidean space), while the zero mean curvature surfaces in Lorentz space have a positive Gauss curvature (in contrast to minimal surfaces in Euclidean space). Falaco Solitons, easily created as topological defects in a swimming pool, are claimed to be experimental artifacts of zero mean curvatures surfaces immersed into 3D spaces of non-Euclidean metric. The topological defects, in the otherwise flat surface of fluid density discontinuity, appear as a pair of zero mean curvature surfaces, with a conical (dimple) singularity at each end. The two conical singularities of the Falaco Soliton pair appear to be connected with a 1D topological defect, or string under tension. The singular conical points are associated with rotation (not translation) about a rotational axis or "a fixed point", and are not mapped globally by affine transitive transformations. In particular, the metric signature of 3D space with matter, and its resultant dynamics, need not be Euclidean. These surfaces of zero mean curvature, which are dominated by rotation, are generated by macroscopic spinors.

First posted 12/19/2005, Updated Feb 28, 2006

A Topological Theory of the Physical Vacuum

This article examines how the physical presence of field energy and particulate matter could influence the topological properties of space time. The theory is developed in terms of vector and matrix equations of exterior differential forms. The topological features and the dynamics of such exterior differential systems are studied with respect to processes of continuous topological evolution. The theory starts from the sole postulate that field properties of a Physical Vacuum (a continuum) can be defined in terms of a vector space domain, of maximal rank, infinitesimal neighborhoods, that supports a Basis Frame as a 4 x 4 matrix of C2 functions with non-zero determinant. The basis vectors of such Basis Frames exhibit differential closure. The particle properties of the Physical Vacuum are defined in terms of topological defects (or compliments) of the field vector space defined by those points where the maximal rank, or non-zero determinant, condition fails. The topological universality of a Basis Frame over infinitesimal neighborhoods can be refined by particular choices of a subgroup structure of the Basis Frame, [B]. It is remarkable that from such a universal definition of a Physical Vacuum, specializations permit the deduction of the field structures of all four forces, from gravity fields to Yang Mills fields, and associate the origin of topological charge and topological spin to the Affine torsion coefficients of the induced Cartan Connection matrix [C] of 1-forms. gr-qc/0602118

Updated July 17, 2005

Topological Fluctuations and thermodynamic irreversibility are caused by the Spinor components of Process Direction fields.

The topological perspective of thermodynamics defines the 1-form of Work as the interior product of the process direction field V with the antisymmetric 2-form F = dA generated by the exterior derivative of the 1-form of Action, A, used to define the physical system. The eigenvectors of the 2-form are either vectors of eigenvalue zero, or isotropic complex Spinors (E. Cartan) of imaginary eigenvalues. If the process is such that the Work 1-form vanishes, then the evolution is Hamiltonian, conservative, and composed of eigenvectors of zero eigenvalue. If the 1-form of Work is not zero, then the process must contain spinor components. It follows that topological fluctuations in kinematics are generated by processes that have spinor components.

Updated July 17, 2005 Propagating Topological Singularities in the lightcone: the Photon A 20 Mb annotated slide show with sound of the Aug 2 SPIE conference presentation.

Updated May 15, 2005 Propagating Topological Singularities in the lightcone: the Photon A pdf file summary of the SPIE presentation. The concepts of topological defects, Spinor eigenvectors, Falaco Solitons and non equilibrium Electromagnetic systems are combined to give a topological model for the classical and quantum features of the Photon.

Updated November 10, 2004 Over the last two years I have given a number of lectures on Falaco Solitons and Non Equilibrium Thermodynamics.
The presentations included colored slide shows that now can be downloaded as PDF files from the following URL,s:
ARW The Arrow of Time - from a Topological Perspective. (NATO 95- Tatranska Lomnica, Slovak Republic, 2002)

coslab Falaco Solitons as Cosmic Strings in a swimming pool. (COSLAB - Bilbao, Spain, 2003)

Vigier3 Cosmology as a turbulent non equilibrium van der Waals gas near its critical point. (Vigier III Conference - Paris 2003)

Euromech448 Falaco Solitons in a stratified Fluid (Euromech 448 - Paris 2004)

AMS_ABQ Non Equilibrium Thermodynamics - from the perspective of Continuous Topological Evolution. (AMS meeting - Albuquerque USA 2004)

WMV A movie (3.5 MB) demonstrating the creation and evolution of the topological defects known as the Falaco Soliton. (Mazan, France 2004)

More detail can be found in the POD books referenced above

Finally, a new bifurcation mechanism explains the creation of Falaco Solitons.

Updated April 4, 2004
A new dynamical bifurcation mechanism finally explains the formation of topological defects experimentally defined in 1986 as Falaco Solitons. The Falaco Solitons are topologically universal phenomena created experimentally by a rotational dynamics in a continuous media with a discontinuity surface, such as that found in a swimming pool. The topologically coherent structure of Falaco Solitons replicates certain features found at all physical scales, from spiral arm galaxies and cosmic strings to microscopic hadrons. The easy to replicate experiment indicates the creation of "stationary" themodynamic states (or solitons) far from equilibrium, which are locally unstable but are globally stabilized. Several exact solutions to the Navier-Stokes equations are demonstrated to admit bifurcations to Falaco Solitons.

Cosmology from the point of view of non-equilibrium thermodynamics.
Updated March 03, 2004
"After measuring dynamical properties of stars in galaxies, star clusters and galactic clusters, astronomers are suggesting that our current understanding of the universe, based upon ponderable gravitating matter, does not appear to agree with the observations. In fact, their almost outrageous conjectures state that ordinary matter seems to account for only a few percent of the total matter in the universe, about 75% consists of dark energy, and about 25% consists of dark matter. Immediately, the theoretical physics community has jumped in with equally outrageous suggestions (similar to the abortive theoretical developments of the "fifth force") that such effects on a galactic scale are due to vacuum quantum fluctuations, to a mass inducing Higgs potential, to the tension of infinitesimal strings, to a cosmological constant modifications of the metrically based Einstein gravitational theory, and many other esoteric theoretical concoctions.

Yet could it be that there is a much simpler model that could account for the various "features" of the "new" Universe? In this article the idea is presented that the universe is a non-equilibrium thermodynamic system with topological features of a universal van der Waals gas.

The turbulent non-equilibrium thermodynamic cosmology of a real gas near its critical point yields an explanation for:

1.The granularity of the night sky as exhibited by stars and galaxies. 2.The Newtonian law of gravitational attraction proportional to 1/rē. 3.The expansion and irreversible dissipation of the universe (4th order curvature effects).
4.The possibility of domains of negative pressure (dark energy) due to a classical Higgs mechanism for aggregates below the critical temperature (3rd order curvature effects)
5.The possibility of domains where gravitational effects (2nd order Gauss curvature effects) appear to be related to entropy and temperature properties of the thermodynamic system.
6.The possibility of cohesion properties (dark matter) due to string or surface tension (1st order Mean curvature effects)
7.Black Holes (generated by Petrov Type D solutions in gravitational theory) are to be related to Minimal Surface solutions to the Universal thermodynamic 4th order Phase function of a non-equilibrium universal van der Waals gas.

All of the above are deduced without explicit mention of the geometric features of metric or connection!!"

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