| Cartan's Corner |
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| Cartan's Corner |
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This article examines how the physical presence of field energy and
particulate matter can be interpreted in terms of the topological properties of
space time. The theory is developed in terms of vector and matrix equations of
exterior differential forms. The theory starts from the sole postulate that
field properties of the Cosmological Vacuum (a continuum) can be defined in
terms of a vector space domain, of maximal rank, infinitesimal, neighborhoods,
where exact differentials are mapped into exterior differential 1-forms, |A>, by
a Basis Frame of C2 functions, [B], with non-zero determinant. The particle
properties of the Cosmological Vacuum are defined in terms of topological
defects (or compliments) of the field vector space, where the non-zero
determinant condition fails. When the exterior differential 1-forms, |A>, are
not uniquely integrable, the fibers can be twisted, leading to possible Chiral
matrix arrays of certain 3-forms of Topological Torsion and Topological Spin.
In addition, there exist Chiral objects constructed from vector arrays of
1-forms that mimic the properties of the Einstein tensor. The abreviated theory
is backed up by a detailed example in Maple format for the isotropic
Schwarzschild metric.
Is the Chiral Universe Rotating?
In electromagnetic systems, chirality can be related to constitutive
properties that link D to B and H to E. Optical activity is associated with the
imaginary part of a chirality linkage, and Fresnel-Fizeau phenomena can be
related to the real part. A combination of Fresnel-Fizeau rotation and Optical
Activity can break the inbound-outbound symmetry of propagating electromagnetic
singularities, an effect that can be measured in dual-polarized ring lasers. As
the chirality concept is related to centers of symmetry, or fixed points of
expansion and rotation, and as the universe appears to be expanding, it is
natural to ask: Is the universe rotating, as well as expanding?
A Thermodynamic Explanation of why electrons in a Bohr orbit do not radiate.
The 1-form of Action that encodes the thermodynamic properties of an orbiting
electron consists of two components. One component represents the acceleration
of contraction to a fixed point -- the inverse r squared law. The second
component represenents the acceleration due to a rotation about the fixed point.
It is easy to show that the Pfaff Topological DImension of the 1-form is in
general 3 for the specified Action 1-form of two components. Hence the
non-equilibrium thermodynamic system is closed, and will exchange radiation to
its environment. However, if the second term representing the rotation is
homogeneous of degree zero, it then corresponds to deRham period integral, which
when integrated around a closed cycle has values with integer ratios. The Pfaff
topological dimension of the thermodynamic 1-form becomes equal to 2, and the
resulting thermodynamic system is an isolated equilibrium system that does not
exchange matter or radiation with its environment. The "quantized" orbiting
electron does not radiate even thought it is accelerated.
Turbulence and the Navier Stokes Equations
The concept of Continuous Topological Evolution, based upon Cartan’s methods
of exterior differential systems, is used to develop a topological theory of
non-equilibrium thermodynamics, within which there exist processes that exhibit
continuous topological change and thermodynamic irreversibility. The technique
furnishes a universal, topological foundation for the partial differential
equations of hydrodynamics and electrodynamics; the topological technique does
not depend upon a metric, connection or a variational principle. Certain
topological classes of solutions to the Navier-Stokes equations are shown to be
equivalent to thermodynamically irreversible processes. The method demonstrates,
by example, how an irreversible dissipative process acting in an Open
non-equilibrium system of Pfaff topological dimension 4 can decay, or create in
finite time, topological defect structures, or Closed systems of Pfaff
topological dimension 3. These Closed non-equilibrium systems admit a
Hamiltonian process which can emulate the geometrical evolution of topological
stationary states far from equilibrium. The theory of Continuous Topological
Evolution gives formal credence, as well as analytic examples, to the Prigogine
conjecture of self-organization in terms of disspative (thermo)dynamics (written
in response to the Clay Institute millenium challenge).
Prigogine'a Thermodynamic Emergence and Continuous Topological Evolution
Irreversible processes can be described in Open non-equilibriumthermodynamic
systems, of topological dimension 4. By means of Continuous Topological
evolution, such processes can cause local decay to Closed non-equilibrium
thermodynamic states, of topological dimension 3. These topologically coherent,
perhaps deformable, regions or states of one or more components appear to
"emerge" as compact 3D Contact submanifolds that can be defined as topological
defects in the 4D Symplectic manifold. These emergent states are still far from
equilibrium, as their topological (not geometrical) dimension is greater than 2.
The 3D Contact submanifold admits evolutionary processes with a unique extremal
Hamiltonian vector component, as well as fluctuation spinor components. If the
subsequent evolution is dominated by the Hamiltonian component, the emergent
topological defects will maintain a relatively long-lived, topologically
coherent, approximately non-dissipative structure. These topologically coherent,
"stationary states" far from equilibrium ultimately will decay, but only after a
substantial "lifetime". Analytic solutions and examples of these processes of
Continuous Topological Evolution give credence, and a deeper understanding, to
the general theory of self-organized states far from equilibrium, as conjectured
by I. Prigogine. Moreover, in an applied sense, universal engineering design
criteria can be developed to minimize irreversible dissipation and to improve
system efficiency in general non-equilibrium situations. As the methods are
based on universal topological, not geometrical, ideas, the general
thermodynamic results apply to all synergetic topological systems. It may come
as a surprise, but ecological applications of thermodynamics need not be limited
to the design of specific hardware devices, but apply to all synergetic systems,
be they mechanical, biological, economical or political. (submitted for
consideration of the WTI Prigogine award)
The powerpoint presentations of a talk, "Non-linear, Topologically Coherent, and Compact Flows Far from Equilibrium" given at the EGU 2007 conference are made available for those interested.
The fundamental idea is that topological thermodynamics predicts the production
of topological defect structures
The Pfaff Topological dimension 3 subdomains are thermodynamic systems that are
far from equilibrium,
Vienna EGU
April 20, 2007 "Part1"
Get the free Adobe 5.0 pdf file reader WARNNG: Adobe Reader 5.0 has symbol problems that are not in Adobe Reader 4.0 and 3.0. PDF files created with older versions of Adobe Acrobat do not necessarily display properly when using Adobe Reader 5.0 (they do display properly using Adobe Reader 4.0). The pdf files on Cartan's corner are gradually being recompiled to work with all versions of the Adobe reader. Please advise me by email of those pdf files that display improperly
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