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R. M. Kiehn

Publications and source records attributed to R. M. Kiehn.

14 recordsLinked to original sources
arXiv2007

Part I. The Cosmological Vacuum from a Topological Perspective

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 systems, which are not constrained by tensor diffeomorphic equivalences. The first postulate defines the field properties (a vector space continuum) of the Cosmological Vacuum in terms of matrices of basis functions that map exact differentials into neighborhoods of exterior differential 1-forms (potentials). The second postulate requires that the field equations must satisfy the First Law of Thermodynamics dynamically created in terms of the Lie differential with respect to a process direction field acting on the exterior differential forms that encode the thermodynamic system. The vector space of infinitesimals need not be global and its compliment is used to define particle properties as topological defects embedded in the field vector space. The potentials, as exterior differential 1-forms, are not (necessarily) uniquely integrable: the fibers can be twisted, leading to possible Chiral matrix arrays of certain 3-forms defined as Topological Torsion and Topological Spin. A significant result demonstrates how the coefficients of Affine Torsion are related to the concept of Field excitations (mass and charge); another demonstrates how thermodynamic evolution can describe the emergence of topological defects in the physical vacuum.

gr-qc↗
arXiv2007

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 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.

math-ph↗
arXiv2006

Thermodynamics and quantum cosmology -- Continuous topological evolution of topologically coherent defects

As a point of departure it is suggested that Quantum Cosmology is a topological concept independent from metrical constraints. Methods of continuous topological evolution and topological thermodynamics are used to construct a cosmological model of the present universe, using the techniques based upon Cartan's theory of exterior differential systems. Thermodynamic domains, which are either Open, Closed, Isolated, or in Equilibrium, can be put into correspondence with topological systems of Pfaff topological dimension 4, 3, 2 and 1. If the environment of the universe is assumed to be a physical vacuum of Pfaff topological dimension 4, then continuous but irreversible topological evolution can cause the emergence of topologically coherent defect structures of Pfaff topological dimension less than 4. As galaxies and stars exchange radiation but not matter with the environment, they are emergent topological defects of Pfaff topological dimension 3 which are far from equilibrium. DeRham topological theory of period integrals over closed but not exact exterior differential systems leads to the emergence of quantized, deformable, but topologically coherent, singular macrostates at all scales. The method leads to the conjecture that dark matter and energy is represented by those thermodynamic topological defect structures of Pfaff dimension 2 or less.

gr-qc↗
arXiv2006

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↗
arXiv2001

Instability patterns, wakes and topological limnit sets

Many hydrodynamic instability patterns can be put into correspondence with a subset of characteristic surfaces of tangential discontinuities. These topological limits sets to systems of hyperbolic PDE's are locally unstable, but a certain subset associated with minimal surfaces are globally stabilized, persistent and non-dissipative. Sections of these surfaces are the spiral scrolls so often observed in hydrodynamic wakes. This method of wake production does not depend explicitly upon viscosity.

physics.flu-dyn↗
arXiv2001

Topology and Turbulence

Over a given regular domain of independent variables {x,y,z,t}, every covariant vector field of flow can be constructed in terms a differential 1-form of Action. The associated Cartan topology permits the definition of four basic topological equivalence classes of flows based on the Pfaff dimension of the 1-form of Action. Potential flows or streamline processes are generated by an Action 1-form of Pfaff dimension 1 and 2, respectively. Chaotic flows must be associated with domains of Pfaff dimension 3 or more. Turbulent flows are associated with domains of Pfaff dimension 4. It will be demonstrated that the Navier-Stokes equations are related to Action 1-forms of Pfaff dimension 4. The Cartan Topology is a disconnected topology if the Pfaff dimension is greater than 2. This fact implies that the creation of turbulence (a state of Pfaff dimension 4 and a disconnected Cartan topology) from a streamline flow (a state of Pfaff dimension 2 and a connected topology) can take place only by discontinuous processes which induce shocks and tangential discontinuities. On the otherhand, the decay of turbulence can be described by continuous, but irreversible, processes. Numerical procedures that force continuity of slope and value cannot in principle describe the creation of turbulence, but such techniques of forced continuity can be used to describe the decay of turbulence.

physics.flu-dyn↗
arXiv2001

Topological-Torsion and Topological-Spin as coherent structures in plasmas

The PDE's of classical electromagnetism can be generated from two exterior differential systems that distinguish topologically the field intensities and potentials, F-dA= 0, from the field excitations and the charge current densities, J-dG = 0. The existence of potentials, A, leads to the independent 3-forms of T-Torsion = A^F and T-Spin = A^G. The exterior derivatives (divergences) of these 3-forms produce anomalies that define the two classic Poincare invariants. The closed integrals of these forms, when deformation invariants of frozen-in fields, define topological coherent structures in the plasma. Solutions when T-Torsion (T-Spin) is zero define transverse magnetic (electric) modes on topological grounds. When the divergence of T-Torsion is not zero there exists a classical mechanism for charge acceleration along the magnetic field lines producing symplectic plasma currents; large temperature gradients along the B field lines can act as a source of stellar plasma jets in neutron stars. In such circumstances, the Torsion vector is uniquely defined by conformal invariance of the Action potentials. Plasma currents in the direction of the Torsion vector leave both A^F and A^G conformally invariant, hence these fields are frozen-in even though the processes are thermodynamically irreversible. The decaying coherent and deformable topological structures associated with such frozen-in fields are persistent and observable artifacts, similar to wakes, that can appear in any plasma domain, such as that which surrounds stars.

physics.plasm-ph↗
arXiv2001

Some closed form solutions to the Navier Stokes equations

An algorithm for generating a class of closed form solutions to the Navier-Stokes equations is suggested, with examples. Of particular interest are those exact solutions that exhibit intermittency, tertiary Hopf bifurcations, flow reversal, and hysteresis.

physics.flu-dyn↗
arXiv2001

Cartan's topological structure

A system of differential forms will establish a topology and a topological structure on a domain of independent variables such that is possible to determine which maps or processes acting on the system are continuous. Perhaps the most simple topology is that generated by the existence of a single 1-form of Action, its Pfaff sequence of exterior differentials, and their intersections. In such a topology the exterior derivative becomes a limit point generator in the sense of Kuratowski. The utilization of such techniques in physical systems is examined. A key feature of the Cartan topology is determined by the Pfaff dimension (representing the minimum number of functions to describe the 1-form generator). In particular, when the Pfaff dimension is 3 or more the Cartan topology becomes a disconnected topology, with the existence of topological torsion and topological parity. Most classical physical applications are constrained to cases where the Pfaff dimension is 2 or less, for such is the domain of unique integrability. The more interesting domain of non-unique solutions requires the existence of topological torsion, and can lead to an understanding of irreversible processes without the use of statistics.

math-ph↗
arXiv2001

Curvature and torsion of implicit hypersurfaces and the origin of charge

A formal correspondence is established between the curvature theory of generalized implicit hypersurfaces, electromagnetism as expressed in terms of exterior differential systems, and thermodynamics. Starting with a generalized implicit surface whose normal field is represented by an exterior differential 1-form, it is possible to deduce the curvature invariants of the implicit surface and to construct a globally closed vector density in terms of the Jacobian properties of the normal field. When the closed vector density is assigned the role of an intrinsic charge current density, and the components of the normal field are assigned the roles of the electromagnetic potentials, the theory is formally equivalent to an exterior differential system that generates the PDE's of both the Maxwell Faraday equations and the Maxwell Ampere equations. The interaction energy density between the potentials and the induced closed charge current density is exactly the similarity curvature invariant of highest degree (N-1) for the implicit surface. Although developed without direct contact with M-brane theory, these ideas of generalized implicit surfaces should have application to the study of p-branes that can have multiple components and envelopes. The theory suggests that gravitational collapse of mass energy density should include terms that involve the interaction between charge-current densities and electromagnetic potentials.

gr-qc↗
arXiv2001

Continuous topological evolution

A non-statistical theory of continuous, but irreversible, evolution can be constructed in terms of the Cartan calculus. The fundamental postulate, for an evolutionary theory which admits irreversible processes, is that the topology of the initial state will be different from the topology of the final state. Several fundamental theorems of continuous evolution are established, yielding a set of global conservation laws for reversible or irreversible processes. As examples, a comparison of the evolution of Topological Torsion and Topological Action is made for hydrodynamic and electromagnetic systems. The relationship between the evolution of Topological Torsion and a thermodynamically irreversible process is established.

math-ph↗
arXiv2001

Chirality and helicity in terms of topological spin and topological torsion

In this article the concept of enantiomorphism is developed in terms of topological, rather than geometrical, concepts. Chirality is to be associated with enantiomorphic pairs which induce Optical Activity, while Helicity is to be associated enantiomorphic pairs which induce a Faraday effect. Experimentally, the existence of enantiomorphic pairs is associated with the lack of a center of symmetry, which is also serves as a necessary condition for Optical Activity. However, Faraday effects may or may not require a lack of a center of symmetry. The two species of enantiomorphic pairs are distinct, as the rotation of the plane of polarization by Optical Activity is a reciprocal phenomenon, while rotation of the plane of polarization by the Faraday effect is a non-reciprocal phenomenon. From a topological viewpoint, Maxwell's electrodynamics indicates that the concept of Chirality is to be associated with a third rank tensor density of Topological Spin induced by the interaction of the 4 vector potentials {A,phi} and the field excitations {D,H}. The distinct concept of Helicity is to be associated with the third rank tensor field of Topological Torsion induced by the interaction of the 4 vector potentials and field intensities {E,B}.

physics.optics↗
arXiv1998

Electromagnetic Waves in the Vacuum with Torsion and Spin

Exact radiative wave solutions to the classical homogeneous Maxwell equations in the vacuum have been found that are not transverse, exhibit both torsion and spin, and for which the second Poincare invariant, E.B, is not zero. Two four component rank 3 tensors of spin current and torsion are constructed on topological grounds. The divergence of each pseudo vector generates the Poincare invariants of the electromagnetic system.

physics.class-ph↗