- STRUCTURE: Connections; Curvature; Torsion; HyperVolume
- Ivanenko and Sardansahvily
- Saul-Paul Sirag on Torsion and Curvature
- Vaz and Rodrigues - Conformal Wavelets

In astro-ph/9707165, Lasenby, Doran, Dobrowski, and Challinor say:

"... **Gauge-theory gravity**,
expressed in the language of Geometric Algebra [**Clifford
Algebra**],

allows very efficient numerical calculation of photon paths. ... We discuss ... applications of a gauge theory of gravity ... The theory employs [two] gauge fields in a flat Minkowski background spacetime to describe gravitational interactions. ... The first of these, h(a), is a position-dependent linear function mapping the vector argument a to vectors. The position dependence is usually left implicit. Its gauge-theoretic purpose is to ensure covariance of the equations under arbitrary local displacements of the matter fields in the background spacetime. The second gauge field, W(a), is a position-dependent linear function which maps the vector a to bivectors. Its introduction ensures covariance of the equations under local rotations of vector and tensor fields, at a point, in the background spacetime. Once this gauging has been carried out, and a suitable Lagrangian for the matter fields and gauge fields has been constructed, we find that gravity has been introduced. ... the theory is formally similar in its equations (hence local behaviour) to the Einstein-Cartan-Kibble-Sciama spin-torsion theory, but it restricts the Lagrangian type and the torsion type (... torsion that is not trivector type leads to minimally coupled Lagrangians giving non-minimally coupled equations for quantum fields with non-zero spin). ...

If we restrict attention to situations where the gravitating matter has no spin, then there are still differences between general relativity and the theory presented here. These differences arise when time reversal effects are important (e.g. horizons), when quantum effects are important, and when topological issues are addressed. ...

... **within the framework of gauge-theory gravity, the
****Kerr singularity****
is composed of a ring of matter, moving at the speed of light, which
surrounds a disk of pure isotropic tension**. ...

... As an interesting aside, we note that self-consistent homogeneous cosmologies, based on a classical Dirac field, require that k = 0 (the universe is spatially flat). ... ".

In the paper Gravity, gauge theories and geometric algebra,
downloadable from the web
page of The Geometric Algebra Research Group at Cavendish Laboratory,
University of Cambridge, Lasenby, Doran, and Gull say: that
"...**fermionic matter would be able to detect the center of the
universe if k=/= 0 [if the univese were not spatially
flat]** ...".

In the paper Effects of Spin-Torsion
in Gauge Theory Gravity, downloadable from the web
page of The Geometric Algebra Research Group at Cavendish Laboratory,
University of Cambridge, Doran, Lasenby, Challinor, and Gull say:
that "... **Within [****Gauge-Theory
Gravity****], ****torsion****
is viewed as a physical field derived from the gravitational gauge
fields**. This viewpoint has some conceptual advantages over that
used in differential geometry, where torsion is regarded as a
property of a non-riemannian manifold. ... for a massive spinning
point-particle, moving in a gravitational background with torsion
...the motion is not generally geodesic, the spin vector is not
Fermi-transported, and the particle couples to the torsion. ...
**spinning point particles see a preferred direction in space due to
the spin of the matter field**. ... with spin there are extra
physical fields present which have observable consequences. ...".

In gr-qc/9910099, Chris Doran says: "...

... is global and involves a time coordinate which represents the
local proper time for free-falling observers on a set of simple
trajectories. ... The Kerr solution ... is global, making it suitable
for studying processes near the horizon. ... **the time coordinate
measured by a family of free-falling observers brings the Dirac
equation into Hamiltonian form ... This form of the equations also
permits many techniques from quantum field theory to be carried over
to a gravitational backgroundwith little modification**. ... ".

As to Gravity, Gockeler and Schucker say, in their book Differential Geometry, Gauge Theories, and Gravity (Cambridge 1987): "... Cartan's viewpoint ... Here ...the connection is kept as an independent variable in addition to the metric. Only the field equations obtained by varying the connection fix the torsion as a function of the spin current. ... Lorentz-invariant spin 1/2 ... matter actions can only be constructed by means of the spin connection w. In the presence of such matter, spin and consequently torsion are non vanishing. ... However, torsion does not propagate, and in the vacuum [Einstein-Cartan theory and Einstein's General Relativity] ... are identical. ... the spin density in the universe is small and its coupling to torsion uniquely fixed by the theory to be the gravitational constant G. As a consequence, both theories are presently indistinguishable experimentally. ... The Einstein-Hilbert action is the only action which leads to vanishing torsion in vacuum upon variation of the connection. [see Yates, R. G., Comm. Math. Phys. 76 (1980) 255] ... In the case of nonvanishing torsion both the Einstein and the energy-momentum tensor have an antisymmetric part. ... If torsion is nonzero, neither side of Einstein's equation is conserved. ... ".

As to U(1) Electromagnetism, Kenichi Horie, in hep-th/9409018 and in his Doctoral Dissertation hep-th/9601066, says: "... A complete geometric unification of gravity and electromagnetism is proposed by considering two aspects of torsion: its relation to spin established in Einstein-Cartan theory and the possible interpretation of the torsion trace as the electromagnetic potential. Starting with a Lagrangian built of Dirac spinors, orthonormal tetrads, and a complex rather than a real linear connection we define an extended spinor derivative by which we obtain not only a very natural unification, but can also fully clarify the nontrivial underlying fibre bundle structure. Thereby a new type of contact interaction between spinors emerges, which differs from the usual one in Einstein-Cartan theory. The splitting of the linear connection into a metric and an electromagnetic part together with a characteristic length scale in the theory strongly suggest that gravity and electromagnetism have the same geometrical origin. ...... the

whole complex connectionresulting from the field equationscan be decomposedinto thevector potential... and aLorentzian connectioncompatible with the metric, this being done by means of pull-back techniques. ... the trueelectromagnetic vector potential isnot given by the torsion but by ...another vector part, Su, of the connection.But

formally, the torsion trace Tu is still related to Auand seems to play a role in electromagnetic phenomena. ... torsion is connected to electromagnetism not physically but only formally. ...

... we interpret electromagnetism purely geometrically and use as the only physical constant a characteristic length l close to the Planck length. ... [the Lagrangian has] three parts Lm, LG, and LY[that] resemble more or less the usual Lagrangian densities of spinorial matter, gravity, and the electromagnetic field, respectively. ... from purely dimensional arguments,we must introduce a squared length l^2 in LY. ... we recognize l^2 as the self-coupling constant of the connection, implying that l is an intrinsic length of the space-time geometry.... In order to fix the length scale l we ... compare ... the usual Maxwell equation ... As expected ... the value of l is of the same magnitude as the Planck length, which indicates the close relation of electromagnetism to space-time geometry and to gravity. ... If we had [taken l to be the Planck length], we would have obtained ... [Electromagnetic Fine Structure Constant] = 1/ 64 pi and ... [Electromagnetic Unit Charge] = 1.32 x 10^(-19) Coulomb.Renormalization procedures could perhaps improve [the result] ... one can easily show that the above mentioned field equations ... are exactly the equations of Einstein-Maxwell theory with an electron. Moreover, with these results the Lagrangian ... can be rewritten as... the usual Einstein-Maxwell Lagrangian. ...",and in his Doctoral Dissertation hep-th/9601066, says: "... the relation between the torsion trace and the vector potential on the tangent frame bundle can be obtained only if a special U(1) gauge is chosen and held fixed on the U(1) bundle. For this reason, the long-standing relation between the torsion trace and the electromagnetic potential is merely a formal consequence of the geometrical background underlying the new theory. ...

... Besides these electromagnetic and geometrical aspects, the new theory also incorporates a spin-spin contact interaction between spinning particles.... The contact interaction ... has its origin in the spin-torsion coupling, by which the spacetime geometry can not only respond to mass-energy via the curvature, but also to spin via torsion. These properties of the spacetime together with the geometric interpretation of electromagnetism ... lead to the conclusion that the

spacetime geometry is able to interact with three basic features of elementary particles: Mass-energy, spin, and electromagnetic charge.Contrary to the ordinary axial current contact interaction of the Einstein-Cartan theory, the

new contact interaction has contributions from both the axial and vector currents of Dirac spinors. This has the effect that now there are no self-interactions among Dirac fields as was the case in the Einstein-Cartan theory. This feature respects the quantum nature (Fermi-Dirac statistics) of elementary fermions already on the classical, i.e. not second-quantized, level, andmakes the new contact interaction more favourable than the ordinary one. ...... In the early universe, when the density of spinning particles exceeded some critical value, the contact interaction also leads to pair creations ... As was noted by Kerlick, the required mass density is more than thirty orders of magnitude smaller than the density required for pair creation via tidal forces caused by the curvature of spacetime. Thus, the torsion-induced pair creation effects are much stronger and more likely than the curvature effects, and must be taken into account in the discussion of the scenario of the early universe. ... so far, no prediction has been made which can be investigated by present astronomical observations. One reason for the uncertainty of the predictions is, of course, that the spin-spin contact interaction is very weak and takes place only in a small time interval during the early epoch of the universe. ...

... in order to observe the ... spin-spin contact interaction ... we must consider a many-particle system. ...

...

for ...[ordinary matter]test particles... in sharp contrast to the common opinion, that the contact interaction is attractive only if the spins are opposed, but repulsive if they are in alignment. ... ,

there is no observable force between aligned spins[and] ...- [
there is an]attractive force between opposite spins...... [

for]a test field describing anti-matter in a background torsion...

- [
there is a]repulsive force between aligned spins[and]...opposite spins do not feel any force acting between them. ...... it follows that the universality of the contact interaction is lost... The energy shifts due to the contact interaction between an ordinary background matter field and a test particle describing ordinary matter differs from the case where the test particle describes an anti-matter field. ...

To study the effects of the spin-spin contact interactions in the early epoch of the universe, where the matter density was enormously high, we must evaluate the thermodynamical average of the interaction energies due to the contact interactions. This is not as straightforward as in the ordinary case, since the contributions from the non-flat metric and the spacetime curvature can not be neglected.

Another interesting point is the following: The

spin-spin contact interactionmodifes the energy-momentum equation of ordinary general relativity by the tensor ... whichhas a form similar to the contribution ... of thecosmological constant... in the Einstein's field equation ... then we ... obtain a cosmological constant, which is proportional to the current-current interaction terms ... . This would imply a time-dependent cosmological constant, whose value would have been very high in the early epoch of the universe, where the matter density has been very high, and whose actual value for the present universe is nearly zero. ...".

As Horie says in hep-th/9601066:
"... Since ... [**the spin-spin contact] interaction occures
only when matter fields overlap with each other**, it is called a
contact interaction. It does not only influence the curvature ... but
also creates a cubic self-interaction in the Dirac equation ...".

**In ****the Compton Radius Vortex
Model****, the matter field of a particle** extends at least
to its Compton Radius, and **might extend to ****the
GravitoEM Induction Region**** (in the case of an electron,
about a micron)**.

**If the particle is part of an Extended Quantum State**,
similar to a Bose-Einstein Condensate,
**the spin-spin contact interaction could extend to as far as the
Extended Quantum State**.

The theory of **Horie** in his Doctoral Dissertation hep-th/9601066
**uses** "... the notion of complex spin geometry. This complex
extension is necessary in order to accomodate the real spin structure
to the **complex** tangent bundle **geometry** used in
[his] theory. ...". Perhaps **his approach is consistent
with the ****D4-D5-E6-E7 physics
model** in which SpaceTime is the Shilov Boundary of a Complex
Bounded Homogeneous Domain.

As Horie says in hep-th/9601066: his "... theory ... [based on] a spinor derivative built from a complex linear connection ... does not contain charged vector boson fields as required for the Weinberg-Salam [U(1)xSU(2) ElectroWeak Force] theory. ...".

If Horie had used a Quaternionic connection, he might have reproduced the Higgs Mechanism as done in the1960s by Finklestein, Jauch, Schiminovich, and Speiser,

**Horie got U(1) Electromagnetism by using a Complex
connection**, and for gauge group, the Automorphism Group of the
Complex Numbers, U(1).

**Horie could have gotten the SU(2) Weak Force by using a
Quaternionic connection**, and for gauge group, the Automorphism
Group of the Quaternions, SU(2).

Geoffrey Dixon, in his book Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics (Kluwer 1994), denotes CxQ by P, and resolves P by two Primitive Idempotents

L+ = (1/2)( 1 + i X )

L- = (1/2)( 1 - i X )

where i is the Complex imaginary and

X is a unit Quaternion imaginary. X can be taken to be the non-Complex algebraic generator of the Quaternions. In the Quaternion basis {1,i,j,k}, the algebraic generators are {1,i,j} and j is the element that maps {i} to {k}.

Dixon shows in Section 3.2 of his book that L+ and L- share U(1xSU(2) symmetry.

**Horie could have gotten the SU(3) Color Force by using an
****Octonionic**** connection**,
and for gauge group, the intersection of Automorphism Group of the
Octonions, G2, with the maximal symmetry group of the Euclidean
signature version of SpaceTime, Spin(6) = SU(4) (the Euclidean
version of the Conformal Group
Spin(2,4) = SU(2,2)). The intersection of 15-dimensional Spin(6)
and 14-dimensional G2 is 8-dimensional SU(3). The SU(3) subgroup
fixes one Octonion direction, which can be taken to the the basis
element E in the Octonion basis {1,i,j,k,E,I,J,K}.

Geoffrey Dixon, in his book Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics (Kluwer 1994), denotes CxQxO by T, and resolves T by four Primitive Idempotents, which are required to be orthogonal Primitive Projection Operators

D0 = (1/4)( 1 + i X )( 1 + i E )

D1 = (1/4)( 1 - i X )( 1 + i E )

D2 = (1/4)( 1 + i Y )( 1 - i E )

D3 = (1/4)( 1 - i Y )( 1 - i E )

where i is the Complex imaginary,

X and Y are unit Quaternion imaginaries which can be taken to be j and k in the Quaternion basis {1,i,j,k}, and

E is the non-Complex and non-Quaternonic algebraic generator of the Octonions. In the Octionion basis {1,i,j,k,E,I,J,K}, the algebraic generators can be taken to be {1,i,j,E} or {1,i,k,E, and E is the element that maps {i,j,k} to {I,J,K}, and is the Octonion element that is fixed by the SU(3) subgroup of the Octonion automorphism group G2.

Dixon shows in Section 3.5 of his book that D0, D1, D2, and D3 share U(1)xSU(2)xSU(3) symmetry, and that, taken together, their total symmetry is U(1)^4 x SU(2)^2 x SU(3), which also appears in the D4-D5-E6-E7 physics model.

**R. M.
Kiehn** says:

"... covariant derivatives can handle rotation and translation,
and bending, but NOT deformation. ... I prefer to use the notion of
the **Lie Derivative** ... [which can handle] things that
can be stretched or compressed or otherwise deformed. ... The Lie
derivative is a transplantation law that does not depend upon metric,
nor connection, but merely upon a direction field. ... The Lie
derivative will include the other types of derivatives, such as the
covariant derivative, when extra constraints are placed upon the
domain. ...

[ TheLie DerivativeL_u is a derivative of degree 0 derived from the Exterior Derivative d of degree 1 and the Inner Derivative i_u of degree -1. L_u is described in Differential Geometry, Gauge Theories, and Gravity, by Gockeler and Schucker (Cambridge 1987) as being

The commutation relations

[ i_u , i_v ] = i_u i_v + i_v i_u = 0

[ d , d ] = d d + d d = 0

[ L_u , d ] = L_u d - d L_u = 0

[ i_u , d ] = i_u d + d i_u = L_u

[ L_u , i_v ] = L_u i_v - i_v L_u = i_[u,v]

show that { L_u , i_u , d } form a graded Lie subalgebra of the graded Lie algebra of all derivations D_k under the bracket [ D_k, D_l ] = D_k D_l - (-1)^(dk dl) D_l D_k, where dk and dl are the degrees of D_k and D_l. ]

... [Neither] the concept of wedge exterior product ...[nor] the concept of the exterior derivative depend upon the notion of a metric space. The exterior operations are defined on a variety for which the constraints of metric have not been defined. The exterior derivative is also independent of a connection. It is not equivalent to the tensor covariant derivative except under special circumstances.

Another operation in the Cartan exterior algebra is useful. It is the concept of the interior product. The interior product is essentially the concept of projective transversality or collinearity. The operations may look like the scalar product of ordinary vector analysis. However it is best to think of the interior product as an operation that linearly takes a p element into a p-1 element of the exterior algebra. The interior product of a (contravariant density) vector field with a 1-form creates a 0-form, or function. The interior product is often utilized in terms of a contravariant vector, but then the result is only well defined with respect to diffeomorphisms.

It is best to think of the interior product as being with respect to "currents", and not with respect to vectors. Currents are N-1 form structures on the N dimensional exterior algebra dual to the 1-form. Currents on the final state are well defined on the initial state with respect to the processes of adjoint pullback and functional substitution. The process is dual to the transpose pullback and functional substitution of p-forms. ...

[ The Lie Derivative combines the Exterior Derivative and the Interior Derivative in much the same way that the Clifford Product combines the Exterior Product and the Interior Product. ]

... in 1932, V. Fock ... deduced the characteristic system for the solutions of Maxwell's equations which are not unique in a neighborhood. He clearly formulated the idea that electromagnetic signals were propagating discontinuities. (The concept of a signal and its precise relationship to electromagnetism was not clearly defined by Einstein) To paraphrase Victor Fock:

"The laws of propagation of light in empty space are thoroughly understood. They find their expression in the well-known equations of Maxwell. However, we are not interested in the general case of light propagation, but only in the propagation of a signal advancing with maximum speed; i.e., the propagation of a wave front. Ahead of the front of the wave all components of the field vanish. Behind it some of them are different from zero. Therefore, some of the field components must be discontinuous at the front. On the other hand, given the field (a solution to the equations) on some surface moving in space, the derivatives of the field are, in general, determined by Maxwell's equations. Hence the value of the field at an infinitely near surface is also (uniquely) determined (by analytical continuation) and discontinuities are impossible. The only case when this is not so is when the form and the motion of the surface satisfies certain special conditions subject to which the values of the derivatives is not determined by the values of the field components themselves. Such a surface is called a characteristic surface, or briefly, a characteristic. Thus discontinuities of the field can occur only on a characteristic, but since there must certainly be discontinuities at a wave front ( the signal ), such a front is clearly a characteristic."

... The key idea in Fock's work is the result that the characteristic solutions remain characteristic solutions (discontinuities remain discontinuities - signals remain signals) only under a limited set of coordinate transformations. The laws of Maxwell as tensor equations are well behaved with respect to ALL diffeomorphisms, but

the characteristic solutions to Maxwell's equations retain their
topological properties only with respect to a restricted class of
transformations. ... singular solutions to Maxwell's Equations of
electrodynamics satisfied the eikonal expression, a quadratic partial
differential equation with signature {+++-}. **Mappings which
preserved the eikonal, taking a discontinuity in E field to a
discontinuity, were of two and only two types**. ...

... **A linear type which Fock proved was the Lorentz group of
transformations**. This result is the foundation of special
relativity. ... for which a finite propagation speed is an invariant
concept. ...

However, ... there also exists **a non-linear transformation
group (the Moebius fractional projective transformations) which also
preserve discontinuities, signals, and other properties of
characteristics**. Such signals are not restricted to a finite
propagation speed. .... the propagation speed of the singularity can
be anything - including infinity. !!! ... In optically active media,
the propagation speed of the discontinuities is faster or slower that
the speed of light, depending on the whether or not the helicity
(circular polarization) is aligned or anti-aligned with the optical
axis. ...".

It is then possible for both the state preparation and measurement apparatus to constrain the results of experiments.

... propositions about the results of measurements can satisfy a non-distributive logic rather than the Boolean logic of classical systems.

Reasonable assumptions about the role of the measurement apparatus leads to an orthomodular lattice of propositions characteristic of quantum logic."

If you require that spacetime be 4-dimensional and asymptotically Lorentzian, and if you interpret that requirement to require that spacetime have, asymptotically, quaternionic structure, and

if you require that the 4-geons also be 4-dimensional manifolds with quaternionic structure, and

if you use the result of Joseph Wolf (J. Math. Mech. 14 (1965) 1033-1047) that the complete simply connected Riemannian symmetric spaces with quaternionic structure are:

- 1 - 4-dimensional Euclidean space
- 2 - S2 x S2 = 2 copies of SU(2)/U(1)
- 3 - CP2 = SU(3)/U(2)
- 4 - S4 = Spin(5)/Spin(4) = Spin(5)/(SU(2)xSU(2))

then, if you compactify 4-dimensional Euclidean space to the 4-torus T4, perhaps you could say that there are 4 types of 4-geons (all of which contain Closed Timelike Curves):

- 1 - T4 (which looks like the torus-wormhole used by Wheeler and Misner (Ann. of Phys. 2 (1957) 525-660, reprinted in Wheeler, Geometrodynamics, Italian Physical Society Topics of Modern Physics Vol. 1, Academic Press 1962) to describe classical electromagnetism with gauge group U(1))
- 2 - S2 x S2 (which has the global SU(2) symmetry of the weak force)
- 3 - CP2 (which has the global symmetry of SU(3) of the color force)
- 4 - S4 (which has the global symmetry Spin(5) = Sp(2) used by MacDowell and Mansouri (Phys. Rev. Lett. 38 (1977) 739) to describe gravity, and which has the geometry of curvature of 4-dimensional spacetime and might look like a bubble of spacetime foam, perhaps of Planck-length size)

With those four types of 4-geons, Hadley's model might be similar to the D4-D5-E6-E7 physics model.

......