These SpaceTimes have Parallel Transport 1-form Connections with zero Curvature:
The Connection 1-form is also called the Gauge Potential.
The Curvature 2-form is a (0,2)-tensor, is a Covariant Derivative of the Connection 1-form, and is defined by 2 ___ considering a parallelogram 1 /__/ 4 3 Curvature is defined by parallel transport of a vector C in two ways, and taking the difference: ___ First transport C along 1 and then 2, by path / Then transport C along 3 and then 4, by path __/ Curvature is the difference between the two, C transported along 12 and C transported along 34. Note that in both cases C is, after transportation, in a well-defined tangent space because the parallelogram is closed.
These Parallel Transport 1-form Connections have Affine Torsion that give Structure Constants for Lie Algebras of Gauge Groups:
To define Affine Torsion
(Affine Torsion as opposed to Topological Torsion),
a (1,2)-tensor which is
the antisymmetric part of the Connection,
start with two vectors A __ and B / at an origin point.
___
First parallel transport B along A to get a vector /
Then parallel transport A along B to get a vector __/
For a manifold with no Torsion,
the end points are the same and when you make a parallelogram
out of the two paths AB and BA, the parallelogram closes,
and you might say that AB - BA = 0.
For parallelizable manifolds,
such as non-abelian Lie groups and S3 and S7,
your parallel transport connection can have zero curvature
but non-zero Afffine Torsion. Affine Torsion gives you
the structure constants of Lie algebras,
for the cases of Lie groups and S3.
Since RP1 is topologically equivalent to
the 1-sphere S1 = U(1) = unit Complex Numbers,
the U(1) Complex phase of particle propagators can
be associated with the Time RP1 of SpaceTime.
Since RP1 is 1-dimensional and U(1) is Abelian,
the associated Affine Torsion is zero.
For S7, Torsion varies with the position on the 7-sphere S7,
so you have to take that into account by considering
that the transport of B along A ends at one point on the S7
and
the transport of A along B ends at a different point on the S7
so that
the one of the two point tangent spaces must be mapped to the other.
Such a map has two parts:
the map from one end point to the other can be thought
of as a path on a second S7 7-sphere;
and
the map from one tangent space to the other can be thought
of as an element of the 14-dim Lie group G2 that is
the automorphism group of the octonions.
Therefore,
to make a Lie group from S7 using its Torsion,
you have to combine (non-trivially) two S7 7-spheres with G2,
producing the 7+1+14 = 28-dim Lie group Spin(8) that
is the double cover of the 8-dim rotation group SO(8).
In the D4-D5-E6-E7 physics model
SpaceTime is parallelizable RP1 x S7..
The 28 infinitesimal generators of Spin(8)
correspond to
l6 U(2,2) gauge bosons that produce Gravity and the Higgs Mechanism
plus
12 gauge bosons of the Standard Model.
All 28 of the Spin(8) generators can be associated with Affine Torsion of the Space S7 of RP1 x S7 SpaceTime. Since U(2,2) = U(1) x SU(2,2),the 16 U(2,2) gauge bosons correspond to
the Space part of the U(1) Complex phase of particle propagators and to the Conformal Group Spin(2,4) = SU(2,2) with 15 generators:1 Dilation; 4 Special Conformal Transformations (Non-linear Mobius Fractional Projective Transformations); 3 Rotations plus 3 Boosts; and 4 Translations.
The 4 Translations and 3 Rotations plus 3 Boosts form the 10-dimensional anti-deSitter group Spin(2,3) that produces Gravity by the MacDowell-Mansouri Mechanism as described by Freund in Chapter 21 of his book Introduction to Supersymmetry (Cambridge 1986), saying: "... [if we do not assume space-inversion invariance] we could have ... a parity-violating gravity. This would [produce] ... solutions of the gravitational field equations without definite space-inversion properties. ... Unlike in Einstein's theory, ... [the MacDowell-Mansouri Mechanism] ... does not require the Riemannian invertibility of the metric. ... [The MacDowell-Mansouri Mechanism] is wider in scope than the ordinary Hilbert-Einstein formulation. ... the solution has torsion ... produced by an interference between parity violating and parity conserving amplitudes. Parity violation and torsion go hand-in-hand. ...".
The Parity-Violating Gravitational Torsion described by Freund is different from the Affine Torsion that gives the Structure Constants of the Lie Algebras of the Gauge Groups. Like Einstein's Gravitational Curvature of SpaceTime, the Gravitational Torsion of SpaceTime is an Effective Deformation of 4-dim Physical SpaceTime in which 4-dim Physical SpaceTime effectively appears to be, not an immutable RP1 x S3, but a Compressible Aether.The Gravitational Torsion is NOT fixed by the theory to be the gravitational constant G, as pointed out by Ivanenko and Saradanashvily, in Physics Reports 94 (1983) 1-45, where they say: "... the gravitation Lg [and] the torsion Ls ... components of a total Lagrangian may be chosen independently of each other, e.g. Lg is the Hilbert-Einstein Lagrangian of [General Relativity, but] ... Ls ...[could be a] Lagrangian of the Yang-Mills type. ... nothing requires that coupling constants of the torsion ... coincide with the gravitation constant ... In particular, torsion ... coupling constants may be chosen much stronger than the gravitational one, which opens the door to the hypothesis about the possibility of strong torsion ... whose effect would be comparable with weak or strong interaction effects. ...".
The 4 Special Conformal Transformations (Non-linear Mobius Fractional Projective Transformations) preserve discontinuities, signals, and other properties of characteristics that are not restricted to a finite propagation speed. They correspond to the Conformal GraviPhotons.
R. M. Kiehn says: "... 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. ... 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 ... for which a finite propagation speed is an invariant concept. ...and 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. ...".William D. Walker, in physics/0001063, has shown "... that electromagnetic near-field waves and wave groups, generated by an oscillating electric dipole, propagate much faster than the speed of light as they are generated near the source, and reduce to the speed of light at about one wavelength from the source. The speed at which wave groups propagate (group speed) is shown to be the speed at which both modulated wave information and wave energy density propagate. Because of the similarity of the governing partial differential equations, two other physical systems (magnetic oscillating dipole, and gravitational radiating oscillating mass) are noted to have similar results. ...".
The 1 Dilation can be fixed to set a Mass/Energy scale, such as the Vacuum Expectation Value of the Higgs Mechanism (about 250 GeV), which is equivalent to setting the Compressibility of the Aether, which in turn allows longitudinal degrees of freedom such as the Mass of SU(2) Weak Bosons.
The Dilation sets the scale of the Higgs VeV at 250 GeV so that general deformations of SpaceTime can take place only above that energy level, while GraviPhoton Special Conformal (Hopf flow) transformations are useful in Conformal deformations of SpaceTime.Incompressibility of the Aether below 250 GeV is only with respect to the 6-dim vector space of the Conformal Group Spin(2,4), so that below 250 GeV you can see Conformal phenomena that appear to show compressibility from the point of view of 3-dim space or 4-dim Minkowski spacetime. Such conformal phenomena include the Fock superluminal solutions of Maxwell's equations that are described by R. M. Kiehn.
The 4 GraviPhoton Special Conformal transformations are like the Moebius linear fractional transformations, that do deform Minkowski spacetime but take hyperboloids into hyperboloids and are the symmetries of superluminal solutions of the Maxwell equations. They are incompressible/linear from the point of view of a 6-dimensional SpaceTime, with 4 spatial dimensions and 2 time dimensions, because the conformal group over Minkowski spacetime is just SU(2,2) = Spin(2,4), the covering group of SO(2,4), and therefore the Lie algebra generators look like those of rotations in a 6-dim vector space of signature (2,4). This is the 4-dim space with 2-dim time suggested by Robert Neil Boyd, in which things look linear (even though from our conventional 3-dim spatial or 4-dim Minkowski point of view they might appear, due to our limited conventional perspective, to be nonlinear). If you regard Physical SpaceTime as the 6-dimensional vector space of Spin(2,4), and Internal Symmetry Space as 4-dimensional CP2, then the total space is 6+4=10-dimensional. With respect to tthe D4-D5-E6-E7 model, that 10-dim space corresponds:
to the 10-dim vector space of the D5 Lie Algebra Spin(2,8); andto the 10-dim element of the decomposition of the 27-dim representation of the E6 Lie Algebra into 10 + 16 + 1 under its D5 subalgebra (see, for example, Lie Algebras in Particle Physics, 2nd edition, by Howard Georgi, Perseus Books (1999), page 308).
From the compact version Spin(6) = SU(4) of the Conformal Group Spin(2,4) = SU(2,2) and its coset space with respect to Spin(5) (the compact version of the anti-deSitter group Spin(2,3)),
it appears that, in the D4-D5-E6-E7 physics model, the 4-dim Special Conformal Transformations correspond to the 4-dim Internal Symmetry Space CP2 and the 1-dim Dilation corresponds to the RP1 (topologically equivalent to S1) Time of SpaceTime.
Since the Standard Model is SU(3) x SU(2) x U(1),the 12 Standard Model gauge bosons correspond to
the Gauge Group Generators, represented in the 4-dim Internal Symmetry Space CP2 = SU(3)/U(2) of the D4-D5-E6-E7 physics model, of:8-dim Color Force SU(3) acting globally in CP2, and locally in 4-dim Physical SpaceTime, where Color SU(3) is confined; 4-dim ElectroWeak U(2) acting locally in CP2 and also locally in 4-dim Physical SpaceTime.
Since U(2) = U(1) x SU(2) = S1 x S3, with U(1) = S1 for Electromagnetism and SU(2) = S3 for the Weak Force:
With respect to the 4-dim Physical SpaceTime RP1 x S3: Since S1 = Spin(2) = U(1) = unit Complex Numbers, photons of U(1) Electromagnetism can be associated with their own U(1) propagator phase. Since U(1) is Abelian, it is associated with zero Affine Torsion. Since S3 = Spin(3) = SU(2) = Sp(1) = unit Quaternions, the SU(2) Weak Force can be associated with non-zero Affine Torsion of the Space S3 of 4-dim Physical SpaceTime, which would be inherited from the Affine Torsion of the Space S7 of 8-dim SpaceTime. Since RP1 is topologically equivalent to S1, U(2) is topologically equivalent to 4-dim Physical SpaceTime RP1 x S3.
The action of Gravity on the 4-dim Physical SpaceTime RP1 x S3 to produce Gravitational Curvature and Gravitational Torsion corresponds to the action of the Higgs Mechanism on the Affine Torsion ElectroWeak U(2) Gauge Bosons to give them mass.
The coupling of Gravitational Torsion to Dirac Spinor Fermion Particles and Antiparticles acts as a Yukawa Coupling to give them mass. At tree level, the Yukawa Coupling gives no mass to the Weyl Fermion Neutrinos and Antineutrinos, which are related by triality to the RP1 of Time and not to the Space S7 of SpaceTime, which are associated with the massless photons of S1 = U(1) Electromagnetism and the massive Weak Bosons of the S3 = SU(2) Weak Force and the Higgs Mechanism, respectively.
In American Journal of Physics 39 (1971) 901-904, David Finkelstein showed that in Unimodular Relativity the Cosmological Constant is an unavoidable Lagrange Multiplier beloging to a constraint that expresses the existence of a Fundamental Volume Element of Spacetime Hypervolume at every point of Spacetime. Unimodular SL(4) is related to SU(2,2) which is isomorphic to the Conformal Group Spin(2,4).
"... the gravitation Lg [and] the torsion Ls ... components of a total Lagrangian may be chosen independently of each other, e.g. Lg is the Hilbert-Einstein Lagrangian of [General Relativity, but] ... Ls ...[could be a] Lagrangian of the Yang-Mills type. ... nothing requires that coupling constants of the torsion ... coincide with the gravitation constant ... In particular, torsion ... coupling constants may be chosen much stronger than the gravitational one, which opens the door to the hypothesis about the possibility of strong torsion ... whose effect would be comparable with weak or strong interaction effects. ...
In recent years torsion has attracted great attention ... The reason that torsion comes to the front lies in the fact that at present we only know two observable space-time characteristics of particles, namely, mass (energy-momentum) and spin. And just energy-momentum and spin of matter turn out to be the sources of metric gravity and torsion, respectively. But because we do not observe any object possessing macrovalues of spin polarization, torsion theory as yet cannot rival with Einstein's theory. ...
Let us consider a system of Dirac massless fermions ... in the Einstein-Cartan space ... one finds ... the familiar Einstein gravitation equation, but with the modified right hand side corresponding to the energy-momentum tensor of nonlinear fermions ... representing the non-linear generalization of the Dirac equation ... due to torsion ... Non-linearities due to torsion arise in other fields of non-vanishing spin, e.g. in electromagnetic ... fields ... At the same time, the question of torsion interaction with gauge fields is not quite yet clear as yet because such an interaction breaks the correspondiing gauge invariance. ...
Another interesting phenomenon ... is that vacuum polarization due to quantized spinor matter induces quadratic terms in the Lagrangian of the Einstein-Cartan field quite like the well-known case of the Einstein gravity field (in the last case such terms can lead to non-singular de-Sitter type inflationary cosmology. The calculation of the one-loop corrections leads to the appearance of counterterms in the Lagrangian, which have the form of the quadratic torsion Lagrangian ... Such phenomena attract attention as a possible mechanism of the origin of induced gravitation and other gauge fields by interactions of matter fields. ...".
"... if a space is curved, it is impossible to compare two distant vectors without some method of parallel transport of vectors throughout the curved space. The amount of curvature is a measure of the mismatch of a vector with a copy of itself which has undergone a complete circuit. ... The parallel transport is provided by a structure which is added to the manifold and is called the connection. In the theory of general relativity, the connection is provided by an object calledthe Christoffel symbol G_ij^k. This is a very compact notation for a set of 40 (= 64 -24) functions on the 4-d spacetime. If the symbol carried two asymmetric lower indices, there would be 64 (= 4 x 4 x 4) functions; but the symmetry of the lower indices reduces the independent functions to 40. The standard Christoffel symbol of general relativity is symmetric in the two lower indicies i,j, and generates a connection called the Levi-Civita connection. However, there are geometries for which an asymmetic Christoffel symbol is employed in addition to the the symmetric Christoffel symbol. The asymmetry is carried by a tensor T called the torsion. We can write:
Thus although the Christoffel symbols are not tensors, their difference is a tensor. In physics, we expect tensors to correspond to measurable quantities. If T is 0, then the torsion is zero, and the symbol must be symmetric. A very special case of parallel transport is called absolute parallelism. While ordinary parallel transport guarantees that the vectors will be rotated only by the curvature along the particular path in the circuit, an absolute parallelism connection guarantees that the vectors will remain unrotated by travel along any circuit that follows vector field flow lines. This implies that there is no curvature for this absolute parallelism connection. However [there] will, in general, be a gap in this circuit caused by a "vertical" motion of the ... moving vector. After making the ciruit, the moving vector and its stay-at-home twin will, end up parallel to each other but separated by this "vertical" gap. This gap is called the torsion. ... The connection structure which provides curvature, is based on the symmetric Christoffel symbol. Thus this connection (called the Levi-Civita connection) has zero torsion. By contrast, the absolute parallelism connection which provides torsion has zero curvature. ... there are good examples of spaces carrying both these connections. These spaces are Lie group manifolds. In fact, later work by Joseph Wolf proved that the only spaces that carry an absolute parallelism (Cartan) connection are Lie groups--with one exception: the seven-sphere S7. ... the only spheres that carry an absolute parallelism are spheres of dimension 1, 3, and 7. And the only spheres that are Lie groups are spheres of dimensions 1 & 3. The Lie group structures of these spheres are called U(1) and SU(2). Moreover, S1 (= U(1)) is the set of all unit complex numbers, while S3 (= SU(2)) is the set of all unit quaternions, and S7 is the set of all unit octonions (or Cayley numbers); it is because octonions are not an associative algebra that S7 fails to be a Lie group; but the octonion structure provides an absolute parallelism on S7. ... it is the left-invariance (or right invariance) of the Lie algebra vector fields the provides absolute parallelism. As Cartan discovered, there are three canonical connections on a Lie group manifold. These three connections are generated by three different actions of the Lie group on itself:
(1) Left action: g --> h g (where g and h are group elements of Lie group G)(2) Right action: g --> g h [(where g and h are group elements of Lie group G)]
(3) Adjoint action: g --> h^(-1) gh (where h^(-1) is the inverse element of h )
... The set of all ... tangent planes together form a vector bundle called the tangent bundle of the Lie group. For the Lie group G, the symbol for the tangent bundle is TG, and it is simply the direct product of the Lie group G and the Lie algebra g. ... In contrast to the case of an ordinary manifold, which is not a Lie group, we say that TG is a trivial bundle because it is direct product of the base space G with the the fiber g, this implies a global trivialization of the bundle structure; moreover, this global trivialization corresponds to the absolute parallelism afforded by the group action on the group manifold and thus on the parallel transport of vectors of the Lie algebra, as described above. The intimate relationship between the Lie group G and the Lie algebra g, has the consequence that the torsion of G ... is simply the Lie product, [x,y], of g ... for the torsion T of a Lie group manifold we can write:
... where the componets of T are the structure constants of the Lie algebra; and X, Y, and Zare Lie algebra elements, i.e., left-invariant vector fields on G. For right invariant vector fields the torsion tensor is the would be -T_ij^k. ... In general, the curvature tensor describing the curvature of the Lie group manifold is the Riemann curvature tensor which can be written in terms of the Lie algebra structure constants:
The Riemann curvature tensor is the tensor generated by the Levi-Civita connection, for which there is zero torsion. Thus on the Lie group manifolds we have two radically different connections: the Cartan asymmetric connection which has torsion but no curvature, and the symmetric Levi-Civita connection which has curvature but no torsion. Given two connections on the same manifold, the difference between the Christoffel symbols is a tensor, called the difference tensor. In the case of these two connections on a Lie group manifold the difference tensor is the contorsion tensor K.
We can write:
where the first Christoffel symbol is the Cartan connection of absolute parallelism (with 64 independent functions); the second ... Christoffel symbol is the Levi-Civita connection (with 40 independent functions ...); and K is the contorsion tensor (with 24 independent functions). The contorsion tensor is also the tensor of Ricci rotation coefficients. ...".
Some unconventional SubLuminal solutions may have uses that relate Electromagnetism to SpaceTime curvature, or, with respect to sound, perhaps sonoluminescence.
......