Tony Smith's Home Page
These SpaceTimes have Parallel Transport 1-form Connections withzero 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 3Curvature is defined by parallel transportof 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 Torsionthat 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 parallelogramout 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 curvaturebut non-zero Afffine Torsion. Affine Torsion gives youthe 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 consideringthat 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 S7so 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 tothe 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; and4 Translations.
The 4 Translations and 3 Rotations plus 3Boosts form the 10-dimensional anti-deSitter group Spin(2,3) thatproduces Gravity by the MacDowell-Mansouri Mechanism as described byFreund in Chapter 21 of his book Introduction to Supersymmetry(Cambridge 1986), saying: "... [if we do not assumespace-inversion invariance] we could have ... a parity-violatinggravity. This would [produce] ... solutions of thegravitational field equations without definite space-inversionproperties. ... Unlike in Einstein's theory, ... [theMacDowell-Mansouri Mechanism] ... does not require the Riemannianinvertibility of the metric. ... [The MacDowell-MansouriMechanism] is wider in scope than the ordinary Hilbert-Einsteinformulation. ... the solution has torsion ... produced by aninterference between parity violating and parity conservingamplitudes. 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 SpecialConformal Transformations (Non-linear Mobius FractionalProjective Transformations) preserve discontinuities, signals,and other properties of characteristics that are not restricted to afinite propagation speed. They correspond to the ConformalGraviPhotons.
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 Dilationcan be fixed to set a Mass/Energy scale, such as the VacuumExpectation Value of the Higgs Mechanism (about 250 GeV), which isequivalent to setting theCompressibility of the Aether, which in turn allows longitudinaldegrees 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); and
to 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 GroupSpin(2,4) = SU(2,2) and its coset space with respect to Spin(5) (thecompact version of the anti-deSitter group Spin(2,3)),
it appears that, in the D4-D5-E6-E7physics model, the 4-dim Special Conformal Transformationscorrespond to the 4-dim Internal Symmetry Space CP2 and the 1-dimDilation corresponds to the RP1 (topologically equivalent to S1) Timeof SpaceTime.
Since the Standard Model is SU(3) x SU(2) x U(1),
the 12 Standard Model gauge bosons correspond tothe 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 forElectromagnetism 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-dimPhysical SpaceTime RP1 x S3 to produce Gravitational Curvature andGravitational Torsion corresponds to the action of the HiggsMechanism on the Affine Torsion ElectroWeak U(2) Gauge Bosons togive them mass.
The coupling of Gravitational Torsion to Dirac SpinorFermion Particles and Antiparticles acts as a Yukawa Couplingto give them mass. At tree level, the Yukawa Coupling gives no massto the Weyl Fermion Neutrinos and Antineutrinos, which are related bytriality 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) WeakForce 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 eachother, e.g. Lg is the Hilbert-Einstein Lagrangian of [GeneralRelativity, but] ... Ls ...[could be a] Lagrangian of theYang-Mills type. ... nothing requires that coupling constants of thetorsion ... coincide with the gravitation constant ... In particular,torsion ... coupling constants may be chosenmuch stronger than the gravitational one, which opens thedoor to the hypothesis about the possibility of strong torsion ...whose effect would be comparable with weak or strong interactioneffects. ...
In recent years torsion has attracted great attention ... Thereason that torsion comes to the front lies in the fact that atpresent we only know two observable space-time characteristics ofparticles, namely, mass (energy-momentum) and spin. And justenergy-momentum and spin of matter turn out to be the sources ofmetric gravity and torsion, respectively. But because we do notobserve 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 theEinstein-Cartan space ... one finds ... the familiar Einsteingravitation equation, but with the modified right hand sidecorresponding 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 otherfields of non-vanishing spin, e.g. in electromagnetic ... fields ...At the same time, the question of torsion interaction with gaugefields is not quite yet clear as yet because such an interactionbreaks the correspondiing gauge invariance. ...
Another interesting phenomenon ... is that vacuum polarization dueto quantized spinor matter induces quadratic terms in the Lagrangianof the Einstein-Cartan field quite like the well-known case of theEinstein gravity field (in the last case such terms can lead tonon-singular de-Sitter type inflationary cosmology. The calculationof the one-loop corrections leads to the appearance of countertermsin the Lagrangian, which have the form of the quadratic torsionLagrangian ... Such phenomena attract attention as a possiblemechanism of the origin of induced gravitation and other gauge fieldsby interactions of matter fields. ...".
"... if a space is curved, it is impossible to compare two distantvectors without some method of parallel transport of vectorsthroughout the curved space. The amount of curvature is a measure ofthe mismatch of a vector with a copy of itself which has undergone acomplete circuit. ... The parallel transport is provided by astructure which is added to the manifold and is called theconnection. In the theory of general relativity, theconnection is provided by an object calledthe Christoffel symbolG_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 asymmetriclower indices, there would be 64 (= 4 x 4 x 4) functions; but thesymmetry of the lower indices reduces the independent functions to40. The standard Christoffel symbol of general relativity issymmetric in the two lower indicies i,j, and generates a connectioncalled the Levi-Civita connection. However, there are geometriesfor which an asymmetic Christoffel symbol is employed in additionto the the symmetric Christoffel symbol. The asymmetry is carriedby a tensor T called the torsion. We can write:
Thus although the Christoffel symbols are not tensors, theirdifference is a tensor. In physics, we expect tensors to correspondto measurable quantities. If T is 0, then the torsion is zero, andthe symbol must be symmetric. A very special case of paralleltransport is called absolute parallelism. While ordinary paralleltransport guarantees that the vectors will be rotated only by thecurvature along the particular path in the circuit, an absoluteparallelism connection guarantees that the vectors will remainunrotated by travel along any circuit that follows vector fieldflow lines. This implies that there is no curvature for this absoluteparallelism connection. However [there] will, in general, bea gap in this circuit caused by a "vertical" motion of the ... movingvector. After making the ciruit, the moving vector and itsstay-at-home twin will, end up parallel to each other but separatedby this "vertical" gap. This gap is called the torsion. ...The connection structure which provides curvature, is based on thesymmetric Christoffel symbol. Thus this connection (called theLevi-Civita connection) has zero torsion. By contrast, theabsolute parallelism connection which provides torsion has zerocurvature. ... there are good examples ofspaces carrying both these connections. These spaces are Lie groupmanifolds. In fact, later work by Joseph Wolf proved that theonly spaces that carry an absolute parallelism (Cartan) connectionare Lie groups--with one exception: theseven-sphere S7. ... the only spheres that carry an absoluteparallelism are spheres of dimension 1, 3, and 7. And the onlyspheres that are Lie groups are spheres of dimensions 1 & 3. TheLie group structures of these spheres are called U(1) and SU(2).Moreover, S1 (= U(1)) is the set of all unit complex numbers, whileS3 (= SU(2)) is the set of all unit quaternions, and S7 is the set ofall unit octonions (or Cayley numbers); it is because octonions arenot an associative algebra that S7 fails to be a Lie group; butthe octonion structure provides an absoluteparallelism on S7. ... it is the left-invariance (or rightinvariance) of the Lie algebra vector fields the provides absoluteparallelism. As Cartan discovered, there are three canonicalconnections on a Lie group manifold. These three connections aregenerated 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 vectorbundle called the tangent bundle of the Lie group. For the Lie groupG, the symbol for the tangent bundle is TG, and it is simply thedirect product of the Lie group G and the Lie algebra g. ... Incontrast to the case of an ordinary manifold, which is not a Liegroup, we say that TG is a trivial bundle because it is directproduct of the base space G with the the fiber g, this implies aglobal trivialization of the bundle structure; moreover, this globaltrivialization corresponds to the absolute parallelismafforded by the group action on the group manifold and thus on theparallel transport of vectors of the Lie algebra, as described above.The intimate relationship between the Lie group G and the Lie algebrag, has the consequence that the torsion of G ... is simply the Lieproduct, [x,y], of g ... for the torsion T of a Lie groupmanifold we can write:
... where the componets of T are the structure constants of theLie algebra; and X, Y, and Zare Lie algebra elements, i.e.,left-invariant vector fields on G. For right invariant vector fieldsthe torsion tensor is the would be -T_ij^k. ... In general, thecurvature tensor describing the curvature of the Lie group manifoldis the Riemann curvature tensor which can be written in terms of theLie algebra structure constants:
The Riemann curvature tensor is the tensor generated by theLevi-Civita connection, for which there is zero torsion. Thus onthe 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 notorsion. Given two connections on the same manifold, thedifference between the Christoffel symbols is a tensor, called thedifference tensor. In the case of these two connections on a Liegroup manifold the difference tensor is the contorsion tensorK.
We can write:
where the first Christoffel symbol is the Cartan connection ofabsolute parallelism (with 64 independent functions); the second ...Christoffel symbol is the Levi-Civita connection (with 40 independentfunctions ...); and K is the contorsion tensor (with 24 independentfunctions). The contorsion tensor is also the tensor of Riccirotation coefficients. ...".
Some unconventional SubLuminal solutions may have uses thatrelate Electromagnetism toSpaceTime curvature, or, with respect to sound, perhaps sonoluminescence.
Tony Smith's Home Page