- Complex SpaceTime
- SpaceTime ShapeChanging
- Using Hopf Structure for coherent GraviPhoton phenomena
- Casimir Vacuum Energy Forces and Conformal GraviPhotons
- Spheres, Rodin Coils, Kron, and Negative Permittivity
- Quaternions and GraviPhotons
- GraviPhoton Force Strength
- Astrophysical Phenomena
- Wavelets and the Conformal Group
- GraviPhotons and the Cosmological Constant
- GraviPhotons and SU(5) GUT Monopoles
- Mannheim Weyl Conformal Theory

**The ****D4-D5-E6-E7model**** coset spaces ****E7****/ (****E6**** x U(1)) and****E6**** / (****D5****x U(1)) and ****D5**** /(****D4**** x U(1)) areConformal Spaces. ** You can continue the chain to D4/ (D3 x U(1)) where D3 is the15-dimensional Conformal Group whose compact version is Spin(6), andto D3 / (D2x U(1)) where D2 is the 6-dimensional Lorentz Group whose compactversion is Spin(4).

Note that D3 inherits a non-linear versionof triality from the full triality of D4.

Electromagnetism, Gravity, andthe ZPF all have in common thesymmetry of the 15-dimensional D3 Conformal Group whose compactversion is Spin(6), as can be seen by the following structures withD3 Conformal Group symmetry:

- Maxwell's equations of Electromagnetism
- the canonical Dirac Lagrangian for massive fermions
- Gravity derived from the Conformal Group using the MacDowell-Mansouri mechanism
- the Quantum Theoretical Hydrogen atom
- the Lie Sphere geometry of SpaceTime Correlations in the Many-Worlds picture

Further, the 12-dimensional Standard Model Lie AlgebraU(1)xSU(2)xSU(3) may be related to the D3 Conformal Group Lie Algebrain the same way that the 12-dimensionalSchrodinger Lie Algebra is related to the D3 Conformal Group LieAlgebra.

The physical 4-dimensional SpaceTime of the D4-D5-E6-E7-E8VoDou Physics model is a 4-dimensional HyperDiamond latticeSpaceTime that is continuously approximated globally by RP1 x S3 andlocally by Minkowski SpaceTime, with Gravity coming from the15-dimensional Conformal Group Spin(2,4) by the MacDowell-Mansourimechanism. The curved SpaceTimeof General Relativity is not considered fundamental, but is producedby by starting with a linear spin-2 field theory in flatspacetime, and then adding higher-order terms to getEinstein-Hilbert gravity. The observed curved SpaceTime is thereforebased on an unobservable flat Minkowski SpaceTime. (See Feynman,Lectures on Gravitation, Caltech 1971 and Addison-Wesley 1995, andsee Deser, Gen. Rel. Grav. 1 (1970) 9-18 as described in Misner,Thorne, and Wheeler, Gravitation, Freeman 1973, pp. 424-425.)

If you were to start, not with locally Minkowski SpaceTime, butwith the curved SpaceTime of General Relativity, then you would seethat the Conformal transformations of Minkowski SpaceTime by the15-dimensional Conformal Group Spin(2,4) corresponds to the Conformaltransfomations of the curved SpaceTime by the infinite-dimensionalConformal subgroup of the group Diff(M4) of General Relativisticcoordinate transformations of the 4-dimensional SpaceTime M4 ofGeneral Relativity, which Conformal subgroup is defined as thoseGeneral Relativistic coordinate transformations that preserveconformal structure and which infinite-dimensional Conformal subgroupcan be called the Weyl Conformal Group. (See Ward and Wells, TwistorGeometry and Field Theory, Cambridge 1991, p. 261.)

The Presentations of the 15 Infinitesimal Causal Symmetries ofSegal's Conformal SpaceTime are given in the following Table 1. fromProc. Nat. Acad. Sci. USA 78 (1981) 5261-5265, by I. E. Segal, H. F.Jakobsen, B. Oersted, S. M. Paneitz, and B. Speh:

In his book Mathematical Cosmology and Extragalactic Astronomy(Academic Press 1976) (pages 72-75, 88-91), Segal says:

"... The suitably scaled 15 linearly independent generators Lij ofsymmetries of unispace [Segal's term for the Conformal RP1 x S3SpaceTime used in the D4-D5-E6-E7 model- from here on on this page I will call it Conformal SpaceTime]... differ from the 11 generators of the group of global conformaltransformations in Minkowski space

[as R becomes infinite, where R is the radius of curvature ofConformal SpaceTime] ...

... The angular momenta Lij ... [ i,j = 1,2; 2,3; 3,1 ]... have ... the same expression both in Minkowski space and in[Conformal SpaceTime] ... The same is true of the boosts ...-iL0,j ... ( j = 1,2,3 ) ... and the infinitesimal scaletransformation [ L-1,4 ] ...

... The scale generator - L-1,4 ... determines a ... scalar field.This ... is most naturally interpreted from a gravitationalstandpoint ...

... two ordered sets, each containing four of the Lij, converge onthe same ... fields in Minkowski space ... in particular, R^(-1)L-1,j and R^(-1) Lj,4 [ for j = 0,1,2,3 ] both ... agree ...[as R becomes infinite] with the [correspondingMinkowski] conventional energy-momentum component. ... Thedifferences

thus are ... representable by a ... vector field, which physicallywould appear most naturally as potentially related to gravitationalphenomena ... ".

At this point, I will depart from the program of Segal and introducesome of my own ideas, so that any errors that appear in what followsare more likely attributable to me than to Segal.

First,** **

Then,

**GraviPhotons** look like:

Virtual Covariant Conventional U(1)Photons, in that they have 4 Components, includingLongitudina/Scalar Components; and

Vector Gravitonsthat can interact with the Imaginary Part of Complex Spacetime.

In the D4-D5-E6-E7 physics model,4-dimensional Physical Spacetime is the ShilovBoundary of an 8-real-dimensional Bounded Complex HomogeneousDomain corresponding to the Hermitian Symmetric Space Spin(2,4) /Spin(4) x Spin(2) = SU(2,2) / S(U(2)xU(2)), which symmetric space isa 15-6-1 = 8-real-dimensional space with Complexstructure, or a 4-Complex-dimensional space.

The 4-real dimensional Shilov Boundaryis RP1 x S3, which is topologically equivalent to S1 x S3, where S1is time and S3 is space. If you suppress one space dimension,Spacetime looks like this

where the gold is the interior 2-diskof the blue 1-sphere S of Time and thered is the interior 4-ball of thegreen 3-sphere S3 of Space.

If the S1 x S3 Physical Spacetime is regarded as the 4-dimensionalReal Part of 8-real-dimensional ComplexSpacetime, then the Imaginary Part of 8-real-dimensional ComplexSpacetime is found in the Interior. The 4 real dimensions of theImaginary Part are generated by:

- 1 vector normal to the 1-sphere S1 pointing into its interior 2-disk;
- 1 vector normal to the 3-sphere S3 pointing into its interior 4-ball;
- 1 vector normal to the 1-sphere S1 pointing to the S3 and its interior; and
- 1 vector normal to the 3-sphere S3 pointing to the S1 and its interior.

**Of the 15 Conformal gravitons**,

the 1 scale generator, or dilation, does not change the geometricrelationship between the Shilov BoundaryReal Part and the Interior Imaginary Part;

but

**the 4 GraviPhotons can go from the ****ShilovBoundary**** Real Part into the Interior ImaginaryPart**.

By the **Harmonic structure of the ****D4-D5-E6-E7physics model**:

any Continuous Function on the Shilov Boundary S1 x S3 Real Part can be extended to a Harmonic Function throughout the entire 8-real-dimensional Complex Spacetime;the Harmonic Functions form a basis in terms of which you can express any Analytic Function in the entire 8-real-dimensional Complex Spacetime; and

if you know an Analytic Function in any neighborhood, no matter how small, you can use Analytic Continuation to know that Analytic Function everywhere in the entire 8-real-dimensional Complex Spacetime.

A couple of notes: you can probably get the same results bylooking at the 8-real-dimensional SpaceTime as a2-Quaternionic-dimensional space (with Fueter's Quaternioinicanalyticity and Quaternionic Cauchy-Riemann equations as described,for example, in On the Role of Division, Jordan and Related Algebrasin Particle Physics, by Feza Gursey and Chia-Hsiung Tze (World 1996))or by looking at the 8-real-dimensional SpaceTime as a1-Octonionic-dimensional space; and, according to YGGDRASIL(journal of paraphysics whose name is "... the "world tree" in Norsemythology. It is also known as the "tree of knowledge," the "tree ofthe universe" and the "the tree of fate." ..."), the idea of lookingat the imaginary part of an 8-real-dimensional SpaceTime was used byElizabeth Rauscher in 1977 and Puthoff, Targ and Edwin May in1979.

As **Vector Gravitons** they can (considered as gauge bosons)be attractive or repulsive and can (considered as contributors to thegeneral relativistic shape of SpaceTime) **transformSpaceTime** in their neighborhood.

The Dilation sets the scale of the Higgs VeV at 250 GeV so thatgeneral deformations of SpaceTime can take place only above thatenergy level, while GraviPhotonSpecial Conformal (Hopf flow)transformations are useful in Conformal deformations ofSpaceTime.

Incompressibility of the Aetherbelow 250 GeV is only with respect to the 6-dim vector space ofthe Conformal Group Spin(2,4), so that below 250 GeV you can seeConformal phenomena that appear to show compressibility from thepoint of view of 3-dim space or 4-dim Minkowski spacetime. Suchconformal phenomena include theFock superluminal solutions of Maxwell's equations that are describedby R. M. Kiehn.

The 4 GraviPhoton Special Conformal transformations are like theMoebius linear fractional transformations, that do deform Minkowskispacetime but take hyperboloids into hyperboloids and are thesymmetries of superluminal solutions of the Maxwell equations. Theyare incompressible/linear from the point of view of a 6-dimensionalSpaceTime, with 4 spatial dimensions and 2 time dimensions, becausethe conformal group over Minkowski spacetime is just SU(2,2) =Spin(2,4), the covering group of SO(2,4), and therefore the Liealgebra generators look like those of rotations in a 6-dim vectorspace of signature (2,4). This is the 4-dim space with 2-dim timesuggested by Robert Neil Boyd, in which things look linear (eventhough from our conventional 3-dim spatial or 4-dim Minkowski pointof view they might appear, due to our limited conventionalperspective, to be nonlinear). If you regard PhysicalSpaceTime as the 6-dimensional vector space of Spin(2,4), andInternal Symmetry Space as 4-dimensional CP2,then the total spaceis 6+4=10-dimensional. With respect to tthe D4-D5-E6-E7model, 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).

The corresponding Complex Linear Fractional Moebius Transformations can be used to visualize the usefulness of GraviPhoton SpaceTime Shape-Changing. For example,the Elliptic Complex Linear Fractional Moebius Transformations can be used to make a Complex version of a type of Alcubierre Warp Drive that moves through space by the Alcubierre mechanism (Classic and Quantum Gravity 11 (1994) L73) in which you "... create a local distortion of spacetime that will produce an expansion behind the spaceship, and an opposite contraction ahead of it. In this way, the spaceship will be pushed away from the earth and pulled towards a distant star by spacetime itself. ...". Since the spacetime around the spaceship is not distorted, the spaceship and its contents feel no G-forces from acceleration.

(Note: My reference to Alcubierre's warp drive isintended to refer only to the general idea of contracting spacetimeahead of a ship and expanding spacetime behind a ship. I do NOT saythat what I propose uses the same physics mechanism by whichAlcubierre actually proposes to accomplish this. In fact, my proposalis NOT equivalent to the mechanism used by Alcubierre. Among thedifferences, I do not have to build a bubble around the spaceship,while Alcubierre does have to do that.)

In the Elliptic Conformal Moebius Map shown above, a Ship around the two focal points on the Spatial Line would contract SpaceTime above it and pull it through the Ship/Points, while pushing away expanding SpaceTime to the bottom.

Consider the 2-dimensional Ship as being built around the twofocal points.

Then consider the 2-dimensional Ship as being a 2-dimensionalcross section of a 2-dim Torus in 4-dimensions (torusimage from 3D-Filmstripby RichardS. Palais):

Then consider the 2-dim Torus as part ofa **HopfFibration** of a 3-sphere S3 in 4-dimensions.

**The ****HopfFibration** is described by William Thurston in his bookThree-Dimensional Geometry and Topology, volume 1, PrincetonUniversity Press 1997, pages 103-108, where he says:

"... Identifying R4 with C2 ... If the coordinates in C2 are z1 and z2, the equation of a unit sphere becomes | z1^2 + z2^2 | = 1. Each complex line ... in C2 intersects S3 in a great circle called a Hopf circle. Since exactly ond Hopf circle psses through each point of S3, the family of Hopf circles fills up S3, and the circles are in one-to-one correspondence with the complex lines of C2, that is, with the Riemann sphere CP1 ... Informally, the three-sphere is a two-sphere's worth of circles. Formally, we get a fiber bundle p : S3 -> S2, with fiber S1. ... We call this structure the Hopf fibration ... Figure 2.31 shows what the Hopf fibration looks like under stereographic projection. ... the vertical axis is the intersection of S3 with the complex line z1 = 0, and the horizontal circle is the intersectioin with z2 = 0. The locus { | z2 |<a }, for any 0 < a < 1, is a solid torus neighborhood of the z2 = 0 circle. Its boundary { | z2 | = a }, a torus of revolution, is filled up by Hopf circles, each winding once around the z1-circle and once around the z2-circle. ... any torus of revolution has cirves winding around in both directions that are geometric circles ... [called] ... Villarceau circles. ......

the maps gt : S3 -> S3 given by multiplication by exp( i t ), for t in R, are isometries and ... leave the Hopf fibration invartiant. Thus S3 has isometries that do't have an axis: the motion near any point is like the motion near any other point. This is one way in which S3 seems rounder than S2. The one-parameter family { gt } iscalled theHopf flow. ...... The descriptions of S3 via quaternions and via complex numbers can be combined. If we look at [the quaternions] as a complex vector space, multiplication on the left by the quaternion i is the same as multiplication by the complex number i, so the vector field Xi(p) = i p , for p in S3, induces the Hopf flow on S3. ... because i plays no special role among the quaternions ... any pure quaternion of unit norm can be used in lieu of i ... Thus there are many Hopf flows and many Hopf foliations on S3. In particular, we can take three mutually orthogonal vector fields Xi, Xj, and Xk to get three mutually orthogonal families of Hopf circles. ...

... If g in S3 is not +/- 1, the transformation x -> g x fits into the unique Hopf flow generated by the Hopf field x -> p x , where p is the unit quaternion in the direction of the g purely imaginary component of g . Similarly when h in S3 is not +/- 1 , the transformation x -> x h fits into a unique Hopf flow, generated by a vector field x -> x p . The two kinds of Hopf flows are distinguished as left-handed or right-handed: the circles near a given circle wind around it in a left-handed sense or in a right-handed sense ... any right-handed Hopf flow commutes with any left-handed Hopf flow. ...". In a given Hopf Fibration, any two Hopf circles are linked.

In terms of the Space S3, the GraviPhotonMoebius Alcubierre Warp Drive looks like a vortex in which theDestination Space is pulled along the Hopf Flow Circles in toward theship and, after the ship passes through it, is expelled behind theship.

Physically, the 4 covariant polarizations (t,x,y,z) of aGraviPhoton moving in the z-direction correspond to:

- t-polarization = Flow in Time circle RP1 = Timelike movement;
- z-polarization = Hopf Flow in Longitudinal Hopf circles (with plane of Hopf circle parallel to z-axis) of Hopf Fibration = Spacelike movement in the z direction; and
- x and y polarizations = trransverse polarization of conventional Far Field GraviPhoton.

The structure is that of a Penrose Twistor, as described in thebook Spinors and Spacetime, volume 2, by Penrose and Rindler(Cambridge 1986), pages 61-62: "... These curves are ... circles ...They twist around one another (hence the term twistor!) in such a waythat every pair of circles is linked ...

... They lie on the set of coaxial tori ... [that] ... arethe rotations about the z-axis of a system of coaxial circles in the(x,z)-plane. From the point of view of the compactified space-time... we should regard the hyperplane t = tau as being compactified(conformally) by a point at infinity. It then becomes topologically athree-dimensional sphere S3 (of which the hyperplane t = tau may beregarded as the stereographic projection). The vector field on S3 iseverywhere nonsingular and nowhere vanishing. The circles constitutewhat is known as a Hopf fibring of S3. With a suitable scaling theybecome Clifford parallels on S3. ... all the circles in thehyperplane thread through the particular (smallest) circle ... theradius of this smallest circle ... is given, generally, ... by thespin divided by the energy ...".

A sequence of Hopf tori, each made up of Hopf-Clifford circles, isshown in a .mpg movie on a UBCweb page. Here are three of the Hopf tori from that movie:

Why Electrons? Because an Electron is a spin 1/2 particle, so thatit is not only a localized particle, but that it is Entangled withand Connected to the Global Universe in which it is located. Spin 1/2particles orientation-entangled with their environment are Fermionsand their intrinsic orientation-entanglement can be mathematicallydescribed by saying that Fermions areQuaternionic.

To see how orientation-entanglement works, look at this picturefrom Gravitation, by Misner, Thorne, and Wheeler (Freeman 1972):

The orientation of the ball is related to the surrounding sphereby the tangle of the strings connecting them. As Louis H. Kauffmansays in his book Knots and Physics (World Scientific Publishing Co.1991), a spin 1/2 particle is like a ball attached to itssurroundings by string.

Is such a GraviPhoton connection between locat particles and the Global Universe a reasonable thing to see in our physical world?Another example is the graviton-based Mach's Principle connection between local things and the Global Universe, illustrated in Introduction to Cosmology, by J. V. Narlikar (2nd ed Cambridge 1993) by the curvature of the surface of water in a bucket that is spinning with respcct to the Global Universe:

In their book Gravitation (Freeman 1973), Misner, Thorne, and Wheeler say: "... Einstein's theory ... identifies gravitation as the mechanism by which matter there influences inertia here ...".

Then consider the 2-dimensional Ship as being a 2-dimensionalcross section of a 2-dim Torus in 4-dimensions (torusimage from 3D-Filmstripby RichardS. Palais):

Then consider the 2-dim Torus as part of a **Hopf Fibration**of a 3-sphere S3 in 4-dimensions.

Then consider Electons, **all in the same coherent EntanglementConnection phase**, flowing coherently along Clifford-Hopf Circles(Clifford-Hopf torus (angle) images from3D-Filmstripby RichardS. Palais)

as shown by the green arrow on thered or white Clifford-Hopf Circle andthus coherently dragging the Rest of the Global Universe from above,into the central hole of the 2-Torus, and then out the bottom in ahigher-dimensional version of the Elliptical Complex Conformal2-dimensional flow.

By symmetry, every Clifford-Hopf Circle Flow(red or white) has a correspondingOpposite Flow (blue)(Clifford-Hopf torus (overhead) images from3D-Filmstripby RichardS. Palais)

and, like all Clifford-Hopf Circles, they are linked with eachother.

Effectively, the Electrons produce GraviPhotons that produce the4-dimensional Conformal Flow. In terms of the Penrose and Rindlerimage,

Such Star-Gate Ring Ships might be regarded as atype of worm-hole, whose stability might be interpretable interms of ghosts. and whoseengineering might involve RodinCoil ring structures.

How could a 2-Torus Ring Ship be engineered?Perhaps by using a fermionic version a Bose-EinsteinCondensate. This could be related to asuggestion of Ivanenko and Vladimirov: Paul Hill (inUnconventional Flying Objects, Hampton Roads 1995) quotes D. D.Ivanenko and Yu. S. Vladimirov (in Matter and Physical Fields - PartOne of The Earth in the Universe, translated 1968 in U.S. Departmentof Commerce Clearinghouse) as saying "... it may be possible forelectron-positron pairs to be transformed not only into photons, butalso into gravitons".

Instead of Frame-Dragging by Einstein Gravity, you haveEntanglement Connection Dragging by flow of Spin 1/2 ElectronSpinors, which is effectively Dragging by Torsionin Einstein-Cartan Gravity.

The GraviPhoton force strength factor is different from thegravitational G of the gravitational force equation F = G mM /r^2,

and is also different from force strength factor of the electricforce equation between two unit electric charges F = e^2 / r^2 ,where e is the electric charge, or amplitude for a single electron toemit or absorb a photon, and e^2 is the electromagnetic finestructure constant 1 / 137.03608.

In the Far Field region of gravity, the modifications for theGraviPhoton force are as follows:

- the geometric part of the GraviPhoton force is equal to the geometric part of the gravitational force strength, which is 1; and
- the mass factor part of the GraviPhoton force is stronger than the mass factor part of the gravitational force by a factor of 1 / e^2 = 137.03608,
- with the gravitational mass factor, in terms of the proton mass, being 1 / (Planck Mass)^2 = 10^(-38) = (10^(-19))^2
- and, as indicated by the astrophysical work of Wesson, the GraviPhoton mass factor, in terms of the proton mass, being 137.03608 / (Planck Mass)^2 = 10^(-36) = (10^(-18))^2.

Unlike gravitation, and like electromagnetism, the GraviPhotonforce will cancel itself out in matter that is randomly oriented. Therotating astrophysical bodies that give evidence (described byWesson**)** for theGraviPhoton force are rotating in a coordinated way such that thecancellation does not occur.

In the Near FieldInduction/Static Region of Strong Gravity, the GraviPhoton forcehas the same strength as the other components of Gravity,which is the maximal value 1.

From the point of view of a neutral Kerr-Newman Black Hole, with coincident outer and inner event horizons, with irreducible mass M, angular momentum J, and charge Q:

Q^2 + (J/M)^2 = M^2

Dividing through by M^2, you getJ^2/M^4 = (J/M^2)^2 = 1 - (Q/M)^2Setting Q/M = x you get

(assuming that p_astro(Wesson) = (G/C) (1/alpha), with naturalunits G = c = 1 so that p_astro(Wesson) = 1 / alpha )

J = sqrt(1 - x^2) M^2 = p_astro(Wesson) M^2 = 137 M^2so that 1 - x^2 = 137^2 and

Q/M = x = sqrt(-137^2) = 137 i = 137 exp(pi/2)

and sqrt(x) = 11.7 exp(pi/4). Since amplitudes are inherently complex, there is no problemwith them being imaginary as well as negative or positive.Since x = Q/M, and sqrt(x) = sqrt( Q/M ), you get the magnitude

|sqrt(M)/sqrt(q)| = |sqrt( M/Q )| = 1/11.7 = 0.0855

Richard Feynman says in his book QED (Princeton 1988):"... e -the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.0854... the inverse of its square: about 137.03... has been a mystery ... all good theoretical physicists put this number up on their wall ...".To see the physical meaning of the ratios Q/M and sqrt(M)/sqrt(q), consider: M^2 corresponds to area and to number of classical q-bits of mass information, represented by real numbers. M, irreducible mass of black hole, corresponds to length (square root of area) and to number of quantum q-bits of mass information, represented by complex numbers with complex phase. sqrt(M) corresponds to spinor (square root of 4-vector) and carries orientation/entanglement with 4-dim spacetime represented by quaternions.Q^2 corresponds to number of classical q-bits of charge information, represented by real numbers. Q, charge of black hole, corresponds to number of quantum q-bits of charge information, represented by complex numbers with complex phase. sqrt(Q) corresponds to square root of charge, which might be called a charge-spinor, represented by quaternions. The ratio sqrt(M) / sqrt(Q) = 0.0855 corresponds to the amplitude for a charge-spinor to emit or absorb a GraviPhoton whose quaternionic property of orientation/entanglement with 4-dim spacetime can alter 4-dim spacetime and its electromagnetic fields. In randomly oriented situations, the GraviPhoton effects cancel out and are not ordinarily observed. In coherent situations (such as macroscopic rotating bodies) the GraviPhoton alterations of 4-dim spacetime and its electromagnetic fields might be observed. The ratio Q / M = 137 corresponds to the fact that the charge Q of a Kerr-Newman black hole carries more quantum information than its irreducible mass M by a factor of about 137.

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