Tony Smith's Home Page

Segal's ConformalTheory




Here are some Obituaries describing aspectsof Segal's work, and references to current work of AlexanderLevichev.


Graviphotonsand Astrophysics

Why don't we see GraviPhotons all the time?

They only exist where either:

as Photons, they are in the Static/Induction Region; or

as Vector Gravitons, they are in a region of SpaceTime where curvature is significant (i.e., a region not much smaller than R).

The SpaceTime regions where curvature is significant include:

( Compare angular momentum /electromagnetic relationships on many size scales. )

As to CosmologicalRegions, in his book Mathematical Cosmology and ExtragalacticAstronomy (Academic Press 1976) (pages 72-75), Segal says:

"... & / &T , the generator of ... [Conformal SpaceTime}temporal evolution [where & denotes partial derivative] ... appears in terms of Minkowski coordinates t,x,y,z [as] ...
&/&T = &/&t + (1/4)( t^2 + X^2 + y^2 +z^2 ) &/&t + ( t/2 )( x &/&x + y &/&y + z &/&z )

... the ... [Conformal SpaceTime] energy is ... positive, and exceeds (in all states) the special relativistic [Minkowski] energy ... the two time scales differ only by at most about 1 part in 10^19 in the course of a year (or less), and similarly for distance scales (scaled by the velocity of light); the difference is thus well beyond the limits of present experimental capabilities. ...".


As to AstrophysicalBlack Hole and Neutron Star Regions,Segal did not discuss them in his work, which was oriented primarilyto cosmology (and some elementary particle theory), but not so muchto structures as large as planets and stars and smaller than theuniverse as a whole. Obviously, BlackHoles are regions in which SpaceTime Curvature is significant,but what other structures might also have significant curvature?

The following diagram, from Figure 1.1 of Black Holes, WhiteDwarfs, and Neutron Stars, by Shapiro and Teukolsky(Wiley-Interscience 1983),

shows that dense Neutron Stars may have significant curvature. Inthe diagram, I added the red line toshow things that were 10% as large in radius as they need to be toform Black Holes of a given density, and thegold line to show things that were 1% aslarge in radius as they need to be to form Black Holes of a givendensity.

As to observations of phenomena that might be related toGraviPhoton phenomena in astrophysical structures, Tom VanFlandern has a web site at URL http://www.metaresearch.orgin which he has a review of the book Seeing Red by HaltonArp. In the review, Van Flandern says that Arp's observationsindicate that redshift is not just a Doppler distance effect, but isat least in part also related to "... an inverse function of the ageof that matter ...".

What does Arp mean by "age of that matter"?

To quote from the review:

"... the O-stars in clusters within our galaxy are redshifted with respect to the B-stars by another 10 km/s or so -- something called the "K-effect" ...

these apparently ejected quasars [have] redshifts ordered inversely with distance from their parent [and they] also tend to line up along the minor axis of the parent galaxy. ... the young-appearing objects with the highest redshifts are aligned on either side of eruptive objects, which implies the ejection of protogalaxies and the association of redshift with youth. Increasing distance from parent leads to brighter, lower redshift objects, so this is the direction of evolution with age. ...

Faraday rotation caused by traversing a magnetized plasma can be measured for quasars. So the amount of such rotation ought to be a distance indicator. But it was then discovered that quasars with redshifts of about 2 had only 1/3 as much Faraday rotation as quasars with redshifts of about 1, when they ought to have had twice as much rotation. By contrast, this is in accord with Arp's model because the redshift z = 2 quasars are intrinsically fainter, and therefore generally seen only at closer distances, than those with z = 1. ...".

How could "age" have any effect on redshift? Consider that the"young" objects with high redshift are closer to the time offormation, and that the process of formation is not just agravitational collapse, but is also a dynamical ElectroMagneticprocess involving very strong EM fields and related plasma effectsthat are not well understood by conventional physics. Such EM effectsmay involve longitudinal photons due to the very large scale of theNear Field in such dynamicalastrophysical conditions, and so might produce redshift even thoughredshift is normally considered to be a "gravity-only"phenomenon.

Elementary Particle Phenomena could beinvolved, especially if you use the ComptonRadius Vortex viewpoint of the D4-D5-E6-E7physics model.


Segal died 30 August 1998 at the age of 79. A number ofobituaries were published in the Notices of the AMS 46 (June/July1999) 659-668.

In one such obituary, Edward Nelson, who had Segal as advisor forhis 1955 Chicago Ph.D., quoted Segal as saying (in a 1992publication):

"... Universal space-time is ... locally conformal to Minkowski space and globally conformal to the Einstein universe E = R1 x S3 ... These developments ... suggest that the fundamental forces of Nature are conformally invariant, but that the state of the Universe breaks the symmetry down to the Einstein isometry group. This provides an alternative to the Higgs mechanism ...

and otherwise has implications for particle physics, including

the elimination of ultraviolet divergencies in representative nonlinear quantum fields,

the formulation of a unified invariant interaction Lagrangian,

assignments of observed elementary particles to irreducible unitary positive-energy representations of the conformal group, and

the correlation of the S-matrix with the action in E of the generators of the infinite cyclic center of the simply-connected form of the conformal group. ... ".

Segal spent much time and effort on a quadratic cosmologicalredshift that he claimed was implicit in the physics of the ConformalGroup. Segal's redshift was described by Bertram Kostant, whohad Segal as advisor for his 1954 Chicago Ph.D., in another of theobituaries in the Notices of the AMS 46 (June/July 1999)659-668.:

"... One particular nilpotent element, in the representation of SU(2,2) associated with solutions of Maxwell's equations, defines the standard operator to determine the frequencies of light waves.

[John Baez, who had Segal as advisor for his 1986 MIT Ph.D.,says: "... That's the generator of Minkowski time translations. ...If you use the Minkowski Hamiltonian everywhere (there's one for eachobserver) you don't get the redshift. ... ".]

But Irving focused on another element with a nonnegative spectrum, an element that was elliptic and not nilpotent, but closely related ... This elliptic element has beautiful mathematical properties, like generating an invariant cone. ... This elliptic element is at the heart of Segal's cosmological theory.

[John Baez says: "... That's the generator of Einstein timetranslations. ... if you use the Einstein Hamiltonian everywhere youdon't get the redshift. ... ".]

What he is saying is that it is the elliptic element that should be used to determine the energy of an electromangnetic wave, and not the nilpotent element.

[John Baez says: "... But if we do that *everywhere*, wesimply get physics on the Einstein universe R x S^3 - no redshift.... ".]

The redshift ... is accounted for by the difference between the elliptic and nilpotent element - negligible locally, but significant at great distances. ...".

[John Baez says: "... Aha: this is the tricky part. How is it"accounted for by the difference" between these elements, exactly?... ".]

I. E. Segal, H. P. Jakobsen, B. Oersted, S. M. Paneitz, and B. Speh, in their article Covariant chronogeometry and extreme distances: Elementary particles (Proc. Nat. Acad. Sci. USA 78 (1981) 5261-5265, at page 5261), say: "... the energy of a photon in ...[ Conformal Unispace ]... splits Lorentz-covariantly into a local and delocalized part. The local part is represented by the conventional energy operator in ...[ Minkowski space-time ]... , which can be regarded as a submanifold of ...[ Conformal Unispace ]... ; the delocalized part drives, essentially as an interaction hamiltonian, a redshift in very good agreement with objective observations on galaxies and quasars. ...".

John Baez says: "... Segal used to say the dynamics of distantstars was governed by the Einstein Hamiltonian but we saw them fromthe Minkowski viewpoint. However, he never clarified how this shouldactually work. ...".

In my opinion, it might work like this:

Bertram Kostant said, in his Segal obituary article:

"... I have it from a highly reliable but unnamed source that there is a growing group of cosmologists who have come to believe that the correct understanding of the redshift is some sort of fusion of the Doppler effect and Irving's theory. So it is not impossible that Irving could turn out to be correct after all. ...".
I asked Bertram Kostant about that quote, and he said that he did not name the source because the source requested anonymity because he (the source) wanted to avoid entanglement in the heated controversy that surrounds Segal's redshift ideas. I think that it is regrettable that interesting scientific ideas become so involved in unpleasant controversy that competent people become are intimidated from discussing such ideas. As to how and why Segal's ideas became controversial, Edward Nelson, in his Segal obituary article, said:
"... Part of the reason lies in Segal's style of scientific exchange - at times is resembles that of Giordano Bruno (later burned at the stake), who very shortly after his arrival in Geneva issued a pamphlet on Twenty Errors Committed by Professor De la Faye in a Single Lesson. But part of the fault lies with cosmologists and particle physicists intent on defending turf. ...".

As to what "... sort of fusion of the Doppler effect and Irving'stheory ..." might turn out to be true, Segal, in the GeneralConclusions section (pages 187-191) of his book MathematicalCosmology and ExtraGalactic Astronomy (Academic 1976) says:

"... In principle, there is no difficulty in combining the chronometric redshift theory with some degree of expansion in accordance with a closed Friedmann model. As long as the rate of expansion is kept sufficiently low ... the basic features of the Friedmann theory and its correlation with cosmology, apart from the redshift itself, would be retained. Such a mixed theory cannot be excluded on a purely statistical basis, and would permit conventional ideas concerning the evolution and age of galaxies to persist in the combined theory without essential change. ...

... it remains to be explored to what extent observational estimates of [such things as] the mass density of the universe ...[and]... energy production in galactic nuclei and quasars ... may be affected by employment of the chronometric ... excess of the unienergy ... over the special relativistic energy ... i.e., of the ... energy corresponding to the difference between the two times involved in the theory (... shown to be positive) ... The amount of energy involved is ... quite substantial ...".

In their tribute to Irving Ezra Segal (Notices of the AMS 48(January 2001) 9-16), Auber Daigneault and Arturo Sangalli say:

"... It is not unthinkable that a rehabilitation of ... a cosmological constant ... /\ ... greater than zero ..., originally related to the radius r of S3 ...[of Segal's]... EU = R x S3 ... by the equation r = /\^(-(1/2)), could ... help Irving Ezra Segal's chronometric cosmology get the attention it deserves. ...".

I want to conclude this section of this web page with anotherquote from Edward Nelson's Segal obituary article

"... It is rare for a mathematician to produce a life work that at the time can be fully and confidently evaluated by no one, but the full impact of the work of Irving Ezra Segal will become known only to future generations."


with a comment that at least one person, AlexanderLevichev, is working on Segal's ideas by conducting aseminar on Newton-Einstein-Segal, studying ways to apply Segal'sideas to ModellingConsciousness, and by puttingon the web in ps format a paper entitled On Mathematical Foundationsand Physical Applications of Chronometry ( whichpaper was published in the book "Semigroups in Algebra, Geometry andAnalysis" ( ed. by Hofmann, Lawson, and Vinberg, Walter de Gruyterand Co, 1995 )).


MOdified Newtonian Dynamics (MOND)

In astro-ph/0112069,Mordehai Milgrom says:

"... In the early 1980s I proposed a modifed-dynamics ... based onthe fact that typical accelerations in galactic systems are manyorders of magnitude smaller than those encountered in the solarsystem. ... This modified dynamics, MOND,introduces a constant with the dimensions of an acceleration, a_0,and posits that standard Newtonian dynamics is a good approximationonly for accelerations that are much larger than a_0. ... the basicpoint of MOND, from which follow most of the main predictions, can besimply put as follows: a test particle at a distance r from a largemass M is subject to the acceleration a given by

a^2 / a_0 = M G r^(-2)

when a << a_0, instead of the standard expression a = M Gr^(-2) , which holds when a >> a_0. ... The value of theacceleration constant a_0 that fits all the data discussed ...[in the paper ]... is about 10^(-8) cm s^(-2) ...

... One immediate result ... is that at a large radius around amass M, the orbital speed on a circular orbit becomes independent ofradius. This indeed was a guiding principle in the construction ofMOND, which took asymptotic flatness of galaxy rotation curves as anaxiom ... When we count the mass of baryonic matter in such systems -in stars, neutral and high-T gas, etc.- the total sum does notprovide enough gravity to explain the observed accelerations in suchsystems within standard physics. If we adhere to standard dynamics,the need for dark matter is the only solution we can conceive. It is,however, possible that the laws of dynamics, proven in thelaboratory and the solar system, cannot be simply applied in therealm of the galaxies. ... Regarding galactic systems other thangalaxies, the comparison of the systematics of the observed massdiscrepancy with the expectations from MOND are shown in Figure 2

in Milgrom (1998) [ astro-ph/9810302, which says:

"... MOND is not a modification at large distances, but at low accelerations - which for a given mass are attained at large distances ... there is no correlation of the discrepancy with system size. ...

... in particular ... the small dwarf spheroidals and LSB discs show large discrepancies, while the large galaxy clusters evince only moderate discrepancies...." ]. ... based on analyzes referenced there.

The agreement is uniform, with one exception: The cores of richx-ray clusters of galaxies show a considerable mass discrepancy,while, according to MOND there shouldn't be any, because theaccelerations there are only of the order of a_0, and not muchsmaller. ( Application of MOND to the clusters at large, say within afew megaparsecs of the center, does predict correctly the massdiscrepancy. ) The resolution, by MOND, will have to be that thesecores harbor large quantities of still undetected baryonic matter,perhaps in the form of dim stars, perhaps as warm gas. Theenvironment, and history, of these cores is so unlike others thatthis would not be surprising. ...

... in interpreting data we equate observed accelerations with thegravitational field ... We can exemplify this point by consideringthe claimed anomaly in the motions of thePioneer 10 and 11 spacecraft. Analysis of their motion haveshown an unexplained effect ...[ see Anderson& al 2001, Study of the anomalous acceleration of Pioneer 10 and11, gr-qc/0104064 ]... that can be interpreted as being dueto an unexplained constant acceleration towards the sun of about 7 x10^(-8) cm s^(-2) , of the order of a_0. ... MOND could naturallyexplain such an anomalous acceleration ... It may well be thatthe modification enters the Pioneers motion, which corresponds tounbound, hyperbolic motions, and the motion of bound, andquasi-circular trajectories in a different way. ...

... It is worth pointing out that in such a modified-gravitytheory, the deep-MOND limit ... the deep-MOND regime ( a<< a_0 ) ... corresponds to a theory that isconformally invariant, asdiscussed in Milgrom (1997) [ gr-qc/9705003which says:

"... The Poisson equation describes many physical problems in linear media such as electrostatics, magnetostatics, steady-state diffusion and other potential flows in the presence of sources and sinks, and, of course, Newtonian gravity. It can be generalized ... to describe, for example, non-linear media with a response coefficient (dielectric constant, permeability, diffusion coefficient, etc.) that is a function of the field strength. ... For example, ... as a modification of Newtonian gravity to replace the dark-matter hypothesis for galactic systems ... In the modified dynamics discussed as an alternative to dark matter, phenomenology requires just this ... conformally invariant (CI) ... Our results here apply then in the large-distance limit of this theory. ...". ]. ...

... this asymptotic rotational speeddepends only on the total mass M via V^4 = M G a_0 . This, accordingto MOND, is the fact underlying the observed Tully-Fisher-typerelations, by which the typical (mean) rotational velocity, V , in adisc galaxy is strongly correlated with the total luminosity of thegalaxy, L, in a relation of the form L ...[ proportional to]... V^alpha . The power alpha is around 3-4, and depends on thewavelength band at which L is measured. The close agreementbetween this TF [ Tully-Fisher-type ] relationand the prediction of MOND isencouraging; but, to test MOND more precisely on this count, onewould have to bridge properly the mass-asymptotic-velocity MONDrelation with the commonly presented luminosity-bulk-velocity TFrelation. One should use the luminosity in a band where it is a goodrepresentative of the stellar mass, take into account not only thestellar mass, as represented by the luminosity, but also thecontribution of gas to the mass, and use the asymptotic velocity, asopposed to other measures of the rotational velocity. It has emergedrecently ... that if one does all this one indeed obtains a tightand accurate relation of the form predicted by MOND. ...".


Universe, Unispace,and Gravitationally Bound Domains

Unconventional Laboratory Experiments onEarth

Gravity at VariousSize-Scales

In his book Mathematical Cosmology and Extragalactic Astronomy(Academic Press 1976) (page 68), Segal says: "... We use the termunispace (short for universal covering space) for theuniversal space ... unispace is conformally an infinite-sheetedcovering of Minkowski space augmented by a light cone at infinity.Similarly, unitime refers to the natural time T in this space....".

 In the D4-D5-E6-E7-E8 VoDouPhysics model, our 4-dimensional Physical SpaceTime Universebegins as a relatively small spatial volume that is entirelyhigh-energy unispace, in which all 15 generators of ConformalSpin(2,4) Unispace, including the 4Conformal GraviPhotongenerators, are effective.


After some expansion, some regions of ourUniverse becomeGravitationally Bound Domains (such as,for example, Galaxies) in which the 4Conformal GraviPhotongenerators are frozen out, forming domains within ourUniverse likeIceBergs in anOcean of Water.

Note that Gravitationally Bound Domains are defined by their gravitational accelerations, not by their size - there is no correlation with system size.

The process of formation of GravitationallyBound Domains releases (like latent heat of fusion whenIcefreezes from Water)the excess of the Conformal Unienergy ofUnispace over the Poincare Energy ofMinkowski space.


Note that, within the Gravitationally BoundDomains there can exist Islands ofUnispace in which all 15 generators ofConformal Spin(2,4) remain effective(such as, for example, Protostar Clouds of Dustand Gas - like Puddles ofWateron an IceBergfloating in an Ocean of Water-

When a Protostar Clouds of Dust and Gas condense to form the Gravitationally Bound Domain of a Star and its Planets, the process releases (like latent heat of fusion when Ice freezes from Water) the excess of the Conformal Unienergy of Unispace over the Poincare Energy of Minkowski space).


During and after more expansion of ourUniverse as a whole, theGravitationally Bound Domains do notexpand within themselves, but are carried apart from each other as inthe familiar example of inflating aBalloon withCoins glued to its surface.

Since our laboratories on our Earth areGravitationally BoundDomains, our conventional experiments on Earth only seethe Poincare Energy of Minkowskispace.


Since the Pioneer spacecraft are notbound to our Solar System,

the PioneerSpacecraft are experiments beyond the Gravitationally Bound Domain ofour Earth-Sun Solar System.

In their Study ofthe anomalous acceleration of Pioneer 10 and 11, gr-qc/0104064,John D. Anderson, Philip A. Laing, Eunice L. Lau, Anthony S. Liu,Michael Martin Nieto, and Slava G. Turyshev say: "... The latestsuccessful precession maneuver to point ...[Pioneer 10]... toEarth was accomplished on 11 February 2000, when Pioneer 10 was at adistance from the Sun of 75 AU. [The distance from the Earth was[about] 76 AU with a corresponding round-trip light time ofabout 21 hour.] ... The next attempt at a maneuver, on 8 July2000, was unsuccessful ... conditions will again be favorable for anattempt around July, 2001. ... At a now nearly constant velocityrelative to the Sun of 12.24 km/s, Pioneer 10 will continue itsmotion into interstellar space, heading generally for the red starAldebaran ... about 68 light years away ... it should take Pioneer 10over 2 million years to reach its neighborhood....

Ecliptic pole view of Pioneer 10, Pioneer 11, andVoyager trajectories. Digital artwork by T. Esposito. NASA ARC Image# AC97-0036-3.

... on 1 October 1990 ... Pioneer 11 ... was [about] 30 AUaway from the Sun ... The last communication from Pioneer 11 wasreceived in November 1995, when the spacecraft was at distance of[about] 40 AU from the Sun. ... Pioneer 11 should pass closeto the nearest star in the constellation Aquila in about 4 millionyears ...

... Calculations of the motion of a spacecraft are made on thebasis of the range time-delay and/or the Doppler shift in thesignals. This type of data was used to determine the positions, thevelocities, and the magnitudes of the orientation maneuvers for thePioneer, Galileo, and Ulysses spacecraft considered in this study.... The Pioneer spacecraft only have two- and three-way S-bandDoppler. ... analyses of radio Doppler ... data ... indicated that anapparent anomalous acceleration is acting on Pioneer 10 and 11... The data implied an anomalous, constant acceleration with amagnitude a_P = 8 x 10^(-8) cm /s^2 , directed towards the Sun ...

... The anomalous acceleration is too large to have goneundetected in planetary orbits, particularly for Earth and Mars. ...For Earth and Mars, delta_r is about -21 km and -76 km. However, theViking data determines the difference between the Mars and Earthorbital radii to about a 100 m accuracy, and their sum to an accuracyof about 150 m. The Pioneer effect is not seen. Further, aperturbation in r produces a perturbation to the orbital angularvelocity ... The determination of the synodic angular velocity ... isaccurate to 7 parts in 10^11 , or to about 5 ms accuracy in synodicperiod. The only parameter that could possibly mask thespacecraft-determined a_R is ( G M_sun ). But a large error herewould cause inconsistencies with the overall planetary ephemeris ...Also, there would be a problem with the advance of the perihelion ofIcarus ... We conclude that the Viking ranging data limit anyunmodeled radial acceleration acting on Earth and Mars to no morethan 0.1 x 10^(-8) cm / s^2 . ... the size of the anomalousacceleration is of the order c H, where H is the Hubble constant...

... The Aerospace Corporation's Compact High Accuracy SatelliteMotion Program (CHASMP) ... analysis of Pioneer 10 data ... showed anunmodeled acceleration in a direction along the radial toward the Sun... The value is ( 8.65 +/- 0.03 ) x 10^(-8) cm / s^2 ... agreeingwith JPL's result ...

... Without using the apparent acceleration, CHASMP shows a steadyfrequency drift of about -6 x 10^(-9) Hz / s, or 1.5 Hz over 8 years(one-way only). ...

... This equates to a clock acceleration, -a_t, of -2.8 x 10^(-18)s / s^2 . The identity with the apparent Pioneer acceleration is a_P= a_t c. ...

... Having noted the relationships

a_P = c a_t

and that of ...

a_H = c H -> 8 x 10^(-8) cm / s^2

if H = 82 km / s / Mpc ...

we were motivated to try to think of any ... "time" distortionsthat might ... fit the CHASMP Pioneer results ... In other words...

Is there any evidence that some kind of "time acceleration"is being seen?

... In particular we considered ... Quadratic TimeAugmentation. This model adds a quadratic-in-time augmentationto the TAI-ET ( International Atomic Time - Ephemeris Time )time transformation, as follows

ET -> ET + (1/2) a_ET ET^2

The model fits Doppler fairly well ...


... There was one [other] model of the ...[timeacceleration]... type that was especially fascinating. This modeladds a quadratic in time term to the light time as seen by the DSNstation:

delta_TAI = TAI_received - TAI_sent ->

-> delta_TAI + (1/2) a_quad (TAI_received^2 - TAI_sent^2)

It mimics a line of sight acceleration of the spacecraft, andcould be thought of as an expanding space model. Note that a_quadaffects only the data. This is in contrast to the a_t ... thataffects both the data and the trajectory. ... This model fit bothDoppler and range very well. Pioneers 10 and 11 ... thenumerical relationship between the Hubble constant and a_P ...remains an interesting conjecture. ...".



In his book Mathematical Cosmology and Extragalactic Astronomy(Academic Press 1976) (pages 61-62 and 72), Segal says:

"... Temporal evolution in ... Minkowski space ... is

H -> H + s I

... unispace temporal evolution ... is ...

H -> ( H + 2 tan(a/2) ) / ( 1 - (1/2) H tan(a/2) ) = H + aI + (1/4) a H^2 + O(s^2)



the Pioneer Doppler anomalous acceleration is anexperimental observation of a system that is not gravitationallybound in the Earth-Sun Solar System, and its results are consistentwith Segal's Conformal Theory.


Rosales and Sanchez-Gomez say, at gr-qc/9810085:

"... the recently reported anomalous acceleration acting on the Pioneers spacecrafts should be a consequence of the existence of some local curvature in light geodesics when using the coordinate speed of light in an expanding spacetime. This suggests that the Pioneer effect is nothing else but the detection of cosmological expansion in the solar system. ... the ... problem of the detected misfit between the calculated and the measured position in the spacecrafts ... this quantity differs from the expected ... just in a systematic "bias" consisting on an effective residual acceleration directed toward the center of coordinates; its constant value is ... H c ... This is the acceleration observed in Pioneer 10/11 spacecrafts. ... a periodic orbit does not experience the systematic bias but only a very small correction ... which is not detectable ... in the old Foucault pendulum experiment ... the motion of the pendulum experiences the effect of the Earth based reference system being not an inertial frame relatively to the "distant stars". ... Pioneer effect is a kind of a new cosmological Foucault experiment, the solar system based coordinates, being not the true inertial frame with respect to the expansion of the universe, mimics the role that the rotating Earth plays in Foucault's experiment ...".

The Rosales and Sanchez-Gomez idea of a 2-phase system in whichobjects bound to the solar system (in a "periodic orbit") are in onephase (non-expanding pennies-on-a-balloon) while unbound (escapevelocity) objects are in another phase (expanding balloon) that"feels" expansion of our universe is very similar to my view of suchthings as described on this page. The Rosales and Sanchez-Gomez papervery nicely unites:

My view, which is consistent with that of Rosales andSanchez-Gomez, can be summarized as a 2-phase model based on Segal'swork which has two phases with different metrics:

If you look closely at the difference between the metrics in thosetwo regions, you see that the full conformal dark energy region givesan "extra acceleration" that acts as a "quadratic in time term" thathas been considered as an explanation of the Pioneer effect by JohnD. Anderson, Philip A. Laing, Eunice L. Lau, Anthony S. Liu, MichaelMartin Nieto, and Slava G. Turyshev in their paper at gr-qc/0104064.

Jack Sarfatti has a 2-phase dark energy / dark matter model thatcan give a similar anomalous acceleration in regions where c^2 /\dark energy / dark matter is effectively present. If there is a phasetransition (around Uranus at 20 AU) whereby ordinary matter dominatesinside that distance from the sun and exotic dark energy / darkmatter appears at greater distances, then Jack's model could alsoexplain the Pioneer anomaly and it may be that Jack's model withordinary and exotic phases and my model with deSitter/Poincare andConformal phases may be two ways of looking at the same thing. As towhat might be the physical mechanism of the phase transition, Jacksays

"... Rest masses of [ordinary matter] particles ... require the smooth non-random Higgs Ocean ... which soaks up the choppy random troublesome zero point energy ...".

In other words in a region in which ordinary matter is dominant,such as the Sun and our solar system, the mass-giving action of theHiggs mechanism "soaks up" the Dark Energy zero point conformaldegrees of freedom that are dominant in low-ordinary mass regions ofour universe (which are roughly the intergalactic voids that occupymost of the volume of our universe). That physical interpretation isconsistent with my view.

It may be that the observation of the Pioneer phase transition atUranus from ordinary to anomalous acceleration is an experimentalresult that gives us a first look at darkenergy / dark matter phenomena that could lead to energy sourcesthat could be even more important than the nuclear energy discoveredduring the past century.


In gr-qc/0104064,Anderson et al say:

"... Beginning in 1980 ... at a distance of 20 astronomical units (AU) from the Sun ... we found that the largest systematic error in the acceleration residuals was a constant bias, aP, directed toward the Sun. Such anomalous data have been continuously received ever since. ...",

so that the transition from inner solar system Minkowskiacceleration to outer Segal Conformal acceleration occurs at about 20AU, which is about the radius of the orbit of Uranus. That phasetransition may account for the unique rotational axis of Uranus,

which lies almost in its orbital plane.

The most stable state of Uranus may be with its rotational axispointed toward the Sun, so that the Solar hemisphere would beentirely in the inner solar system Minkowski acceleration phase andthe anti-Solar hemisphere would be in entirely in the outer SegalConformal acceleration phase.

Then the rotation of Uranus would not take any material from onephase to the other, and there would be no drag on the rotation due tomaterial going from phase to phase.

Of course, as Uranus orbits the Sun, it will only be in that moststable configuration twice in each orbit, but an orbit in theecliptic containing that most stable configuration twice (such as itspresent orbit) would be in the set of the most stable ground states,although such an effect would be very small now.

However, such an effect may have been been more significant on thelarge gas/dust cloud that was condensing into Uranus and therefore itmay have caused Uranus to form initially with its rotational axispointed toward the Sun.

In the pre-Uranus gas/dust cloud, any component of rotation thatcarried material from one phase to another would be suppressed by thedrag of undergoing phase transition, so that, after Uranus condensedout of the gas/dust cloud, the only remaining component of Uranusrotation would be on an axis pointing close to the Sun, which is whatwe now observe.

In the pre-Uranus gas/dust cloud, any component of rotation thatcarried material from one phase to another would be suppressed by thedrag of undergoing phase transition, so that, after Uranus condensedout of the gas/dust cloud, the only remaining component of Uranusrotation would be on an axis pointing close to the Sun, which is whatwe now observe.

Much of the perpendicular (to Uranus orbital plane) angularmomentum from the original gas/dust cloud may have been transferred(via particles "bouncing" off the phase boundary) to the cloudsforming Saturn (inside the phase boundary) or Neptune (outside thephase boundary, thus accounting for the substantial (relative toJupiter) deviation of their rotation axes from exact perpendicularity(see above image from Universe, 4th ed, by William Kaufmann, Freeman1994).


According to Utilizing Minor Planets to Assess the GravitationalField in the Outer Solar System, astro-ph/0504367,by Gary L. Page, David S. Dixon, and John F. Wallin:

"... the great distances of the outer planets from the Sun and the nearly circular orbits of Uranus and Neptune makes it very difficult to use them to detect the Pioneer Effect. ... The ratio of the Pioneer acceleration to that produced by the Sun at a distance equal to the semimajor axis of the planets is 0.005, 0.013, and 0.023 percent for Uranus, Neptune, and Pluto, respectively. ... Uranus' period shortens by 5.8 days and Neptune's by 24.1, while Pluto's period drops by 79.7 days. ... an equivalent change in aphelion distance of 3.8 x 10^10, 1.2 x 10^11, and 4.3 x 10^11 cm for Uranus, Neptune, and Pluto. In the first two cases, this is less than the accepted uncertainty in range of 2 x 10^6 km [ or 2 x 10^11 cm ] (Seidelmann 1992). ... Pluto['s] ... orbit is even less well-determined ... than the other outer planets. ... .... [C]omets ... suffer ... from outgassing ... [ and their nuclei are hard to locate precisely ] ...".


According to agoogle cache of anIndependent UK 23 September 2002 article by Marcus Chown:

"... The Pioneers are "spin-stabilised", making them a particularly simple platform to understand. Later probes ... such as the Voyagers and the Cassini probe ... were stabilised about three axes by intermittent rocket boosts. The unpredictable accelerations caused by these are at least 10 times bigger than a small effect like the Pioneer acceleration, so they completely cloak it. ...".


Can we carry out Unconventional Laboratory Experiments onEarth that let us use all 15 generators of ConformalSpin(2,4), including the 4 Conformal GraviPhotons?

To do that, we would have to construct an Island of Unispace- a process would require putting into the domain an amount of energyat least equal to the excess of the Conformal Unienergy ofUnispace for the domain over thePoincare Energy of Minkowski space forthe domain - like adding the latent heat of fusion when Icemelts to form Water).

How much energy must be added?

In his book Mathematical Cosmology and Extragalactic Astronomy(Academic Press 1976) (page190-191), Segal says:

"... in the very natural form of ...Weyl's conformally oriented theory ... presented by Veblen ... there intervene both a scalar and a vector field, such as are provided locally by the chronometric theory in the R -> infinity limit. ... the mere modification of a Minkowski metric by the introduction of these two fields ... is sufficient to imply all the observational consequences of general relativity (cf., e.g., Hawking and Ellis, 1973). ...

... The mechanism of energy production in galactic nuclei and quasars ... consists basically of the transformation of the excess of the unienergy (1/i) (&/&T ) over the special relativistic energy (1/i) (&/&t), i.e., of the new form of energy corresponding to the difference between the two times involved in the theory (... shown to be positive) into elementary particle processes. ...

... the fundamental local dynamical variables of the chronometric theory, energy, momenta, etc., differ from those of special relativistic field theory by terms of order 1/R, or less, where R is the radius of the universe in conventional laboratory units. ... the subgroup ... O(2,3) ... facilitates mass splitting ... The role of ... the D'Alembertian ... [] would be taken over by the Casimir operator for O(2,3), which differs essentially from [] ...[ of the Conformal Group Spin(2,4) = SU(2,2) ]... only by terms of order 1/R. ...".

and at pages 89-90:

"... The suitably scaled 15 linearly independent generators Lij of symmetries of unispace ... differ from the 11 generators of the group of global conformal transformations in Minkowski space by terms of order 1/R^2, as R -> infinity. ...

... It might appear anomalous that 15 vector fields converge to 11 vector fields. What happens is that two ordered sets, each consisting of four of the Lij, converge to the same conformal fields in Minkowski space. ...

... define the (physically scaled) uni-energy-momentum vector to have the form

( - i (1/R) L(-1,0) , - i (1/R) L(-1,1) , i (1/R) L(-1,2) , i (1/R) L(-1,3) )

... (1/R) L(-1,j) and (1/R) L(j,4) both differ from ej &/&x'j by O(1/R^2), and agree in the limit R -> infinity with the conventional energy-momentum component ej &/&x'j . The differences L(-1,j) - L(j,4) thus are locally approximate absolute constants of the motion. As such they are locally approximately representable by a slowly-varying vector field, which physically would appear most naturally as potentially related to gravitational phenomena as in the Weyl-Veblen theory, but possibly related also to microscopic processes as internal quantum numbers. ...".

For the design of our Unconventional Laboratory Experiment hereon Earth, R would be the size of the domain of the experiment, sothat 1/R ( as well as 1/R^2 ) would be about 1/R = 1/1 = 1,and the excess of the ConformalUnienergy of Unispace for the domainover the Poincare Energy of Minkowskispace for the domain would be about the total Poincare Energyof the total domain of the experiment, so that:

we would have to double the energy of the domain of theUnconventional Laboratory Experiment in order to construct an Islandof Unispace and get to useall 15 generators of ConformalSpin(2,4), including the 4 Conformal GraviPhotons,to build such things asWarp Drives, ComplexHyperspace Drives, etc.


Effectively, that means levitating the UnconventionalLaboratory Experiment so that it is no longer part of aGravitationally Bound Domain. It must be freed, not only from beingGravitationally Bound to Earth, but also from being GravitationallyBound to the Sun and to the Milky Way Galaxy.

A useful configuration might be 4 Rodincoils arranged in the geometry of a FullerVector Eqilibrium Cuboctahedron.

Each Rodin coil would have the geometry of a Hopffibration of a 3-sphere S3 by Clifford circles.


There should be 4 coils, one for each of the 4 physical dimensionsof SpaceTime, and they should be configured as 4 axes that are 3-dimprojections of the 4-dim coordinate axes of the 4-dim 24-cell,i.e., as 4 axes of  Fuller'sVector Equilibrium, the cuboctahedron,


The 4 Rodin Coil Clifford-Hopf Tori areindicated by the cyan,magenta,yellow, andgreen hexagonal "equators" of acuboctahedron

which configuration is a 3-dimensional projection of the4-dimensional 24-cell

Each of the 12 vertices of the cuboctahedron corresponds to 2 ofthe 24 vertices of the 24-cell, which 24 vertices are the vertices ofa green cuboctahedron, ared octahedron, and ablue octahedron.

Here is a different perspective illustration, patterned after Fig.172 of Geometry and the Imagination (Anschauliche Geometrie) by DavidHilbert and S. Cohn-Vossen (Chelsea 1952), with 12green vertices of a centralcuboctahedron, 6 red vertices of aninner octahedron, and 6 blue vertices ofan outer octahedron:

The 4 Rodin Coil Clifford-Hopf Tori of thecyan,magenta,yellow, andgreen hexagonal "equators" of acuboctahedron appear on the 24-cell as four (cyan,magenta,yellow, andgreen ) hexagonal rings:

Since the 24-cell is self-dual, the 4 hexagonal rings can also beseen as 4 rings of 6 octahedral faces.

Three such rings of octahedra are of the same type as the oneillustrated here

where the common triangular faces of the 6 octahedra in the ringare shaded.

The fourth ring, corresponding to thegreen ring of 6 octahedra, all of whichare in the central cuboctahedron, is of the type similarlyillustrated here

The three similar rings can be thought of as space-like, while thedistinct fourth ring can be thought of as timelike.

Note that the 6th octahderon of the distinct ring consists of theexterior of the outer octahedron, going all the way to infinity.

Note also that if you were to try to construct a Rodincoil, you would have to be careful that the loops are orientedcorrectly with respect to each other as shown in the 24-cellillustrations, and that the cuboctahedron projection illustrationobscures some of that orientation information.

Also, you should realize that flows through the Rodin coil wouldbe described by the dual 24-cell to the 24-cell used in constructionof the windings of the Rodin coil, and

that flow within a Rodincoil would be internal for the 3 similar rings, but would be a flowfrom the outside of the Rodin coil through the center of the Rodincoil and exiting into the opposite outside, through the center of thedistinct fourth ring.


As noted by Johan E.Mebius, "... The 96 edges of the 24-cell make up 16 regularhexagons. ...". The Clifford-Hopf configuration shown above uses 4x6= 24 of the 96 edges, and 4 of the 16 regular hexagons. The remaining72 edges, or 12 hexagons, fall into three more distinct Clifford-Hopfconfigurations of 24 edges, or 4 hexagons, each.

You can think of the four Clifford-Hopf configurations ascorresponding to four possible orientations in 4-dimensionalspacetime of Clifford-Hopf fibrations of the 3-sphere S3.

The Clifford-Hopf configurations are a finite version of theHopf fibration of S3, as mentioned by Coxeter (inhis book Regular Complex Polytopes, 2nd ed, Cambridge 1991, pp. 53and 204), where Coxeter says:

"... T. F. Banchoff (Senechal and Fleck ...1988, Shaping Space: A Polyhedral Approach, Birkhauser, Boston andBasel .... p. 229) found a ... finite version of the Hopffibration, ... a fibration of {3,4,3} [the 24-cell] into 4hexagons represented by the vertices of {3,3}. ...".


The 24-cell and cuboctahedral Clifford-Hopf geometry is used inthe conformal graviphoton physics of the4-dimensional HyperDiamond Feynman CheckerboardLattice of the D4-D5-E6-E7-E8 VoDouPhysics model.


Rodin coil experiments such as those described above may beuseful in building StarGateRing Ships and in constructionof star-gate worm-holes, whose stability might beinterpretable in terms of ghosts,and in theutilization on Earth of /\ > 0 Zero-Point Energy.



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