Less Mundane Explanation of Pioneer Anomaly from Q-Relativity

2007·0.09 MB·

by  Florentin Smarandache

Volume1 PROGRESSINPHYSICS January, 2007
Less Mundane Explanation of Pioneer Anomaly from Q-Relativity
FlorentinSmarandache andVicChristianto
∗ †
∗DepartmentofMathematics,UniversityofNewMexico,Gallup,NM87301,USA
E-mail:[email protected]
†Sciprint.org—aFreeScientificElectronicPreprintServer, http://www.sciprint.org
E-mail:[email protected]
There have been various explanations of Pioneer blueshift anomaly in the past few
years;nonethelessnoexplanationhasbeenofferedfromtheviewpointofQ-relativity
physics. In the present paper it is argued that Pioneer anomalous blueshift may
be caused by Pioneer spacecraft experiencing angular shift induced by similar Q-
relativity effect which may also affect Jupiter satellites. By taking into consideration
“aether drift” effect, the proposed method as described herein could explain Pioneer
blueshift anomaly within 0.26% error range, which speaks for itself. Another new
∼
proposition of redshift quantization is also proposed from gravitational Bohr-radius
which is consistent with Bohr-Sommerfeld quantization. Further observation is of
courserecommendedinordertorefuteorverifythisproposition.
1 Introduction tional phenomena. Einstein himself apparently recognizes
thislimitation[8a,p.61]:
Inthepastfewyears,itisbecomingwell-knownthatPioneer
“...allbodiesofreference K shouldbegivenprefer-
spacecraft has exhibited an anomalous Doppler frequency 0
ence in this sense, and they should be exactly equiva-
blueshifting phenomenon which cannot be explained from
lent to K for the formation of natural laws, provided
conventional theories, including General Relativity [1, 4].
thattheyareinastateof uniformrectilinearandnon-
Despite the nature of such anomalous blueshift remains un-
rotarymotion withrespectto K.”(ItalicbyEinstein).
known, some people began to argue that a post-einsteinian
gravitationtheorymaybeinsight,whichmaybeconsidered Therefore, it shall be clear that the restriction of non-
as further generalisation of pseudo-Riemannian metric of rotary motion remains a limitation for all considerations by
generalrelativitytheory. relativity theory, albeit the uniform rectilinear part has been
Nonetheless, at this point one may ask: Why do we re- relaxedbygeneralrelativitytheory.
quire a generalization of pseudo-Riemannian tensor, instead After further thought, it becomes apparent that it is re-
of using “patch-work” as usual to modify general relativity quired to consider a new kind of metric which may be able
theory? A possible answer is: sometimes too much path- torepresenttherotationalaspectsofgravitationphenomena,
work doesn’t add up. For instance, let us begin with a and by doing so extends the domain of validity of general
thought-experiment which forms the theoretical motivation relativitytheory.
behindGeneralRelativity,anelevatorwasputinfree-falling In this regard, the present paper will discuss the afore-
motion [8a]. The passenger inside the elevator will not feel mentioned Pioneer blueshift anomaly from the viewpoint of
any gravitational pull, which then it is interpreted as formal Q-relativity physics, which has been proposed by Yefremov
analogue that “inertial acceleration equals to gravitational [2]inordertobringintoapplicationthequaternionnumber.
acceleration” (Equivalence Principle). More recent experi- Despite the use of quaternion number in physical theories
ments (after Eo¨tvo¨s) suggest, however, that this principle is is very scarce in recent years — apart of Pauli matrix —
onlyapplicableatcertainconditions. it has been argued elsewhere that using quaternion number
Furtherproblemmayariseifweask:whatiftheelevator one could expect to unify all known equations in Quantum
also experiences lateral rotation around its vertical axis? Mechanics into the same framework, in particular via the
Does it mean that the inertial acceleration will be slightly known isomorphism between Dirac equation and Maxwell
higherorlowerthangravitationalpull?Similarlyweobserve equations[5].
that a disc rotating at high speed will exert out-of-plane Another problem that was often neglected in most treat-
field resemble an acceleration field. All of this seems to ises on Pioneer spacecraft anomaly is the plausible role of
indicate that the thought-experiment which forms the basis aether drift effect [6]. Here it can be shown that taking
of General Relativity is only applicable for some limited this effect into consideration along with the aforementioned
conditions,inparticularthe F =mdv part(becauseGeneral Q-relativity satellite’s apparent shift could yield numerical
dt
Relativity is strictly related to Newtonian potential), but it prediction of Pioneer blueshift within 0.26% error range,
∼
maynotbeabletorepresenttherotationalaspectsofgravita- whichspeaksforitself.
42 F.Smarandache,V.Christianto.MundaneExplanationofPioneerAnomalyfromQ-Relativity
January,2007 PROGRESSINPHYSICS Volume 1
We also suggest a new kind of Doppler frequency shift Because of antisymmetry of the connection (generalised
whichcanbepredictedusingNottale-typegravitationalBohr- angular velocity) the dynamics equations can be written in
radius, by taking into consideration varying G parameter as vectorcomponents,byconventionalvectornotation[2]:
described by Moffat [7]. To our knowledge this proposition
of new type of redshift corresponding to gravitational Bohr- m ~a+2Ω~ ~v+Ω~ ~r+Ω~ Ω~ ~r =F~. (4)
× × × ×
radiushasneverbeenconsideredbeforeelsewhere. The(cid:16)refore,fromequation(4)oner(cid:0)ecogniz(cid:1)e(cid:17)sknowntypes
Further observation is of course recommended in order
of classical acceleration, i.e. linear, coriolis, angular, centri-
toverifyorrefutethepropositionsoutlinedherein.
petal. Meanwhile it is known that General Relativity intro-
duces Newton potential as rigid requirement [2a, 6b]. In
2 Some novel aspects of Q-relativity physics. Pioneer
otherwords,wecanexpect—usingQ-relativity—topredict
blueshiftanomaly
neweffectsthatcannotbeexplainedwithGeneralRelativity.
From this viewpoint one may consider a generalisation
In this section, first we will review some basic concepts of
ofMinkowskimetricintobiquaternionform[2]:
quaternion number and then discuss its implications to qua-
ternionrelativity(Q-relativity)physics[2].Thenwediscuss dz =(dx +idt )q , (5)
k k k
Yefremov’s calculation of satellite time-shift which may be
observedbyprecisemeasurement[3].Wehoweverintroduce withsomenovelproperties,i.e.:
a new interpretation here that such a satellite Q-timeshift is temporalintervalisdefinedbyimaginaryvector;
already observed in the form of Pioneer spacecraft blueshift •
space-time of the model appears to have six dimen-
anomaly. •
sions(6D);
Quaternionnumberbelongstothegroupof“verygood”
vector of the displacement of the particle and vector
algebras: of real, complex, quaternion, and octonion [2].
•
of corresponding time change must always be normal
While Cayley also proposed new terms such as quantic, it
toeachother,or:
islessknownthantheabovegroup.Quaternionnumbercan
be viewed as an extension of Cauchy imaginary plane to dxkdtk =0. (6)
become[2]:
It is perhaps quite interesting to note here that Einstein
Q a+bi+cj+dk, (1)
≡ himselfapparentlyonceconsideredsimilarapproach,bypro-
where a, b, c, d are real numbers, and i, j, k are imaginary posingtensorswithRiemannianmetricwithHermitiansym-
quaternionunits.TheseQ-unitscanberepresentedeithervia metry [8]. Nonetheless, there is difference with Q-relativity
2 2matricesor4 4matrices[2]. describedabove,becauseinEinstein’sgeneralisedRiemann-
× ×
It is interesting to note here that there is quaternionic ianmetricithas8-dimensions,ratherthan3d-spaceand3d-
multiplicationrulewhichacquirescompactform: imaginarytime.
OneparticularlyinterestingfeatureofthisnewQ-relativ-
1q =q 1=q , q q = δ +ε q , (2)
k k k j k − jk jkn n ity(orrotationalrelativity)isthatthereisuniversalcharacter
where δ and ε represent 3-dimensional symbols of of motion of the bodies (including non-inertial motions),
kn jkn
Kronecker and Levi-Civita, respectively [2]. Therefore it which can be described in unified manner (Hestenes also
could be expected that Q-algebra may have neat link with considersClassicalMechanicsfromsimilarspinorlanguage).
pseudo-Riemannian metric used by General Relativity. Inte- Forinstanceadvancedperihelionofplanetscanbedescribed
restingly, it has been argued in this regard that such Q-units intermofsuchrotationalprecession[2].
canbegeneralisedtobecomeFinslergeometry,inparticular InspiredbythisnewQ-relativityphysics,itcanbeargued
withBerwald-Moormetric.ItalsocanbeshownthatFinsler- that there should be anomalous effect in planets’ satellite
Berwald-Moormetricisequivalentwithpseudo-Riemannian motion. In this regard, Yefremov argues that there should
metric, and an expression of Newtonian potential can be be a deviation of the planetary satellite position, due to
foundforthismetric[2a]. discrepancybetweencalculatedandobservedfromtheEarth
It may also be worth noting here that in 3D space Q- motion magnitudes characterizing cyclic processes on this
connectivityhascleargeometricalandphysicaltreatmentas planetornearit.Heproposes[2]:
movableQ-basiswithbehaviourofCartan3-frame[2].
ωV V
It is also possible to write the dynamics equations of Δϕ e p t, (7)
≈ c2
Classical Mechanics for an inertial observer in constant Q-
or
basis.SO(3,R)-invarianceoftwovectorsallowtorepresent
ωV V
thesedynamicsequationsinQ-vectorform[2]: Δϕ0 e p t0. (8)
≈− c2
d2 Therefore, given a satellite orbit radius r, its position
m (x q )=F q . (3)
dt2 k k k k shift is found in units of length Δl = rΔϕ. His calculation
F.Smarandache,V.Christianto.MundaneExplanationofPioneerAnomalyfromQ-Relativity 43
Volume1 PROGRESSINPHYSICS January, 2007
Satellites Cyclefrequencyω,1/s AngularshiftΔϕ,00/100yrs LinearshiftΔl,km/100yrs Linearsizea,km
Phobos(Mars) 0.00023 18.2 54 20
Deimos(Mars) 0.00006 4.6 34 12
Metis(Jupiter) 0.00025 10.6 431 40
Adrastea(Jupiter) 0.00024 10.5 429 20
Amalthea(Jupiter) 0.00015 6.3 361 189
Table 1: The following table gives values of the effect for five fast satellites of Mars and Jupiter. Orbital linear velocities are: of the
EarthV =29.8km/s,ofMarsV =24.1km/s,ofJupiterV =13.1km/s;thevalueofthelightvelocityisc=299793km/s;observation
E P P
periodischosen100years.CourtesyofA.Yefremov,2006[3].
for satellites of Mars and Jupiter is given in Table 1. None- Interestinglythisangularshiftcanbeexplainedwiththe
theless he gave no indication as to how to observe this same order of magnitude from the viewpoint of Q-satellite
anomalouseffect. angularshift(seeTable1),inparticularforJupiter’sAdrastea
Inthisregard,weintroducehereanalternativeinterpreta- (10.500 per100years).Thereishowever,alargediscrepancy
tion of the aforementioned Q-satellite time-shift effect by attheorderof50%fromtheexpectedangularshift.
Yefremov,i.e.thiseffectactuallyhassimilareffectwithPio- It is proposed here that such discrepancy between Q-
neer spacecraft blueshift anomaly. It is known that Pioneer satellite angular shift and expected angular shift required
spacecraft exhibits this anomalous Doppler frequency while to explain Pioneer anomaly can be reduced if we take into
entering Jupiter orbit [1, 4], therefore one may argue that consideration the “aether drift” effect [6]. Interestingly we
thiseffectiscausedbyJupiterplanetarygravitationaleffect, can use experimental result of Thorndike [6, p.9], saying
whichalsomaycausesimilareffecttoitssatellites. that the aether drift effect implies a residual apparent Earth
DespitetheapparentcontradictionwithYefremov’sown velocity is v =15 4km/sec. Therefore the effective V
obs e
±
intention, one could find that the aforementioned Q-satellite inequation(8)becomes:
time-shiftcouldyieldanaturalexplanationofPioneerspace-
V =v +V =44.8km/sec. (12)
craftblueshiftanomaly.Inthisregard,Taylor[9]arguesthat e.eff obs e
there is possibility of a mundane explanation of anomal- UsingthisimprovedvalueforEarthvelocityinequation
ous blueshift of Pioneer anomaly (5.99 10 9Hz/sec). The (8),onewillgetlargervaluesthanTable1,whichforAdras-
× −
all-angle formulae for relativistic Doppler shift is given teasatelliteyields:
by[9a,p.34]: ωV V V
Δϕ = e.eff pt= e.effΔϕ=15.93500/100yrs. (13)
(1 βcosφ) obs c2 V
v0 =v0γ − , (9) e
1 β2 Using this improved prediction, the discrepancy with
−
requiredangularshiftonly(15.89400 per100years)becomes
where β=v/c.Byneglectingpthe 1 β2 termbecauseof
0.26%, which speaks for itself. Therefore one may con-
−
lowvelocity,onegetsthestandardexpression: ∼
p cludethatthislessmundaneexplanationofPioneerblueshift
anomalywithQ-relativitymaydeservefurtherconsideration.
v =v γ(1 βcosφ). (9a)
0 0
−
Thederivativewithrespectto φis: 3 AnewtypeofredshiftfromgravitationalBohrradius.
Possibleobservationinsolarsystem.
dv
0 =v γβsinφ, (10)
0
dφ
In preceding paper [10, 11] we argued in favour of an alter-
where ddvφ0 =5.99×10−9Hz/sec, i.e. the observed Pioneer native interpretation of Tifft redshift quantization from the
anomaly. Introducing this value into equation (10), one gets viewpointofquantizeddistancebetweengalaxies.Amethod
requirementofaneffecttoexplainPioneeranomaly: can be proposed as further test of this proposition both at
solar system scale or galaxies scale, by using the known
dφ= arcsin(5v.99γ×β10−9Hz) =1.4×10−12deg/sec. (11) quantizedTifftredshift[14,15c,16]:
0
δr δz. (14)
≈ H
Therefore, we can conclude that to explain 5.99 10 9
× − Inthisregards,weusegravitationalBohrradiusequation:
Hz/sec blueshift anomaly, it is required to find a shift of
emission angle at the order 1.4 10 12degree/sec only (or GM
× − r =n2 . (15)
around15.89400 per100years). n v2
0
44 F.Smarandache,V.Christianto.MundaneExplanationofPioneerAnomalyfromQ-Relativity
January,2007 PROGRESSINPHYSICS Volume 1
Insertingequation(15)into(14),thenonegetsquantized Another new proposition of redshift quantization is also
redshiftexpectedfromgravitationalBohrradius: proposed from gravitational Bohr-radius which is consistent
with Bohr-Sommerfeld quantization. It is recommended to
H GM
z = n2 (16) conduct further observation in order to verify and also to
n c v2
0 explorevariousimplicationsofourpropositionsasdescribed
whichcanbeobservedeitherinsolarsystemscaleorgalax- herein.
ies scale. To our present knowledge, this effect has never
beendescribedelsewherebefore.
Acknowledgment
Therefore,itisrecommendedtoobservesuchanaccele-
rated Doppler-freequency shift, which for big jovian planets The authors are would like to thank to Profs. C. Castro,
this effect may be detected. It is also worth noting here E. Scholz, T. Love, and D. L. Rapoport for valuable discus-
that according to equation (16), this new Doppler shift is sions. Special thanks to Prof. A. Yefremov for sending his
quantized. recentcalculationofJupitersatellite’sQ-time-shifteffect.
At this point one may also take into consideration a
propositionbyMoffat,regardingmodificationofNewtonian
References
accelerationlawtobecome[7]:
1. Anderson J.D., Campbell J.K., & Nieto M.M. arXiv: astro-
G M exp( μ r)
a(r)=− ∞r2 +K r−2 φ (1+μφr) (17) pqhc//0500670085028.7; [1a] Nieto M.M. & Anderson J.D. arXiv: gr-
where 2. YefremovA. HypercomplexNumbersinGeometryandPhys-
G =G 1+ M0 . (17a) ics, 2004, v.1(1), 105; [2a] Pavlov D.G. arXiv: math-ph/
∞ (cid:20) rM (cid:21) 0609009.
Thereforeequation(16)mayberewrittentobecome: 3. Yefremov A. Private communication, October 2006. Email:
[email protected].
H GM M H GM 4. Laemmerzahl C. & Dittus H. Clocks and gravity, from quan-
z n2 1+ 0 χ n2 (18)
n ≈ c v2 M ≈ c v2 tum to cosmos. UCLA, 2006, http://www.physics.ucla.edu/
0 (cid:20) r (cid:21) 0 quantum_to_cosmos/q2c06/Laemmerzahl.pdf
where nisinteger(1,2,3,...)and: 5. ChristiantoV.EJTP,2006,v.3,No.12,http://www.ejtp.com.
6. Consoli M. arXiv: physics/0306094, p.9; [6a] Consoli M. et
M
χ= 1+ 0 . (18a) al.arXiv:gr-qc/0306105;[6b]arXiv:hep-ph/0109215.
M
(cid:20) r (cid:21) 7. MoffatJ.arXiv:astro-ph/0602607.
Tousetheaboveequations,onemaystartbyusingBell’s
8. Einstein A. Ann. Math., 1945, v.46; [8a] Einstein A. Relati-
suggestionthatthereisfundamentalredshift z=0.62which vity: the special and general theory. Crown Trade Paperback,
is typical for various galaxies and quasars [14]. Assuming NewYork,1951,pp.61,66–70.
we can use equation (16), then by setting n=1, we can
9. TaylorS.arXiv:physics/0603074;[9a]EngelhardtW.Apeiron,
expecttopredictthemassofquasarcentreorgalaxycentre. 2003,v.10,No.4,34.
Then the result can be used to compute back how time-
10. Smarandache F. & Christianto V. Progress in Physics, 2006,
variation parameter affects redshift pattern in equation (18).
v.4,27–31.
In solar system scale, time-varying radius may be observed
11. Smarandache F. & Christianto V. Progress in Physics, 2006,
intheformofchangingAstronomicalUnit[4].
v.4,37–40.
This proposition, however, deserves further theoretical
12. Smarandache F. & Christianto V. Progress in Physics, 2006,
considerations. Further observation is also recommended in
v.2,63–67.
ordertoverifyandexplorefurtherthisproposition.
13. Fischer U. arXiv: cond-mat/9907457; [13a] arXiv: cond-mat/
0004339.
4 Concludingremarks
14. Bell M.B. arXiv: astro-ph/0111123; [14a] Bell M.B. arXiv:
astro-ph/0305112;[14b]BellM.B.arXiv:astro-ph/0305060.
In the present paper it is argued that Pioneer anomalous
15. HumphreysR.TJArchive,v.16,http://answersingenesis.org.
blueshift may be caused by Pioneer spacecraft experiencing
16. Mu´neraH.Apeiron,1998,v.5,No.3–4.
angular shift induced by similar Q-relativity effect which
17. ZurekW.(ed.)Proc.EuroconferenceinFormationandInter-
may also affect Jupiter satellites. By taking into considera-
action of Topological Defects, Plenum, 1995; arXiv: cond-
tion aether drift effect, the proposed method as described
mat/9502119.
herein could predict Pioneer blueshift within 0.26% error
∼ 18. VolovikG.arXiv:cond-mat/0507454.
range,whichspeaksforitself.Furtherobservationisofcourse
recommendedinordertorefuteorverifythisproposition.
F.Smarandache,V.Christianto.MundaneExplanationofPioneerAnomalyfromQ-Relativity 45