JournalofModernPhysics,2020,11,1856-1873
https://www.scirp.org/journal/jmp
ISSNOnline:2153-120X
ISSNPrint:2153-1196
MotioninCliffordSpace
MagdE.Kahil
1,2
1
FacultyofEngineering,ModernSciencesandArtsUniversity,Giza,Egypt
2
EgyptianRelativityGroup,Cairo,Egypt
Howtocitethispaper:Kahil,M.E. (2020)
MotioninCliffordSpace.JournalofModern
Physics,11,1856-1873.
https://doi.org/10.4236/jmp.2020.1111116
Received:October22,2020
Accepted:November17,2020
Published:November20,2020
Copyright
c
2020byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbi pulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehiculapurus.Fusceacnisl aliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociisnatoque penatibus etmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoin nisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubia nostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretium nonummyfringilla, tortorlacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumac metusdictum tempus.Namut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligula nec orci.Aliquam purus turpis,aliquam
id, ornarevitae, porttitor non,wisi.Maecenas
luctusportalorem.Donecvitae ligulaeu ante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclacinia auctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuada egestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis, felis.Morbinullanibh, pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nunc sitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel, metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac, mauris.Maecenas etelit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat, wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiam fermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero, malesuadavitae, ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurussedsem
tempor rhoncus.Utquam dui, fringillaat, dic-
tum eget,ultricies quis, quam.Etiam semest,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sed scelerisquesus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareante nuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisi ultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsum anisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisi commodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputate elit.Pellentesque
sem.Praesent eusapien.Duiselit magna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimet commodomalesuada, tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultricies accumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eu sapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisse consequat.Mauris adui
et tortormattis pretium.Sednullametus, vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Ut blanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesqueodio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inporta congue,velit leofacilisis odio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,velposuere nibh lacusnec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulumlibero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sit ameteuismodlectus feliset
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pedecursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,variuseu, commodo sed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sedpellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuer variusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velitat elit vehicula porttitor.Phasel-
lusnislmi, sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsit amet,consectetueradipiscing elit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreet eget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Maurisenim sem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisquelacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverra nislat
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedid lec-
tus sitamet purus faucibus vehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac, nunc.Maurismalesuada malesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin, ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplatea dictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristique quis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris portadui, vitae aliquamarcu turpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellus aliquam,tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementumdui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu, euismodac, tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna, blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnon arcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbi ipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsum vel sem.Nul-
laenim.Proinnonummy felisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum, justoat dignissim
auctor,wisi odio lobortisarcu, sedsollicitudin
felisfeliseuneque.Praesent atlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nulla quammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudin hen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in, dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, duiut
dignissim auctor, orcilibero fringilla massa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tinciduntfermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netus etmalesuada fames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipit sitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringilla tempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitantmorbi tristique senectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnased enim.Donecegeteros ut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semper at,metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputate purus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedvel diam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,non porttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vel lectusut me-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatisvitae, vulputatea,pharetra sit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid, mi.Nullamnec nibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnullapulvinar justo,a euismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus, eusodalesmagna risusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integer purus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursus arcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit, dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In egetrisus luctus wisiconva-
llis pulvinar.Vivamus sapien risus,temporin,
viverra in, aliquetpellentesque, eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisquedignissim, lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligula suscipiturna,vitae portaodioer-
at quissapien.Proinluctus leo iderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fusce vehicula, tortor etgravidaporttitor, me-
tusnibhconguelorem,ut tempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit, imperdiet vitae,faucibus nec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisse ornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit, gravidaac, pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehicula at,conguesitamet,
sem.Sedsagittispedeanisl.Sedtincidunt o-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nulla ettellus nonfelis imperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsit ametnibhnec purusdignissim mat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpis diamfeugiat ar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Nam ligulaarcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisus atmetuspulvinar malesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsum primisinfaucibusorci
luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nuncbibendum augueluctus
diam.Etiam nec sem.Sed eros turpis,facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajusto vehiculadiam,eu condimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitamet est.Vivamuspulvinarleo non
sapien.Aliquamerat volutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet, mattis suscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibh ullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt, leovitae tristiquefa-
cilisis,ipsum wisiinterdumsem,dapibus sem-
pernulla velit vel lectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim, sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibus orciluctus et ultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augue erosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
Praesentsedestnonnibhtempusvenenatis.
Praesentrhoncus.Curabitursagittisestsit
ametneque.Sedcommodomalesuadalectus.
Phasellusenimtellus,temporut,tristiqueeu,
aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamus pulvinar.Aliquamerat volutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodales risus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
Namnullalacus,bibendumsitamet,sagit-
tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibh pretiumtellus,at dignissimleo eros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsum primisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoa justo.Quisquenec au-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenim sitametrisus.Curabiturvitae velit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeu nulla ut nulla interdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
Clifford algebraasanapproach of geometrizationofphysics plays a
vitalroleinunificationofmicro-physicsandmacro-physics,which
leads toexamine theproblemof motionfor different objects.Equa-
tions ofcharged andspinning ofextended objects arederived.Their
correspondingdeviationequationsasanextensionofgeodesicsand
geodesicdeviationofvectorsinRiemanniangeometryhavebeen
developedincaseofCliffordspace.
Keywords
CiffordSpace,Poly-Vectors-Geodesics,GeodesicDeviation,
SpinningObjects,ExtendedObjects
1.CliffordSpace:AimsandProspects
Motionofobjectsisregardedasamirrortoidentifythebehaviorof
field equationson manifolds.This may give rise to examinethe trajec-
toriesofdifferentparticlestoensuretheexistenceofanytheoryand
itsviability.Fromthisperspective,weoughttostudytheproblem
ofspinningobjectsindepth,asitisveryclosetothereality,rather
thanexaminingitssimplicitybymeansofdeterminingtheequation
ofmotionoftestparticles,i.e.thegeodesicequation.Thespinning
object has beenstudied by many authors long time ago, Mathisson[1]
startedtheidea;Papapetrouamendeditscontent[2]andthenitwas
developedtoincludechargedobjectsbyDixon[3],whichledmany
oftheirfollowerstoobtainthecorrespondingequationsofmotionof
movingobjectsindifferenttypesofgeometries[4–9].Notonlythese
pathequationsbutalsotheirdeviationequationsplayafundamental
roleinregulatingthestabilityofobjects[10].Thisismandatoryin
caseofexaminingtheperturbationproblemofanobjectorbitinga
gravitational field.Yet,such a descriptionmaybe inneed toberevis-
ited thoroughly for sake of unification of physics.Since the problem of
unification ofall fieldsof natureisa farfetched goal, itis wiseenough
to search fordifferent methodsand conceptsthat enableus toachieve
thisgoaloneday.
DOI:10.4236/jmp.2020.1111116Nov.20,20201856JournalofModernPhysics
M.E.Kahil
Accordingly,examiningtheproblemofunificationmaybefound
bystudyingthemotionofparticlessubjecttothesefields.Thismay
giverisetosearchfordifferenttypesofgeometriesorphilosophiesto
achievethistask.Oneofproposalsistoreconsideraspecificgeom-
etrybasedstrings,1loop,2-loops,3-loops,etc.Ingeneralitcanbe
describedasp-branesratherthanpoints.Thisshowsvectorsascan-
didates of poly-vectors to represent objects in nature as extended ones
ratherthanpoint-like.Consequently,ithasbeenfoundthatClifford
Algebrais agood candidateto describe thesequantities, leading tothe
conceptofCliffordspace(C-space)[11].Fromthisperspective,and
relyingtoexaminetheproblemofmotionforthoseextendedobjects,
Pezzaglia(1997)performedanintroductorysteptowardthistaskby
studying the equations of motion [11,12].Throughout his work, he fo-
cussedonthenatureofthebasesof C-spacethat arenon-orthogonal,
whichwidensthescopetorevealsomeobscurerelationsthatareal-
readyuntested.Suchanapproachhasbeengivenanadditionalrole
forquantitieslikethetetradformalismasatoolfordefiningskew
symmetricquantitiesliketorsionofspace-time,andspinconnection
innon-Riemanianngeometries.Thistendencywouldbeabletore-
vise theabsolute parallelismapproach [13–15], aswell as thePoincare
gaugetheoriesofgravity[16,17].Moreover,itgivesaclearvision
todetecttheeffectofspinofspace-timeasappearedinskewpart
ofg
µν
[12].Accordingly,theprinciplegeometrizationofphysicshas
beenappliedinanextensivewayratherthanbeforeintheconventional
point-likemanifoldtheories[18].
Consequently,theconceptofamanifoldcomposingpointstoiden-
tifyspace-timeisamendedtoinclude,lines,ares,volumes,...etc.
Accordingtothisdescriptivevisionparticlesmaybedefinedasex-
tendedobjects,ratherthanpoint-likeones[19].Thisisperformed
bymeansofapplyingtheCliffordalgebra,whichisregardedaspan-
dimensionalcontinuumorCliffordspace(C-space).Duetothistype
ofclassificationitisworth mentioning that,CastroandPavsic(2005)
revisitedCliffordspacetoperformitsextendedrelativity[18].This
theoryhastwofundamentalparameters:thespeedoflightcanda
lengthscalewhichcanbesettothePlancklength.Thepolyvector
coordinatesx
µ
,x
µν
,x
µνρ
...arenowconnectedwithbasispoly-vector,
bi-vector,tri-vector,...r-vector,thegeneratorsγ
µ
1
.......γ
µ
r
of
theCliffordalgebra,includingtheCliffordalgebraunitelement[20].
From this perspective, it can be found that stringsandp-branesare
expressedinthefollowingway:
Foraclosedstring(1-loop)whichisembeddedinatargetflat
backgroundofD-dimensionswhoseprojectionisappearedwithin
coordinate-planesintermsofvariablesx
µν
.Similarly,aclosedmem-
brane(a2-loop)maybedescribedbyanti-symmetricvariablesx
µνρ
representingthecorresponding3-volumeenclosedbythethe2-loop
[21].
Theaimofthe presentworkis toextendtheCastro-Pavsicapproach
ofpoly-vectorsinC-spacetoderivemodifiedequationsofmotionfor
extended objects and spinning ones.The significance of the derivation
oftheseequationsisin examiningtheexistenceof thenotionofmass,
chargeandspinofanextendedobjectassociatedwithpoly-vectors
and comparethem withtheconventional onesas describedwithin the
contextofvectors.
Thepaperisorganizedasfollows:Section(2)displaysbrieflythe
geometryofC-spaceanditsimplicationsinphysics.Section(3)per-
DOI:10.4236/jmp.2020.11111161857JournalofModernPhysics
M.E.Kahil
forms the trajectories of poly-vectors in C-space.Section (4) discusses
therelationshipbetweengeodesicsandgeodesicdeviationsofpoly-
vectorsandtheirlinkwithchargedandspinningpoly-vectors.Sec-
tion (5) displays a modified Lagrangian which enables us to determine
bothchargedandspinningpoly-vectorequationsaswellastheirde-
viationequations.Section(6)discussestheimpactofdetermining
theseequationsinrevisitingtheoldnotationsofproblemofmotion
usingtheconventionalmethodsofapoint-likemanifoldofdescribing
space-time.
2.C-Space:ABriefIntroduction
Following the geometrization scheme of physics, a new elegant formal-
ismhasbeenperformed,whichmayexplorenewhiddenphysicsto
showaninsightfulvisionforrevisitingtheoldnotationsofphysics.
SuchatrendisacrystalclearusingCliffordAlgebrawhichleadsto
expressmanyphysicalquantitiesinacompactform[22].Duetothe
richnessofCliffordalgebra,scalars,vectors,bi-vectorsandr-vectors
areexpressed inone formcalledClifford aggregateorpoly-vector.The
coordinatesofpoly-vectorscontain, vectors,areas,volumes ...etcex-
pressedasfollowsThepoly-vectorX
M
isdefinedasfollows[23]
X
M
X
µ
1
µ
2
µ
3
....
.(2.1)
Accordingly,itcanbeshownthatpoly-vectorcoordinatesofC-
spaceareparameterizednotonlyby1-vectorcoordinatesx
µ
butalso
bythe2-vectorcoordinatesx
µν
,3-vectorcoordinatesx
µνρ
...etccalled
holographiccoordinates,suchthat
X
M
=s1+x
µ
γ
µ
=x
µν
γ
µ
γ
ν
+x
µνρ
γ
µ
γ
ν
γ
ρ
+x
µνρσ
γ
µ
γ
ν
γ
ρ
γ
σ
,(2.2)
wherethecomponentsistheCliffordscalarcomponentsofapoly-
vectorvaluedcoordinates.
Thus,inC-spacepropertimeintervalmaybedescribedasin
Minkowskispace[18]
(dS)
2
= (ds)
2
+dx
µ
dx
µ
+dx
µν
dx
µν
+.....(2.3)
2.1.C-Space:UnderlyingGeometry
ApointinC-Spaceisdefinedasasetofholographiccoordinates
(s,x
µ
,x
µν
,...)formingthecoordinatesofapoly-vector.Eachone
is expressed within bases {γ
A
}= {1
a
1
a
1
a
2
a
1
a
2
a
3
....}, a
1
<a
2
<
a
3
<a
4
<a
5
<..., r= 1,2,3....,whereγ
a
1
a
2
a
3
...
= a
1
a
2
a
3
....[23].
Itiswellknownthatthelocalbasisγ
µ
,isrelatedtothetetradfield
e
a
µ
suchthat
γ
µ
= e
a
µ
γ
a
.
AnelementofaCliffordalgebraisasuperposition,calledClifford
aggregateorpoly-vector:
A= a+
1
1!
a
µ
γ
µ
+
1
2!
a
µν
γ
µ
γ
ν
+..........
1
n!
a
µ
1
.....µ
n
γ
µ
1
......γ
µ
n
.(2.4)
ThedifferentialofC-spaceisdefinedasfollows[21]
dA=
∂A
∂X
B
dX
B
,(2.5)
DOI:10.4236/jmp.2020.11111161858JournalofModernPhysics
M.E.Kahil
i.e
dA=
∂A
∂s
+
∂A
∂x
µ
dx
µ
+
∂A
∂x
µν
dx
µν
+......(2.6)
IfonetakesA= γ
µ
,then
µ
=
∂γ
∂s
+
∂γ
µ
∂x
µ
dx
µ
+
∂γ
µ
∂x
µν
dx
µν
+......,(2.7)
whichbecomes
µ
=
∂γ
∂s
α
µν
dx
ν
α
µ[ν,ρ]
dx
νρ
+......,(2.8)
andcanbereducedto
µ
=
∂γ
∂s
α
µν
dx
ν
+
1
2
R
α
βνρ
dx
νρ
+......,(2.9)
whereR
α
βνρ
isthecurvatureofspace-time.
Also,itcanbefoundthatforanarbitrarypoly-vectorA
M
DA
M
Dx
µν
= [D
µ
,D
ν
]A
M
.(2.10)
where
D
Dx
µν
isthecovariantderivativewithrespecttoaplanex
µν
,
suchthat
Ds
Dx
µν
= [D
µ
,D
ν
]s= K
ρ
µν
ρ
s,(2.11)
suchthatK
ρ
µν
isthetorsiontensor.
Da
α
Dx
µν
= [D
µ
,D
ν
]a
α
= R
α
ρµν
a
ρ
+K
ρ
µν
D
ρ
a
α
,(2.12)
Yet,thistypeoftorsion(2.12)canrelatedtothenotionoftorsionas
mentionedby Hammondascommutator thepotential associatedbya
prescribedscalarfieldφ[25]
K
α
µν
=
1
2
(δ
α
µ
φ
δ
α
ν
φ
).(2.13)
Thus,wecanfigureoutthatthetorsionasdefinedinC-space
(Riemannian-type)bymeansoftheparameter(s)smayactlikean
independentscalarfielddefinedintheusualcontextoftheRiemann-
Cartangeometry.
FromexaminingEquation(2.11)andEquation(2.12),onecanfind
the existence of torsiontensor evenin the presence of symmetric affine
connection apart fromits conventionalnotation definition of being the
antisymmetricpartofanaffineconnectionasinthecontextofnon-
Riemanniangeometries[13–15].Accordingly,owingtoC-spaceone
mayrealizethedisputebetweenthereliabilityoftorsionpropagating
ornon-propagatingthiscanberesolvedby meansof ofwe describe tor-
sionas aresultofcovariantdifferentiation ofareasof holographiccoor-
dinates andnon-propagating asbeingdefined asanti-symmetric parts
ofanaffineconnectionofpoly-vectorsorvectors,ifoneutilizesinthe
internalorexternalcoordinateanditscorrespondingnon-symmetric
affineconnection.
Consequently,wecanregardthatthegeometrydescribedwithin
thecoordinatesofpoly-vectordescribednotonlyRiemannianbutal-
soanon-Riemannian,i.e.thecompositionofaRiemannianaffine
DOI:10.4236/jmp.2020.11111161859JournalofModernPhysics
M.E.Kahil
connectionforthepoly-vectorisnotnecessarilyRiemannianaswell.
ThismayshedsomelighttofindoutthataRiemannianPoly-vector
affine connection andcurvature (external coordinatecapitalLatin let-
ters) may be described by non Riemannian quantities as describing its
holographiccoordinates(internalcoordinate(Greekletters).
Accordingly,inC-spaceitisconvenienttodistinguishtwoframe
fields[21]:
(i)Coordinateframefield,whosebaseselementsγ
M
,M=
1,2,3....2
n
dependonthepositionofXinC-spacesuchthattheir
expressionsasthewedgeproductsofvectorscannotbepreserved
globally.Thus,onewrites
γ
M
γ
µ
1
....µ
r
.(2.14)
Thescalarproductoftwosuch basiselementsgivesthemetrictensor
oftheC-space
γ
M
N
= G
MN
,(2.15)
(ii)Orthonormalframefield,whosebaseselementsγ
(A)
,(A)=
1,2,3,...2canateverypointbeexpressedasawedgeproduct,
γ
(A)
γ
a
1
....a
r
= γ
a
1
γ
a
2
γ
a
3
......γ
a
r
,(2.16)
Thescalarproductofthelatterbasiselementsgives
γ
(A)
(B)
= η
AB
(2.17)
whereηisthemetrictensorinflatspace.
Thederivativeofapoly-vectorisclassifiedasfollows[24]:
(i)Scalarfield:Actingonascalarfield,itbehavesasanordinary
partialderivative,i.e.
M
φ=
∂φ
∂X
M
.(2.18)
(ii) Coordinatebases: Actingoncoordinatebasiselements, itgives
M
γ
N
= Γ
Q
MN
γ
Q
(2.19)
whereΓ
Q
MN
isan affineconnection.The commutator ofthederivatives
actingonγ
J
givestheRiemanntensorinC-space:
[
M
,∂
N
]γ
J
= R
K
.JMN
γ
K
(2.20)
(iii)Orthonormalbases(localframefield):Thesetypesofbases
turntobeasfollows
M
γ
(A)
=
N
A.M
γ
(B)
(2.21)
suchthatΩ
N
A.M
actsasitsappropriatespinconnection.
Thus,thecommutatorofthederivativesactingonγ
A
gives
[
M
,∂
N
]γ
A
= R
B
.AMN
γ
B
.(2.22)
Thus,foranaggregatepoly-vectorAonefindsthat:
M
(A
N
γ
N
) =
M
A
N
γ
N
+A
N
M
γ
N
=(
M
A
N
N
MK
A
K
)γ
N
(D
M
A
N
)γ
N
,(2.23)
where D
M
A
N
is thecovariant derivative actingon theplyvector com-
ponentsA
N
.
DOI:10.4236/jmp.2020.11111161860JournalofModernPhysics
M.E.Kahil
2.2.C-spaceandtheTetradField
Therelationshipbetweenthetwobasesarerelatedby thetetradfield
E
A
associatedwithpoly-vectors[24]
γ
M
= E
(A)
M
γ
(A)
,(2.24)
whereE
A
M
istheC-spacevielbein,suchthat
G
MN
= E
(A)
M
E
(B)
N
η
AB
.(2.25)
Suchadescriptiondoesnotprecludethenon-Riemannianversionof
poly-vectors,whichisasteptorevisitthedefinitionofthefunda-
mentalquantities thatplayanactiveroleofdescribingsuchatypeof
geometry.
Accordingly,itisvitaltonotethatthecovariantderivativeofC-
spaceassociatedwithVielbeinofpoly-vectortakesthefollowingcon-
dition[20]
N
E
C
M
Γ
P
MN
E
C
P
E
A
M
C
A.N
= 0,(2.26)
Consequently,thecovariant derivativeactingonthetetradfieldisin-
variant undergeneral coordinate transformation by means ofΓ
M
NQ
and
localLorentzinvarianceasexpressedintermsofΩ
C
A.N
representing
thespinconnectioninC-space.
ThecurvatureofC-spaceisdefined,asusually,bythecommutator
ofderivativesactingonbasispoly-vectors[23]:
[D
M
,D
N
]γ
J
= R
K
MNJ
γ
K
,(2.27)
or
[D
M
,D
N
]γ
(A)
= R
(B)
MNJ
γ
(B)
,(2.28)
whereD
M
isitsassociatecovariantderive.
Meanwhile,introducingthereciprocalbasispoly-vectorsγ
M
and
γ
A
satisfying
(γ
M
)
γ
N
= δ
M
N
,(2.29)
(γ
(A)
)
γ
(B)
= δ
(A)
(B)
.(2.30)
Thecomponentsofcurvatureinthecorrespondingbasesaredefined
asfollows
R
K
MNJ
=
M
Γ
K
NJ
N
Γ
K
MJ
L
NJ
Γ
K
ML
Γ
L
MJ
Γ
K
NL
(2.31)
or
R
K
MN(A)
=
M
K
(A)N
N
K
(A)M
+Ω
(C)
(A)N
K
(C)M
(C)
(A)M
K
(C)N
.(2.32)
ThelatterexpressionisageneralizationtoC-spaceoftheanalogous
expression in Riemanniangeometry for manifoldsof point-like objects
[26].
Moreover,there isacounterpart versionofnon-vanishingcurvature
andtorsionasdefinedinnon-Riemanniangeometry,inC-spaceleading
todefinetorsionofpoly-vectorstobecome
Λ
J
MN
=
¯
Γ
J
MN
¯
Γ
J
NM
.(2.33)
DOI:10.4236/jmp.2020.11111161861JournalofModernPhysics
M.E.Kahil
Also,thecontortionofpoly-vectorsΩ
BCM
isgivenby
BCM
=
1
2
E
(A)
M
(∆
[(A)(B)](C)
[(B)(C)](A)
+∆
[(C)(A)](B)
)(2.34)
where∆
[(A)(B)](C)
isdefinedastheRiccicoefficientofrotation[23]
[(A)(B)](C)
= E
M
(A)
E
N
(B)
(
M
E
N(C)
M
E
N(C)
).
Moreover,anothercovariantderivativeassociatedwithnon-
symmetricaffineconnection
¯
Γ
M
NK
isdefinedasfollows
X
M
|N
=
N
X
M
+
¯
Γ
M
NS
X
S
,(2.35)
suchthat
¯
Γ
M
NS
X
S
= Γ
M
NS
X
S
+∆
M
NS
(2.36)
X
M
|NS
X
M
|SN
=
¯
R
M
QNS
X
Q
Q
NS
X
M
|Q
.(2.37)
suchthat
R
K
MNJ
=
M
¯
Γ
K
NJ
N
¯
Γ
K
MJ
+
¯
Γ
L
NJ
¯
Γ
K
ML
¯
Γ
L
MJ
¯
Γ
K
NL
.(2.38)
Duetorichnessofthesequantitiesthistypeworkwillbegoingto
examinethebehaviorofextendedobjectssubjectshavingsensitivity
tothesequantitiesinourfuturework;whileinourpresentworkwe
focusonderivingtheequationsofmotionandtheirdeviationpaths
fordifferentextendedobjectsspinningandchargedforpoly-vectors
definedwithinthecontextofRiemannian-likeC-Spaceasexplained
inthefollowingsections.
2.3.C-Space:AnArenaofUnifyingPhysics
Itisworthmentioningthat,physicalobjectsconsideredasmatter
inspace-time,canbeintheformofmembranes(brane)ofvarious
dimensions(p-branes)[18]
Fromthisperspective,thenotationofCliffordaggregateasde-
scribedin(2.1),itmaybeimportanttorevisittheconventionalno-
tationsofmassandchargeinthepresenceofC-spacetobe
M= m1+p
µ
γ
µ
+S
µν
γ
µν
,(2.39)
and
= e1+A
µ
γ
µ
+F
µν
γ
µν
,(2.40)
whereMandaremassandchargeofextendedobjectsrespectively.
Nevertheless,itisvitaltobenotedthatinCliffordSpace,there
isastrikingvirtue,unliketheconventionalstringtheorywhichex-
pressedwithin26dimensionsinwhichgaugefieldsaredescribedas
compactdimensions;whileinC-spacethereisonly16non-compact
dimensions.Consequently, Pavsic(2005)generalized theconventional
4-dimensionalspace-timeinto16dimensional`alaKlauza-Kleintheo-
ry[23].
(i) There is no need for extra-dimensions of its corresponding space-
time.
(ii)Thereisnoneedto havecompactextra-dimensions.Theextra-
dimensionsofC-space,namelyx
µ
,x
µν
,x
µνρ
,andx
µνρσ
describethe
extendedobjects,thereforetheyareregardedasphysicaldimensions.
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(iii)ThenumberofcomponentsG
µ
˜
M
,µ=0,1,2,3and
˜
M6=µis
12,whichisthesameasthenumberofgaugefieldsinthestandard
models.
Thus,thelineelementofpoly-vectorsinC-spacebecome
|dX|
2
= dS
2
= G
MN
dX
M
dX
N
,(2.41)
i.e.
dS
2
= ds
2
+L
2
dx
µ
dx
µ
+L
4
dx
µν
x
µν
+L
6
dx
µνρ
x
µνρ
+....,
whereG
MN
= E
M
E
N
isC-spacemetric,ListhePlancklength[12].
Consequently,Equation(2.43)canbeexpressedas
dS
2
=c
2
dt
2
(1+
L
2
c
2
dx
µ
dt
dx
µ
dt
+
L
4
c
2
dx
µν
dt
dx
µν
dt
+
L
6
c
2
dx
µνρ
dt
dx
µνρ
dt
+....)(2.42)
with takinginto account thatthe parametersisdescribed onlyin the
ordinaryspace-timei.e.(ds)
2
= g
µν
dx
µ
dx
ν
[22].
Meanwhile,inC-space,CastroandPavsic(2005),showedthat,
therearesomeprinciplesmustberevisitedespeciallythespeedof
lightwhichisnolongertheupperlimittobereached,butanoth-
ercombinationofconstantsmadeofPlanck’slengthinthefollowing
way[18]:
(i)Maximum1-vectorspeed
dx
µ
ds
= 3×10
8
ms
1
,
(ii)Maximum2-vectorspeed
dx
µν
ds
= 1.6×10
35
m
2
s
1
,
(iii)Maximum3-vectorspeed
dx
µνρ
ds
= 7.7×10
62
m
3
s
1
,
(iv)Maximum4-vectorspeed
dx
µνρ
ds
= 7.7×10
96
m
4
s
1
...etc.
Also,itisworthmentioningthatbymeansofC-spaceaparticle
asobservedfrom4-dimensionalspace,thatcanbespeedofparticles
maybemorethantheconventionalspeedoflightduetoinvolvement
of holographiccoordinatesx
µν
,x
µνρ
...etc;butdoesnot exceedsthe
modifiedspeedoflightduetoC-space.Thiscanleadustoconsider
therewillbespecificupperlimittoobjectsduetop-bransasp=
0,1,2,3,...
TheLineelementofpoly-vectorsinC-spaceisdefined asfollows[23]:
dS
2
= G
MN
dX
M
×dX
N
,(2.43)
providedthatthematrixG
MN
isdefinedas
G
MN
=
g
µν
G
µ
¯
N
G
¯
G
¯
M
¯
N
.(2.44)
Thesedegreesoffreedomareinprinciplenothiddenbywhichwe
describetheextendedobjects,thereforewedonotneedtocompact
dimensionsofitsinternalspace.
ThemetricofC-spaceG
MN
issubdividedintoG
µν
=g
µν
which
relatestogravity,whilegaugefieldsG
µ
¯
M
,whereµ6=
¯
Massumes12
possible values, excluding the four valuesof ν= 0,1,2,3, andother 12
gaugefields,tobedefinedasfollows:1photon,3weakgaugebosons
and8gluonsdescribedA
µ
,A
a
µ
,a=1,2,3andbyA
c
µ
c=1,1,3,...8.
respectively.
ItcanbefoundthatthenumberofmixedcomponentsinG
µ
¯
M
=
(G
µ1
,G
µ[αβ]
,G
µ[αβγ]
,G
µ[αβγδ]
)ofCliffordmetriccoincideswiththe
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numberofgaugefieldsinthestandardmodel.Forafixedµ,there
are12mixedcomponentsofG
µ
¯
M
and12gaugefieldsA
a
µ
,W
c
µ
,A
c
µ
.
Thiscoincidenceisfascinatinganditmayindicatethattheknown
interactionsareincorporatedin curved Cliffordspace.Thenumberof
mixedmetricG
µ
¯
M
is12thesamenumbersasthenumberofgauge
fieldsinthestandardmodel.
Inaddition,therearealsointeractionduetocomponents G
¯
M
¯
N
but
donothavethepropertiesofYang-Millsfields.Thus,Pavsic(2006)
hasconsidereditasametricofaninternalspace[24],whichmaybe
aglimpsetoexpressbi-metrictheoryofgravitywithinthecontextof
C-space.Suchastudywillbeexaminedinourfuturework.
3.TrajectoriesofPoly-VectorsinC-Space
3.1.EquationsofMotioninC-Space
Pavsic[24]consideredtheclassicalactionforapointparticleinC-
space:
I[X
µ
,G
MN
] =
Z
q
[G
MN
˙
X
M
˙
X
N
)]+
κ
16π
Z
d[x]R.(3.45)
Theaboveclassicalactionisacombinedactionforpathequations
andgravitationalfieldequationsrespectively.Oneobtainsitspath
equationbytakingthevariationwithrespecttoX
C
todervecorre-
spondinggeodesicequationi.e.
1
p
˙
X
2
d
˙
X
M
p
˙
X
2
M
NK
˙
X
M
˙
X
N
˙
X
2
= 0.(3.46)
Also,fromthesamefunction(3.47)bytakingthevariationwith
respecttoG
MN
oneobtainsitscorrespondingfieldequation
R
MN
1
2
G
MN
R= 8πκ
Z
δ
(C)
(xX)
˙
X
M
˙
X
N
,(3.47)
whereδ
(C)
isthedeltafunctioninC-space.Thelatterequationcan
beexpressedas
R
MN
1
2
G
MN
R= 8πT
MN
,(3.48)
whereT
MN
istheenergy-momentumtensorasdefinedinC-space.
3.2.C-SpaceandProblemofMotion:Pezzaglia’s
Approach
As an introductory step to the C-space formalism obtained by Castro-
Pavsic[18],itisworthtomentionthatPezzagliahadpresenteda
speculative visionabout theneed toutilize Cliffordspace, the problem
ofunificationcanbepassedbystagesofcomposingscalar,vectors
andbi-vectorsinoneformwithtakingintoconsiderationthatthe
correspondingbasesvectorsaredescribedinnon-orthonormalcurved
spaceinthefollowingway[11].
γ
µ
ν
1
2
{γ
µ
γ
ν
}=
1
2
{γ
µ
γ
ν
+γ
ν
γ
µ
}= g
µν
,(3.49)
whereg
µν
isthemetrictensorwhichmaybeafunctionofposition.
Theouterproductoftwodifferentbasesvectorsyieldsanewobject
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calledabi-vector,whichisthebasisofaplanespannedbytwobasis
vectors
γ
µν
= γ
νµ
=γ
µ
γ
ν
1
2
[γ
µ
γ
ν
] =
1
2
(γ
µ
γ
ν
γ
ν
γ
µ
) = s
µν
,(3.50)
wheres
µν
actsastheskewsymmetricpartofthemetricg
µν
i.e.
g
[µν]
= s
µν
.
Thisillustrationcanbeclarifiedbydescribingthemomentumpoly-
vectorandspinbi-vectorangularmomentum
M
1
λ
p
µ
γ
µ
+
1
λ
2
S
µν
γ
µν
,(3.51)
suchthatλisauniversallengthconstant
p
µ
= m
dx
µ
ds
,
and
S
µν
= m
da
µν
ds
,
whereS
µν
,inwhichisdefinedtobeabi-vectorcoordinate.Inflat
spaceonefindsitscorrespondingequationofmotion
˙
M
µ
=
dM
µ
ds
= 0,(3.52)
i.e.
˙p
µ
=
dp
µ
ds
= 0,(3.53)
and
˙
S
µν
=
dS
µν
ds
= 0.(3.54)
Itisworthmentioningthatthemodifiedmomentumwhichinclude
both linearandspinangularmomentum may give risetodescribethe
momentumoftheextendedobjectnotconsideritasatestparticle
butadipoleonehasanimpactonitsspinningcase.Thus,studying
momentum withtakingits spinningmotionmaygive riseto regardthe
spinning of theparticleincreases its mass asshown in Equation (3.51)
andEquation(3.54)[12].Now,thearisingquestionisthesituation
oftheseequationsinthe presenceofgravitational field,which may be
expressedasfollows
d
2
x
µ
ds
2
µ
νρ
dx
ν
ds
dx
ρ
ds
µ
ν[ρ,δ]
dx
ν
ds
da
ρδ
ds
= 0(3.55)
tobecomeinthefollowingway
d
2
x
µ
ds
2
µ
νρ
dx
ν
ds
dx
ρ
ds
+
1
2m
R
µ
νρδ
dx
ν
ds
S
ρδ
= 0(3.56)
and
dS
µν
ds
µ
σρ
S
σν
dx
ρ
ds
ν
σρ
S
µσ
dx
ρ
ds
= 0,(3.57)
D
2
x
µ
Ds
2
=
1
2m
R
µ
νρσ
S
ρσ
dx
ν
ds
,(3.58)
and
DS
µν
Ds
=
da
αβ
ds
µ
α[β,σ]
S
σν
ν
α[β,σ]
S
µσ
) = 0,(3.59)
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M.E.Kahil
i.e.
DS
µν
Ds
=
1
2m
S
αβ
(R
µ
αβσ
S
σν
+R
ν
αβσ
S
µσ
),(3.60)
suchthat
D
Ds
istheconventionalcovariantderivativeasdefinedin
RiemannianSpace.ItcanbefiguredoutthattheEquation(3.60)
actsexactlyasthePapapetrouequationforshort;whileextraterms
intermsofcurvatureandspintensorareaddedinEquation(3.61).
Following this streamof thinking in caseof electro-magnetic fieldin
thepresenceofflatspace,onemyfindoutthat
d
2
x
µ
ds
2
=
e
m
F
µ
ν
dx
ν
ds
,(3.61)
and
˙
S
µν
=
dS
µν
ds
=
e
m
F
[µ
ρ
S
ν]ρ
,(3.62)
whereF
µν
istheelecromangnetictensor,which turnstobeincaseof
gravitationalfieldas
D
2
x
µ
Ds
2
=
1
2m
R
µ
νρδ
S
ρδ
dx
ν
ds
+
e
m
F
µν
dx
ν
ds
,(3.63)
and
DS
µν
Ds
=
e
m
F
[µ
ρ
S
ν]ρ
+
1
2
da
αβ
ds
(R
µ
αβσ
S
σν
+R
ν
αβσ
S
µσ
),(3.64)
suchthat
DS
µν
Ds
=
e
m
F
[µ
ρ
S
ν]ρ
+
1
2m
S
αβ
(R
µ
αβσ
S
σν
+R
ν
αβσ
S
µσ
).(3.65)
Thus,itcanbeshownthatEquation(3.66)actsastheDixonequa-
tion,extratermsintermsofcurvatureandspintensorareaddedin
Equation(3.67).Fromstudyingtheseequations,wecanseethein-
volvementof bi-vector to be affected by additional factors ofcurvature
ofthespace-timeandspintensor.Thismaygiverisetomakesure
thatincaseoftri-vectors andothermulti-vectorsthese effectswill be
added.Suchanargumentmayimposethenecessitytofindageneral
formulacapabletoembodyalltheseeffects.Itwasfoundthatusing
poly-vectorcoordinatesmayrepresentalltheseformsinacompact
way.Soitisessentialtodisplaytheunderlyinggeometryofpoly-
vectors priorto examiningthe correspondingequations ofmotion and
thedeviationones.
3.3.TheBazanskiApproachforPoly-Vectors
EquationsofgeodesicandgeodesicdeviationequationsinRiemanni-
angeometryarerequiredtoexaminemanyproblemsofmotionfor
differenttestparticlesingravitationalfields.Thisledmanyauthors
toderive themby variousmethods,oneofthe mostapplicableonesis
theBazanskiapproach[27]inwhichfromonesingleLagrangianone
canobtainsimultaneouslyequationofgeodesicandgeodesicdevia-
tionswhichhasbeenappliedindifferenttheoriesofgravity[4–9],and
[28–30].Thus,byanalogythistechniqueincaseofPoly-vectorsto
become[31],
L= G
MN
U
M
DΨ
N
DS
,(3.66)
where,G
MN
isthemetrictensor,U
M
=
dX
M
dS
,isa unittangent poly-
vectorofthepathwhoseparameterisS,andΨ
ν
isthedeviation
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poly-vectorassociatedtothepath(S),
D
DS
isthecovariantderivative
withrespecttoparameterS.
Applyingthe EulerLagrangeequation ,bytaking thevariation with
respecttothedeviationpoly-vectorΨ
C
,
d
dS
∂L
˙
Ψ
C
∂L
Ψ
C
= 0,(3.67)
toobtainthegeodesicequation
DU
C
DS
= 0,(3.68)
andtakingthevariationwithrespectthetheunitpoly-vectorU
C
,
d
dS
∂L
∂U
C
∂L
∂x
C
= 0,(3.69)
toobtainthegeodesicdeviationequation,
D
2
Ψ
E
DS
2
= R
E
ABC
U
A
U
B
Ψ
C
,(3.70)
whereR
A
BCD
isRiemann-Christoffeltensordescribedby poly-vectors.
3.4.GeodesicandGeodesicDeviationofC-Space:
AnAlternativeApproach
Equationsgeodesicdeviationofpoly-vectorarederivedinawayof
modifyingtheconventionalBazanskimethod,usingthecommuta-
tion relations anda condition between deviation poly-vector andpoly-
vector as well.This method hasbeen worked successfully toderivede-
viation equation forextended objects with wobbling as shown in [6–9].
Applyingtheusualcommutationrelation[32]onEquation(3.70)
weobtain:
D
D
DA
D
DS
D
DS
DA
D
D
=
¯
R
D
BCE
A
B
U
C
Ψ
E
,(3.71)
providedthat
DU
A
DS
=
DΨ
A
D
.(3.72)
Thus,weobtainitscorrespondinggeodesicdeviationequations
D
2
Ψ
C
DS
2
= R
C
BDE
U
B
U
D
Ψ
E
.(3.73)
4.TransformationofGeodesicPoly-Vectors
toChargedandSpinningPoly-Vectors
4.1.From GeodesicPoly-VectortoChargedObject
Poly-Vector
Inthisapproachwearegoingtoutilizethefamousapplicationfor
transforminggeodesicstospinningobjectdifferently[32]tobecome
reliableforchargedobject.Inthiscase,onecanassumethatcharged
objecttravelsonadeviatedtrajectorybelongstofamilyofdeviation
parameterssuchthat
W
C
= U
C
+
M
DΨ
C
DS
,(4.74)
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M.E.Kahil
whereW
C
=
dX
C
d
¯
Ξ
.
Thus,differentiating bothsidesby
D
D
¯
S
andtakingthat
dS
d
¯
Ξ
= 1,one
obtains,
DW
C
D
¯
Ξ
=
D
DS
[U
C
+
M
DΨ
C
DS
]
dS
d
¯
Ξ
.(4.75)
Now,ifweconsiderelectro-magnetictensorasfollows
F
AB
U
B
= (R
A
BDE
U
D
Ψ
E
)U
B
,(4.76)
thenEquation(4.75)becomes
DW
C
=
M
F
C
.B
U
B
dS
d
¯
Ξ
(4.77)
whichbecomestheequivalentLorentzforce(Chargedobject)in
CurvedCliffordSpace
DW
C
DΞ
=
M
F
AB
W
B
(4.78)
4.2.FromGeodesicPoly-VectortoSpinning
Poly-Vectors
Ifweconsiderspinningpoly-vectorsareregardedasgeodesicstrans-
ferredfromonepathdefinedbyaparameterSintoanotherone
¯
S,
suchatranslationmaycausetheobjectbehaveasaspinningparticle
rather moving asatest particle.Wesuggest thefollowing poly-vector
V
C
definedinthefollowingway
V
C
= U
C
+β
DΨ
C
DS
.(4.79)
Bytakingthecovariantderivativeonboth sidesweobtain,with taking
intoaccount
dS
d
¯
S
= 1,
DV
C
D
¯
S
=
D
DS
[U
C
+β
DΨ
C
DS
]
dS
d
¯
S
.(4.80)
Assumingthatβ=
¯
S
M
,and
˜
Sisthemagnitudeofthespintensoras
definedby
S
AB
=
˜
S[U
A
Ψ
B
U
B
Ψ
A
].(4.81)
Accordingly,substitutingEquations(3.70),(3.72)and(4.83)into
(4.82),providedthat
dS
d
¯
S
= 1,wegetaftersomemanipulations:
DU
C
DS
=
1
2m
R
C
BDE
U
B
S
DE
,(4.82)
which isthecorresponding Papapetrou equation,for short,in C-space.
5.TheBazankiApproachforExtended
ObjectsofC-Space
5.1.EquationsofChargedandChargedDeviation
EquationsinC-Space
The Bazanski approach to obtain an equation of a chargedobject may
befoundasfollows:
L= G
AB
W
A
DΦ
B
DΞ
+
M
F
AB
W
A
Φ
B
.(5.83)
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Taking the variation with respect to Φ
C
and W
C
we obtain after some
manipulationsonefinds
DW
C
DS
= F
CB
W
B
,(5.84)
and
D
2
Φ
C
D
¯
S
2
= R
C
BDE
W
E
W
D
Φ
E
+
M
(F
CB
W
B
)
;D
Φ
D
.(5.85)
5.2.SpinningandSpinningDeviationEquationsof
C-Space
We suggesttheequivalent Bazanski Lagrangian forderiving theequa-
tionsforspinningandspinningpoly-vectorstobe
L= G
AB
P
A
DΨ
B
D
¯
S
+S
AB
DΨ
AB
D
¯
S
+F
A
Ψ
A
+M
AB
Ψ
AB
,(5.86)
suchthat
P
A
= mU
A
+U
β
DS
AB
DS
,
whereP
µ
isthemomentumpoly-vectorF
µ
=
1
2
R
µ
νρδ
S
ρδ
U
ν
,R
α
βρσ
is
the Riemanncurvature,
D
D
¯
S
is thecovariant derivative with respect to
a parameter
¯
S,S
αβ
is thespin poly-tensor,and M
µν
= P
µ
U
ν
P
ν
U
µ
suchthatU
α
=
dx
α
ds
istheunittangentpoly-vectortothegeodesic
one.Inasimilarwayasperformedin(5.83),bytakingthevariation
withrespecttoΨ
µ
andΨ
µν
simultaneouslyin(5.86)oneobtains
DP
M
D
¯
S
= F
M
,(5.87)
and
DS
MN
D
¯
S
= M
MN
.(5.88)
Usingthefollowingidentity
A
D
;NH
A
D
;HN
= R
D
BNH
A
B
,(5.89)
onbothEquation(5.87)andEquation(5.88),suchthatA
D
isanar-
bitrarypoly-vector.Also,multiplyingbothsideswitharbitrarypoly-
vectors,U
H
andΨ
B
aswellasusingthefollowingcondition[33]
U
A
;H
Ψ
H
= Ψ
A
;H
U
H
,(5.90)
andΨ
A
isitsdeviationpoly-vectorassociatedtotheunitpoly-vector
tangentU
A
.Andinasimilarwecanfindoutthat:
S
AB
;H
Ψ
H
= Ψ
AB
;H
U
H
.(5.91)
Consequently,oneobtainsthecorrespondingdeviationequations
whichareinspiredfromtheworkingsofMohseni[34]
D
2
Ψ
A
DS
2
=
¯
R
A
BHC
P
B
U
H
Φ
C
+F
A
;H
Ψ
H
,(5.92)
and
D
2
Ψ
AB
D
¯
S
2
= S
[BD
R
A]
DHC
U
H
Ψ
C
+M
AB
;H
Ψ
H
.(5.93)
DOI:10.4236/jmp.2020.11111161869JournalofModernPhysics
M.E.Kahil
5.3.TheGeneralizedDixonEquationinC-Space
Inthispart,wesuggestthefollowingLagrangianwhichenablesusto
obtainthespinningchargedpoly-vectorinCliffordspace.
L= G
AB
P
A
DΨ
B
DS
+S
AB
DΨ
AB
DS
+
˜
F
AB
Ψ
A
U
B
+
˜
M
AB
Ψ
AB
,(5.94)
suchthat
˜
F
AB
=
1
m
(eF
AB
+
1
2
R
ABCD
S
CD
)
and
˜
M
AB
= (P
A
U
B
P
B
U
U
+F
AC
S
C
.B
F
AC
S
.C
A.
).
TakingthevariationwithrespecttoΨ
µ
andΨ
µν
simultaneouslyin
(5.94)weobtain
DP
M
DS
=
M
F
M
B
U
B
+
1
2M
R
M
NEQ
S
EQ
U
N
,(5.95)
DS
MN
DS
= P
M
U
N
P
N
U
M
+F
MC
S
.N
C
F
CN
S
M
.C
.(5.96)
Thus,applyingthesamelawsofcommutationasillustratedin(5.92)
and(5.93)weobtaintheircorrespondingdeviationequations
D
2
Ψ
A
DS
2
=R
A
BHC
P
B
U
H
Φ
C
+(
M
F
M
B
U
B
+
1
2m
R
M
NEQ
S
EQ
U
N
)
;H
Ψ
H
,(5.97)
and
D
2
Ψ
AB
DS
2
=S
[BD
R
A]
DHC
U
H
Ψ
C
+(P
M
U
N
P
N
U
M
+F
MC
S
.N
C
F
CN
S
M
.C
)
;H
Ψ
H
.(5.98)
6.ConcludingRemarks
Inourapproach,wehaveobtainedtherelevantequationsofspinning
andspinningofchargedpoly-vectors,aswellastheirdeviationequa-
tionsinC-spacei.e.(4.84),(5.97)and(5.98).Thistypeofwork
isregardedanextensiontoapreviouswork,obtainingequationsof
spinningandspinningobjectsinRiemannianandnon-Riemannian
geometries,usingtheBazanskiapproach[4–9].
Throughout,thisstudy,ithasbeenfoundthenecessitytoregard
extendedobjects,themostreliableonestoexpresstheactualnature
of objects, rather thanrelying toa point-like system.Dueto thistran-
sition, theassociatebuilding block of space-time has toberevisited in
thecontextofC-space.Thisapproachhasserved toredefinedifferent
notationsandquantities likemassandchargeofany objectasshownin
(2.41)and(2.42).Notonly thedefinitionsofmassand charge forany
object are neededtobe revised,but alsothenotation ofthe torsion of
space-time asdefinedin theinternalcoordinates (2.13)is obtaineddue
toimposingtherulesofdifferentiationofCliffordspaceasshownin
equations (2.10),(2.11) and(2.12); whiletheir correspondingexternal
coordinates (poly-vector) are purely Riemannaian.This isdueto tak-
ingthecovariantderivative affectedoriented areasx
µν
ofholographic
coordinatesasshownin(2.10),and(2.11)from(2.12).Consequently,
DOI:10.4236/jmp.2020.11111161870JournalofModernPhysics
M.E.Kahil
themeaningoftorsionisbecomingassimilarasbeingobtainedin
termsofcommutationrelationofdifferentialoperatoronapotential
scalar field as expressed by Hammond [25] on dealing with internal co-
ordinates.While,theconventionaldefinitionoftorsion(2.35)inthe
contextofnon-symmetricaffineconnectionisalsopreserved,ifone
appliestheassociategeometrydescribingtheexternalcoordinatesas
anon-Riemanianngeometry[13–17].
Moreover,wehaveextendedourstudytoderivetheequivalent
chargedspinningequationsandtheirdeviationonesforpoly-vectors
in C-space, tobe considered itscorresponding generalizedDixon equa-
tion(5.97)and(5.98)inwhichtheircorrespondingdeviations(5.99)
and(5.100)areobtained.Thisapproachmayhelptoexaminethe
stabilityofextendedobjectsinC-space.
Tosumup,theproblemofmotioninC-spacebecomesaparadigm
shift towardsidentifyingthe behaviorof objects in adeterministic way
whichcanbedetectedintermsofbothinternalcoordinatesandtheir
external ones.Such atendency may inspire many scientists tofind out
such aunified theorydescribing both micro-physics andmacro-physics
inoneform.
ConflictsofInterest
Theauthordeclaresnoconflictsofinterestregardingthepublication
ofthispaper.
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