Using the Hubble Telescope to Determine the Split of a Cosmological Object s Redshift into its Gravitational and Distance Parts

Transcription

1 Apion, Vol. 8, No. 2, Apil Using th Hubbl Tlscop to Dtmin th Split of a Cosmological Objct s dshift into its Gavitational and Distanc Pats Phais E. Williams Engtic Matials sach and Tsting Cnt 801 Loy Plac, Socoo, NM 8780 Th Hubbl Tlscop should mak it possibl to spaat th dshift of light fom any cosmological objct into that dshift du to th gavitational potntial diffnc btwn th mission and cption points and that potion du th distanc btwn ths points by compaing th dshifts masud on th Eath s sufac to dshifts masud by th Hubbl Tlscop. This would allow a mapping of th cosmological objcts and an incasd undstanding of gavitational conditions of ths objcts. Kywods: gavitational dshift, cosmological dshift, Hubbl tlscop, Wyl gaug facto, gaug function, unit of action, vaiabl unit of action 2001 C. oy Kys Inc.

2 Intoduction Apion, Vol. 8, No. 2, Apil I n 1911 Einstin usd tim dilation in th Spcial Thoy of lativity to pdict that th atomic spctal lins should b shiftd towads th d nd of th colo spctum [1]. This pdiction has sinc bn vifid by xpimnt using both sola and tstial gavitational potntial diffncs [2-5]. Hubbl masud th dshift of numous stas and galaxis and notd that ths displayd an almost dict lationship btwn th dshift and th cosmological distanc of th adiating objct fom th Eath [6]. Ths xpimntal findings w lat suppotd by solutions to Einstin s Gnal lativistic fild quations, such as th obtson- Walk solution [7]. Howv, ths xpimntal and thotical considations povid no way to sot out how much of th cosmological objct s dshift is du to th gavitational potntial diffnc btwn th mitting objct and th Eath and how much is du solly to th distanc btwn thm. Th xists a nw thoy, calld th Dynamic Thoy [8-14], which pdicts significant diffncs btwn th dshifts that a masud on th Eath s sufac and thos that should b masud by th Hubbl Tlscop in its obit. Futh, th mann of th pdiction is such that compaison of th dshifts masud at th Eath s sufac and by th Hubbl Tlscop sts up two quations in th two unknowns; th cosmological distanc to th objct and its gavitational potntial. To display how th nw pdictions diff fom th old lt s fist look at th old pdictions. Einstin s gavitational potntial diffnc dshift is givn by G M M z - -, 2 c 2001 C. oy Kys Inc.

3 Apion, Vol. 8, No. 2, Apil wh th subscipts and dnot th mitting and civing gavitational potntials. Th dshift du to th cosmological distanc is th lina lation H z L, c wh H is Hubbl s constant and L is th cosmological distanc btwn th mitting objct and th Eath. By adding ths two dshift pdictions w find G M M H z L, 2 c c which shows that th only diffnc this pdiction would hav btwn masumnts at th Eath s sufac and by th Hubbl Tlscop would com fom th chang in th diffnc in th gavitational potntial of cption du to th obital hight of th tlscop. Howv, bcaus of th small siz of th Eath in compaison to th cosmological objcts this diffnc would b ngligibl. Fo xampl, fo th Sun this ffct is som 10 5 pcnt. Fo objcts lag than th Sun th diffnc is vn lss. Th Dynamic Thoy is basd upon th laws of classical thmodynamics and it has bn shown that ths fundamntal laws qui Einstin s postulat of th constancy of th spd of light [13]. Givn this sult it is no supis that Einstin s spcial thoy of lativity quickly follows. What is lss obvious, but also quid, is that ths basic laws povid a dsciption of physical phnomna in fiv dimnsions of spac, tim and mass [14]. W shall now povid a bif outlin of th thotical backgound lading to th dshifts pdiction within this nw thoy stating b stating th adoptd laws. (1) 2001 C. oy Kys Inc.

4 Apion, Vol. 8, No. 2, Apil Fist Law (Consvation of Engy) Th concpt of consvation of ngy is fundamntal to all banchs of physics and is th bginning of thmodynamics and mchanics [8,10,12,13]. In tms of gnalizd coodinats o indpndnt vaiabls, th notion of wok, o mchanical ngy, is considd lina foms of th typ 1 n i dw F i( q,..., q, u 1,..., u n ) dq (i 1,2,...,n), wh th focs F i may b functions of th vlocitis (dq i /dt u i ) as wll as th coodinats q i and th summation convntion is usd. A systm may acqui ngy by oth mans in addition to th wok tms; such ngy acquisition is dnotd d E. Th systm ngy, which psnts th ngy possssd by th systm, is considd to b 1 n 1 n U( q,..., q, u,..., u ). With ths concpts, thn th Fist Law, which is th gnalizd Law of Consvation of Engy, has th fom de du dw du F dq i i ( i 1,..., n). In th Fist Law th dimnsionality is n + 1 and is dtmind by th systm considd. Scond Law Th statmnt of th Scond Law is mad using th axiomatic statmnt povidd by th Gk mathmatician Caathéodoy [15] who psntd an axiomatic dvlopmnt of th Scond Law of thmodynamics that may b applid to a systm of any numb of vaiabls. Th Scond Law may thn b statd as follows: 2001 C. oy Kys Inc.

5 Apion, Vol. 8, No. 2, Apil In th nighbohood (howv clos) of any quilibium stat of a systm of any numb of dynamic coodinats, th xist stats that cannot b achd by vsibl E consvativ ( d E 0) pocsss o motions. sults of th Scond Law Th Scond Law quis that an intgating dnominato must xist fo th Fist Law and that this intgating facto must b a function of vlocity only fo mchanical systms. Using th intgating dnominato th xpssion fo th Fist Law may b wittn d E (u) f( σ φ ) d σ. Sinc f(σ)dσ is an xact diffntial, th quantity 1/ φ ( u) is an intgating dnominato fo d E. Th univsal chaact of φ (u) maks it possibl to dfin an absolut spd in th sam mann as is don in thmodynamics whn dfining th absolut tmpatu. Th dfinition of th absolut spd quis constant spd motions b considd. All Galilan fams of fnc will display this pocss as on of constant spd. Futh, if all fnc fams a to b of qual status thn obsvs in all Galilan fnc fams must sha th d E 0 constant spd motion quivalntly. Futhmo, ach obsv will hav th sam valu fo th absolut spd o ls on of th fams will njoy a pivilgd natu. Thn th absolut spd is uniqu fo all Galilan fams of fnc. Th is on such spd alady known and that spd is th spd of light, c. Thfo, th absolut spd must b th spd of light and th sam fo all Galilan obsvs. This is Einstin s postulat. Thus, th fist two laws qui Einstin s postulat concning th spd of light C. oy Kys Inc.

6 Apion, Vol. 8, No. 2, Apil Sinc σ is an actual function of u and q, th ight-hand mmb is an xact diffntial, which may b dnotd by ds; and wh S is th mchanical ntopy of th systm. de ds φ (u) Gomty With th abov laws and th dfinition of th ntopy an xpssion fo th gnalizd Clausius inquality may b wittn and usd to spcify th stability condition δ F which lads to th quadatic fom and i U - i q - φδ S > 0. 2 i j 0 ( ds) hij dq dq ; j, k 01,,2,..., n, wh q S / F0 h jk 2001 C. oy Kys Inc. 2 U j k q q. Th lmnt of ac lngth may b paamtizd using th local tim as ds cdt. Howv, Clausius Inquality dos not lad to a singl vaiational pincipal on tim ath it lads to two vaiational pincipals, on quiing th minimization of F Engy and on quiing th maximization of th ntopy fo isolatd systms fo which d E 0. Th diffntial of th ntopy is on th ight hand sid of this quadatic fom so that th fom must b solvd fo th diffntial xpssion of ntopy in od to us th ntopy vaiational pincipal. Whn this is don w find that ( dq ) ( ) o 2 α α β f c dt + 2h cdq dt h dq dq f ( d ) oα αβ σ

7 Apion, Vol. 8, No. 2, Apil This shows th quimnt fo two mtic spacs coupld by a gaug function, f. Sinc th Scond Law quis that th ntopy is a total divativ on may suspct that th ntopy spac will b an intgabl spac and this is indd th cas whn th Scond Law is applid to th mtic cofficints. In addition, on finds that th scond spac, which w might call an ngy spac bcaus of th ti to th Fist Law, must b a Wyl spac. Thfo, w find that th gaug function acts as a gomtical intgating facto coupling th non-intgabl Engy spac to th intgabl ntopy spac. Th appaanc of Wyl chaact of th Engy spac allows th us of London s wok that shows that null tajctois in a Wyl spac must b dscibd by th quations of quantum mchanics [16]. In th Dynamic Thoy, th ncssity of considing null tajctois coms in a vy natual way. Fo instanc, in thmodynamics th dsi to consid stabl stats would caus on to look fo isntopic stats. This is of cous a null tajctoy in th ntopy spac, howv, fo non-zo gaug functions this condition is also a null tajctoy in th ngy, Wyl, spac. By th Scond Law th diffntial chang of ntopy can nv b ngativ fo an isolatd systm so that dq 0 0. Thfo, th ntopy mtic is positiv dfinit. Fo ngativ gaug functions th ngy spac will b ngativ dfinit and, thfo, complx. Th a an infinit numb of null tajctois fo a complx spac and ths a givn by th quantum stats. This may b mo asily sn by considing th displacmnt of th lmnt of ac lngth in th ngy spac that must tak on th Wyl displacmnt fom of k d(d ε ) φ dq (d ε) k wh th φ k a th gaug potntials and a th logaithmic divativ of th squa oot of th gaug function. Thn th 2001 C. oy Kys Inc.

8 Apion, Vol. 8, No. 2, Apil isntopic condition that th intgal of Equation (10) qui that ( d ε) ( dε and, thfo, so that and ) 0 φ j φ jdq 1 j dq j 2πiN. This is th quantum condition that London usd to div Schöding s quations of quantum mchanics. H th quantum condition is quid by th isntopic stat spcification. Whn on fist bgins to study th thmodynamics of stam systms on wits th Fist Law as d E du + Pdv F dx α α, α 1,2,3. Th ight hand sid of this statmnt of th Fist Law contains fiv unknown vaiabls. Th accptd mthod of ducing th numb of unknowns is to, fist, stat that th mass dnsity can always b wittn as a function of spac and tim thby ducing th numb of additional indpndnt quations ndd to fou. Ths fou quations a pointd out to b an quation of stat and th th mchanical laws of motion fom Nwtonian physics. Th pocdu outlind abov fo obtaining th quations of th mtics may also b usd in fiv dimnsions and thn th dpndnc o indpndnc of th mass dnsity upon spac and tim may b dtmind as th pdictd phnomna ag o disag with xpinc. This lads on to a fiv dimnsional ntopy mtic of spac-tim-mass. H also on finds th appaanc of th two spacs coupld by a gaug function fo an isolatd systm. In this cas th gaug function is a function of th sam fiv vaiabls. Th Dynamic Thoy maks its pdiction of dshifts stating fom this fiv-dimnsional gomty of spac-tim-mass in which th gaug function poducs th filds. This gaug function is a function 2001 C. oy Kys Inc.

9 Apion, Vol. 8, No. 2, Apil C. oy Kys Inc. of spac, tim, and mass and it dtmins th unit of action in th atomic stats as may b sn fom th Quantum Poisson backts whn covaiant diffntiation is usd, o [ ] Ψ Ψ x l s j + g i,p x s jl kl k j δ ħ wh th vcto cuvatu would appa in th Chistoffl symbols insid th backts whil th gaug function is a multiplicativ facto in th g kl. Thn whn th vcto cuvatu is ngligibl th Quantum Poisson backts bcom [ ] Ψ Ψ δ jk k j f i p, x ħ wh it may b sn that th unit of action is Diac δ tims th gaug function. It may b shown [12] by using th gaug fild quations that th functional fom of th gaug function must b. k(a+ bt)m xp f - By quiing th photon ngy to b consvd w hav v hf v hf which poducs th dshift xpssion - 1. ) (a+ bt M - ) (a+bt M k z - - xp By xpanding th ight-hand sid as a pow sis and compaing th fist od appoximation to th classical xpssion in Eqn. (1) th constants k, a, and b may b valuatd. By stting 0 t and c L t / w find ou dshift xpssion bcoms xp 2 1, + M M c HL M M c G z

10 Apion, Vol. 8, No. 2, Apil wh M/ is th gavitational potntial at ith th point of mission o th point of cption and th subscipt stands fo mission whil th subscipt stands fo cption. Th and M without subscipts a th valus of th adius and mass of th Eath spctivly. Bcaus of th distancs involvd th appoximation << may b usd fo both th mitt and th civ. Thn th appoximation of Eqn (2) bcoms G M M ln(z+1) - - c M HL + c M 2 (3) so that on th Eath s sufac w would hav th appoximation ln( z G M +1) - 2 c M HL - + c ES (4) whil, if w w to obtain xpimntal dshifts using th Hubbl Tlscop whil in obit at an obital hight of h w would hav G M M HL ln( z HT + 1) 2 ( ) ( ) ( ). c + h + c (5) + h Equation (4) and (5) psnt two quations in th two unknowns, L and M /. Th solution of ths quations is givn by th quations M 2 c ( ) ( h) ln( zht 1) ln( zes 1 hg ) (6) and GM { ( ES ) ( HT )( ) 2 } c L ln z + 1 ln z (7) H h c On may thn s how compaing th dshifts obtaind fom th Eath s sufac with thos takn at a hight abov it will allow th dtmination of distanc to, and th gavitational potntial of, a 2001 C. oy Kys Inc.

11 Apion, Vol. 8, No. 2, Apil cosmological objct. Th ability to obtain solutions fom th two quations fo dshifts at diffnt civing gavitational potntials dos not xist in oth pdictions of dshifts. It is bcaus of th appaanc of th atio of gavitational potntials in Eqn (2) that allows th Dynamic Thoy to mak th distanc and potntial pdictions. Fo objcts with a lag gavitational potntial compad to that of th Eath th majo chang in ln(z + 1) coms fom th atio of h to + h. Fo an obit hight of 380 mils this atio is This mans that th xpctd chang in th masud dshift fom th Eath s sufac to th Hubbl Tlscop obit is of th od of a fw pcnt. A studnt suvy of books poting dshifts in th optical ang puts th xpimntal o btwn a fw pcnt and na 30 pcnt. Thfo, ca nds to b takn o fquncis sought which hav lss xpimntal o. fncs [1] A. Einstin, Üb dn Einfluss d Schwkaft auf di Ausbitung ds Lichts, Annaln d Physik 35 (1911). [2] J. Bault, Thsis, Pincton Univsity, Pincton, N.J., (1960). [3].E. Dick, lativity, Goups and Topology, Expimntal lativity, in B. Dwitt (d). (1964). [4] T.E. Canshaw, J.P. Schiff, and A.B. Whithad, Masumnt of th Gavitational dshift Using th Mössbau Effct, Phys. v. Ltts 4 (1960) [5].V. Pound, and G.A. bka, Gavitational dshift in Nucla sonanc, Phys. v. Ltts 3 (1959) [6] E.P. Hubbl, Th Luminosity Function of Nbula, II, Astophys. J. 84 (1936) [7] H.P. obtson, Kinmatics and Wold Stuctu, Astophys. J. 82 (1935) C. oy Kys Inc.

12 Apion, Vol. 8, No. 2, Apil [8] P.E. Williams, On a Possibl Fomulation of Paticl Dynamics in Tms of Thmodynamic Concptualizations and th ol of Entopy in it, thsis, U.S. Naval Postgaduat School, (1976). [9] P.E. Williams, Th Pincipls of th Dynamic Thoy, sach pot EW-77-4, U.S. Naval Acadmy, (1977). [10] P.E. Williams, Th Dynamic Thoy: A Nw Viw of Spac, Tim, and Matt, Los Alamos Scintific Laboatoy pot LA MS, (Fb. 1980). [11] P.E. Williams, Th Possibl Unifying Effct of th Dynamic Thoy, Los Alamos Scintific Laboatoy pot LA MS, (May 1983). [12] P.E. Williams, Th Dynamic Thoy: A Nw Viw of Spac-Tim-Matt, copyight (1993) ( [13] P.E. Williams, Thmodynamic Basis fo th Constancy of th Spd of Light, Modn Physics Ltts A 12 No. 35 (1997) [14] P.E. Williams, Quantum Masumnt, Gavitation, and Locality in th Dynamic Thoy, Symposium on Causality and Locality in Modn Physics and Astonomy: Opn Qustions and Possibl Solutions, Yok Univsity, Noth Yok, Canada, (August 25-29, 1997). [15] C. Caathéodoy, Math. Ann. 67 (1909) 355. [16] F. London, Quantum Mchanisch Dulung d Thoi, Physik 42 (1927) C. oy Kys Inc.