Apein, Vl. 15, N. 3, July 008 94 New Time Dependent Gavity Displays Dak Matte and Dak Enegy Effets Phais E. Williams Williams Reseah 1547 W. Dming Ln. Sun City West AZ, 85375 It is shwn that a time dependent gavitatinal field that is getting weake with time will pdue the effets measued f bth the tangential velity in the ams f spial galaxies and f the high z supenvas. These esults shw that the effets that have led t the hypthesis f Dak Matte and Dak Enegy may me fm the same basi physial phenmena, namely that gavity is getting weake as a funtin f time, and nt fm the existene f exti matte. Keywds: distanes and ed shifts, dak matte, dak enegy, they Intdutin Muh has been witten, hypthesized, and alulated n the subjet f Dak Matte and Dak Enegy. Hweve, nne nside a time dependent gavitatinal field. A gavitatinal field that gets weake with time will display galaxy dynamis espnding t a muh stnge field befe sending light fm spae twads the Eath 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 95 that an nly be eeived many light yeas late. The theetial basis f suh a time dependent gavitatinal field has aleady been pesented [1][][3]. Thee elements f this they apply t the ptential explanatin f Dak Matte and Dak Enegy. These elements inlude: 1. The they is a five dimensinal gauge they with Weyl gemety [4]. This means that the fields within the they ae gauge fields. Hweve, the they is nt anthe Kalusa-Klein type f they in that the fifth dimensin desibes a eal physial ppety, mass density, and, theefe, is nt hidden bsued by sme mathematial tehnique. The five dimensinality f the gauge they equies that the gavitatinal field be time dependent.. Quantum Mehanis is equied by estiting the Weyl sale fat within the gauge they t have nly a value f unity. This was nted by Shödinge [5] befe he published his wave equatins and late it was shwn by Lndn [6] that this estitin equied Shödinge s wave equatins. This quantizatin equies that the gauge ptentials be nn-singula [1]. 3. The fundamental Weyl gemety equies that the Pissn bakets and the unit f atin be dependent upn the gauge funtin [1]. This vaiable unit f atin leads t a elatin detemining the ed shift f light ming t Eath fm distant stas [1]. These thee aspets f the new they suffie t ffe a diffeent view f the data fm whih the hypthesis f dak matte and dak enegy have evlved. 008 C. Ry Keys In. http://edshift.vif.m
Dak Matte Apein, Vl. 15, N. 3, July 008 96 Data wheein the tangential velities f stas in the ams f a spial galaxy diffeed fm Newtnian peditins wee fist epted nealy seventy yeas ag [7]. A fundamental they suppting these data has nt heetfe been given, thugh empiial theies have been pesented. The best f these theies is the Mdified Newtnian Dynamis (MOND) [8][9][10]. The they pesented hee had its beginning in 1974 and nly eently has it been applied t the dynamis f spial galaxies. Newtnian unifm iula mtin equates the gavitatinal aeleatin t the entipetal aeleatin s that GMm mv =. (1) A time-dependent, nn-singula gavitatinal field, suh as the Dynami They pedits, altes Equatin (1) t GMm ( 1 H0τ ) mv 1 e =, () whee H 0 is Hubble s nstant and GM (3) as detemined by planetay bits. F this time dependent gavitatinal field the gavitatinal aeleatin ating n an am f a galaxy feels is due t the gavitatinal field f the mass M at a pevius time. This pevius time is given by the time that it takes f the field t tavel fm the site f the gavitatinal field t the pint n the am unde nsideatin. This means that when all the mass is nsideed t be at the ente f the galaxy the time that entes int 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 97 Equatin () is τ = s that, when >> the velity f the am f the galaxy wuld be given by ( τ ) GM 1 H0 1 H0 v = = GM +. (4) Equatin (4) shws a vey diffeent haate than the expessin f the velity f the time independent gavitatinal field. This expessin shws that the velity f the galaxy ams shuld nt be expeted t dp ff as the time independent Newtnian gavitatinal field des. We may lk at the 5-dimensinal appah f the Dynami They by lking at the Lagangian 1 1 1 e L = m ( τ) + m + m( θ) + GMm( 1 Hτ) (5) whee the univese time, τ, is teated as anthe vaiable and t is the lal time. The univese time, τ, bemes a gemetial dinate that makes the pblem lal-time independent in five dimensins. The time Lagange equatin may then be witten as d L L 0 d [ m ] H e = = τ m ds τ +. (6) τ dt F a spheially symmeti field the adial equatin is d L L 0 d = = [ m ] m θ GMm ( 1 Hτ) 1 e dt + dt The thid Lagange equatin bemes.(7) 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 98 d E E d = 0 = m θ dt θ θ dt. (8) F the pblem f spial galaxy behaviu we may assume the << and wite the equatins f mtin as H τ =, (9) GM θ = ( 1Hτ) 1 (10) and θ + θ = 0. (11) If we nw lk at unifm iula mtin we find that Equatin (9) bemes H d τ H τ = = nstant = (1) dt s that this may be integated t get H τ = τ ( tt) (13) whih may be integated again t get H H τ = τ t + τ t t +. (14) Als f the assumed unifm iula mtin Equatin (10) may be witten as 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 99 GM v = ( 1 Hτ ) 1 (15) whee v is the tangential velity f the unifm iula mtin. Putting Equatin (14) int Equatin (15) btains GM H H v 1 H t H τ τ t t 1 = + +. (16) We must keep in mind thee ae tw times t be nsideed. Fist thee is the time it takes f the gavitatinal hange t tavel fm the ente f the galaxy t the pint f measuement in the galaxy am. The send time is f the light signal t tavel fm the galaxy t the Eath. Let us set τ = 0and t = 0 at the pint in time when the light left the sta n its way twad Eath. Nw u Equatin (16) bemes GM H v 1+ t H τ t 1. (17) Time uns fm the time the gavitatinal signal left the ente f the galaxy at t =, (18) whee is the distane fm the ente f the galaxy. Using Equatin (18) in Equatin (17) we find GM H τ H GM H τ v 1+ 1 1 +. (19) Nw we need t establish a value f τ. Lk at the enegy at time t=0 and τ=0 with >>, 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 300 1 ( ) 1 GMm E = m τ + mv (0) This may be ewitten as E v = ( τ ) + (1) m Sine the tangential velities ae nn-elativisti this equies that E τ = +. () m Equatin () shws that the initial nditins establish the pint at whih the tangential velities begin t diffe fm thse pedited by Newtnian gavity. In the absene f a means f evaluating the initial nditins we may tun t expeimental esults. Fist, suppse we wite the aeleatin in the ams f the galaxy as H a = an 1 t H τ t. (3) an 1 + H τ We nw use the data that shws the aeleatin begins t deviate fm Newtnian when the aeleatin dps t a value f 1. x 10-10 m/se s that GM an = 1. 10 Then equiing GM 10 10 1. 10. (4) 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 301 τ = (5) H sets a value f τ in keeping with the data. Equatin (3) bemes a an 1 + (6) whee we see the sht ange Newtnian aeleatin and the lng ange aeleatin pedited by MOND. It shuld be nted that the appximate lineaity f the tangential velity with espet t time f Equatins (17) and (19) displays an independene f the time it takes f light t tavel fm the galaxy t Eath. This appaent independene f time masks the fat that the gavitatinal stength f the galaxy, elative t the uent eph, depends upn the time f light tavel t Eath. Dak Enegy Data displaying evidene that pvided the beginning f the hypthesized dak enegy was fist pesented in 1998 [11]. T date n fundamental they has had suess in explaining these data. The univese expansin fat is taken fm geneal elativity and is a 4 p = πg ρ + 3 a 3. (7) The mean density and pessue ae uently taken t inlude dak enegy and ae taken t bey the lal nsevatin f enegy elatin 3 a ρ = ρ + p. (8) a 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 30 The fist integal f Equatins (7) and (8) is the Fiedman equatin 8 a = πgρa + nstant. (9) 3 But nside what happens if ne wishes t mpae this with the smlgy pdued by the nn-singula, time dependent, gavitatinal gauge ptential. Then Equatin (7) bemes 3 d x 4π x ρ(t)gmg mg = 1 ( 1 H ) x e dt 3 x x τ, (30) 4π = G ρ(t)x 1 ( 1 H τ) e x 3 x whee τ is the univese time. Nw let us eplae x with the -mving dinate x=r(t) whee R(t) is the sale fat f the univese and is the -mving distane dinate as is dne in the standad mdel. When we als nmalize the density t its value at the pesent eph, ρ, by ρ(t)=ρ R -3 (t) we btain d R 4πGρR = 1 ( 1 H τ ) R. e (31) dt 3 R If we multiply Equatin (31) by dr/dt and integate with espet t time we find d R 4 G R π ρ dt = ( 1 Hτ ) RRdt 3 dt. (3) R R 4πGρ = ( 1 Hτ ) RdR 3 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 303 We nw need t knw hw t integate the ight hand side f Equatin (3). Suppse we nside the time it takes f light t tavel fm the distant sta t Eath, t = a, whee a is the distane fm the sta t Eath and the minus sign mes fm lking bakwads in time. The adius f the univese nw has tw pats. The fist pat is the adius f the univese when the light left the sta n its juney t the Eath. Let this time be R. Thus we see that R = R + a (33) and R = a (34) Futhe, fm nsideatins f the dak matte it was detemined that the wld time was given by HGM HGM τ = τ t + τ t t R + R. (35) When we set bth initial times t ze and use the value f GM U, (36) Equatin (35) bemes HU τ = t + τ t. (37) R Nw we find that Equatin (3) may be witten as Ra + ( + H τ R) a 8πGρ a = 6 1 + K. (38) 3 + ( HU + H τ) a 3 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 304 If we set the nstant f integatin, K, t ze, then Equatin (38) bemes 1 H τr 1 HU H τ 3 a = HΩ M Ra + 1 + a + + a.(39) 6 whee we have used the definitins 3H ρ ρ, and Ω M. (40) 8π G ρ In Equatin (39) we find that the mass density tem splits int thee tems f a time-dependent gavitatinal field. F a time-independent gavitatinal field thee was nly ne tem. An inteesting aspet f Equatin (39) is that the tw new mass tems bth invlve the same time dependene fat as the ne that auses the tangential velity f the ams f spial galaxies t diffe fm Newtnian behaviu. That is t say that shuld the tw new tems pvide a basis f the uent expeimental evidene f dak enegy it mes fm the same sue as the basis f dak matte. The time dependene f the gavitatinal field explains bth phenmena. Cnside Equatin (39) again and add the usual tem f adiatin s that we find R 1 H τr 1 HU H τ a 1 Ω M + + + + a H a = 6 a (41) +ΩRO whee we did nt add a tem f the smlgial nstant. If this is t mpae with the usual expessin we uld wite 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 305 ( 1 z) ( 1 z) ( 1 z) 4 ( 1 z) 3 3 3 a Ω M + +Ω DM + +Ω DE + = H a +Ω RO + 008 C. Ry Keys In. http://edshift.vif.m (4) wheein the sum f the tems ae taken t be unity at z=0 and the integatin nstant has been taken t be ze. Equatin (41) and (4) wuld equie R 1 H τr 1 HU H τ Ω M + 1 a 1 RO a + + + +Ω =.(43) 6 The elatin between the ed shift and a is a( tbs ) R + a 1 + z = = (44) a( tem ) R By putting Equatin (44) int (41) we find 3 1 H τ a 1 HGM H τ a 1 Ω M z + + a 4 + + 4 = H z 6 a (45) +ΩRO The fat that these tems have expessins elating them agues that thei elative values may be detemined. F example, if Ω RO is taken t be small mpaed with the mass tems and Ω M is set at the typial value f 0.5, then we wuld equie HUa z 1 6z τ =. (46) Ha ( 3+ z)
Apein, Vl. 15, N. 3, July 008 306 Sine the sue f the light being measued left its igin sme time afte the univese mpleted the expnential inflatinay expansin ealy in univese time, we wuld have z37 ( z) τ =. (47) Ha ( 3+ z) Putting this bak int Equatin (45) we find ( z) ( ) a 1 z z 7 = H Ω M z+ + +ΩRO. (48) a ( 3+ z) 3+ z Thee ae thee tems f the mass with diffeent funtins f z. Nw we wuld have ( 3+ z) Ω M = = 0.5 (49) 4( 3+ z) as set abve and we an then evaluate eah tem when z=0. Let us assiate the middle tem with Ω M, the fist tem with Ω DMO and the emaining tem with Ω DEO. We wuld then have the values 1 z ( Ω M] = 1 z= 0 Ω M = ( 3+ z) z= 0 Ω = Ω z = 0 (50) ( DM ] ( ] z= 0 M z= 0 z( 7 z) ( DE ] z= 0 M ( 3+ z) Ω = Ω = 0 f z=0. Ou veall equatin wuld then be z= 0 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 307 ( z) ( ) a 1 z z 7 = H Ω Mz +Ω M +Ω M +Ω RO,(51) a ( 3+ z) 3+ z whee Ω M vaies as ( 1 + z) 3. Cmpaing with Expeiment The expansin f the univese means the distane between tw distant galaxies vaies with time as L t a t. (5) ( ) ( ) The ate f hange f the distane is the speed dl a v = Hl, H dt = = a (53) whee H is the time dependent Hubble paamete. A methd f measuing f the expansin f the univese mes fm measuing the shift f fequenies f light, the ed shift, ming fm distant stas. The bseved wave length,, f a featue in the spetum that had wavelength e at emissin is given by the elatin a( t ) 1 + z = =. (54) e a( te) When the velity is given by z then Hubble s law is witten as z = Hl (55) fm whih we see that HL z z =, H = L. (56) 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 308 An additinal featue f the new they pesented hee is the expessin f the ed shift f light fm distant stas. This has been shwn t be [1] M e R R e Δ G Me R Me e HL zexp = = exp 1, e R R + (57) e M E R E whee the subsipt designates values at the time and pint f eeptin, the subsipt e epesents values at the time and pint f emissin and the quantities, M E and R E, epesent the mass and mean adius f the Eath. We have als used the subsipt exp n the ed shift t indiate that it is the expeimental value f ed shift measued at the eeiving latin. Thee ae tw pats t the ed shift. One pat is due t the gavitatinal fields at the pints and time f emissin and eeptin and the the pat is due t the tavel time between emissin and eeptin. This is the pat that invlves the expansin f the univese. Theefe let us ewite Equatin (57) as M e R R e G Me R Me e HL zexp = exp exp 1, R R (58) e M E R E and then eaange it t get 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 309 M e E R R e HL G Me R Me e E z = lg ( 1 z exp ) R R + + (59) e M R whih is the ed shift f the univese expansin. Tw simplifiatins may nw be made. Fist, in many ases the gavitatinal mpnent f the expeimental ed shift may be igned. Sendly, if we ae nly using the ed shift data measued at the Eath s sufae then Equatin (59) edues t HL z = lg ( 1 + zexp ). (60) This is the ed shift value t be used in the expansin velity f the univese, Equatin (51). s that we may wite 3 0.5( 1 + lg ( 1 + zexp )) lg ( 1 + z ) exp a 3 1 lg ( 1 zexp ) + H 0.5( 1 lg ( 1 zexp )) RO a = + + + +Ω ( 3+ lg( 1+ zexp )) 3 z( 7 lg( 1+ zexp )) + 0.5( 1+ lg ( 1 + zexp )) ( 3+ lg( 1+ zexp )). (61) If ne an simultaneusly measue the ed shift and the distane t the bjet then Equatin (56) gives a value f Hubble s paamete 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 310 that may be used in Equatin (53) t get the univese expansin velity. Standad Candles One easn f hsing the Type Ia supenva in the univese expansin eseah is the assumptin that the mass f this type supenva ae all the same; ughly the Chandasekha Limit mass f 1.39 sla masses. Hweve, a time dependent gavitatinal field auses this limit t hange with time. This may be seen by nsideing the Newtnian equatin f hydstati equilibium knwn as the Tlman-Oppenheime-Vlkv [TOV] equatin, dp GM ()1 ( Hτ ) ρ =. (6) d The gavitatinal field that is hlding the sta tgethe against the intenal pessue is diminishing in time. This means that the limiting mass ineases in time. Supenva fund lse t Eath will have me mass and theefe geate luminsity, than me distant supenva. A edutin in luminsity fm the assumed nstany wuld shw up in an analysis by making the me distant supenva appea futhe away than it eally is. The natual nlusin, based n the time-independent gavitatinal field that pdues the nstant Chandasekha limiting mass, wuld be that the expansin f the univese is aeleating. Using the Viial Theem develpment by Cllins [13] wh aives at the Chandasekha limiting mass with the equatin R MSun > 8 00 GM M 008 C. Ry Keys In. http://edshift.vif.m 4 9 (63)
Apein, Vl. 15, N. 3, July 008 311 the time dependent gavitatinal field equies that this elatin beme R MSun > 8 00 GM M. (64) ( 1 Hτ ) This gives the limiting mass as ( 1 ) 9 4 M = M H τ, (65) L Ch whee M Ch is the Chandasekha limiting mass. By diffeentiating Equatin (65) with espet t univese time we find the limiting mass f the type Ia supenvae t hange ading t dm 9 ( 1 ) 13 L H = M 4 Ch Hτ dτ 4. (66) Cnlusins A time-dependent gavitatinal field, that gets weake in time, shws the physial effets f this past, stnge field in the dynamis f spial galaxies. This weakening gavitatinal field als shws up in the analysis f the distanes t, and ed shift f light fm, supenvas. Hee it adds tems t the univese expansin velity elatins that ae nt pesent in the analysis f time-independent fields. It als hanges the luminsity f the supenvas that wee assumed t have nstant luminsity. These effets f the time dependent gavitatinal field emve the need f hypthesizing new matte enegy t explain these effets. Thee have been many attempts in the past t find diffeent slutins t Einstein s field equatins and t shw hw an expanding univese may be viewed in diffeent ways. Ptins f the abve may 008 C. Ry Keys In. http://edshift.vif.m 4 9
Apein, Vl. 15, N. 3, July 008 31 be emindes f pi appahes. Theefe, it may pve useful t pint ut what is new in this atile. Fundamentally thee ae thee things that ae new in this atile. Fist, the fifth dimensin is nsideed t be a eal physial entity. All five dimensinal theies that I knw f in the past, whethe by Kalusa-Klein, Einstein with his many llabats, and thes, did nt nside the fifth dimensin t be eal and, theefe, equied seveal tems in the esulting gauge field equatins t be ze. Hee these tems ae nn-ze and equie that the gavitatinal ptential and field be time-dependent. Send, this atile uses the Weyl Gauge Piniple as its basis f quantum they and this equies that the gavitatinal ptential be a nn-singula ptential. These tw things equie the gavitatinal field t be a time-dependent, nn-singula, gauge field nt seen peviusly. The thid aspet f the atile is that the Weyl Gauge Piniple equies that the unit f atin be dependent upn the gauge funtin. This equies the ed-shift fm distant bjets t have an expnential dependene upn bth the time and distane between emissin and eeptin. The new ed-shift elatin bemes imptant in bth dak matte and dak enegy peditins beause bth phenmena ae witnessed by ed-shifted light. The time dependene, weakening, f the gavitatinal field is the maj fat in pediting effets intepeted as dak matte. The time dependene f the gavitatinal field als pvides the maj fat in peditins with espet t dak enegy as it is espnsible f the diminishing f the luminsity f the distant supenvas used as standad andles and the expessin f the expansin f the univese. Refeenes: [1] Williams, P.E., Mehanial Entpy and its Impliatins, Entpy, 3, 76-115. http://www.mdpi.g/entpy/list01.htm#new 008 C. Ry Keys In. http://edshift.vif.m
Apein, Vl. 15, N. 3, July 008 313 [] Williams, P. E., 00, Enegy and Entpy as the Fundaments f Theetial Physis, Entpy, 4, 18-141. http://www.mdpi.g/entpy/htm/e404018.htm [3] Williams, P. E., 007, Altenate Cmmuniatins f Spae Tavel, Spae Tehnlgy and Appliatins Intenatinal Fum (STAIF-007), Albuqueque, NM. [4] Weyl, H., 1918, Spae Time Matte. [5] Shödinge, E., 19, On a Remakable Ppety f the Quantum-Obits f a Single Eletn, Zeit. F. Phys. 1. [6] Lndn, F., 197, Quantum-Mehanial Intepetatin f Weyl s They, Zeit. F. Phys. 4. [7] Zwiky, F., 1937. On the Masses f Nebulae and f Clustes f Nebulae, Astphysial Junal, Vl. 86, N. 3. [8] Milgm, M., 1983a, ApJ 70, 365. [9] Milgm, M., 1983b, ApJ 70, 371. [10] Milgm, M., 1983, ApJ 70, 384. [11] Riess, et. Al., 1998, Obsevatinal Evidene fm Supenvae f an Aeleating Univese and a Csmlgial Cnstant, Astn.J. 116, 1009-1038. [1] Williams, P.E., 001b, Using the Hubble Telespe t Detemine the Split f a Csmlgial Objet s Redshift in its Gavitatinal and Distane Pats, Apein, Vl. 8, N., http://edshift.vif.m/junalfiles/v08nopdf/v08nwil.pdf [13] Cllins II, 003, Viial Theem in Stella Astphysis, http://ads.havad.edu/bks/1978vtsa.bk/ 008 C. Ry Keys In. http://edshift.vif.m