New Time Dependent Gravity Displays Dark Matter and Dark Energy Effects

Transcription

1 Apein, Vl. 15, N. 3, July New Time Dependent Gavity Displays Dak Matte and Dak Enegy Effets Phais E. Williams Williams Reseah 1547 W. Dming Ln. Sun City West AZ, 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.

2 Apein, Vl. 15, N. 3, July 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.

3 Dak Matte Apein, Vl. 15, N. 3, July 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.

4 Apein, Vl. 15, N. 3, July 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 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.

5 Apein, Vl. 15, N. 3, July 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.

6 Apein, Vl. 15, N. 3, July 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 (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.

7 Apein, Vl. 15, N. 3, July ( ) 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 m/se s that GM an = Then equiing GM (4) 008 C. Ry Keys In.

8 Apein, Vl. 15, N. 3, July 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.

9 Apein, Vl. 15, N. 3, July 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 C. Ry Keys In.

10 Apein, Vl. 15, N. 3, July 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 = K. (38) 3 + ( HU + H τ) a C. Ry Keys In.

11 Apein, Vl. 15, N. 3, July 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 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.

12 Apein, Vl. 15, N. 3, July ( 1 z) ( 1 z) ( 1 z) 4 ( 1 z) a Ω M + +Ω DM + +Ω DE + = H a +Ω RO C. Ry Keys In. (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 = 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)

13 Apein, Vl. 15, N. 3, July 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= C. Ry Keys In.

14 Apein, Vl. 15, N. 3, July ( 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.

15 Apein, Vl. 15, N. 3, July 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.

16 Apein, Vl. 15, N. 3, July 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.

17 Apein, Vl. 15, N. 3, July 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 (63)

18 Apein, Vl. 15, N. 3, July 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

19 Apein, Vl. 15, N. 3, July 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, C. Ry Keys In.

20 Apein, Vl. 15, N. 3, July [] Williams, P. E., 00, Enegy and Entpy as the Fundaments f Theetial Physis, Entpy, 4, [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., 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, [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., [13] Cllins II, 003, Viial Theem in Stella Astphysis, C. Ry Keys In.