Gravitation as Geometry or as Field

Transcription

1 Journal of Appld Mathmatcs and Physcs, 7, 5, ISSN Onln: ISSN Prnt: Gravtaton as Gomtry or as Fld Waltr Ptry Mathmatcal Insttut of th Unvrsty Dussldorf, Dussldorf, Grmany How to ct ths papr: Ptry, W (7) Gravtaton as Gomtry or as Fld Journal of Appld Mathmatcs and Physcs, 5, Rcvd: January, 7 Accptd: Aprl 7, 7 Publshd: Aprl 3, 7 Copyrght 7 by author and Scntfc Rsarch Publshng Inc Ths work s lcnsd undr th Cratv Commons Attrbuton Intrnatonal Lcns (CC BY 4) Opn Accss Abstract Gnral rlatvty (GR) and gravtaton n flat spac-tm (GFST) ar covarant thors to dscrb gravtaton Th mtrc of GR s gvn by th form of propr-tm and th mtrc of GFST s a flat spac-tm form dffrnt from that of propr-tm Th sourc of GR s th mattr tnsor and th Enstn tnsor dscrbs th gravtatonal fld Th sourc of GFST s th total nrgymomntum ncludng gravtaton Th fld s dscrbd by a non-lnar dffrntal oprator of ordr two n dvrgnc form Th rsults of th two thors agr for wak gravtatonal flds to th ordr of masurabl accuracy It s wll-known that homognous, sotropc, cosmologcal modls of GR start from a pont sngularty of th unvrs, th so calld bg bang Th dnsty of mattr s nfnt Thrfor, our obsrvabl bg unvrs mpls an xpanson of spac, n partcular an nflatonary xpanson n th bgnnng Doubts ar statd bcaus nfnts don t xst n physcs An xplanaton to th prsnt, controvrsal dscusson of xpandng acclratng or non-acclratng unvrs as wll as non-xpandng unvrs s gvn GFST starts n th bgnnng from a homognous, sotropc unvrs wth unformly dstrbutd nrgy and no mattr In th cours of tm mattr s cratd out of nrgy whr th total nrgy s consrvd Thr s no sngularty, no bg bang Th spac s flat and non-xpandng Kywords Gravtaton, Cosmology, Flat Spac, No Sngularty, No Bg Bang, Non-Expandng Unvrs Introducton Enstn s gnral thory of rlatvty s at prsnt th most accptd thory of gravtaton Th thory gvs for wak gravtatonal flds, agrmnt wth th corrspondng xprmntal rsults But th rsults for homognous, sotropc, cosmologcal modls mply dffcults So, th unvrs starts from a pont sngularty, th unvrs starts from a pont wth nfnt dnsty of mattr Th DOI: 436/jamp75476 Aprl 3, 7

2 W Ptry obsrvd unvrs s vry bg Hnc, th spac of th unvrs must xpand vry quckly whch mpls th ntroducton of an nflatonary unvrs n th bgnnng Thr ar controvrsal dscussons about th unvrs, g s th unvrs acclratng or not GFST uss a psudo-eucldan gomtry and th propr tm s dfnd smlar to that of gnral rlatvty, spac-tm and propr tm ar dffrnt from on anothr GFST starts from an nvarant Lagrangan whch gvs by standard mthods, th fld quatons of gravtaton Th sourc s th total nrgy-momntum tnsor ncludng gravtaton Th nrgy-momntum of gravtaton s a tnsor Th fld s dscrbd by non-lnar dffrntal quatons of ordr two n dvrgnc form Th thory s gnrally covarant Th gravtatonal quatons togthr wth th consrvaton law of th total nrgy-momntum gv th quatons of moton for mattr Th applcaton of th thory mpls for wak gravtatonal flds th sam rsults as GR to xprmntal accuracy, g gravtatonal rd shft, dflcton of lght, prhlon prcsson, radar tm dlay, post-nwtonan approxmaton, gravtatonal radaton of a two-body systm and th prcsson of th spn axs of a gyroscop n th orbt of a rotaton body But thr ar also dffrncs of th rsults of ths two thors GFST gvs non-sngular, cosmologcal modls Th covaranc of GFST and th xstnc of non-sngular cosmologcal modls mply th possblty to ntrprt th solutons as xpandng or as non-xpandng spac yldng an acclratng rsp non-xpandng unvrs GFST may g b found n th book [] and n th ctd rfrncs Addtonally, non-sngular, cosmologcal modls ar g gvn n th artcls [] [3] [4] [5] [6] Subsquntly, homognous, sotropc, cosmologcal modls wll b summarzd Lt us us th psudo-eucldan gomtry Th rsultng unvrs s nonsngular undr th assumpton that th sum of th dnsty paramtrs s gratr than on, g a lttl bt gratr than on It starts wthout mattr and wthout radaton and all th nrgy s gravtatonal nrgy Mattr and radaton mrg from ths nrgy by vrtu of th consrvaton of th total nrgy Th spac s flat and th ntrprtaton of a non-xpandng spac s natural But t s also possbl to stat an xpanson of spac by a sutabl transformaton as consqunc of gnral covaranc of th quatons Mattr and radaton ar gnratd from th bgnnng of th unvrs and th unvrs bcoms hot A crtan tm aftr th bgnnng mattr and radaton dcras and th unvrs convrgs to dark nrgy as tm gos to nfnty Hnc, a unvrs gvn by GFST appars mor natural than that rcvd by GR whch gvs sngular soluton wth nfnt dnsts Th gomtry of GR s n gnral non-eucldan but th obsrvd unvrs mpls a flat spac GR s wll-known n contrast to GFST Thrfor, GFST and rsultng cosmologcal modls ar shortly summarzd n th nxt two sctons All ths rsults can b found n th artcl [5] Scton contans GFST; Scton 3 contans cosmologcal modls; Scton 4 contans GR and Scton 5 stats GFST Cosmologcal modls of GR and GFST ar compard wth on anothr 863

3 W Ptry GFST Th thory of GFST s shortly summarzd Th mtrc s th flat spac-tm gvn by ( ds) = η x () d j whr ( η j ) s a symmtrc tnsor Psudo-Eucldan gomtry has th form ( j ) (,,, ) 3 Hr, ( x ) ( x, x, x ) η = () = ar th Cartsan coordnats and 4 x = ct Lt η = dt η j (3) j Th gravtatonal fld s dscrbd by a symmtrc tnsor ( g j ) Lt ( g ) b dfnd by and put smlarly to (3) g g Th propr tm τ s dfnd by k kj = δ (4) j G = dt g j (5) ( d ) Th Lagrangan of th gravtatonal fld s gvn by c τ = g dx dx j (6) j G mn k jl j kl L( G) = gjgklg g/ mg/ n g/ mg/ n η whr th bar/dnots th covarant drvatv rlatv to th flat spac-tm mtrc () Th Lagrangan of dark nrgy (gvn by th cosmologcal constant Λ ) has th form Lt L ( Λ) 8Λ G = η (7) (8) 4 κ = 4πk c (9) and of mattr of a prfct flud ar whr κ s th gravtatonal constant Thn, th mxd nrgy-momntum tnsor of gravtaton, of dark nrgy and of mattr of a prfct flud ar Hr, holds by (6) G r km ln kl mn T( G) = gkl gmng g/ j g/ r g/ j g/ r δ jl( G) j + 8κ η ρ, p and T ( Λ) δ L( Λ) j (a) = j (b) 6κ k = ( ρ + ) + δ (c) T M p g u u pc j jk j u dnot dnsty, prssur and four-vlocty of mattr It 864

4 W Ptry Dfn th covarant dffrntal oprator G kl m Dj = g g jmg/ l η c j = g uu () of ordr two Thn, th fld quatons for th potntals ( g j ) hav th form j k Dj δ jdk = 4κTj (3) whr th total nrgy-momntum s th sum of th nrgy-momntum tnsors of mattr, gravtaton and cosmologcal constant,, T = T( G) + T( M) + T( Λ) (4) j j j j Dfn th symmtrc nrgy-momntum tnsor j k j g T( M) T M k / k () = (5) Thn, th quatons of moton n covarant form ar k kl T( M) = g / k / kl T M (6) In addton to th fld Equatons (3) and th quatons of moton (6) th consrvaton law of th total nrgy-momntum holds, k T k / = (7) Th fld quatons of gravtaton ar formally smlar to thos of GR whr T j s th nrgy-momntum wthout that of gravtaton snc th nrgy-momntum of gravtaton s not a tnsor for GR Furthrmor, th dffrntal oprator s th Enstn tnsor whch may gv a non-eucldan gomtry Th rsults of ths chaptr may b found n th book [] and n many othr artcls of th author, as g n [5] 3 Homognous, Isotropc, Cosmologcal Modls In ths chaptr GFST s appld to homognous, sotropc, cosmologcal modls Th psudo-eucldan gomtry () wth () s usd Th mattr tnsor s gvn by prfct flud wth vlocty u = =,,3 (8) and prssur p and dnsty ρ wth p = pm + pr, ρ = ρ + ρ whr th ndcs m and r dnot mattr and radaton Th quatons of stat for mattr (dust) and radaton ar pm =, pr = ρr () 3 Th potntal ar by vrtu of (8) and th homognty and sotropy a ( t) ( = j =,, 3) gj = h( t) ( = j = 4) () ( j) m r (9) 865

5 W Ptry Th four-vlocty s by Equaton (8) and Equaton (6) ( u ) ( ch ) =,,, () Lt t = b th prsnt tm and assum as ntal condtons at prsnt, ( ),, a = h = a = H h = h ρ = ρ ρ = ρ (3), m m r r whr th dot dnots th tm drvatv, H s th Hubbl constant and h s a furthr constant, ρ m and ρ r dnot th prsnt dnsts of mattr and radaton It follows from (6) undr th assumpton that mattr and radaton do not ntract ρ = ρ h, ρ = 3 p = ρ ah (4) m m r r r Th fld Equaton (3) mpls by th us of () th two nonlnar dffrntal quatons whr 3 d 3 a 4 Λ a ah = κc ρm + ρr +, dt a 3 κ c h d Λ dt h 8κc κc h 3 3 h 4 a ah = 4κc ρm + ρr + L ( G) L G Th xprsson 6κ L G of th total nrgy gvs 6 6 a ah h 3 a ah h = ah + + c (5a) (5b) (6) s th dnsty of gravtaton Th consrvaton law 3 ρ + ρ Λ a m r c + L G c 6κ + κ h = λ (7) whr λ s a constant of ntgraton Equatons (5), (6) and (7) gv by th us of th ntal condtons (3) 4 h a 4κc λt + ϕ = 6 + (8) 4 h a κc λt + ϕ t + wth Intgraton of (8) ylds ϕ = H h H (9) 3 4 ah = κcλt + ϕ t+ (3) Equaton (7) gvs for th prsnt tm t = by th us of th ntal condtons (3) 4 8 Λc ( 8κc λ ϕ ) = 4 π k ρm + ρr + H 3 3 8πk (3) 866

6 W Ptry It follows from (7) by th us of th standard dfnton of th dnsty paramtrs of mattr, radaton and th cosmologcal constant wth th abbrvaton K th dffrntal quaton = Ω +Ω +Ω Ω (3) m r Λ m a H 3 6 = Ω mk +Ω ra +Ω ma +ΩΛa 4 ( κ λ ϕ ) a c t + t + Th ntal condton s by (3) (33a) a = (33b) Th soluton of (33) wth (3) dscrbs a homognous, sotropc, cosmologcal modl by GFST Rlaton (3) can b rwrttn n th form 4 κc λ ϕ H H 8 = ΩmK (34) A ncssary and suffcnt condton to avod sngular solutons of (33) s K > (35) whch ylds 4 κc λt ϕt + + > (36) for all t Hnc, condton (35) mpls a non-sngular soluton for all t, w gt a non-sngular cosmologcal modl It xsts a t < t = such that Put a a( t ) a t = (37) = thn t follows from (33a) wth t = t 3 6 Ω a +Ω a +Ω a =Ω K (38) r m m m It holds for all t Subsquntly assum a a t > (39) a a = (4) Thn w gt by vrtu of (38) K (4) It follows from (3) by vrtu of (4) Ω +Ω +Ω = +Ω, (4) K r m Λ m th sum of th dnsty paramtrs s a lttl bt gratr than on Hnc, a( t ) starts from a postv valu, dcrass to a small postv valu, and thn ncrass for all t Th propr tm from th bgnnng of th unvrs tll tm t s t τ ( t) h ( t) d t (43) = 867

7 W Ptry Th dffrntal Equaton (33a) s rwrttn by th us of (3) n th form a ΩmK Ωr Ωm = H Ω Λ a h a a a (44) Hnc, th dffrntal quaton for th functon aa by th us of th propr tm s da ΩmK Ωr Ωm = H Λ a dτ + + +Ω a a a (45) Ths dffrntal quaton s by vrtu of (4) and a not too small functon a( t ) dntcal wth that of GR for a flat homognous, sotropc unvrs Thrfor, away from th bgnnng of th unvrs, th rsult for th unvrs agrs for GFST wth that of GR Undr th abov statd assumptons and Ω r = th dffrntal Equaton (33) can analytcally b solvd It follows that a( t ) starts from a small postv valu at and thn t dcrass for ncrasng ttto a > at t Fnally t ncrass for t > t to nfnty as t gos to nfnty Rlaton (3) gvs postv valus h( t ) for all t h( t ) starts from nfnty at, dcrass to a postv valu and thn t ncrass to nfnty as t gos to nfnty Th longr calculatons ar omttd and thy can b found n th artcl [3] Th dffrntal Equatons (44) and (45) show that th condton (35) s mportant to avod sngularts GR gvs K = whch ylds th sngularty of th modl (bg bang) W ntroduc n addton to th propr tm τ th absolut tm t by dt = dt = d τ (46) a t h t a t Ths gvs for th propr tm n th unvrs ( cd ) = a ( t) dx ( dct ) τ = 3 Rlaton (47) mpls that th absolut valu of th lght-vlocty s qual to vacuum lght-vlocty c for all tms t Th ntroducton of th absolut tm t n th dffrntal Equaton (45) gvs whr dx dnots th Eucldan norm of th vctor dx ( d x,d x,dx ) H da = Ω +Ω +Ω +ΩΛ dt a 3 6 ( mk ra ma a ) Assum that a lght ray s mttd at dstanc r at tm t rsp at tm t + dt and t s rcvd by th obsrvr at tm t rsp at tm t + dt Thn, t follows t t t + dt t + dt Ths two quatons mply r= ct d = ct t, r= ct d = ct + dt t d t (47) (48) 868

8 W Ptry dt = d t Th ag of th unvrs snc th mnmal valu of a( t ) masurd wth absolut tm t tll now da t = t = a= aa Ω K +Ω a +Ω a+ω a t d d d t a dt H a aa ( mk ( r m ) a ) H a Λ H d Ω + Ω +Ω +Ω 3 6 ( m r m Λ ) Thrfor, th ag of th unvrs masurd wth absolut tm s gratr than H ndpndnt of th dnsty paramtrs, thr s no ag problm W wll now calculat th rd shft of lght mttd from a dstant objct at rst and rcvd by th obsrvr at prsnt tm It s usful to ntroduc th absolut tm Assum that an atom at a dstant objct mts a photon at tm t It follows from rlaton (46) d τ = a t d t (49) Thrfor, th nrgy of th mttd photon s dt E ~ g44 ~ a( t ) E dτ Th nrgy of th photon movng to th obsrvr n th unvrs s constant by vrtu of (47), by th constant lght vlocty Thn, th corrspondng rcvd frquncy s ν = a( t ) ν (5) whr ν s th frquncy mttd at th obsrvr from th sam atom Th rd shft s gvn by z = ν ν = a( t ) (5) Lght mttd at dstanc r at tm t and rcvd at r = at tm t has by th constant vlocty of lght th rlaton r = c( t t ) Ths gvs by Taylor xpanson of a( t ) n rlaton (5) r d a( t ) r z = H + H c H dt c Dffrntaton of quaton (48) ylds by nglctng small xprssons Ths gvs th rd shft formula Ω m +ΩΛ d a t H dt r 3 r = + Ωm z H H c 4 Th dtald calculatons of Formula (5) can b found n th book [] Hghr ordr Taylor xpanson gvs hghr ordr rd shft approxmatons Th rd shft s alrady drvd n th artcl [] wthout Dopplr Effct but c (5) 869

9 W Ptry only by gravtaton 4 Gnral Rlatvty Th thory of gnral rlatvty as wll as th rsultng cosmologcal modls s wll-known Astronomcal obsrvatons show that th unvrs s flat Thrfor, only flat spac of gnral thory s statd Th curvatur of th unvrs must b zro by th cosmologcal prncpl Ths mans that th sum of th dnsty paramtrs s qual to on Th strong gravtatonal fld n th nghbourhood of th sngularty mpls a hgh curvatur whch contradcts to a flat unvrs, wth curvatur zro Ths problm s solvd partly by th ntroducton of an nflatonary xpanson Hnc, thr gnral rlatvty s not corrct or th cosmologcal prncpl s not vald 5 Fld thory of Gravtaton GFST s a fld thory whch dscrbs gravtaton as a fld n flat spac-tm Th thory s covarant and t s studd n th book [], n th ctd rfrncs thr n and n th artcls [] [3] and [4] Ths thory gvs for wak gravtatonal flds to th lowst ordr of accuracy (masurabl accuracy) th sam rsults as gnral rlatvty But thr ar dffrncs to gnral rlatvty for strong gravtatonal flds, g for th unvrs n th bgnnng Th sourc of th fld quatons of gravtaton s th total nrgy-momntum ncludng that of gravtatonal fld whch s a tnsor for ths thory Th unvrs starts wthout mattr n th bgnnng and conssts only of (gravtatonal) nrgy In th cours of tm mattr and radaton ar cratd whr th total nrgy s consrvd Sngularts don t xst undr th assumpton that th sum of th dnsty paramtrs s gratr than on (at last a lttl bt gratr whch s subsquntly assumd) Hnc, thr s no bg bang Modls wth and wthout cosmologcal constant ar studd n th book [] By th us of th psudo-eucldan gomtryas mtrc th soluton ylds a non-xpandng unvrs Th rd shft of dstant objcts n a non-xpandng unvrs was alrady gvn n artcl [] It s worth to mnton that by vrtu of th covaranc of th thory th non-sngular rsults can b ntrprtd n a non-xpandng and n an xpandng spac Th spac of th thory s flat ndpndnt of th dnsty paramtrs Th prsntly assumd dnsty paramtr of mattr s 3 To avod sngular solutons of th cosmologcal modl th dnsty paramtr of th cosmologcal constant must b 7 such that th sum of th two valus s a lttl bt gratr than on Th prsnt dscusson of th unvrs about non-xpandng or xpandng wth acclraton can b solvd by GFST bcaus non-xpandng spac sms to b th natural ntrprtaton but th ntrprtaton as xpandng spac s also possbl GR dmands by vrtu of th pont -sngularty an acclraton of th unvrs Artcl [] contans furthr dffrncs of th two thors A thory of gravtaton n flat spac-tm (GFST) s gvn Th fld s a tnsor of rank whch s dscrbd on a flat spac-tm mtrc, g th psudo-eucld- 87

10 W Ptry an gomtry Th fld quatons hav as sourc th total nrgy-momntum tnsor nclusv that of gravtaton whch s a tnsor Th consrvaton of th total nrgy-momntum tnsor mpls th quatons of moton and rvrs Th thory s gnrally covarant and th rsults of GFST and gnral rlatvty (GR) agr for wak flds to th lowst ordr of approxmaton Homognous, sotropc, cosmologcal modls of GFST ar studd n th psudo-eucldan gomtry Assumng that th sum of th dnsty paramtrs s a lttl bt gratr than on th rsultng cosmologcal modls ar non-sngular In th bgnnng of th unvrs no mattr xsts, all th nrgy s gravtaton In th cours of tm mattr and radaton ar gnratd from gravtatonal nrgy Th total nrgy s consrvd Th spac s flat and non-xpandng Crtan tm aftr th bgnnng th rsults of th two thors hghly agr wth on anothr undr th assumpton that th unvrs s flat Th gnral covaranc of th thory gvs th possblty to ntrprt th rsults n a non-xpandng or n an xpandng unvrs 6 Conclusons GFST s a fld thory lk Elctrodynamcs and GR s gomtry For wak flds, th two thors gv approxmatly th sam rsults undr th assumpton that th unvrs s flat Astrophyscal obsrvatons show that th unvrs s flat Cosmologcal modls of GR mply a sngularty n th bgnnng of th unvrs wth nfnt mattr dnsty (bg bang) Hnc, n th nghbourhood of th sngularty, thr s a hgh curvatur, spac s not flat n th nghbourhood of th sngularty Th cosmologcal prncpl mpls that spac s vrywhr flat Hnc, w gt a contradcton to GR or to th cosmologcal prncpl Th unvrs starts from a pont-sngularty Thrfor, spac must xpand or vn nflatonary xpand by vrtu of th bg obsrvd unvrs GFST s gnrally covarant, th spac can b ntrprtd as non-xpandng or as xpandng Th dnsty paramtr of mattr s at prsnt assumd to b 3 Thrfor, th dnsty paramtr of dark nrgy s 7 wth th assumpton that th sum of th dnsty paramtr s a lttl bt gratr than on to mply non-sngular cosmologcal modls Cosmologcal modls of GR hav a flat spac undr th assumpton that th sum of th dnsty paramtrs s qual to on Thrfor, GR and GFST gv about th sam valus for th dnsty paramtrs But n th bgnnng of th unvrs, th solutons of GR and GFST ar qut dffrnt Thr xsts a sngularty (bg bang) by GR and th soluton of GFST s vrywhr dfnd and rgular, no bang It s worth to mnton that sngularts ar physcally not allowd Rfrncs [] Ptry, W (4) A Thory of Gravtaton n Flat Spac-Tm Scnc PG 4 [] Ptry, W (98) Cosmologcal Modls wthout Sngularts Gnral Rlatvty and Gravtaton, 3, [3] Ptry, W (99) Nonsngular Cosmologcal Modl wth Mattr Craton and En- 87

11 W Ptry tropy Producton Gnral Rlatvty and Gravtaton,, [4] Ptry, W (997) On th Hubbl Law n a Nonxpandng Nonstatonary Unvrs wth Cosmologcal Constant Astrophyscs and Spac Scnc, 54, [5] Ptry, W (3) Cosmology wth Bounc by Flat Spac-Tm Thory of Gravtaton and a Nw Intrprtaton Journal of Modrn Physcs, 4, -5 [6] Ptry, W (4) Gravtaton n Flat Spac-Tm and Gnral Rlatvty Journal of Appld Mathmatcs Physcs,, 5-54 [7] Lrnr, E (5) Evdnc for a Non-Expandng Unvrs: Surfac Brghtnss Data from HUDF AIP Confrnc Procdngs, 8, 6 arxv: astro-ph/596 [8] Ptry, W (3) Modfd Hubbl Law Physcs Essays, 6, [9] Ptry, W (5) Craton of a Non-Expandng, Non-Sngular Unvrs Journal of Modrn Physcs, 6, [] Ptry, W (6) Comparng Gravtaton n Flat Spac-Tm wth Gnral Rlatvty Journal of Modrn Physcs, 7, [] Ptry, W (7) Is th Unvrs Rally Expandng? arxv: Submt or rcommnd nxt manuscrpt to SCIRP and w wll provd bst srvc for you: Accptng pr-submsson nqurs through Emal, Facbook, LnkdIn, Twttr, tc A wd slcton of journals (nclusv of 9 subjcts, mor than journals) Provdng 4-hour hgh-qualty srvc Usr-frndly onln submsson systm Far and swft pr-rvw systm Effcnt typsttng and proofradng procdur Dsplay of th rsult of downloads and vsts, as wll as th numbr of ctd artcls Maxmum dssmnaton of your rsarch work Submt your manuscrpt at: Or contact jamp@scrporg 87