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A t ree-dime si a ua tum a uum de sate is i tr du ed as a u dame ta medium r m w i ra it emer es i a e metr - dr d ami imit I t is a r a , t e ur ature s a e-time ara teristi e era re ati it is btai ed as a ei ue a m re u dame t turiab e e er de sit ua tum a uum w i as a rete si a mea i T e u tuati s t e ua tum a uum e er de sit su est a i teresti s uti r t e dark e e b em Keywords: ur ature a uum, dark e er s a e, e era re ati it , e er de sit ua tum PACS numbers: 04 ; 04 20- ; 04 50 Kd; 04 0 -m D. ISCALETTI, A. SORLI 1. INTRODUCTION The 20th ce tury the retica physics br u ht the ide u i ied ua tum acuum as a u dame ta medium subte di the bser ab e rms ter, e er y a d space-time. The ti a empty space de id a y physica pr rties has bee rep aced with that a ua tum acuum state, de i ed t be the r u d ( west e er y de sity state a c ecti ua tum ie ds. A cu iar a d tru y ua tum mecha ica eature the ua tum ie ds the acuum is that they e hibit er -p i t uctuati s e ery i space, e e i re i s which are de id ter a d radiati . These er -p i t uctuati s the ua tum ie ds, as we as ther acuum phe me a ua tum ie d the ry, i e rise t a e rm us acuum e er y de sity. The e iste ce a physiccuum ca be c sidered as the m st imp rta t c se ue ce c temp rary ua tum ie d the ries, such as the ua tum e ectr dy amics, the ei ber -Sm- ash w the ry e ectr weak i teracti s a d the ua tum chr m dy amics str i teracti s. These ua tum ie d the ries imp y that ari us c tributi s t the acuum e er y de sity e ist: the uctuati s characteri i the er -p i t ie d, the uctuati s characteri i the ua tum chr m dy amic e e sub uc ear physics, the uctuati s i ked with the i s ie d, as we as rhaps ther c tributi s r m p ssib e e isti s urces utside the Sta dard M de ( r i sta ce, ra d U i ied The ries, stri the ries, etc. . O the ther ha d, there is structure withi the Sta dard M de which su ests re ati s betwee the di ere t c tributi s t the ua tum acuum e er y de sity, a d it is there re cust mary t assume that the t tcuum e er y de sity is, at east, as ar e as a y these i di idua c tributi s. As re ards the r e the di ere t c tributi s t the acuum e ery de sity, the reader ca i d a detai ed ysis, r e amp e, i the par 1 by Ru h a d Zi ker a e , wh studied the c ecti betwee the acuum c cept i ua tum ie d the ry a d the c ceptua ri i the c sm ica c sta t pr b em, a d i the par 2 by Timashe , wh e ami ed the p ssibi ity c sideri the physiccuum as a u i ied system er i the pr cesses taki p ace i micr physics a d macr physics, which ma i ests itse a space-time sca es, r m sub uc ear t c sm ica . The rea istic c cept the acuum ca be c sidered as the u tie isiti card which c mp etes a d c mp eme ts Ei stei s the ry re ati ity. Re ati ity the ry iews space-time as a re ati e a d dy amic ma i d, i teracti with ter a d e er y. It is the back r u d i st which the e e ts the ma i est w r d u d. But the ri i s this back r u d are t acc u ted r i re ati ity the ry: space-time is simp y i e t ether with ter a d e er y. I e era re ati ity, the sta dard i terpretati phe me a i ra itati a ie ds is i terms a u dame ta y cur ed space-time. we er, SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y this appr ach eads t we -k w pr b ems i e aims t i d a u i yi picture which takes i t acc u t s me basic ascts the ua tum the ry. I rder t esca this situati impasse, se eruth rs ad cated thus a ter ati e ways i rder t treat ra itati a i teracti , i which the space-time ma i d ca be c sidered as a emer e ce the deest pr cesses situated at the u dame ta e e ua tum ra ity. I this re ard, the ermi a pr p sa Sachar was deduci ra itati as a metric e asticity space, which c sists i a e era i ed rce pp si the cur i space 3 (the reader c s see the re ere ce 4 r a re iew this c cept . Sachar s m de starts r m the i terpretati the acti space-time as the e ect ua tum uctuati s the acuum i a cur ed space. Other i teresti appr aches are aisch s a d Rueda s m de 5 , re ardi the i terpretati i ertia mass a d ra itati a mass as e ects a e ectr ma etic ua tum acuum, Puth s p ari ab e acuum m de ra itati a d, m re rece t y, a m de de e d by C s i based u tra-weak e citati s i a c de sed ma i d i rder t describe ra itati a d i s mecha ism -9 . U der the c structi a these m de s there is pr bab y e u der yi u dame ta bser ati : as i ht i Euc id space de iates r m a strai ht i e i a medium with ariab e de sity, a e ecti e cur ature mi ht ri i ate, u der pp rtu e c diti s, r m the same physict-space acuum. I this par, by wi the phi s phy that is at the basis these appr aches, we su est a m de a three-dime si a (3D ua tum acuum i which e era re ati ity emer es as the hydr dy amic imit s me u der yi the ry a m re u dame ta micr sc pic structure space-time. Acc rdi t this m de , the cur ature space-time characteristic e era re ati ity ca be c sidered as a heic ue a m re u dame tctua e er y de sity ua tum acuum which has a c crete physica mea i . I the uter i ter aactic space, ame y i the abse ce eria b ects, the e er y de sity the 3D ua tum acuum is de i ed by the wi re ati : mp c 2 l p3 (1 th. The mP is P a ck s mass, c is the i ht sed a d l p is P a ck s e ua tity (1 is the ma imum a ue the ua tum acuum e er y de sity a d physica y c rresp ds t the t t era e umetric e er y de sity, wed t a the re ue cy m des p ssib e withi the isib e si e the u i erse, e pressed by c14 2 G4 4,641266 10113 J / m3 109 Kg / m3 , (2 D. ISCALETTI, A. SORLI bei P a ck s reduced c sta t, G the u i ersa ra itati c sta t. I the uter i ter ctic space cur ature space is er a d its e er y de sity c rresp ds t the a ue (2 . O the ther ha d, i the picture Rueda s a d aisch s i terpretati the i ertia mass as a e ect the e ectr ma etic ua tum acuum 5 , the prese ce a partic e with a ume V0 e s r m the acuum e er y withi this ume e act y the same am u t e er y as is the partic e s i ter a e er y (e ui a e t t its rest mass . O the basis Rueda s a d aisch s resu ts, here we assume that each e eme tary partic e is ass ciated with uctuati s the ua tum acuum which determi e a dimi ishi the ua tum acuum e er y de sity. There re, e ca say that i the prese ce a eria b ect the cur ature space i creases a d c rresp ds physica y t a m re u dame ta dimi ishi the e er y de sity the ua tum acuum, which, i the ce tre the eria b ect, is i e by re ati m , V qvE (3 m a d V bei the mass a d ume the b ect 10 . ere, we pr p se that the ua tum acuum e er y de sity is the u dame ta , u tie physica rea ity characteri i the ra itati a space. The physica pr rty mass is c sidered as a sec dary t ica y rea ity with resct t the e er y de sity ua tum acuum: the de sity a i e eria b ect is pr duced by a cha e the ua tum acuum e er y de sity the basis e uati qvE (4 qvE qvE This par is structured i the wi ma er. I chapter 2 we wi e p re i what se se e era re ati ity ca be see as the hydr dy amic imit a u der yi ua tum acuum c de sate ha i ua ti ed eatures. I chapter 3 we wi sh w h w the cha es the e er y de sity the 3D ua tum acuum i e rise t the cur ature space-time characteristic e era re ati ity a d which is simi ar t the cur ature pr duced by a dark e er y de sity. i a y, i chapter 4 we wi yse the m ti a eria b ect i the cur ed space determi ed by the cha es a d uctuati s the ua tum acuum e ery de sity. SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y 2. SPACE AS T E EOMETRO- YDRODYNAMIC LIMIT O A 3D QUANTUM UUM CONDENSATE AVIN A DISCRETE NATURE Taki acc u t Sachar S R s assumpti 1 dx 16 G that the acti gR , spacetime (5 R is the i aria t Ricci te s r, is iewed as a cha e i the acti ua tum uctuati s acuum i a cur ed space a d c sideri the c siste t hist ries appr ach ua tum mecha ics 11-13 , acc rdi t which the ua tum e uti ca be see as the c here t surp siti irtua i e rai ed hist ries, e era re ati ity ca be i terpreted as the hydr dy amic imit a u der yi the ry micr sc pic structure space, m re precise y a 3D ua tum acuum c de sate wh se m st u i ersa physica pr rty is its e er y de sity. A i e- rai ed hist ry ca be de i ed by the a ue a ie d x at the p i t x a d has ua tum amp itude eiS , S is the c assiccti c rresp di t the c sidered hist ry. The ua tum i ter ere ce betwee tw irtua hist ries d B ca be ua ti ied by a dec here ce u cti a : * B A B i S (6 that i es the c arse- rai ed hist ries c rresp di t the bser ati s i c assica w r d. The ua tum amp itude r a c arse- rai ed hist ry is the de i ed by: eiS ca be c sidered as a i ter u cti that se ects which i erai ed hist ries are ass ciated t the same surp siti with theie ati e phases. The dec here ce u cti a r a c up e c arse- rai ed hist ries is the : i S * A B i which the hist ries d B assume the same a ue at a i e time i sta t the uture, dec here ce i dicates that the di ere t hist ries c tributi t the u ua tum e uti ca e ist i di idua y, are characteri ed by ua tum amp itude a d that the system u der es a i ri a d D. ISCALETTI, A. SORLI predictabi ity de radati 13 (i this se se the system bec mes st chastic a d dissipati e . By app yi the rma ism ( t hydr dy amics ariab es 14 , Ei stei s stress-e er y te s r ca be e pressed thr u h the wi rat r: xA xB . (9 I e uati (9 is a e eric ie d rat r de i ed at tw p i ts that eads t the c ser ati aw : ; (10 mea i that the dec hered ua tities, sh wi a c assica beha i r, are the l m c ser ed es. It ca be sh w that, r cti , the S l lm wi re ati h ds ^ ^ T A , T^BA ^ B K n x A , x B D T ,T D K x ,x F F n A l i Kn Kn n m lm iK m n lm iK n T n T n ^ A x A , x B ,T B Bx A , x B A i T , T (11) (11 i which we ha e used the i te ra represe tati de t d the CTP i dices bei the c sedtime path tw -partic e irreducib e acti . l.m.n 1, 2 , The c ser ati imp ies that the dec here ce u cti a has ma i- mum a ues i c rresp de ce the hydr dy amic ariab es , p that, i tur , are the m st readi y dec hered a d ha e the hi hest pr babi ity t bec me c assica . By app yi the ab e pr cedure t Ei stei s te s r G a y ^ emer es betwee the c ser ati aw r T a d the Bia chi ide tity G ; 0 which imp ies the dec here ce a d the emer e ce the hydr dy amics ariab es the e metry. I this se se e era re ati ity ca be c sidered as e metr hydr dy amics a d the m st readi y dec hered ariab es are th se ass ciated t the ar est i ertia represe ti the c ecti e ariab es e metry. I e era re ati ity must be re arded as a e metr -hydr dy amic imit a u der yi micr sc pic back r u d e has c ecti e ariab es, a d the aws er i macr -c assica space-time are e pressed i terms c ecti e ariab es, the precise characteri ati this u der yi back r u d a d thus the ua ti ati the e era -re ati istic metric r the c ecti ariab es wi y resu t i the disc ery the e citati s i the e metry a d t SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y its ua tum micr -structures. I we c sider the c ecti e hydr dy amics a d p apari i the stress-e er y te s r T , the the ua ti aariab es ti has se se whe med the ie d u cti x r m which they are c structed a d t a d p themse es. The situati is simi ar t that re ardi c de sed ter physics i which the ua ti ati c ecti e e citati s eads t ph s a d t t the at mic structure ter. I the iew e era re ati ity as e metr -hydr dy amic imit a u der yi back r u d, there is there re a imp rta t y betwee ua tum t c assica tra siti ra ity a d the beha i r c de sed ter. M re er, i e the c ecti e ariab es (the metric a d the c ecti s i e era re ati ity), h w ca we characteri e the micr sc pic structure the u der yi back r u d, ame y what ca we say ab ut the ua tum micr -structure r m which the c ecti e ariab es deri e I this re ard, a p ssib e strate y is starti r m a suitab e the ry ua tum micr sc pic structure a d studyi its pre isi s i the wa e e th w e er y imit. ppr ach this ki d has bee rece t y su ested, r e amp e, by C s i -9 , wh has i tr duced a physiccuum i te ded as a sur uid medium a B se c de sate e eme tary spi ess ua ta wh se -ra e uctuati s, a c arse- rai ed sca e, resemb e the Newt ia p te tia , yie di the irst appr ii t the metric structure c assica e era re ati ity. I y with C s i s m de , taki i t c siderati the -wa e e th m des, here ra ity is i duced by the u der yi ie d the acuum. I weak x which describes the de sity uctuati s ra itati a ie ds, a c arse- rai ed sca e, the u der yi ide ti ied with the Newt ia p te tia UN GN Mi , i ie d x ca be (12) ame y x . (13) A i teresti ar ume t which a ws us t characteri e the ua tum micr sc pic structure the u der yi back r u d e erati ra ity ca be deri ed r m the ua tum u certai ty pri cip e 15 a d r m the hyp theses spacetime discrete ess at the P a ck sca e. I particu ar, i re ard t the ra u arity space-time a d its i k with ra ity, i the pars 16-19 N sh wed that the ua tum uctuati s space-time ma i est themse es i the rm u certai ties i the e metry space-time a d thus the structure the space-time am ca be i erred r m the accuracy with which we ca measure its e metry. By c sideri a mappi the e metry space-time r a spherica D. ISCALETTI, A. SORLI ume radius l er the am u t time T 2l / c it takes i ht t cr ss the ume, i N s appr ach the a era e separati betwee ei hb uri ce s space c rresp ds t the a era e mi imum u certai ty, a d thus t the accuracy i the measureme t a dista ce l, i e by l 2 l lP2/ 3 . (14) A i teresti asct N s ua tum am m de ies i its h raphic eatures i the se se that here, dr ppi the mu tip icati e act der 1, 2 a spatia re i si e l ca c tai m re tha l 3 / llP2 l / lP ce s a d thus a ma imum umber bits i ri l / lP i a reeme t with the h raphic pri cip e 20-25 which imp ies that, a th u h the w r d ar u d us apars t ha e three spatia dime si s, its c te ts cctua y be e c ded a tw -dime si a sur ace, ike a h ram. By app yi the discrete ess hyp thesis N s m de , ame y the act that we ca t make x sma er tha the e eme tary e th (14) 1: t eise ber s u certai ty re ati 2/ l lP 3 (15) r the p siti x a d m me tum p (16) 2 p e btai s that, i p i creases, the e pressi p x as a u cti must c tai a term direct y pr p rti a t p that c u terb ces the term pr p rti a t i p , is: . By wi 26 , a p ssib e ch ice, at the irst rder p 2 p 2 2/ 3 l 2/ 3lP4/ 3 i which the act r i the sec d term the ri ht ha d side is se ected by mea s dime si r ume ts. The e pressi ca be iewed as the e era i ed ersi the u certai ty pri cip e i a discrete space-time. a us imitati h ds i time. SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y By a simi aeas i time u certai ty as: e ca btai the c rresp di ersi 2 E ET02 , 2 1 1 2 2 / 3 l 1lP2 is the e eme c tary time. I the appr ach pr p sed i this artic e, the ew terms apari i e uati s a d ha e a ery scia mea i : they represe t the i tri sic u certai ty space-time due t the prese ce a partic e a i e e er ym me tum deri i r m pp rtu e cha es the ua tum acuum e er y de sity . Thus, the prese ce ter de sity (4) qvE qvE E is the e er y u certai ty a d T0 m di ies the e metry space-time. I act, the e er y E pc c tai ed i a r m ter de sity (4) m di ies the e te re i si e L a d deri i si this re i m u t: L 2 l l p2/ 3T0 E (19) O the basis e uati (19), the cur ature space-time ca be re ated t the prese ce e er y a d m me tum i it. I ther w rds, i the appr ach here su ested, e ca say that the cha es the ua tum acuum e er y de sity ass ciated with the prese ce ter de sity (4) c rresp d t a u der yi micr sc pic back r u d e metry dei ed by e uati (19). M re er, taki i t acc u t that i N s m de the h raphic space-time am de i ed by e uati (14) ca be re ated t the c smic sca e i the a era e mi imum u certai ty (14) c rresp ds t a ma imum e er y de sity r a sphere radius l that d es tc llP (20) apse i t a b ack h e, ame y RH lP (21) RH is the ubb e radius (which is the critica c smic e er y de sity as bser ed), he ce deri es that the terms i e uati s a d represe ti the i tri sic u certai ty space-time due t the cha es the ua tum acuum e er y de sity ca be themse es re ated t the c smic sca e. I par- D. ISCALETTI, A. SORLI ticu ar, by taki acc u t e uati (21), e uati m ica sca e rescti e y bec me x p 2 p 2 2 s a d at the c s- 2 RH 2lP4 , (22) (23) 2 E ET02 , 2 1 1 2 2 / 3 RH 1lP2 . i a y, e uac ti (19) describi the i k betwee the u der yi micr sc pic structure space-time a d the cur ature space-time, at the c sm ica e e may be e pressed as E is the e er y u certai ty a d T0 RH l p2/ 3T0 E (24) N w, a ter sh wi h w the ua tum micr sc pic structure the u der yi back r u d e erati ra ity ca be characteri ed a d the imp rta t i k this micr sc pic structure with the c smic sca e, the e t u dame ta step is t make e p icit the r e the ua tum acuum e er y de sities i e by e uati s (1) a d (3) (i particu ar, i rder t deri e the critica c smic e er y de sity (21) as bser ed). 3. T E C AN ES O T E 3D QUANTUM UUM ENER Y DENSITY AS T E ORI IN O T E CURVATURE O SPACE-TIME The P a ck e er y de sity (2) is usua y c sidered as the ri i the dark e er y a d thus a c sm ica c sta t, i the dark e er y is supp sed t be wed t a i terp ay betwee ua tum mecha ics a d ra ity. we er, the bser ati s are c mpatib e with a dark e er y DE 10 26 Kg / m3 (25) a d thus e uati s (2) a d (25) i e rise t the s -ca ed c sm ica c sta t pr b em because the dark e er y (25) is 123 rders ma itude wer tha (2). I rder t s e this pr b em, a i teresti e p ti r the actu ue (25) which i kes the uctuati s the ua tum acuum has re- SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y ce t y bee su ested by Sa t s 2 -29 . Acc rdi t this appr ach, ua tum acuum uctuati s determi e a cur ature space-time a d, u der p ausib e hyp theses withi ua ti ed ra ity, a re ati betwee the tw -p i t c rre ati u cti the acuum uctuati s a d the space-time cur ature was btai ed. The ua tum acuum uctuati s ca be ass ciated with a cur ature space-time simi ar t the cur ature pr duced by a dark e er y de sity, the basis the e uati DE 70G 0 C s sds (26) which states that the p ssib e a ue the dark e er y de sity is the pr duct Newt c sta t, G, times the i te ra the tw -p i t c rre ati u cti the acuum uctuati s de i ed by C r1 r2 1 ^ r1 , t ^ r2 , t 2 ^ r2 , t ^ r1 , t , (27) a e er y de sity rat r such that its acuum e ctati is er whi e the acuum e ctati the s uare it is t er . The c rre ati u cti (27) determi es a s the ra itati a e er y ass ciated with the acuum uctuati s acc rdi t the e uati grav ^ bei 4 G C r12 r12 dr12 . (2 ) rmu a 11 , (29) M re er, dime si a ysis eads t Ze d ich s Gm 2 1 , rC rC3 DE es a parameter, m, with di( rC / mc bei C mpt s radius) which i me si s a mass. I i Ze d ich s ri i a m de , e uati (29) repr duces the bser ed a ue the dark e er y de sity r a mass m 7,6 10 29 Kg that is ab ut 1/20 times the pr t mass r ab ut 0 times the e ectr mass, Sa t s appr ach d es t a w t deri e the a ue m, but i side his appr ach it is p ausib e t assume that acuum uctuati s hi h e er y, i i ery massi e partic es, w u d t be pr bab e. ere, ur aim is t sh w that the cur ature space-time ass ciated with a dark e er y de sity ca be i terpreted as a c se ue ce m re u dame ta cha es the 3D ua tum acuum e er y de sity , i ther qvE qvE w rds it ca be physica y de i ed as the heic ue the 3D ua tum acuum e er y de sity (wh se u der yi micr sc pic structure is characteri ed D. ISCALETTI, A. SORLI by a e metry e pressed by e uati s (17)-(19) a d by e uati s (22-(24) at the c sm ica e e ). I this re ard, be re a , we assume that the e ctati a ue the stress-e er y te s r rat r the ua tum ie ds (9) at a y p i t i es the ter e er y ass ciated with the ter (bary ic p us dark) e er y de sity, which is determi ed by cha es a d uctuati s the 3D ua tum acuum e er y de sity, with ut a y additi a c tributi t the acuum. This assumpti a ws us t btai the c rrect riedma -R berts - a ker metric ds 2 dt 2 a t dr 2 r 2d (30) i which the recessi the dista t ies ca be ca cu ated i terms the i k the measurab e ubb e c sta t a d the dece erati parameter with ew time a d radia c rthe time-de de t parameter a t ), by i tr duci di ates r ' a d t ' i e by re ati s By i serti r' a t' O r '3 , t t' r '2 da t ' 2a t ' dt ' O r '4 . (31) (31) i t (30) e btai s the e uati ds 1 1 G 3 r ' dr ' r' d 1 r '2 dt '2 (32) DE r '2 dr '2 r '2 d e uati G 3 G 1 3 2 DE r '2 dt '2 , the riedma DE G 1 3 2 (33) DE ha e bee take i t acc u t i the sec d e ua ity, a a t' , a da t ' , dt ' d 2a t ' , is the de sity ter i e by e uati (4), DE is the dt '2 de sity wed t a p ssib e e iste ce dark ter. I re ere ce t e uati (32), the assumpti that the e ctati a ue the stress-e er y te s r rat r the ua tum ie ds (9) i es the ter stress-e er y de sity determi ed by uctuati s the 3D ua tum acuum e er y de sity, mea s that a SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y 44 qvE 00, (34) bei the ua tum state the u i erse c rresp di t the a ue the ie d x de i i a i e i e- rai ed hist ry. This su ests t e press the stress-e er y te s r (9) c rresp di t the ua tum acuum uctuati s as ^ I, (35) ^ ua tum acuum uctua I is the ide tity rat r. The e iste ce ^ is er by de i iti , e has ti s imp y that, despite the e ctati T ^ x T y (36) e era . N w, i rder t deri e e uati (32), taki i t acc u t Sa t s resu ts, i side ur m de it is reas ab e t assume that the u der yi ua tum acuum c de sate ca be characteri ed by c sideri the metric the ua tum acuum de i ed by re ati ^ ds 2 the c e icie ts (i p ar c ^ g dx dx , rdi ates) are (37) ^ g 00 ^ g 00 ^ ^ 1 h00 , g11 1 h11 , g 22 ^ ^ 1 00 , g11 2 ^ ^ ^ ^ 1 h11 , g11 111 , g 33 2 r 2 si 2 ^ ^ g ^ r 1 h22 , g 33 r sin ^ ^ 1 h33 , g ^ 1 h33 ,^ for (38) (3 ) mu tip icati e ery term times the u it rat r is imp icit a d, at the rder O r 2 , e has ^ 0 e cept h00 G 3 G 3 qvE DE r2 a d (39) 11 ^ h qvE DE 1 qvE 2 r , 2 sta ds qvE (a d the uctuati d t a u der yi the ua tum acuum c rresp micr sc pic e metry de i ed D. ISCALETTI, A. SORLI by e uations (17), (18) and (19)). In irtue of the uanti ed eometry defined by e uations (17), (18) and (19), the metric (37), at a fundamenta e e , has to be considered as a uanti ed metric. As re ards the uanti ed metric (37), it is important to remark that in the approach de e od by Santos in 28 , by writin the uantum coefficients of the metric as (38), ^ 0 e cept h00 ^ and h11 DE 8 G 3 1 r2 , 2 DE r2 (39a) , in the appro iion of the second order ^ h stands for ^ in the (sma ) tensor h , it is possib e to deri e the components of uantum Einstein e uations of the form ^ G 8 G . c4 (40) is e pressed ^ In Santos approach, the uantum Einstein tensor orator G ^ in terms of the orators h , by reso in these (non- inear coup ed partia ) differentia orator e uations (40) in order to obtain the uantum metric coeffi^ cients g in terms of inte ra s in o in the stress-ener y tensor orator and ^ fina y ca cu atin the e ctations of the metric coefficients g in terms of inte ra s in o in the e ctations of the stress-ener y tensor orator. The reader can find detai s of these ca cu ations, for e amp e, in the abo e reference 28 and in 29 . ere, we under ine that, in ana o y with Santos resu ts, due to the fact that the re ations between the metric coefficients and the ter stress-ener y tensor are non- inear, the e ctation of the uanti ed metric (37) of the acuum condensate is not the same as the metric of the e ctation of the ter tensor (9). The difference i es rise to a contribution of the acuum f uctuations which reproduces the effect of a cosmo o ica constant. Moreo er, we wi assume that, ^ when the distance r , one has g , is the Minkowski metric. By startin from the uanti ed metric (37) whose coefficients are defined by re ations (38) and (39), one can obtain the components of the uantum Einstein e uations (40) on the basis of the assumption that they are simi ar to the c assica SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y counterparts. In particu ar, the e ctation a ue of the (orator) metric pa^ rameter h11 may be written in the form 11 11 11 (41) name y it is the sum of two e pressions, one containin the ter density produced by the chan es of the uantum acuum ener y density, and the other indicatin the acuum density f uctuations, . In e uation (41), by mode in the ter density of the uni erse by means of a constant, the ter term can be e pressed as 11 M 2GM r 2G 2 M 2 , r2 (42) r3 qvE r 3 , which a rees with the second order e - pansion of the we -known Schwar schi d so ution g11 2GM 1 r (43) Moreo er, takin into account e uation (3), here the dark ener y density DE can be associated with opportune f uctuations of the 3D uantum DE qvE acuum ener y density defined by re ation DE qvE mDE c 2 , V DE (44) in the o ume V mDE bein the mass correspondin to the dark ener y and thus DE qvE DE (45) In this way, takin into account that accordin to Santos resu ts, the acuum contribution aparin in e uation (41), to order G 2 , is 11 600G 2 r 2 0 C s sds , (46) D. ISCALETTI, A. SORLI r bein a distance which is estied to fu fi r / s 1040 , in our mode the acuum contribution may be e pressed as 11 150 G r V 1 1 , l l3 (47) (48) V DE qvE and, takin into account e uation (26), Santos inte ra of the two-point correation function has been assimi ated to the f uctuations of the uantum acuum ener y density (44) on the basis of e uation V 1 1 l l3 C r12 r12 dr12 . (49) Therefore, the tota e ctation a ue (41) becomes, to orde 2 11 c qvE r 2 150 1 G 2 r 2 V2 1 1 . l l3 (50) ence, a comparison with the riedmann e uations (33), takin account of re ations (26) and (46), eads to the fo owin e uation DE 35G V 2 V 1 1 , l l3 (51) name y DE (52) which states the e ui a ence of the cur ature of space-time produced by the chan es of the uantum acuum ener y density and the one determined by a constant dark ener y density. This means that in the approach based on e uations (37)-(52), the chan es and f uctuations of the uantum acuum ener y density enerate a cur ature of space-time simi ar to the cur ature produced by a dark ener y density. Moreo er, it is interestin to obser e that, whi e in SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y Santos mode , the dark ener y is associated with the two-point corre ation function of the acuum f uctuations (on the basis of e uation (26)), in the approach su ested by the authors of this artic e, the dark ener y is direct y determined by f uctuations of the uantum acuum ener y density on the basis of e uation (52). It must be emphasi ed that here the f uctuations of the uantum acuum ener y density p ay the same ro e of Santos two-point corre ation function. In other words, there is an e ui a ence between the f uctuations of the uantum acuum ener y density and the two-point corre ation function: in the approach here su ested, the f uctuations of the 3D uantum acuum enery density act as a two-point corre ation function on the basis of re ation c4 4 V C s sds. (53) Moreo er, introducin e uation (52) into e uation (39), the e ctation a ues of the coefficients of the uanti ed metric (30) become and ^ 0 e cept h00 qvE qvE r2 (54) 11 2c 2 r2 , name y turn out to dend direct y on the chan es of the uantum acuum ener y density. As a conse uence, one can say that the chan es and f uctuations of the uantum acuum ener y density, throu h the uanti ed metric (37) of the uantum acuum condensate whose coefficients are defined by e uations (38) and (54) (and whose under yin microscopic eometry is described by e uations (17)-(19) and, at the cosmo o ica e e , by e uations (22)-(24)) can be considered the ori in of the cur ature of space-time characteristic of enera re ati ity. In other words, one can say that the cur ature of space-time may be considered as a heic ue which emer es from the uanti ed metric (37) and thus from the chan es and f uctuations of the uantum acuum ener y density (on the basis of e uations (38) and (54)). In synthesis, accordin to the iew su ested in this chapter, the uanti ed metric (37) associated with the chan es and f uctuations of the uantum acuum ener y density, on the basis of e uations (38) and (54), can be considered as the u tie isitin card of enera re ati ity. D. ISCALETTI, A. SORLI 4. ABOUT T E MOTION O A ERIAL OB ECT IN T E CURVED SPACE-TIME Now, et us see how the cur ature of space-time correspondin to the chan es and f uctuations of the uantum acuum ener y density acts on a test partic e of mass m0 , in other words how the motion of a eria ob ect in a back round characteri ed by chan es of its ener y density can be treated heica y. hen a eria ob ect correspondin to a i en diminishin of the uantum acuum ener y density mo es, this diminishin of the uantum acuum ener y density by irtue of its ink with the uantum acuum condensate defined by e uations (54) (and whose under yin microscopic eometry is described by e uations (17)-(19) and, at the cosmo o ica e e , by e uations (22)-(24)) causes a shadowin of the ra itationa space which determines the motion of other eria ob ects present in the re ion under consideration. In the approach here su ested, the shadowin of the ra itationa space determined by riab e density of uantum acuum tries inspiration from the idea of the po ari abi ity of the acuum in the icinity of a mass (or other massener y concentrations) introduced by Puthoff s po ari ab e mode of ra itation 6 . In order to interpret and reproduce the cur ature of space-time Puthoff postu ated the fo owin re ation for the ariab e po ari ation of the acuum caused by the presence of a mass K 0 E, (55) E is the e ectric fie d, K is the (a tered) die ectric constant of the acuum (typica y a function of position) due to ( enera re ati istic-induced) acuum po ari abi ity chan es under consideration. Puthoff s e uation (55) estabishes that the presence of e ectroma netic ener y or massi e ob ects moduates the acuum po ari ation in a inear fashion. The acuum die ectric constant K constitutes the u tie isitin card of Puthoff s mode . Its effects on the arious measurement processes that characteri e enera re ati ity are the fo owin : the e ocity of i ht chan es from c to c/K, the time inter a s chan e from t0 to t0 K (which indicates that for K 1, name y in a ra itationa potentia , the time inter a s between c ock ticks is increased, that is the c ock runs s ower), the en ths of rods chan e from r0 to r0 / K . In Puthoff s mode , the cur ature of space for e amp e in the icinity of a p anet or a star is associated with the effects on measurement processes of en ths and time inter a s that take p ace for K 1. Such an inf uence on the measurin processes due to induced po ari abi ity chan es in the acuum near SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y the body under consideration eads to the enera -re ati istic concept that the presence of a body inf uences the metric . Tryin inspiration from Puthoff s idea of po ari abi ity of the acuum in our mode we assume that the shadowin (po ari ation) of the 3D uantum acuum can be e pressed by the e uation Eg , (56) is a factor which represents the re ati e y smmount of the a tered rmitti ity of the free space (with resct to the situation in which the ener y density of uantum acuum is i en by e uation (1)) and Eg H eg V qvE 35Gc 2 V 2 4 1 ^ 2 (57) can be defined as the ra itostatic fie d determined by both density of ter G and density of dark ener y (here H eg is the basic ra itodynamic con stant) 2. The ra itostatic fie d is inked with the uantum acuum condensate defined by e uations (54) (and whose under yin microscopic eometry is described by e uations (17)-(19) and, at the cosmo o ica e e , by e uations (22)-(24)) throu h re ation Eg 3H egV ^ 1 00 2 r. 8 G r (58) The tot ran ian density for ter-fie d interactions in the po ari ed acuum is i en by re ation Ld m0 c 2 K 1 v c/K qA v r0 2 1 Bg 2 K 0 K 0 Eg 2 K2 1 c/K K t (59) In ana o y with Sacharo s ermina proposa of treatment of ra itation as metric e asticity of space 3 . D. ISCALETTI, A. SORLI , re the ra itationa potentia s, B is the ra itoma netic fie d defined by Bg ( J H eg J , r3 (60) V 35Gc 2 V DE v , S bein the spin qvE qvE 2 4 an u ar momentum of the eria ob ect determined by the diminishin of the c4 . It must uantum acuum ener y density under consideration) and 32 G be emphasi ed that a so the ra itoma netic fie d (60), by irtue of the ink of the orbitn u ar momentum of the eria ob ect determined by the diminishin of the uantum acuum ener y density with the uantum acuum condensate defined by e uations (54) e pressed by L S, L 3V 00 v , 8 G (61) is itse f associated with the uanti ed metric of the uantum acuum condensate. Now, in ana o y with Puthoff s po ari ab e acuum mode of ra itation 6 , ariation of the action functiona with resct to the test partic e ariab es eads to the fo owin e uation of motion of a test eria ob ect of mass m0 in the po ari ed 3D uantum acuum: 1 2 1 v c/ v c/ d dt m0 1 v c/ m0 c 2 Eg v Bg m0 c 2 (62) E uation (62) shows that there are two forces actin onto the test partic e of mass m0 : the Lorent force due to the uantum acuum ener y density surroundin the ob ect and a second term representin the die ectric force proportiona to the radient of the shadowin of uantum acuum (56). The importance of this second term ies in the fact that thanks to it one can account for the ra itationa potentia , either in Newtonian or enera re ati istic form. It mi ht be SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y interestin to note that with m0 0 and v , as wou d be the case for a photon, the def ection of the tra ectory is twice as the def ection of a s ow mo in massi e partic e. This is an important indication of conformity with enera re ati ity. Variation of the action functiona with re ard to the ariab e eads to the e pression of the eneration of the shadowin of the ra itationa space within enera re ati ity, owed to the presence of both ter and fie ds. The e uation has three ri ht-hand side terms: 1 c/ t2 (63) ere P represents the chan e in space shadowin by the mass density associated with the ob ect of mass m0 , with the ecto as the distance from the system mass centre: 1 1 v c/K v c/K P K m0 c 2 K r0 . (64) is the chan e caused by the ener y density of the fie ds (57) and (60) determined by the diminishin of the uantum acuum ener y density: 2 1 Bg 2 0 Eg 2 . (65) itse f: is the chan e caused by the uantum acuum shadowin ener y density 1 c/ (66) In the case of a static ra ity fie d of a spherica mass distribution (a p anet or a star), the so ution of e uation (63) has a simp e e ponentia form: eGM / rc (67) D. ISCALETTI, A. SORLI M V qvE . The so ution (67) can be appro ied by e pandin it into a series: e 2 GM / rc 2GM rc 2 1 2GM 2 rc 2 (68) This so ution reproduces (to the appropriate order) the usua enera re ati istic Schwar schi d metric predictions in the weak fie d imit conditions (i.e. so ar system). Accordin to this mode , it is important to under ine that a so partic es without mass (for e amp e, photons) ha e an indirect inf uence on the uantum acuum ener y density. In fact, because of e uation (65) a so a photon wi add a contribution to the effecti e cur ature of space-time associated with the fie ds (57) and (60). This resu t turns out to be a so in accordance with enera theory of re ati ity, both mass and ener y cause the cur ature of space-time. Moreo er, with the obtained so ution (67) or (68) re ardin the factor measurin the po ari abi ity of the uantum acuum in the presence of ter, one can ana y e the ra itationa red shift characteristic of enera re ati ity, and find inside this approach a more detai ed form in order to obtain the fre uency shift of the photon emitted by an atom on the surface of a star of mass M and radius R. ust ike in Puthoff s mode , the photon detected far away from the star wi apaed shifted by the fo owin amount: GM , Rc 2 (69) GM 1 . The photon, after ha in c imbed up the Rc 2 ra ity potentia of the star, wi retain its ac uired fre uency unchan ed, and the chan e in fre uency can be tested oca y by comparin it with photons emitted by the same ty of atoms at the same temrature, but within the weak ra ity fie d of the aboratory. ith that same resu t it is a so possib e to ana yse the amount of the bendin of i ht rays from a distant star passin near a massi e body, ike in the c assic enera re ati ity test rformed by Eddin ton s e dition durin the so ar ec ipse in May 1919. The i ht ray from a distant star, whi e passin c ose to the Sun, wi e rience a radua s owin of wa efront e ocity comin towards the Sun, and a radua increasin e ocity in ea in the Sun s ra ity fie d. Because increases c oser to a massi e body ( 1 ), the e ocity of i ht wi ary as c / . The part of the wa efront c oser to the Sun wi thus e rience a we ha e assumed SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y reater s owdown than the part of the wa efront passin further away. This is seen from the Earth as an apparent shift of the position of the star c ose to the Sun s disk ed e in the outward direction. In enera re ati ity s terms, this def ection is a measure of oca space-time cur ature. e are interested in ca cu atin the tota bendin an e. Because in case of the Sun the tota def ection is sma ( 2 arc-seconds) we can app y the usua ow an e appro iions throu hout the ca cu ation. And because of the same reason we wi not make a bi mistake if we appro ie the ariab e e ocity of i ht to the first order term of the series e pansion (66) of : v c c 2GM 1 rc 2 c 1 2GM . rc 2 (70) In this re ation the radius- ecto denotes the distance of the wa efront from the centre of the Sun as it tra e s by from to , with the minimum dis R is the Sun s radius, and is the minimum distance tance of R from the Sun s surface. By assi nin z to the distance of the wa efront a on the ), the radius- ector becomes ine of si ht (rndicu ar to R z 2 , so the e uation (70) can be written as: 1 R c 1 (71) The differentia e ocity of i ht, assumin R , is then (72) As the wa efront tra e s a distance dz vdt , the differentia e ocity a on the path of i ht resu ts in an accumu ated wa efront path difference z : vdt z2 dz. (73) This resu ts in an accumu ated ti t an e of: z/ dz. (74) D. ISCALETTI, A. SORLI By inte ratin e uation (74) o er the entire path yie ds: (75) 4 GM . Rc 2 By insertin G 6,672 10 11 Nm 2 Kg 2 , 1,9891 1030 Kg , and R 6,96 10 m , we obtain 1,75 arc-seconds, which is e act y the a ue predicted by Einstein s enera theory of re ati ity in 1915, and e rimenta y erified by Eddin ton in 1919 (between 1.2 and 1.9 arc-seconds, main y because of the imrfect optics of the portab e te escos used). Moreo er, as re ards the e uations of motion (62) and (63), it is important to emphasi e that, accordin to this approach, the modification of the uantum acuum ener y density determinin both the ter density and dark ener y density and the action of the shadowed uantum acuum on another eria ob ect are phenomena direct y determined by the fie ds (60), (64), (65) and (66). This imp ies that no time is needed to transmit the inforion from a eria ob ect to the surroundin re ion in order to shadow the ra itationa space because the chan e of the uantum acuum ener y density is a ready there as it is associated with the fie ds (60), (64), (65) and (66) (what proptes from point to point is ust the actua effects of this chan e); and, on the other hand, that no time is needed to transmit the inforion from the shadowed ra itationa space to another eria ob ect in order to cause its mo ement. ina y, accordin to the iew proposed here, the 3D uantum acuum as a direct medium for the transmission of ra itation estab ished by e uations (64), (65) and (66) can e press in an e e ant heica way the rscti e about the non-e istence of ra itationa wa es. In this re ard, it seems compatib e with some Loin er s resu ts accordin to which ra itationa wa es are on y hypotheticnd do not e ist in the physica wor d 30, 31 . On the other hand, despite se erttempts of research about the ra itationa fie d rformed since the 1960s (see for e amp e the reference 32 ), ra itationa wa es ha e not yet been detected. As under ined by Schorn in the par 33 , To search foa itationa wa es in boratory, c assica or uantum mechanica detectors can be used. Despite the e riments of eber (1960 and 1969) and many others (Bra inski et a ., 1972; Dre er et a ., 1973; Le ine and arwin, 1973; Tyson, 1973; Maischber er et a ., 1991; Abramo ici et a ., 1992; and Abramo ici et a ., 1996) and theoretica ca cu ations and estiions (Bra inski and Rudenko, 1970; arry et a ., 1996; and Schut , 1997), ra itationa wa es ha e ne er been obser ed direct y in aboratory . It is a so interestin to obser e that recent NASA research confirms that uniersa space is f at with on y a 0.4 mar in of error which is a stron indication that cur ature of space in enera theory of re ati ity is on y a heica description of ener y density of uni ersa space which ori inates in a more fun- SPACE-TIME CURVATURE O ENERAL RELATIVITY AND ENER Y damenta ener y density of uantum acuum 34 . NASA measurements re ardin the eometry of uni ersa space turn out to be comp ete y in a reement with the approach de e od in this par. 5. CONCLUSIONS A mode of a three-dimensiona uantum acuum has been proposed in which the cur ature of space-time emer es, in the hydrodynamic imit of some under yin theory of a microscopic structure of space-time, as a heic ue of a more fundamentctua ener y density of uantum acuum. The f uctuations of the uantum acuum ener y density enerate a cur ature of space-time simi ar to the cur ature produced by a dark ener y density and produce a shadowin of the ra itationa space which determines the motion of other eria ob ects present in the re ion under consideration. In this approach, the interestin rscti e is oned that the three-dimensiona uantum acuum acts as a direct medium of ra itation: at a fundamenta e e , no time (as duration) is needed to transmit ra ity force. A i en eria ob ect diminishes ener y density of uantum acuum which enerates cur ature of space-time. ra ity does not act direct y between massi e ob ects, ra ity acts in the uantum acuum: the chan es of the uantum acuum ener y density cause cur ature of space-time which enerate ra itationttraction between massi e bodies. This iew does not re uire e istence of hypothetica ra iton as a carrier of ra ity. RE ERENCES 1. 2. 3. 4. 5. 6. 7. 8. Ru h S. E. and Zinkerna e . 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Annales UMCS, Physica – de Gruyter
Published: Mar 1, 2015
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