A N N A L E S
U N I V E R S I T A T I S M A R I A E C U R I E - S K à 2 ' 2 : S K A
L U B L I N – P 2 L 2 N I A
V2L L;V SECTI2 AAA 2014
SPACE-TIME CURVATURE 2) *ENERAL RELATIVITY
AN' ENER*Y 'ENSITY 2) A T+REE-'IMENSI2NAL
QUANTUM VACUUM
'aYide )isFaOetti*, Amrit SRrOi
SSaFeLiIe IQstitute, Via RRQFaJOia, 35-104 SaQ LRreQ]R iQ CamSR (PU , ItaO\ sSaFeOiIeiQstitute#JmaiOFRm; sRrOiamrit#JmaiOFRm
*CRrresSRQdiQJ autKRr, e-maiO address sSaFeOiIeiQstitute#JmaiOFRm
ABSTRACT
A tKree-dimeQsiRQaO TuaQtum YaFuum FRQdeQsate is iQtrRduFed as a IuQda-meQtaO medium IrRm wKiFK JraYit\ emerJes iQ a JeRmetrR-K\drRd\QamiF Oim-it IQ tKis aSSrRaFK, tKe FurYature RI sSaFe-time FKaraFteristiF RI JeQeraO reOa-tiYit\ is RbtaiQed as a matKematiFaO YaOue RI a mRre IuQdameQtaO aFtuaO Yaria-bOe eQerJ\ deQsit\ RI TuaQtum YaFuum wKiFK Kas a FRQFrete SK\siFaO meaQiQJ TKe IOuFtuatiRQs RI tKe TuaQtum YaFuum eQerJ\ deQsit\ suJJest aQ iQterestiQJ sROutiRQ IRr tKe dark eQerJ\ SrRbOem
Keywords: FurYature RI sSaFe, JeQeraO reOatiYit\, eQerJ\ deQsit\ RI TuaQtum
YaFuum, dark eQerJ\
1. INTRODUCTION
The 20th ceQtury theRreticaO physics brRuJht the idea RI a uQiIied TuaQtum Yacuum as a IuQdameQtaO medium subteQdiQJ the RbserYabOe IRrms RI matter, eQerJy aQd space-time. The QRtiRQ RI aQ ³empty´ space deYRid RI aQy physicaO prRperties has beeQ repOaced with that RI a TuaQtum Yacuum state, deIiQed tR be the JrRuQd (ORwest eQerJy deQsity state RI a cROOectiRQ RI TuaQtum IieOds. A pecuOiar aQd truOy TuaQtum mechaQicaO Ieature RI the TuaQtum IieOds RI the Yacuum is that they e[hibit ]erR-pRiQt IOuctuatiRQs eYerywhere iQ space, eYeQ iQ reJiRQs which are deYRid RI matter aQd radiatiRQ. These ]erR-pRiQt IOuctuatiRQs RI the TuaQtum IieOds, as weOO as Rther ³Yacuum pheQRmeQa´ RI TuaQtum IieOd theRry, JiYe rise tR aQ eQRrmRus Yacuum eQerJy deQsity.
The e[isteQce RI a physicaO Yacuum caQ be cRQsidered as the mRst impRrtaQt cRQseTueQce RI cRQtempRrary TuaQtum IieOd theRries, such as the TuaQtum eOec-trRdyQamics, the :eiQberJ-SaOam-*OashRw theRry RI eOectrRweak iQteractiRQs aQd the TuaQtum chrRmRdyQamics RI strRQJ iQteractiRQs. These TuaQtum IieOd theRries impOy that YariRus cRQtributiRQs tR the Yacuum eQerJy deQsity e[ist: the IOuctuatiRQs characteri]iQJ the ]erR-pRiQt IieOd, the IOuctuatiRQs characteri]iQJ the TuaQtum chrRmRdyQamic OeYeO RI subQucOear physics, the IOuctuatiRQs OiQked with the +iJJs IieOd, as weOO as perhaps Rther cRQtributiRQs IrRm pRssibOe e[ist-iQJ sRurces Rutside the StaQdard MRdeO (IRr iQstaQce, *raQd UQiIied TheRries, striQJ theRries, etc. . OQ the Rther haQd, there is QR structure withiQ the StaQdard MRdeO which suJJests QR reOatiRQs betweeQ the diIIereQt cRQtributiRQs tR the TuaQtum Yacuum eQerJy deQsity, aQd it is thereIRre custRmary tR assume that the tRtaO Yacuum eQerJy deQsity is, at Oeast, as OarJe as aQy RI these iQdiYiduaO cRQ-tributiRQs. As reJards the rROe RI the diIIereQt cRQtributiRQs tR the Yacuum eQer-Jy deQsity, the reader caQ IiQd a detaiOed aQaOysis, IRr e[ampOe, iQ the paper >1@ by RuJh aQd ZiQkerQaJeO, whR studied the cRQQectiRQ betweeQ the Yacuum cept iQ TuaQtum IieOd theRry aQd the cRQceptuaO RriJiQ RI the cRsmRORJicaO cRQ-staQt prRbOem, aQd iQ the paper >2@ by TimasheY, whR e[amiQed the pRssibiOity RI cRQsideriQJ the physicaO Yacuum as a uQiIied system JRYerQiQJ the prRcesses takiQJ pOace iQ micrRphysics aQd macrRphysics, which maQiIests itseOI RQ aOO space-time scaOes, IrRm subQucOear tR cRsmRORJicaO.
The reaOistic cRQcept RI the Yacuum caQ be cRQsidered as the uOtimate Yisit-iQJ card which cRmpOetes aQd cRmpOemeQts EiQsteiQ¶s theRry RI reOatiY-ity. ReOatiYity theRry Yiews space-time as a reOatiYe aQd dyQamic maQiIROd, iQ-teractiQJ with matter aQd eQerJy. It is the ³backJrRuQd´ aJaiQst which the eYeQts RI the maQiIest wRrOd uQIROd. But the RriJiQs RI this backJrRuQd are QRt ac-cRuQted IRr iQ reOatiYity theRry: space-time is simpOy ³JiYeQ´ tRJether with mat-ter aQd eQerJy. IQ JeQeraO reOatiYity, the staQdard iQmat-terpretatiRQ RI pheQRmeQa iQ JraYitatiRQaO IieOds is iQ terms RI a IuQdameQtaOOy curYed space-time. +RweYer,
this apprRach Oeads tR weOO-kQRwQ prRbOems iI RQe aims tR IiQd aQ uQiIyiQJ pic-ture which takes iQtR accRuQt sRme basic aspects RI the TuaQtum theRry. IQ Rrder tR escape this situatiRQ RI impasse, seYeraO authRrs adYRcated thus aOterQatiYe ways iQ Rrder tR treat JraYitatiRQaO iQteractiRQ, iQ which the space-time maQiIROd caQ be cRQsidered as aQ emerJeQce RI the deepest prRcesses situated at the IuQ-dameQtaO OeYeO RI TuaQtum JraYity. IQ this reJard, the JermiQaO prRpRsaO RI SacharRY was RI deduciQJ JraYitatiRQ as a ³metric eOasticity´ RI space, which cRQsists iQ a JeQeraOi]ed IRrce RppRsiQJ the curYiQJ RI space >3@ (the reader caQ aOsR see the reIereQce >4@ IRr a reYiew RI this cRQcept . SacharRY¶s mRdeO starts IrRm the iQterpretatiRQ RI the actiRQ RI space-time as the eIIect RI TuaQtum IOuc-tuatiRQs RI the Yacuum iQ a curYed space. Other iQterestiQJ apprRaches are +aisch¶s aQd Rueda¶s mRdeO >5@, reJardiQJ the iQterpretatiRQ RI iQertiaO mass aQd JraYitatiRQaO mass as eIIects RI aQ eOectrRmaJQetic TuaQtum Yacuum, PuthRII¶s pROari]abOe Yacuum mRdeO RI JraYitatiRQ >@ aQd, mRre receQtOy, a mRdeO deYeO-Rped by CRQsROi based RQ uOtra-weak e[citatiRQs iQ a cRQdeQsed maQiIROd iQ Rrder tR describe JraYitatiRQ aQd +iJJs mechaQism >-9@. UQder the cRQstructiRQ RI aOO RI these mRdeOs there is prRbabOy RQe uQderOyiQJ IuQdameQtaO RbserYatiRQ: as OiJht iQ EucOid space deYiates IrRm a straiJht OiQe iQ a medium with YariabOe deQsity, aQ ³eIIectiYe´ curYature miJht RriJiQate, uQder RppRrtuQe cRQditiRQs, IrRm the same physicaO IOat-space Yacuum.
IQ this paper, by IROORwiQJ the phiORsRphy that is at the basis RI these ap-prRaches, we suJJest a mRdeO RI a three-dimeQsiRQaO (3D TuaQtum Yacuum iQ which JeQeraO reOatiYity emerJes as the hydrRdyQamic Oimit RI sRme uQderOyiQJ theRry RI a mRre IuQdameQtaO micrRscRpic structure RI space-time. AccRrdiQJ tR this mRdeO, the curYature RI space-time characteristic RI JeQeraO reOatiYity caQ be cRQsidered as a mathematicaO YaOue RI a mRre IuQdameQtaO actuaO eQerJy deQsity RI TuaQtum Yacuum which has a cRQcrete physicaO meaQiQJ. IQ the Ruter iQterJa-Oactic space, QameOy iQ the abseQce RI materiaO RbMects, the eQerJy deQsity RI the 3D TuaQtum Yacuum is deIiQed by the IROORwiQJ reOatiRQ:
2 3 , p pE p m c l U (1 where mP is POaQck¶s mass, c is the OiJht speed aQd l is POaQck¶s OeQJth. The p
TuaQtity (1 is the ma[imum YaOue RI the TuaQtum Yacuum eQerJy deQsity aQd physicaOOy cRrrespRQds tR the tRtaO aYeraJe YROumetric eQerJy deQsity, Rwed tR aOO the IreTueQcy mRdes pRssibOe withiQ the YisibOe si]e RI the uQiYerse, e[-pressed by 14 113 3 9 3 2 4 4,641266 10 / 10 / , pE c J m Kg m G U | # = (2
=
beiQJ POaQck¶s reduced cRQstaQt, G the uQiYersaO JraYitatiRQ cRQstaQt. IQ the Ruter iQterJaOactic space curYature RI space is ]erR aQd its eQerJy deQsity cRr-respRQds tR the YaOue (2 . OQ the Rther haQd, iQ the picture RI Rueda¶s aQd +aisch¶s iQterpretatiRQ RI the iQertiaO mass as aQ eIIect RI the eOectrRmaJQetic TuaQtum Yacuum >5@, the preseQce RI a particOe with a YROume V0 e[peOs IrRmthe Yacuum eQerJy withiQ this YROume e[actOy the same amRuQt RI eQerJy as is the particOe¶s iQterQaO eQerJy (eTuiYaOeQt tR its rest mass . OQ the basis RI Rueda¶s aQd +aisch¶s resuOts, here we assume that each eOemeQtary particOe is assRciated with IOuctuatiRQs RI the TuaQtum Yacuum which determiQe a dimiQ-ishiQJ RI the TuaQtum Yacuum eQerJy deQsity. ThereIRre, RQe caQ say that iQ the preseQce RI a materiaO RbMect the curYature RI space iQcreases aQd cRrre-spRQds physicaOOy tR a mRre IuQdameQtaO dimiQishiQJ RI the eQerJy deQsity RI the TuaQtum Yacuum, which, iQ the ceQtre RI the materiaO RbMect, is JiYeQ by reOatiRQ 2 , qvE pE m c V U U (3
m aQd V beiQJ the mass aQd YROume RI the RbMect >10@. +ere, we prRpRse that
the TuaQtum Yacuum eQerJy deQsity is the IuQdameQtaO, uOtimate physicaO reaO-ity characteri]iQJ the JraYitatiRQaO space. The physicaO prRperty RI mass is cRQsidered as a secRQdary RQtRORJicaOOy reaOity with respect tR the eQerJy deQ-sity RI TuaQtum Yacuum: the deQdeQ-sity RI a JiYeQ materiaO RbMect is prRduced by a chaQJe RI the TuaQtum Yacuum eQerJy deQsity RQ the basis RI eTuatiRQ
2 , qvE mat c U U ' (4
where
'U
qvEU
pEU
qvE.
This paper is structured iQ the IROORwiQJ maQQer. IQ chapter 2 we wiOO e[-pORre iQ what seQse JeQeraO reOatiYity caQ be seeQ as the hydrRdyQamic Oimit RI aQ uQderOyiQJ TuaQtum Yacuum cRQdeQsate haYiQJ TuaQti]ed Ieatures. IQ chapter 3 we wiOO shRw hRw the chaQJes RI the eQerJy deQsity RI the 3D TuaQtum Yacu-um JiYe rise tR the curYature RI space-time characteristic RI JeQeraO reOatiYity aQd which is simiOar tR the curYature prRduced by a ³dark eQerJy´ deQsity. )iQaO-Oy, iQ chapter 4 we wiOO aQaOyse the mRtiRQ RI a materiaO RbMect iQ the curYed space determiQed by the chaQJes aQd IOuctuatiRQs RI the TuaQtum Yacuum eQer-Jy deQsity.
2. SPACE AS T+E *EOMETRO-+YDRODYNAMIC LIMIT
O) A 3D QUANTUM VACUUM CONDENSATE
+AVIN* A DISCRETE NATURE
TakiQJ accRuQt RI SacharRY¶s assumptiRQ that the actiRQ RI spacetime
1 , 16 S R dx gR G S
³
(5where R is the iQYariaQt Ricci teQsRr, is Yiewed as a chaQJe iQ the actiRQ RI TuaQtum IOuctuatiRQs RI Yacuum iQ a curYed space aQd cRQsideriQJ the cRQ-sisteQt histRries apprRach RI TuaQtum mechaQics >11-13@, accRrdiQJ tR which the TuaQtum eYROutiRQ caQ be seeQ as the cRhereQt superpRsitiRQ RI YirtuaO IiQe–JraiQed histRries, JeQeraO reOatiYity caQ be iQterpreted as the hydrRdyQam-ic Oimit RI aQ uQderOyiQJ theRry RI ³mhydrRdyQam-icrRscRphydrRdyQam-ic´ structure RI space, mRre pre-ciseOy RI a 3D TuaQtum Yacuum cRQdeQsate whRse mRst uQiYersaO physicaO prRperty is its eQerJy deQsity.
A IiQe-JraiQed histRry caQ be deIiQed by the YaOue RI a IieOd )
x at the pRiQt
x
aQd has TuaQtum ampOitude < )> @
eiS> @), whereS
is the cOassicaOac-tiRQ cRrrespRQdiQJ tR the cRQsidered histRry. The TuaQtum iQterIereQce betweeQ twR YirtuaO histRries A aQd B caQ be TuaQtiIied by a ³decRhereQce´ IuQctiRQaO:
>
@
> @ > @
* > @ > @, i S A S B
F A B A B
D ) ) | < ) < ) |e ) ) (6
that JiYes the cRarse-JraiQed histRries cRrrespRQdiQJ tR the RbserYatiRQs iQ cOassicaO wRrOd. The TuaQtum ampOitude IRr a cRarse-JraiQed histRry is theQ deIiQed by:
> @
iS> @
, F D eZ
Z
<³
) ) (where
Z
caQ be cRQsidered as a ³IiOter´ IuQctiRQ that seOects which IiQe-JraiQed histRries are assRciated tR the same superpRsitiRQ with their reOatiYe phases. The decRhereQce IuQctiRQaO IRr a cRupOe RI cRarse-JraiQed histRries is theQ:>
@
> @ > @> @ > @
* , i S A S B F A B F A F B A B DZ Z
D) D ) e ) )Z
)Z
)³
(iQ which the histRries )A aQd )B assume the same YaOue at a JiYeQ time iQ-staQt RI the Iuture, where decRhereQce iQdicates that the diIIereQt histRries cRQtributiQJ tR the IuOO TuaQtum eYROutiRQ caQ e[ist iQdiYiduaOOy, are character-i]ed by TuaQtum ampOitude aQd that the system uQderJRes aQ iQIRrmatiRQ aQd
predictabiOity deJradatiRQ >13@ (iQ this seQse the system becRmes stRchastic aQd dissipatiYe . By appOyiQJ the IRrmaOism ( tR hydrRdyQamics YariabOes >14@, EiQsteiQ¶s stress-eQerJy teQsRr caQ be e[pressed thrRuJh the IROORwiQJ RperatRr:
ˆ , .
A B A B
TPQ x x * )PQ x ) x (9
IQ eTuatiRQ (9
*
PQ is a JeQeric IieOd RperatRr deIiQed at twR pRiQts that Oeads tR the ³cRQserYatiRQ Oaw´:;
ˆ 0
TQ
PQ (10
meaQiQJ that the decRhered TuaQtities, shRwiQJ a cOassicaO behaYiRr, are the cRQserYed RQes. It caQ be shRwQ that, IRr aQ actiRQ
> @
l l mlm
S ) ) ' ) , the IRO-ORwiQJ reOatiRQ hROds
, , , , , , , ˆ ,ˆ , l n A B n A B m n A A B B A B n A B A B lm i K x x x x iK x x T x x i T x x T x x A B l F F n A B F D T T D KPQ x x D e PQ PQ e PQ PQ e PQ PQ PQ PQ ª º ª º ) '¬ * ¼ ) :¬ ¼ ª º ) | ¬ ¼ ³ ³ (11
iQ which we haYe used the iQteJraO represeQtatiRQ RI deOta aQd the CTP iQdices . . 1,2
l m n , : beiQJ the cORsed–time path twR-particOe irreducibOe actiRQ. The cRQserYatiRQ RI TˆPQ impOies that the decRhereQce IuQctiRQaO has ma[i-mum YaOues iQ cRrrespRQdeQce RI the hydrRdyQamic YariabOes
U
, pthat, iQ turQ, are the mRst readiOy decRhered aQd haYe the hiJhest prRbabiOity tR becRme cOassicaO. By appOyiQJ the abRYe prRcedure tR EiQsteiQ¶s teQsRr
G
PQ aQ aQaORJy emerJes betweeQ the cRQserYatiRQ Oaw IRr TˆPQ aQd the BiaQchi ideQtity; 0
GQ PQ
which impOies the decRhereQce aQd the emerJeQce RI the hydrRdyQamics Yaria-bOes RI the JeRmetry. IQ this seQse JeQeraO reOatiYity caQ be cRQsidered as JeRmetrR–hydrRdyQamics aQd the mRst readiOy decRhered YariabOes are thRse assRciated tR the OarJest ³iQertia´ represeQtiQJ the cROOectiYe YariabOes RI JeRme-try.
II JeQeraO reOatiYity must be reJarded as a JeRmetrR-hydrRdyQamic Oimit RI aQ uQderOyiQJ ³micrRscRpic´ backJrRuQd where RQe has cROOectiYe YariabOes, aQd the Oaws JRYerQiQJ macrR-cOassicaO space-time are e[pressed iQ terms RI cROOec-tiYe YariabOes, the precise characteri]atiRQ RI this uQderOyiQJ backJrRuQd aQd thus the TuaQti]atiRQ RI the JeQeraO-reOatiYistic metric Rr the cRQQectiRQ Yaria-bOes wiOO RQOy resuOt iQ the discRYery RI the e[citatiRQs iQ the JeRmetry aQd QRt
>
@
> @ > @³
³
)l i)l'Kn xAxB*n xAxB lm)m iKn xAxBTn xAxB | i:TA xAxB TB xAxB F B A n F B A FT T D K x x D e e e D ˆ ,ˆ PQ , PQ , PQ , PQ , PQ , ˆPQ , ,ˆPQ , PQ PQ (11)RI its TuaQtum micrR-structures. II we cRQsider the cROOectiYe hydrRdyQamics YariabOes U aQd p appeariQJ iQ the stress-eQerJy teQsRr
T
PQ, theQ the TuaQti]a-tiRQ has seQse wheQ perIRrmed RQ the IieOd IuQcTuaQti]a-tiRQ )x IrRm which they are cRQstructed aQd QRt RQ U aQd p themseOYes. The situatiRQ is simiOar tR that reJardiQJ cRQdeQsed matter physics iQ which the TuaQti]atiRQ RI cROOectiYe e[ci-tatiRQs Oeads tR phRQRQs aQd QRt tR the atRmic structure RI matter. IQ the Yiew RI JeQeraO reOatiYity as JeRmetrR-hydrRdyQamic Oimit RI aQ uQderOyiQJ backJrRuQd, there is thereIRre aQ impRrtaQt aQaORJy betweeQ TuaQtum tR cOassicaO traQsitiRQ RI JraYity aQd the behaYiRr RI cRQdeQsed matter. MRreRYer, JiYeQ the cROOectiYe YariabOes (the metric aQd the cRQQectiRQs iQ JeQeraO reOatiYity), hRw caQ we characteri]e the micrRscRpic structure RI the uQderOyiQJ backJrRuQd, QameOy what caQ we say abRut the TuaQtum micrR-structure IrRm which the cROOectiYe YariabOes deriYe" IQ this reJard, a pRssibOe strateJy is RI startiQJ IrRm a suitabOe theRry RI TuaQtum micrRscRpic structure aQd studyiQJ its preYisiRQs iQ the ORQJ waYeOeQJth–ORw eQerJy Oimit. AQ apprRach RI this kiQd has beeQ receQtOy suJ-Jested, IRr e[ampOe, by CRQsROi >-9@, whR has iQtrRduced a physicaO Yacuum iQteQded as a superIOuid medium – a BRse cRQdeQsate RI eOemeQtary spiQOess TuaQta – whRse ORQJ-raQJe IOuctuatiRQs, RQ a cRarse-JraiQed scaOe, resembOe the NewtRQiaQ pRteQtiaO, yieOdiQJ the Iirst apprR[imatiRQ tR the metric structure RI cOassicaO JeQeraO reOatiYity. IQ aQaORJy with CRQsROi¶s mRdeO, takiQJ iQtR cRQsid-eratiRQ the ORQJ-waYeOeQJth mRdes, here JraYity is iQduced by the uQderOyiQJ IieOd )
x which describes the deQsity IOuctuatiRQs RI the Yacuum. IQ weak JraYitatiRQaO IieOds, RQ a cRarse-JraiQed scaOe, the uQderOyiQJ IieOd )
x caQ be
ideQtiIied with the NewtRQiaQ pRteQtiaO
, i N N i i M U G r r ) |
¦
G G (12) QameOy. gPQ gPQ ª¬) x º¼ (13)
AQ iQterestiQJ arJumeQt which aOORws us tR characteri]e the TuaQtum micrR-scRpic structure RI the uQderOyiQJ backJrRuQd JeQeratiQJ JraYity caQ be deriYed IrRm the TuaQtum uQcertaiQty priQcipOe >15@ aQd IrRm the hypRtheses RI space-time discreteQess at the POaQck scaOe. IQ particuOar, iQ reJard tR the JraQuOarity RI space-time aQd its OiQk with JraYity, iQ the papers >16-19@ NJ shRwed that the TuaQtum IOuctuatiRQs RI space-time maQiIest themseOYes iQ the IRrm RI uQcer-taiQties iQ the JeRmetry RI space-time aQd thus the structure RI the space-time IRam caQ be iQIerred IrRm the accuracy with which we caQ measure its JeRme-try. By cRQsideriQJ a mappiQJ RI the JeRmetry RI space-time IRr a sphericaO
YROume RI radius l RYer the amRuQt RI time T 2 /l c it takes OiJht tR crRss the YROume, iQ NJ¶s apprRach the aYeraJe separatiRQ betweeQ QeiJhbRuriQJ ceOOs RI space cRrrespRQds tR the aYeraJe miQimum uQcertaiQty, aQd thus tR the accuracy iQ the measuremeQt RI a distaQce l, JiYeQ by
2
1/3 1/3 2/3
2 / 3 P
l l l
G t S . (14)
AQ iQterestiQJ aspect RI NJ¶s TuaQtum IRam mRdeO Oies iQ its hRORJraphic Ieatures iQ the seQse that here, drRppiQJ the muOtipOicatiYe IactRr RI Rrder 1, a spatiaO reJiRQ RI si]e l caQ cRQtaiQ QR mRre thaQ 3
2
2
/ P / P
l ll l l ceOOs aQd
thus a ma[imum Qumber RI bits RI iQIRrmatiRQ
2
/ P
l l iQ aJreemeQt with the hRORJraphic priQcipOe >20-25@ which impOies that, aOthRuJh the wRrOd arRuQd us appears tR haYe three spatiaO dimeQsiRQs, its cRQteQts caQ actuaOOy be eQcRded RQ a twR-dimeQsiRQaO surIace, Oike a hRORJram.
By appOyiQJ the discreteQess hypRthesis RI NJ¶s mRdeO, QameOy the Iact that we caQQRt make
'
x
smaOOer thaQ the eOemeQtary OeQJth (14)1:2
1/3 1/3 2/3
2 / 3 P
x S l l
' t (15)
tR +eiseQberJ¶s uQcertaiQty reOatiRQ IRr the pRsitiRQ
'
x
aQd mRmeQtum 'p2 x p ' t ' = (16) RQe RbtaiQs that, iI p' iQcreases, the e[pressiRQ RI 'x as a IuQctiRQ RI 'p
must cRQtaiQ a term directOy prRpRrtiRQaO tR p' that cRuQterbaOaQces the term prRpRrtiRQaO tR
1
p
' . By IROORwiQJ >26@, a pRssibOe chRice, at the Iirst Rrder iQ 'p, is:
2
2/3 2/3 4/3 2 / 3 2 2 P p x l l p S ' ' t ' = = (1)
iQ which the IactRr iQ the secRQd term RI the riJht haQd side is seOected by meaQs RI dimeQsiRQaO arJumeQts. The e[pressiRQ (1) caQ be Yiewed as the ³JeQeraOi]ed´ YersiRQ RI the uQcertaiQty priQcipOe iQ a discrete space-time.
By a simiOar reasRQiQJ RQe caQ RbtaiQ the cRrrespRQdiQJ YersiRQ RI (1) IRr time uQcertaiQty as:
2 0 , 2 2 ET t E ' ' t ' = = (1)
where E' is the eQerJy uQcertaiQty aQd
2
1/3 1/3 2/3 0
1
2 / 3 P
T l l
c S is the
eOemeQ-tary time. IQ the apprRach prRpRsed iQ this articOe, the Qew terms appeariQJ iQ eTuatiRQs (1) aQd (1) haYe a Yery speciaO meaQiQJ: they represeQt the ³iQ-triQsic´ uQcertaiQty RI space-time due tR the preseQce RI a particOe RI a JiYeQ eQerJy–mRmeQtum deriYiQJ IrRm RppRrtuQe chaQJes RI the TuaQtum Yacuum eQerJy deQsity '
U
qvEU
pEU
qvE. Thus, the preseQce RI matter RI deQsity (4) mRdiIies the JeRmetry RI space-time. IQ Iact, the eQerJy E| pc cRQtaiQed iQ a reJiRQ RI si]e L aQd deriYiQJ IrRm matter RI deQsity (4) mRdiIies the e[teQ-siRQ RI this reJiRQ RI aQ amRuQt:2
1/3 1/3 2/3 0 2 / 3 . 2 p l l T E L S ' # = (19)
OQ the basis RI eTuatiRQ (19), the curYature RI space-time caQ be reOated tR the preseQce RI eQerJy aQd mRmeQtum iQ it.
IQ Rther wRrds, iQ the apprRach here suJJested, RQe caQ say that the chaQJes RI the TuaQtum Yacuum eQerJy deQsity assRciated with the preseQce RI matter RI deQsity (4) cRrrespRQd tR aQ uQderOyiQJ micrRscRpic backJrRuQd JeRmetry de-IiQed by eTuatiRQ (19).
MRreRYer, takiQJ iQtR accRuQt that iQ NJ¶s mRdeO the hRORJraphic space-time IRam deIiQed by eTuatiRQ (14) caQ be reOated tR the cRsmic scaOe iI the aYeraJe miQimum uQcertaiQty (14) cRrrespRQds tR a ma[imum eQerJy deQsity
2 3 llP
U
S
(20) IRr a sphere RI radius l that dRes QRt cROOapse iQtR a bOack hROe, QameOy2 3 R lH P U S (21) where RH is the +ubbOe radius (which is the criticaO cRsmic eQerJy deQsity as
RbserYed), heQce deriYes that the terms iQ eTuatiRQs (1) aQd (1) represeQtiQJ the ³iQtriQsic´ uQcertaiQty RI space-time due tR the chaQJes RI the TuaQtum Yacuum eQerJy deQsity caQ be themseOYes reOated tR the cRsmic scaOe. IQ
par-ticuOar, by takiQJ accRuQt RI eTuatiRQ (21), eTuatiRQs (1) aQd (1) at the cRs-mRORJicaO scaOe respectiYeOy becRme
2
2/3 2/3 4/3 2 / 3 , 2 2 H P p x R l p S ' ' t ' = = (22) 2 0 , 2 2 ET t E ' ' t ' = = (23) where E' is the eQerJy uQcertaiQty aQd
2
1/3 1/3 2/3
0
1
2 / 3 H P
T R l
c S . )iQaOOy,
eTua-tiRQ (19) describiQJ the OiQk betweeQ the uQderOyiQJ micrRscRpic structure RI space-time aQd the curYature RI space-time, at the cRsmRORJicaO OeYeO may be e[pressed as
2
1/3 1/3 2/3 0 2 / 3 . 2 H p R l T E L S ' # = (24)
NRw, aIter shRwiQJ hRw the TuaQtum micrRscRpic structure RI the uQderOy-iQJ backJrRuQd JeQeratuQderOy-iQJ JraYity caQ be characteri]ed aQd the impRrtaQt OiQk RI this micrRscRpic structure with the cRsmic scaOe, the Qe[t IuQdameQtaO step is tR make e[pOicit the rROe RI the TuaQtum Yacuum eQerJy deQsities JiYeQ by eTua-tiRQs (1) aQd (3) (iQ particuOar, iQ Rrder tR deriYe the criticaO cRsmic eQerJy deQ-sity (21) as RbserYed).
3. T+E C+AN*ES O) T+E 3D QUANTUM VACUUM ENER*Y
DENSITY AS T+E ORI*IN O) T+E CURVATURE O) SPACE-TIME
The POaQck eQerJy deQsity (2) is usuaOOy cRQsidered as the RriJiQ RI the dark eQerJy aQd thus RI a cRsmRORJicaO cRQstaQt, iI the dark eQerJy is suppRsed tR be Rwed tR aQ iQterpOay betweeQ TuaQtum mechaQics aQd JraYity. +RweYer, the RbserYatiRQs are cRmpatibOe with a dark eQerJy
26 3
10
/
DE
Kg m
U
#
(25)aQd thus eTuatiRQs (2) aQd (25) JiYe rise tR the sR-caOOed ³cRsmRORJicaO cRQ-staQt prRbOem´ because the dark eQerJy (25) is 123 Rrders RI maJQitude ORwer thaQ (2). IQ Rrder tR sROYe this prRbOem, aQ iQterestiQJ e[pOaQatiRQ IRr the actu-aO Yactu-aOue (25) which iQYRkes the IOuctuatiRQs RI the TuaQtum Yacuum has
re-ceQtOy beeQ suJJested by SaQtRs >2-29@. AccRrdiQJ tR this apprRach, TuaQtum Yacuum IOuctuatiRQs determiQe a curYature RI space-time aQd, uQder pOausibOe hypRtheses withiQ TuaQti]ed JraYity, a reOatiRQ betweeQ the twR-pRiQt cRrreOa-tiRQ IuQccRrreOa-tiRQ RI the Yacuum IOuctuacRrreOa-tiRQs aQd the space-time curYature was RbtaiQed. The TuaQtum Yacuum IOuctuatiRQs caQ be assRciated with a curYature RI space-time simiOar tR the curYature prRduced by a ³dark eQerJy´ deQsity, RQ the basis RI the eTuatiRQ
0 70 DE G C s sds U f #
³
(26)which states that the pRssibOe YaOue RI the ³dark eQerJy´ deQsity is the prRduct RI NewtRQ cRQstaQt, G, times the iQteJraO RI the twR-pRiQt cRrreOatiRQ IuQctiRQ RI the Yacuum IOuctuatiRQs deIiQed by
1 2
1 2
2 1 1 ˆ ˆ ˆ ˆ , , , , , 2 C r rG G vac U r tG U r tG U r tG U r t vacG (27) ˆ
U
beiQJ aQ eQerJy deQsity RperatRr such that its Yacuum e[pectatiRQ is ]erR whiOe the Yacuum e[pectatiRQ RI the sTuare RI it is QRt ]erR. The cRrreOatiRQ IuQctiRQ (27) determiQes aOsR the JraYitatiRQaO eQerJy assRciated with the Yac-uum IOuctuatiRQs accRrdiQJ tR the eTuatiRQ2 12 12 12 0 4 . gravc G C r r dr U S f
³
(2)MRreRYer, dimeQsiRQaO aQaOysis Oeads tR ZeOdRYich¶s IRrmuOa >11@,
2 2 3 1 , DE C C Gm c r r U | (29)
(rC =/mc beiQJ CRmptRQ¶s radius) which iQYROYes a parameter, m, with di-meQsiRQs RI a mass. II iQ ZeOdRYich¶s RriJiQaO mRdeO, eTuatiRQ (29) reprRduces the RbserYed YaOue RI the dark eQerJy deQsity IRr a mass RI m|7,6 10 29Kg that
is abRut 1/20 times the prRtRQ mass Rr abRut 0 times the eOectrRQ mass, SaQ-tRs¶ apprRach dRes QRt aOORw tR deriYe the YaOue RI m, but iQside his apprRach it is pOausibOe tR assume that Yacuum IOuctuatiRQs RI hiJh eQerJy, iQYROYiQJ Yery massiYe particOes, wRuOd QRt be prRbabOe.
+ere, Rur aim is tR shRw that the curYature RI space-time assRciated with a dark eQerJy deQsity caQ be iQterpreted as a cRQseTueQce RI mRre IuQdameQtaO chaQJes RI the 3D TuaQtum Yacuum eQerJy deQsity 'UqvE UpE UqvE, iQ Rther wRrds it caQ be physicaOOy deIiQed as the mathematicaO YaOue RI the 3D TuaQtum Yacuum eQerJy deQsity (whRse uQderOyiQJ micrRscRpic structure is characteri]ed
by a JeRmetry e[pressed by eTuatiRQs (17)-(19) aQd by eTuatiRQs (22-(24) at the cRsmRORJicaO OeYeO). IQ this reJard, beIRre aOO, we assume that the e[pectatiRQ YaOue RI the stress-eQerJy teQsRr RperatRr RI the TuaQtum IieOds (9) at aQy pRiQt JiYes the matter eQerJy assRciated with the matter (baryRQic pOus dark) eQerJy deQsity, which is determiQed by chaQJes aQd IOuctuatiRQs RI the 3D TuaQtum Yacuum eQerJy deQsity, withRut aQy additiRQaO cRQtributiRQ tR the Yacuum. This assumptiRQ aOORws us tR RbtaiQ the cRrrect )riedmaQQ-RRbertsRQ-:aOker metric
2
2 2 2 2
ds dt ª¬a t º¼ dr r d: (30)
iQ which the recessiRQ RI the distaQt JaOa[ies caQ be caOcuOated iQ terms RI the OiQk RI the measurabOe +ubbOe cRQstaQt aQd RI the deceOeratiRQ parameter with the time-depeQdeQt parameter a t
), by iQtrRduciQJ Qew time aQd radiaO cRRr-diQates
r
'
aQd 't JiYeQ by reOatiRQs''
' ,3 r r O r a t
2 4 ' ' ' ' . 2 ' ' da t r t t O r a t dt (31)
By iQsertiQJ (31) iQtR (30) RQe RbtaiQs the eTuatiRQ
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 ' ' ' 1 ' ' 1 1 ' ' ' 1 ' ' , 3 mat DE 3 2 mat DE a a ds r dr r d r dt a a G G r dr r d r dt S U U S U U ª § · º ª § · º « ¨ ¸ » : « ¨ ¸ » © ¹ © ¹ ¬ ¼ ¬ ¼ ª º ª º : § · ¨ ¸ « » « » ¬ ¼ ¬ © ¹ ¼ (32)
where the )riedmaQQ eTuatiRQs
2 , 3 mat DE a G a S U U ª º « » ¬ ¼ 1 3 2 mat DE a G t a S § U U · ¨ ¸ © ¹ (33)
haYe beeQ takeQ iQtR accRuQt iQ the secRQd eTuaOity, a a t{
' ,
' ' da t a dt { ,
2 2 ' ' d a t a dt {
,
U
mat is the deQsity RI matter JiYeQ by eTuatiRQ (4),U
DE is the deQsity Rwed tR a pRssibOe e[isteQce RI dark matter. IQ reIereQce tR eTuatiRQ (32), the assumptiRQ that the e[pectatiRQ YaOue RI the stress-eQerJy teQsRr Rp-eratRr RI the TuaQtum IieOds (9) JiYes the matter stress-eQerJy deQsity deter-miQed by IOuctuatiRQs RI the 3D TuaQtum Yacuum eQerJy deQsity, meaQs that4 4 2 ˆ qvE; ˆ 0 IRr 00, T T c Q P U PQ ' < < < < | z (34)
< beiQJ the TuaQtum state RI the uQiYerse cRrrespRQdiQJ tR the YaOue RI the IieOd )
x deIiQiQJ a JiYeQ IiQe-JraiQed histRry. This suJJests tR e[press the stress-eQerJy teQsRr (9) cRrrespRQdiQJ tR the TuaQtum Yacuum IOuctuatiRQs as
ˆvac ˆ ˆ ˆ,
TPQ {TPQ < TPQ < I (35)
where ˆI is the ideQtity RperatRr. The e[isteQce RI TuaQtum Yacuum IOuctua-tiRQs impOy that, despite the e[pectatiRQ RI Tˆvac
PQ is ]erR by deIiQitiRQ, RQe has
ˆvac ˆvac 0
TPQ x TOV y
< < z (36)
iQ JeQeraO.
NRw, iQ Rrder tR deriYe eTuatiRQ (32), takiQJ iQtR accRuQt SaQtRs¶ resuOts, iQ-side Rur mRdeO it is reasRQabOe tR assume that the uQderOyiQJ TuaQtum Yacuum cRQdeQsate caQ be characteri]ed by cRQsideriQJ the metric RI the TuaQtum Yacu-um deIiQed by reOatiRQ
2
ˆ ˆ ,
ds g dx dxP Q
PQ (37)
where the cReIIicieQts (iQ pROar cRRrdiQates) are
00 ˆ00 ˆ 1 g h , gˆ11 1 hˆ11, gˆ11 1 hˆ11, 2 2
33
ˆ
33ˆ
siQ
1
g
r
-
h
, ˆ ˆgPQ hPQ IRrP Q
z
, (3)where muOtipOicatiRQ RI eYery term times the uQit RperatRr is impOicit aQd, at the Rrder
2 O r , RQe has ˆ 0 hPQ e[cept 2 00 2 ˆ 3 qvE DE G h r c
U
S
§'U
· ¨ ¸ © ¹ aQd 2 11 2 1 ˆ 3 2 qvE DE G h r cU
S
§U
' · ¨ ¸ © ¹ , (39)where ˆhPQ staQds IRr < ˆhPQ < (aQd the IOuctuatiRQs RI the TuaQtum Yacuum
qvE pE qvE
U
U
U
' cRrrespRQd tR aQ uQderOyiQJ micrRscRpic JeRmetry deIiQed
00 00 1 ˆ ˆ h g , gˆ11 1hˆ11, gˆ22 r2
1hˆ22
,
33
2 2 33 sin 1 ˆ ˆ r h g - , gˆPQ hˆPQ for PzQ (38)
by eTuations (17), (18) and (19)). In Yirtue of the Tuanti]ed Jeometry defined by eTuations (17), (18) and (19), the metric (37), at a fundamentaO OeYeO, has to be considered as a Tuanti]ed metric.
As reJards the Tuanti]ed metric (37), it is important to remark that in the ap-proach deYeOoped by Santos in >28@, by writinJ the Tuantum coefficients of the metric as (38), where ˆ 0 hPQ e[cept
2 00 8 ˆ 3 mat DE G h S U U r and 2 11 8 1 ˆ 3 DE 2 mat G h S §¨U U ·¸r © ¹ , (39a)
where
ˆh
PQ stands for < ˆhPQ < , in the appro[imation of the second order in the (smaOO) tensor ˆhPQ, it is possibOe to deriYe the components of Tuantum Einstein eTuations of the form4 8 ˆ G ˆ . G T c PQ PQ S (40) In Santos¶ approach, the Tuantum Einstein tensor operator GˆPQ is e[pressed
in terms of the operators ˆhPQ, by resoOYinJ these (non-Oinear coupOed partiaO) differentiaO operator eTuations (40) in order to obtain the Tuantum metric coeffi-cients
ˆg
PQ in terms of inteJraOs inYoOYinJ the stress-enerJy tensor operator andfinaOOy caOcuOatinJ the e[pectations of the metric coefficients
ˆg
PQ in terms of inteJraOs inYoOYinJ the e[pectations of the stress-enerJy tensor operator. The reader can find detaiOs of these caOcuOations, for e[ampOe, in the aboYe reference >28@ and in >29@.+ere, we underOine that, in anaOoJy with Santos¶ resuOts, due to the fact that the reOations between the metric coefficients and the matter stress-enerJy tensor are non-Oinear, the e[pectation of the Tuanti]ed metric (37) of the Yacuum con-densate is not the same as the metric of the e[pectation of the matter tensor (9). The difference JiYes rise to a contribution of the Yacuum fOuctuations which reproduces the effect of a cosmoOoJicaO constant. MoreoYer, we wiOO assume that, when the distance ro f, one has
ˆg
PQo
K
PQ , whereK
PQ is the Minkowski metric.By startinJ from the Tuanti]ed metric (37) whose coefficients are defined by reOations (38) and (39), one can obtain the components of the Tuantum Einstein eTuations (40) on the basis of the assumption that they are simiOar to the cOassicaO
counterparts. In particuOar, the e[pectation YaOue of the (operator) metric pa-rameter h may be written in the form ˆ11
11 11 11
ˆ ˆ ˆ
mat vac
h h h
< < < < < < , (41)
nameOy it is the sum of two e[pressions, one containinJ the matter density produced by the chanJes of the Tuantum Yacuum enerJy density, and the other indicatinJ the Yacuum density fOuctuations,
U
vac. In eTuation (41), by modeO-OinJ the matter density of the uniYerse by means of a constant, the matter term can be e[pressed as 2 2 11 2 2 2 ˆ , mat GM G M h r r < < # (42) where 3 3 2 4 4 3 3 qvE mat M r r cU
SU
S
' , which aJrees with the second order e[-pansion of the weOO-known Schwar]schiOd soOution1 11 2 1 GM . g r § · ¨ ¸ © ¹ (43) MoreoYer, takinJ into account eTuation (3), here the dark enerJy density
DE
U
can be associated with opportune fOuctuations DE qvEU
' of the 3D Tuantum Yacuum enerJy density defined by reOation
2 DE DE qvE m c V U ' , (44) DE
m beinJ the mass correspondinJ to the dark enerJy
U
DE in the YoOume V and thus 2 . DE qvE DE cU
U
' (45)In this way, takinJ into account that accordinJ to Santos¶ resuOts, the Yacuum contribution appearinJ in eTuation (41), to order G2, is
11 ˆ vac h < < # 2 2
0 600G r
³
fC s sds, (46)r beinJ a distance which is estimated to fuOfiO r s/ |1040, in our modeO the
Yacuum contribution may be e[pressed as
11 ˆ vac h < < # 2 2 2 2 3 1 1 1 150 DE , qvE V G r c U l l S §¨© ' ·¸¹ (47) where 2 DE qvE l V c c U § ' · ¨ ¸ © ¹ = (48)
and, takinJ into account eTuation (26), Santos¶ inteJraO of the two-point corre-Oation function has been assimiOated to the fOuctuations of the Tuantum Yacuum enerJy density (44) on the basis of eTuation
2 2 3
1 1
DE qvEV
c
U
l l
§
'
·
¨
¸
©
¹
12 12 12 0 4
S
f³
C r r dr . (49)Therefore, the totaO e[pectation YaOue (41) becomes, to order
r
2 2 11 2 8 ˆ 3 qvE G h r cS
'U
< < # 2 2 2 2 3 1 1 1 150 DE . qvE V G r c U l l S § · ¨ ' ¸ © ¹ (50) +ence, a comparison with the )riedmann eTuations (33), takinJ account ofreOations (26) and (46), Oeads to the foOOowinJ eTuation
2 DE
c
U
#
2 2 3 35 1 1, 2 DE qvE G V V cU
l lS
§ ' · ¨ ¸ © ¹ (51) nameOy DEU
# 6 2 4 2 35 2 DE qvE Gc V V cU
S
§ ' · ¨ ¸ © ¹ = (52) which states the eTuiYaOence of the curYature of space-time produced by thechanJes of the Tuantum Yacuum enerJy density and the one determined by a constant dark enerJy density. This means that in the approach based on eTua-tions (37)-(52), the chanJes and fOuctuaeTua-tions of the Tuantum Yacuum enerJy density Jenerate a curYature of space-time simiOar to the curYature produced by a ³dark enerJy´ density. MoreoYer, it is interestinJ to obserYe that, whiOe in
Santos¶ modeO, the dark enerJy is associated with the two-point correOation function of the Yacuum fOuctuations (on the basis of eTuation (26)), in the ap-proach suJJested by the authors of this articOe, the dark enerJy is directOy de-termined by fOuctuations of the Tuantum Yacuum enerJy density on the basis of eTuation (52). It must be emphasi]ed that here the fOuctuations of the Tuantum Yacuum enerJy density pOay the same roOe of Santos¶ two-point correOation function. In other words, there is an eTuiYaOence between the fOuctuations of the Tuantum Yacuum enerJy density and the two-point correOation function: in the approach here suJJested, the fOuctuations of the 3D Tuantum Yacuum ener-Jy density act as a two-point correOation function on the basis of reOation
6 4 4 2 4 DE qvE c V c
U
S
§ ' · ¨ ¸ © ¹ = 0 C s sds. f #
³
(53)MoreoYer, introducinJ eTuation (52) into eTuation (39), the e[pectation YaO-ues of the coefficients of the Tuanti]ed metric (30) become
ˆ 0 hPQ e[cept 6 2 2 00 2 4 2 8 35 ˆ 3 2 qvE DE qvE G Gc V h r c V c U S U S §' § · · ' ¨ ¨ ¸ ¸ ¨ © ¹ ¸ © = ¹ and 6 2 2 11 2 4 2 8 35 ˆ 3 2 2 qvE DE qvE G Gc V h r c V c U S U S § ' § · · ' ¨ ¨ ¸ ¸ ¨ © ¹ ¸ © = ¹ , (54)
nameOy turn out to depend directOy on the chanJes of the Tuantum Yacuum enerJy density. As a conseTuence, one can say that the chanJes and fOuctua-tions of the Tuantum Yacuum enerJy density, throuJh the Tuanti]ed metric (37) of the Tuantum Yacuum condensate whose coefficients are defined by eTua-tions (38) and (54) (and whose underOyinJ microscopic Jeometry is described by eTuations (17)-(19) and, at the cosmoOoJicaO OeYeO, by eTuations (22)-(24)) can be considered the oriJin of the curYature of space-time characteristic of JeneraO reOatiYity. In other words, one can say that the curYature of space-time may be considered as a mathematicaO YaOue which emerJes from the Tuanti]ed metric (37) and thus from the chanJes and fOuctuations of the Tuantum Yacuum enerJy density (on the basis of eTuations (38) and (54)). In synthesis, accord-inJ to the Yiew suJJested in this chapter, the Tuanti]ed metric (37) associated with the chanJes and fOuctuations of the Tuantum Yacuum enerJy density, on the basis of eTuations (38) and (54), can be considered as the uOtimate YisitinJ card of JeneraO reOatiYity.
4. ABOUT T+E MOTION O) A MATERIAL OB-ECT
IN T+E CURVED SPACE-TIME
Now, Oet us see how the curYature of space-time correspondinJ to the chanJ-es and fOuctuations of the Tuantum Yacuum enerJy density acts on a tchanJ-est particOe of mass
m
0, in other words how the motion of a materiaO obMect in a backJround characteri]ed by chanJes of its enerJy density can be treated mathematicaOOy. :hen a materiaO obMect correspondinJ to a JiYen diminishinJ of the Tuantum Yacuum enerJy density moYes, this diminishinJ of the Tuantum Yacuum enerJy density – by Yirtue of its Oink with the Tuantum Yacuum condensate defined by eTuations (54) (and whose underOyinJ microscopic Jeometry is described by eTuations (17)-(19) and, at the cosmoOoJicaO OeYeO, by eTuations (22)-(24)) – causes a shadowinJ of the JraYitationaO space which determines the motion of other materiaO obMects present in the reJion under consideration.In the approach here suJJested, the shadowinJ of the JraYitationaO space de-termined by a YariabOe density of Tuantum Yacuum tries inspiration from the idea of the poOari]abiOity of the Yacuum in the Yicinity of a mass (or other mass-enerJy concentrations) introduced by Puthoff¶s poOari]abOe modeO of JraYitation >6@. In order to interpret and reproduce the curYature of space-time Puthoff pos-tuOated the foOOowinJ reOation for the YariabOe poOari]ation of the Yacuum caused by the presence of a mass
0 ,
D K EG
H
G (55)where EG is the eOectric fieOd, K is the (aOtered) dieOectric constant of the um (typicaOOy a function of position) due to (JeneraO reOatiYistic-induced) Yacu-um poOari]abiOity chanJes under consideration. Puthoff¶s eTuation (55) estab-Oishes that the presence of eOectromaJnetic enerJy or massiYe obMects modu-Oates the Yacuum poOari]ation in a Oinear fashion. The Yacuum dieOectric con-stant K constitutes the uOtimate YisitinJ card of Puthoff¶s modeO. Its effects on the Yarious measurement processes that characteri]e JeneraO reOatiYity are the foOOowinJ: the YeOocity of OiJht chanJes from c to c/K, the time interYaOs chanJe from 't0 to 't0 K (which indicates that for K!1, nameOy in a JraYi-tationaO potentiaO, the time interYaOs between cOock ticks is increased, that is the cOock runs sOower), the OenJths of rods chanJe from 'r0 to 'r0/ K . In
Puthoff¶s modeO, the curYature of space – for e[ampOe in the Yicinity of a pOan-et or a star – is associated with the effects on measurement processes of OenJths and time interYaOs that take pOace for K!1. Such an infOuence on the measurinJ processes due to induced poOari]abiOity chanJes in the Yacuum near
the body under consideration Oeads to the JeneraO-reOatiYistic concept that the presence of a body ³infOuences the metric´.
TryinJ inspiration from Puthoff¶s idea of poOari]abiOity of the Yacuum in our modeO we assume that the shadowinJ (poOari]ation) of the 3D Tuantum Yacuum can be e[pressed by the eTuation
0 g,
DG
NH
EG (56)where
N
is a factor which represents the reOatiYeOy smaOO amount of the aOtered permittiYity of the free space (with respect to the situation in which the enerJy density of Tuantum Yacuum is JiYen by eTuation (1)) and6 2 2 4 2 2 35 1 ˆ 2 DE g eg qvE qvE V Gc V E H r c U S c U r § § · · ¨¨ ' ¨ ' ¸ ¸¸ © ¹ © ¹ G = (57) can be defined as the JraYitostatic fieOd determined by both density of matter
and density of dark enerJy (here
2 eg
G H
c is the basic JraYitodynamic
con-stant)2. The JraYitostatic fieOd is Oinked with the Tuantum Yacuum condensate defined by eTuations (54) (and whose underOyinJ microscopic Jeometry is described by eTuations (17)-(19) and, at the cosmoOoJicaO OeYeO, by eTuations (22)-(24)) throuJh reOation 00 2 3 ˆ 1 ˆ. 8 eg g H V E h r G r
S
G (58)The totaO OaJranJian density for matter-fieOd interactions in the poOari]ed Yacuum is JiYen by reOation
2 2 2 3 2 0 0 0 0 1 1 / 2 g d g B m c v L q qA v r r K E c K K K G P H § § · · § · ¨ ¸ ¨ ¸ ) ¨ ¸ ¨ © ¹ ¸ © ¹ © ¹ G G G
2 2 2 2 1 / K K K c K t O § §w · · ¨¨ ¨w ¸ ¸¸ © ¹ © ¹ , (59)
2 In anaOoJy with SacharoY¶s JerminaO proposaO of treatment of JraYitation as ³met-ric eOasticity´ of space >3@.
where
), AG are the JraYitationaO potentiaOs, BG is the JraYitomaJnetic fieOd defined by 3 g eg J B H r G G , (60) (where JG L SG G, 6 2 2 4 2 35 2 DE qvE qvE V Gc V L r v c U S c U § § · · u¨¨ ' ¨ ' ¸ ¸¸ © ¹ © ¹ G G = , S G
beinJ the spin anJuOar momentum of the materiaO obMect determined by the diminishinJ of the Tuantum Yacuum enerJy density under consideration) and
4 32 c G
O
S
. It mustbe emphasi]ed that aOso the JraYitomaJnetic fieOd (60), by Yirtue of the Oink of the orbitaO anJuOar momentum of the materiaO obMect determined by the dimin-ishinJ of the Tuantum Yacuum enerJy density with the Tuantum Yacuum con-densate defined by eTuations (54) e[pressed by
00 3 ˆ , 8 V L r h v G
S
u G G (61)is itseOf associated with the Tuanti]ed metric of the Tuantum Yacuum conden-sate.
Now, in anaOoJy with Puthoff¶s poOari]abOe Yacuum modeO of JraYitation >6@, Yariation of the action functionaO with respect to the test particOe YariabOes Oeads to the foOOowinJ eTuation of motion of a test materiaO obMect of mass
m
0 in the poOari]ed 3D Tuantum Yacuum:2 3/ 2 2 0 2 0 0 2 2 1 / . 1 2 1 / / g g v m v m c d m c E v B c dt v v c c N N N N N N N ª º § · « » ¨ ¸ « » u © ¹ « » § · § · « ¨ ¸ » ¨ ¸ « © ¹ » © ¹ ¬ ¼ G G G G (62)
ETuation (62) shows that there are two forces actinJ onto the test particOe of mass m0: the Lorent] force due to the Tuantum Yacuum enerJy density sur-roundinJ the obMect and a second term representinJ the dieOectric force propor-tionaO to the Jradient of the shadowinJ of Tuantum Yacuum (56). The importance of this second term Oies in the fact that thanks to it one can account for the JraYi-tationaO potentiaO, either in Newtonian or JeneraO reOatiYistic form. It miJht be
interestinJ to note that with m0 o0 and v c
N
o , as wouOd be the case for a photon, the defOection of the traMectory is twice as the defOection of a sOow moYinJ massiYe particOe. This is an important indication of conformity with JeneraO reOatiYity.
Variation of the action functionaO with reJard to the
N
YariabOe Oeads to the e[pression of the Jeneration of the shadowinJ of the JraYitationaO space within JeneraO reOatiYity, owed to the presence of both matter and fieOds. The eTuation has three riJht-hand side terms:2 2 2 2 1 . 4 / t P Q R c
N
N
N
N
N
N
O
N
w ª¬ º¼ w (63)+ere P
N
represents the chanJe in space shadowinJ by the mass density associated with the obMect of mass m0, with the Yector rG as the distance from the system mass centre:2 2 3 0 0 2 1 / . 1 / v m c c K P K r r K v c K G § · ¨© ¸¹ § · ¨© ¸¹ G G (64)
Q
N
is the chanJe caused by the enerJy density of the fieOds (57) and (60) determined by the diminishinJ of the Tuantum Yacuum enerJy density:2 2 0 0 1 . 2 g g B Q
N
NH
ENP
§ · ¨ ¸ © ¹ (65)R
N
is the chanJe caused by the Tuantum Yacuum shadowinJ enerJy density itseOf:2 2 2 2 1 . / R t c
O
N
N
N
N
N
§ §w · · ¨¨ ¨ ¸ ¸¸ w © ¹ © ¹ (66) In the case of a static JraYity fieOd of a sphericaO mass distribution (a pOanetor a star), the soOution of eTuation (63) has a simpOe e[ponentiaO form:
2 /
GM rc
e
where V 2qvE
M c
U
'. The soOution (67) can be appro[imated by e[pandinJ it into a series: 2 2 2 / 2 2 2 1 2 1 . 2 GM rc GM GM e rc rc
N
§¨ ·¸ © ¹ ! (68) This soOution reproduces (to the appropriate order) the usuaOJeneraO-reOatiYistic Schwar]schiOd metric predictions in the weak fieOd Oimit conditions (i.e. soOar system).
AccordinJ to this modeO, it is important to underOine that aOso particOes with-out mass (for e[ampOe, photons) haYe an indirect infOuence on the Tuantum Yac-uum enerJy density. In fact, because of eTuation (65) aOso a photon wiOO add a contribution to the effectiYe curYature of space-time associated with the fieOds (57) and (60). This resuOt turns out to be aOso in accordance with JeneraO theory of reOatiYity, where both mass and enerJy cause the curYature of space-time.
MoreoYer, with the obtained soOution (67) or (68) reJardinJ the factor
N
measurinJ the poOari]abiOity of the Tuantum Yacuum in the presence of matter, one can anaOy]e the JraYitationaO red shift characteristic of JeneraO reOatiYity, and find inside this approach a more detaiOed form in order to obtain the freTuency shift of the photon emitted by an atom on the surface of a star of mass M and radius R. -ust Oike in Puthoff¶s modeO, the photon detected far away from the star wiOO appear red shifted by the foOOowinJ amount:0 2 0 0 GM Rc
Z Z
Z
Z
Z
' | , (69) where we haYe assumed GM2 1Rc . The photon, after haYinJ cOimbed up the
JraYity potentiaO of the star, wiOO retain its acTuired freTuency unchanJed, and the chanJe in freTuency can be tested OocaOOy by comparinJ it with photons emitted by the same type of atoms at the same temperature, but within the weak JraYity fieOd of the Oaboratory.
:ith that same resuOt it is aOso possibOe to anaOyse the amount of the bendinJ of OiJht rays from a distant star passinJ near a massiYe body, Oike in the cOassic JeneraO reOatiYity test performed by EddinJton¶s e[pedition durinJ the soOar ecOipse in May 1919. The OiJht ray from a distant star, whiOe passinJ cOose to the Sun, wiOO e[perience a JraduaO sOowinJ of waYefront YeOocity cominJ towards the Sun, and a JraduaO increasinJ YeOocity in OeaYinJ the Sun¶s JraYity fieOd. Because
N
increases cOoser to a massiYe body (N ! ), the YeOocity of OiJht wiOO 1 Yary asc
/
N
. The part of the waYefront cOoser to the Sun wiOO thus e[perience aJreater sOowdown than the part of the waYefront passinJ further away. This is seen from the Earth as an apparent shift of the position of the star cOose to the Sun¶s disk edJe in the outward direction. In JeneraO reOatiYity¶s terms, this de-fOection is a measure of OocaO space-time curYature. :e are interested in caOcuOat-inJ the totaO bendcaOcuOat-inJ anJOe. Because in case of the Sun the totaO defOection is smaOO (
M
2 arc-seconds) we can appOy the usuaO Oow anJOe appro[imations throuJhout the caOcuOation. And because of the same reason we wiOO not make a biJ mistake if we appro[imate the YariabOe YeOocity of OiJht to the first order term of the series e[pansion (66) ofN
:2 2 2 1 . 2 1 c c GM v c GM rc rc N § · | | ¨ ¸ © ¹ (70)
In this reOation the radius-Yector r denotes the distance of the waYefront from the centre of the Sun as it traYeOs by from
f
tof
, with the minimum dis-tance ofR
G
where R is the Sun¶s radius, andG
is the minimum distance from the Sun¶s surface. By assiJninJ z to the distance of the waYefront aOonJ the Oine of siJht (perpendicuOar toR
G
), the radius-Yector becomes2 2
r R
G
z , so the eTuation (70) can be written as:2 2 2 2 1 1 GM v c c R G z § · ¨ ¸ | ¨ ¸ © ¹ . (71)
The differentiaO YeOocity of OiJht, assuminJ G , is then R
3/ 2 2 2 2 2 . GM R v c R z
G
' (72)As the waYefront traYeOs a distance dz|vdt, the differentiaO YeOocity aOonJ the path of OiJht resuOts in an accumuOated waYefront path difference z' :
3/ 2 2 2 2 2 . GM R z vdt dz c R z
G
' ' | (73)This resuOts in an accumuOated tiOt anJOe of:
3/ 2 2 2 2 2 / GM R . z dz c R z
G
M
| 'G
| (74)By inteJratinJ eTuation (74) oYer the entire path
f f
z
yieOds: 2 4 . GM Rc S M | (75) By insertinJ 11 2 26,672 10
G
Nm Kg
, 301,9891 10
M
Kg
, and 8 6,96 10R m, we obtain
M
1,75 arc-seconds, which is e[actOy the YaOue predicted by Einstein¶s JeneraO theory of reOatiYity in 1915, and e[perimentaOOy Yerified by EddinJton in 1919 (between 1.2 and 1.9 arc-seconds, mainOy because of the imperfect optics of the portabOe teOescopes used).MoreoYer, as reJards the eTuations of motion (62) and (63), it is important to emphasi]e that, accordinJ to this approach, the modification of the Tuantum Yacuum enerJy density determininJ both the matter density and dark enerJy density and the action of the shadowed Tuantum Yacuum on another materiaO obMect are phenomena directOy determined by the fieOds (60), (64), (65) and (66). This impOies that no time is needed to transmit the information from a materiaO obMect to the surroundinJ reJion in order to shadow the JraYitationaO space be-cause the chanJe of the Tuantum Yacuum enerJy density is aOready there as it is associated with the fieOds (60), (64), (65) and (66) (what propaJates from point to point is Must the actuaO effects of this chanJe); and, on the other hand, that no time is needed to transmit the information from the shadowed JraYitationaO space to another materiaO obMect in order to cause its moYement.
)inaOOy, accordinJ to the Yiew proposed here, the 3D Tuantum Yacuum as a direct medium for the transmission of JraYitation estabOished by eTuations (64), (65) and (66) can e[press in an eOeJant mathematicaO way the perspectiYe about the non-e[istence of JraYitationaO waYes. In this reJard, it seems compatibOe with some LoinJer¶s resuOts accordinJ to which JraYitationaO waYes are onOy hypo-theticaO and do not e[ist in the physicaO worOd >30, 31@. On the other hand, de-spite seYeraO attempts of research about the JraYitationaO fieOd performed since the 1960s (see for e[ampOe the reference >32@), JraYitationaO waYes haYe not yet been detected. As underOined by Schorn in the paper >33@, ³To search for JraYita-tionaO waYes in a Oaboratory, cOassicaO or Tuantum mechanicaO detectors can be used. Despite the e[periments of :eber (1960 and 1969) and many others (BraJinskiM et aO., 1972; DreYer et aO., 1973; LeYine and *arwin, 1973; Tyson, 1973; MaischberJer et aO., 1991; AbramoYici et aO., 1992; and AbramoYici et aO., 1996) and theoreticaO caOcuOations and estimations (BraJinski and Rudenko, 1970; +arry et aO., 1996; and Schut], 1997), JraYitationaO waYes haYe neYer been obserYed directOy in Oaboratory´.
It is aOso interestinJ to obserYe that recent NASA research confirms that uni-YersaO space is fOat with onOy a 0.4 marJin of error which is a stronJ indication that curYature of space in JeneraO theory of reOatiYity is onOy a mathematicaO description of enerJy density of uniYersaO space which oriJinates in a more
fun-damentaO enerJy density of Tuantum Yacuum >34@. NASA measurements reJard-inJ the Jeometry of uniYersaO space turn out to be compOeteOy in aJreement with the approach deYeOoped in this paper.
5. CONCLUSIONS
A modeO of a three-dimensionaO Tuantum Yacuum has been proposed in which the curYature of space-time emerJes, in the hydrodynamic Oimit of some underOyinJ theory of a microscopic structure of space-time, as a mathematicaO YaOue of a more fundamentaO actuaO enerJy density of Tuantum Yacuum. The fOuctuations of the Tuantum Yacuum enerJy density Jenerate a curYature of space-time simiOar to the curYature produced by a ³dark enerJy´ density and produce a shadowinJ of the JraYitationaO space which determines the motion of other materiaO obMects present in the reJion under consideration. In this approach, the interestinJ perspectiYe is opened that the three-dimensionaO Tuantum Yacuum acts as a direct medium of JraYitation: at a fundamentaO OeYeO, no time (as dura-tion) is needed to transmit JraYity force. A JiYen materiaO obMect diminishes enerJy density of Tuantum Yacuum which Jenerates curYature of space-time. *raYity does not act directOy between massiYe obMects, JraYity acts in the Tuan-tum Yacuum: the chanJes of the TuanTuan-tum Yacuum enerJy density cause curYa-ture of space-time which Jenerate JraYitationaO attraction between massiYe bod-ies. This Yiew does not reTuire e[istence of hypotheticaO JraYiton as a ³carrier´ of JraYity.
RE)ERENCES
1. RuJh S. E. and ZinkernaJeO +. (2002) Studies in +istory and PhiOosophy of Modern Physics 33, 4, 663-705.
2. TimasheY S. ). (2011) Physical vacuum as a system manifesting itself on various scales – from nuclear physics to cosmology, ar;iY:1107.pdf >Jr-Tc@.
3. SacharoY A. D. (1967) DokOady Akad. Nauk S.S.S.R. 177, 1, 70-71.
4. Misner C. :., Thorne K. S. and :heeOer -. S. (1971) Gravitation, )reeman, New York.
5. Rueda A. and +aisch B. (2005) Gravity and the quantum vacuum inertia hypothesis, ar;iY:Jr-Tc0504061Y3.
6. Puthoff +. E. (2002) )oundations of Physics 32, 6, 927-943. 7. ConsoOi M. (2009) Physics Letters B 672, 3, 270-274.
8. ConsoOi M. (2009) Ultraweak excitations of the quantum vacuum as physical models of gravity, ar;iY:0904.1272Y2 >Jr-Tc@.
9. ConsoOi M. (2011) The vacuum condensates: a bridge between particle physics to gravity?. in: Vision of oneness, I. Licata and A. SakaMi (eds), Aracne Editrice, Roma.
10. SorOi A. (2012) -ournaO of AdYanced Physics 1, 1, 110-112. 11. ZeOdoYich Yu B. (1967) Zh. Eksp. Teor. )i]. Pis¶ma 6, 883-884. 12. SacharoY A. D. (2000) *eneraO ReOatiYity and *raYitation 32, 365-367.
13. CaOiJiuri L. M. (2014) The emergence of spacetime and gravity: Entropic of Geometro-Hydrodynamic process? A comparison and critical review, Quantum Matter, speciaO Issue ³The Tuantum worOd and the Jeometry of space´ 3, 3, 246-252.
14. :iOc]ek ). (2012) Origins of Mass, http://ar[iY.orJ/pdf/1206.7114.pdf. 15. *hao S. (2010) Why gravity is fundamental, ar;iY:1001-3029Y3. 16. NJ Y. -ack (2007) Physics Letters B 657, 10-14.
17. NJ Y. -ack (2008) Entropy 10, 441-461.
18. NJ Y. -ack (2010) Holographic quantum foam, ar;iY:1001.0411Y1 >Jr-Tc@. 19. NJ Y. -ack (2011), Various facets of spacetime foam, ar;iY:1102.4109.Y1 >Jr-Tc@. 20. *. ¶t +ooft (1993) in: SaOamfestschrift; A. AOi et aO. (eds), :orOd Scientific
PubOishinJ, SinJapore.
21. Susskind L. (1995) -ournaO of MathematicaO Physics 36, 6377-6396. 22. Bekenstein -. D. (1973) PhysicaO ReYiew D 7, 2333-2346.
23. +awkinJ S. (1975) Communications in MathematicaO Physics 43, 199-220. 24. *iddinJs S. B. (1992) PhysicaO ReYiew D 46, 1347-1352.
25. Bousso R. (2002) ReYiews in Modern Physics 74, 825-874.
26. Ma]ur P. O. and MottoOa E. (2004) Proc. Nat. Acad. Sci. 101, 26, 9545-9550. 27. Santos E. (2009) Quantum vacuum fluctuations and dark energy, ar;iY:0812.
4121Y2 >Jr-Tc@.
28. Santos E. (2011) InternationaO -ournaO of TheoreticaO Physics 50, 7, 2125-2133. 29. Santos E. (2010) Dark energy as a curvature of space-time induced by quantum
vacuum fluctuations ar;iY:1006.5543.
30. LoinJer A. (1998) The Gravitational Waves Are Fictitious Entities, http://[[[.OanO.JoY/abs/astro-ph/9810137.
31. LoinJer A. (2004) The Gravitational Waves Are Fictitious Entities-II, http://ar[iY.orJ/Yc/astro-ph/papers/9904/9904207Y1.pdf.
32. CiufoOini I. and *orini V. (2004) Gravitational Waves, Theory and Experiment (An Overview), http://bookmarkphysics.iop.orJ/fuOObooks/0750307412/ciufoOinioYer.pdf. 33. Schorn +.--. (2001) InternationaO -ournaO of TheoreticaO Physics 40, 8, 1427-1452. 34. NASA, http://map.Jsfc.nasa.JoY/uniYerse/uniBshape.htmO (2013).
