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SHMEIWSEIS KOSMOLOGIAS

MIQALHS TSAMPARLHS 1

AJHNA 2013

1Email: [email protected]

Keflaio 1

KosmografÐa

1.1 To tupikì kosmikì sÔsthma suntetagmènwn

Sthn kosmologÐa jewroÔme ìti to sÔmpan (dhlad h Ôlh sto sÔmpan) mporeÐ na jewrhjeÐ wcèna fusikì sÔsthma to opoÐo qarakthrÐzoume me mia seir apì upojèseic ¸ste na eÐnai dunat h`melèth' tou sta plaÐsia miac jewrÐac Fusik c. To sÔnolo twn upojèsewn pou pragmatopoioÔnaut n th dunatìthta to onomzoume Kosmologik Arq (Cosmological Principle). Hje¸rhsh miac tètoiac Arq c eÐnai anagkaÐa gia polloÔc lìgouc merikoÐ apì touc opoÐouc eÐnai:a. To sÔmpan eÐnai to monadikì fusikì sÔsthma tou eÐdouc toub. Oi mìnoi parathrhtèc eÐmaste emeÐc, ra den eÐnai dunat h epal jeush h apìrriyh twnapìyewn kai twn sumperasmtwn mac gi autìg. EÐmaste mèrh tou fusikoÔ sust matoc epomènwc h antÐlhy mac kai oi parathr seic mac eÐnaiapotèlesma thc eswterik c dom c tou.

H arqik upìjesh pou eisgei h Kosmologik Arq afìr ta stoiqeÐa tou sÔmpantoc.

1.1.1 To kosmikì reustì

To sÔmpan (dhlad h Ôlh kai ta pedÐa pou parathroÔme / sungoume) mporeÐ na jewrhjeÐ wc ènakosmikì aèrio to opoÐo:a. PlhroÐ ìlo to sÔmpan me ton Ðdio trìpo pou ta mìria enìc aerÐou plhroÔn to q¸ro tou doqeÐousto opoÐo brÐskontaib. Ta toma tou aerÐou eÐnai oi galaxÐec.g. Sth shmerin tou fsh to kosmikì aèrio èqei dÔo idiìthtec:

1. EÐnai omogenèc. H omogèneia tou kosmikoÔ aerÐou prèpei na katanohjeÐ ìpwc h omogèneiaenìc aerÐou, dhlad perigraf me makroskopikèc idiìthtec me metablhtèc - paramètrouc poumetr¸ntai se perioqèc tou q¸rou me distash polÔ megalÔterh apì th mèsh diadrom twnatìmwn/morÐwn tou aerÐou. Sto sÔmpan h antÐstoiqh mèsh diadrom eÐnai perioqèc diamètrou108 èwc 109 et¸n fwtìc, oi opoÐec eÐnai arket meglec ¸ste na perièqoun sm nh (clusters)galaxi¸n.

2. H katanom twn galaxi¸n (sthn Ðdia klÐmaka diamètrwn) gÔrw apì kje galaxÐa stontrisdistato q¸ro eÐnai isìtroph, isodÔnama, sfairik summetrik .

3

4 KEFALAIO 1. KOSMOGRAFIA

H perigraf thc Fusik c apaiteÐ suntetagmènec sust mata anaforc. H epìmenh upìjeshthc Kosmologik c Arq c afor thn Ôparxh enìc tètoiou sust matoc suntetagmènwn.

1.1.2 To tupikì kosmologikì sÔsthma suntetagmènwn

Uprqei èna tupikì kosmologikì sÔsthma suntetagmènwn (cosmic standard co-ordinate system) to opoÐo eÐnai to sÔsthma suntetagmènwn pou pragmatopoioÔme tic para-thr seic mac. To sÔsthma autì qrhsimopoieÐtai wc to apìluto sÔsthma anaforc, sto opoÐoanafèrontai ìlec oi kosmologikèc kin seic kai apoteleÐ to sÔsthma suntetagmènwn thckosmografÐac. To sÔsthma autì kataskeuzetai wc akoloÔjwc.

1. H arq tou sust matoc eÐnai sto kèntro tou galaxÐa mac llo shmeÐo sto galaxÐa mac(Milky Way).

2. Oi qwrikèc grammèc suntetagmènwn kajorÐzontai wc akoloÔjwc:

(aþ) Oi dieujÔnseic twn qwrik¸n suntetagmènwn sthn arq twn axìnwn kajorÐzontai apìtreic anexrthtec dieujÔnseic, oi opoÐec orÐzontai me tic fainìmenec dieujÔnseic (linesof sight) kpoiwn tupik¸n makrin¸n galaxi¸n apì to galaxÐa mac.Apì thn parat rhshèqei prokÔyei ìti oi tupikoÐ galaxÐec èqoun sugkekrimènh stajer jèsh sthn ourniasfaÐra. Epiplèon den eÐnai ìloi oi galaxÐec tupikoÐ galaxÐec.

(bþ) Oi grammèc suntetagmènwn ston 3-q¸ro kajorÐzontai apì tic gewdesiakèc (wc procth metrik pou ja kajoristeÐ), oi opoÐec sundèoun thn arq me touc epilegmènouctupikoÔc galaxÐec

(gþ) Oi qwrikèc apostseic kat m koc twn gramm¸n suntetagmènwn pou anafèrame kajo-rÐzontai apì tic fainìmenec lamprìthtec (apparent luminosity) ( me llakatllhla mètra apìstashc)twn antÐstoiqwn tupik¸n galaxi¸n ìpwc parathroÔntaisthn arq

3. Oi qwrikèc suntetagmènec enìc opoioud pote shmeÐou sto sÔmpan kajorÐzontai wc procautì to qwrikì sÔsthma suntetagmènwn me thn tom twn antÐstoiqwn qwrik¸n gramm¸noi opoÐec orÐzoun to qronikì shmeÐo. To kosmikì sÔsthma suntetagmènwn eÐnai sunki-noÔmeno (comoving) giatÐ se autì to sÔsthma suntetagmènwn oi tupikoÐ galaxÐec (ìqiìloi) èqoun stajerèc qwrikèc suntetagmènec.

4. Th qronik suntetagmènh tou tupikoÔ kosmologikoÔ sÔsthmatoc suntetagmènwn onom-zoume kosmologikì qrìno (cosmic time). O qrìnoc autìc orÐzetai me èna kosmikìrolìi to opoÐo mporeÐ na eÐnai opoiod pote kosmologikì bajmwtì pedÐo tou opoÐou oi timècmetabllontai kat suneq kai monos manto trìpo. Oi arijmhtikèc timèc tou bajmwtoÔpedÐou se kje qwrikì shmeÐo mac dÐnoun tic qronikèc stigmèc twn gegonìtwn se autì toqwrikì shmeÐo. H parat rhsh deÐqnei ìti tètoia pedÐa ufÐstantai kai eÐnai p.q. h energeiak puknìthta tou sÔmpantoc (proper energy density), h jermokrasÐa mèlanoc s¸matoc thcaktinobolÐac upobjrou (back body radiation temperature) kai lla.

To sÔsthma suntetagmènwn pou kataskeuzetai apì tic anwtèrw qwrikèc suntetagmèneckai th qronik suntetagmènh onomzoume tupikì kosmikì sÔsthma suntetagmènwn(cosmic standard coordinate system).

1.2. QWROQRONIKH PERIGRAFH 5

1.2 Qwroqronik perigraf

1.2.1 To kosmologikì reustì

H qwroqronik perigraf apoteleÐ to plaÐsio thc JewrÐac thc Genik c Sqetikìthtac, epomènwcprokeimènou to kosmikì reustì na perigrafeÐ qwroqronik prèpei ta difora stoiqeÐa tou naantistoiqhjoÔn me gewmetrik antikeÐmena.

JewroÔme ìti to upìbajro sto opoÐo ja perigrafeÐ to kosmologikì reustì eÐnai ènac q¸roc(Riemann), tessrwn diastsewn me metrik gab me qarakt ra (Lorentz), ton opoÐo onomzoumeqwrìqrono.

SÔmfwna me thn Kosmologik Arq h Ôlh sto sÔmpan perigrfetai apì to kosmologikìreustì, (cosmic fluid) . Sth JewrÐa thc Genik c Sqetikìthtac to kosmologikì reustì eÐnaimia dèsmh qronik¸n kosmik¸n gramm¸n (timelike congruence), oi opoÐec eÐnai gewdesiakèc thcmetrik c tou qwroqrìnou, kai èqoun ta akìlouja qarakthristik:

1. EÐnai qronikèc wc proc th jewroÔmenh metrik

2. ApoteloÔn tic oloklhrwtikèc grammèc enìc peira diaforÐsimou dianusmatikoÔ pedÐou. Hdiaforisimìthta tou pedÐou apaiteÐtai prokeimènou na uprqei pnta mia arket mikr pe-rioq sthn opoÐa oi grammèc den tèmnontai. Epomènwc topik kje gramm tou reustoÔkajorÐzetai monos manta apì treic arijmoÔc pou eÐnai oi qwrikèc suntetagmènec enìc o-poioud pote shmeÐou touc.

3. Oi kosmikèc grammèc twn shmeÐwn tou reustoÔ eÐnai afinik parametrismènec kampÔlec

4. H sÔndesh thc sqetikistik c perigraf c tou kosmologikoÔ reustoÔ me thn parat rhshgÐnetai apì thn antistoÐqhsh twn kosmik¸n gramm¸n tou reustoÔ me touc kosmikoÔc pa-rathrhtèc. Oi parathrhtèc autoÐ orÐzontai apì tic apait seic:

a. H tetrataqÔthta touc na eÐnai to monadiaÐo efaptìmeno kat m koc twn qronik¸nkosmik¸n gramm¸n thc Ôlhc me mza (dhlad ìqi ta fwtìnia)

b. H tetrataqÔthta touc eÐnai oi tetrataqÔthtec twn tupik¸n galaxi¸n.

O idiìqronoc aut¸n twn kosmik¸n parathrht¸n jewroÔme ìti tautÐzetai me thn afinik par-metro kat m koc twn kosmik¸n gramm¸n. To qwroqronikì reustì onomzoume reustì twn(sqetikistik¸n) parathrht¸n.

H fusik melèth thc kosmologÐac afor th melèth tou reustoÔ twn (sqetikistik¸n) parath-rht¸n. H melèth aut mporeÐ na gÐnei plèon se epÐpedo Diaforik c GewmetrÐac kai na melet seikpoioc autì pou onomzoume Sqetikistik KosmologÐa. (Relativistic Cosmology)

Sto shmeÐo autì èqoume dh ta pr¸ta mèsa na efarmìsoume sumpersmata thc Diaforik cGewmetrÐac, ta opoÐa ja mac stajeropoi soun to jewrhtikì all kai to parathrhsiakì montèlothc KosmologÐac.

Apì th Diaforik GewmetrÐa eÐnai gnwstì ìti apì èna shmeÐo enìc q¸rou Riemman kjedieÔjunsh orÐzei sthn perioq tou shmeÐou mia monos manth gewdaisiak . Autì shmaÐnei ìti sekje gegonìc sto qwrìqrono o kje kosmikìc parathrht c eÐnai monos manta orismènoc apì thntetrataqÔtht tou. IsodÔnama autì shmaÐnei ìti en se èna shmeÐo dojoÔn ta trÐa kateujÔnontasunhmÐtona thc dieÔjunshc thc qwrik c taqÔthtac tou parathrht tìte kje shmeÐo thc kosmik c


gramm c tou parathrht qarakthrÐzetai monos manta apì ton idiìqronì tou. Praktik autìsunepgetai ìti kje kosmik gramm tou kosmikoÔ reustoÔ kajorÐzetai apì èna shmeÐo stoqwrìqrono, mia bajmwt posìthta, s èstw, kai treic paramètrouc y1, y2, y3. Gia to lìgo autìgrfoume th dèsmh twn kosmologik¸n parathrht¸n wc xa(s, yµ). H parat rhsh aut faÐnetaiaj¸a sto shmeÐo autì ìmwc, ìpwc ja deÐxoume argìtera, eÐnai basikì stoiqeÐo thc sÔndeshc thcjewrÐac me thn parat rhsh.

1.3 H Arq Sqetikìthtac thc KosmologÐac

To anwtèrw senrio den perièqei akìma upojèseic sqetik me ta difora fusik pedÐa kai kurÐwcto pedÐo barÔthtac, to opoÐo èqoume upojèsei ìti perigrfetai apì th metrik tou qwroqrìnou.Oi upojèseic autèc prèpei na eÐnai pli gewmetrikèc kai na diatupwjoÔn ètsi ¸ste na mporeÐ nasundejeÐ h parat rhsh me th jewrÐa. H basik mejodologÐa pou èqoume gia autèc tic peript¸seiceÐnai oi summetrÐec thc metrik c (me thn eureÐa ènnoia).

To basikì stoiqeÐo thc sqetikistik c Fusik c eÐnai h isodunamÐa twn susthmtwn sunte-tagmènwn ìpwc aut ekfrzetai apì thn arq tou sunalloi¸tou, dhlad thn apaÐthsh ìti tafusik megèjh kaj¸c kai oi exis¸seic thc Fusik c ja perigrfontai apì tanustèc ( genikìteraapì gewmetrik antikeÐmena wc proc mia sugkekrimènh omda metasqhmatism¸n). Sth jewrÐathc Genik c Sqetikìthtac aut h omda twn metasqhmatism¸n afor ìlouc touc metasqhmati-smoÔc suntetagmènwn sto qwrìqrono. Onomzetai omda apeikìnishc pollaplìthtac(manifold mapping group) kai eÐnai autì pou pollèc forèc anafèretai isodÔnama ìti h JewrÐathc Genik c Sqetikìthtac isqÔei gia tuqaÐo sÔsthma suntetagmènwn.

H efarmog thc JewrÐac thc Genik c Sqetikìthtac sth melèth twn kosmologik¸n fainomènwnparousizei mia idiomorfÐa. Prgmati sthn kosmologÐa oi parathr seic gÐnontai mìno apì ènanparathrht (emc) kai se èna sÔsthma suntetagmènwn, to kosmikì sÔsthma suntetagmènwn poukataskeusame sto edfio 1.1. Epomènwc den uprqei toulqiston ènac deÔteroc parathrht c,o opoÐoc ja epalhjeÔsei ja aporrÐyei ta sumpersmat / parathr seic mac. Autì sunep-getai ìti den eÐnai dunat h diatÔpwsh miac (Kosmologik c) Arq c Sqetikìthtac, h opoÐa jajemeli¸sei th diadikasÐa bsh thc opoÐac mia parat rhsh ja eÐnai apodekt aporriptèa. Prg-mati kje Arq Sqetikìthtac kajorÐzei touc draneiakoÔc ' parathrhtèc miac jewrÐac, dhlad touc parathrhtèc oi opoÐoi pistopoioÔn thn ntikeimenikìthta' twn parathr sewn sta plaÐsiathc dedomènhc jewrÐac1. Epomènwc h kje Arq Sqetikìthtac eÐnai to jemèlio pnw sto opoÐobasÐzetai h jewrÐa thc Fusik c pou thn qrhsimopoieÐ. Gia pardeigma h Arq thc Sqetikìthtactou GalilaÐou orÐzei touc Neut¸neiouc adraneiakoÔc parathrhtèc pnw stouc opoÐouc anaptÔs-setai h Neut¸neia Fusik jewrÐa. 'Omoia h Arq thc Sqetikìthtac tou Einstein kajorÐzei toucsqetikistikoÔc adraneiakoÔc parathrhtèc gia touc opoÐouc diatup¸netai h JewrÐa the Eidik cSqetikìthtac. H `posotikopoÐhsh' kje Arq c Sqetikìthtac gÐnetai mèsw twn isommetri¸n thcmetrik c thc jewrÐac, toulqiston gia tic kajierwmènec jewrÐec thc Fusik c. Th bajÔterhgewmetrik shmasÐa thc isometrÐac miac metrik c ja anaptÔxoume sto epìmeno edfio.

1Δεν υπάρχει απόλυτη αλήθεια με την έννοια ότι δεν υπάρχει μια και μοναδική θεωρία της ϕυσικής, η οποίααπορρέει από την ΑΠΟΛΥΤΗ ΑΡΧΗ ΣΧΕΤΙΚΟΤΗΤΑΣ και ερμηνεύει όλα τα ϕυσικά ϕαινόμενα. Μέχρι ναβρεθεί - πράγμα που προσωπικά πιστεύω ότι δε θα γίνει ποτέ - όλες οι αρχές σχετικότητας που διατυπώνουμε καιοι συνακόλουθες θεωρίες που αναπτύσσουμε θα αϕορούν ένα συγκεκριμένο υποσύνολο ϕυσικών ϕαινομένων καιπαρατηρητών.

1.3. H ARQH SQETIKOTHTAS THS KOSMOLOGIAS 7

H jewrÐa thc Genik c Sqetikìthtac den èqei mia Arq Sqetikìthtac kai toÔto giatÐ eÐnaimia gewmetrik jewrÐa thc Fusik c, dhlad eÐnai mia platfìrma (upìbajro) pnw sth opoÐa k-poioc mporeÐ na kataskeuzei jewrÐec fusik c jewr¸ntac mia Arq Sqetikìthtac. Autìc eÐnaikai o lìgoc pou h jewrÐa thc Genik c Sqetikìthtac apodèqetai OLOUS touc parathrhtèc wcadraneiakoÔc parathrhtèc isodÔnama èqei omda isometri¸n ìlouc touc dunatoÔc metasqh-matismoÔc (dhlad den epilègei omda, apl lèei ìti uprqei omda kai eÐnai h omda ìlwn twnomdwn Lie).

Epomènwc prokeimènou h kosmologÐa na gÐnei mia jewrÐa fusik c qreizetai mia Arq Sqe-tikìthtac. H anuparxÐa Ôparxhc parathrht¸n pèra apì emc kaj¸c kai h adunamÐa ektèleshcpeiramtwn mac odhgeÐ sth monadik epilog pou eÐnai h diatÔpwsh miac Arq c Sqetikìthtac, hopoÐa ja basÐzetai sth dik mac upokeimenik antÐlhyh kai mìnon. All autì den mac enoqleÐmia kai den uprqei deÔteroc parathrht c, o opoÐoc ja epalhjeÔsei ja amfisbht sei ta sum-persmata mac, en¸ tautìqrona mac epitrèpei na anaptÔxoume thn KosmologÐa wc èna kldo thcFusik c. To tÐmhma eÐnai ìti ja sthriqtoÔme sthn al jeia tou enìc pou eÐnai antÐjeto sto pioshmantikì epiqeÐrhma thc Fusik c, dhlad to apoproswpopoihmèno thc episthmonik c mejìdou.Den gÐnetai ìmwc na knoume diaforetik kai apì to na mhn knoume tÐpota proqwroÔme se miamelèth pou Ðswc odhg sei se tragik ljh kai ektim seic, all mporeÐ na beltiwjeÐ stadiak kaise teleutaÐa anlush afor antikeÐmena (galaxÐec kai sm nh galaxi¸n) ta opoÐa eÐnai pèra apìkje diastatik fantasÐws mac.

H plèon kajierwmènh Arq thc Sqetikìthtac sthn KosmologÐa s mera apoteleÐ mèroc miacsunolik c upìjeshc, h opoÐa onomzetai Kosmologik Arq , kai mporeÐ na diatupwjeÐ wc ako-loÔjwc2

Arq thc Sqetikìthtac thc KosmologÐac: Mèroc ISto kosmikì sÔsthma suntetagmènwn xi, èstw, èna tuqaÐo fusikì mègejoc, to opoÐo peri-grfetai apì ton tanust 3T I

J , èqei suntetagmènec TIJ (xi). Tìte èna llo sÔsthma kosmik¸n

suntagmènwn x′i j a jewreÐtai ìti eÐnai isodÔnamo me to kosmikì sÔsthma suntetagmènwnwc proc kpoia omda (Lie) metasqhmatism¸n suntetagmènwn, en oi suntetagmènec tou ÐdioufusikoÔ megèjouc T ′I′

J ′ (xi′) sto xi′ sÔsthma suntetagmènwn ikanopoioÔn th sqèsh:

T ′I′J ′ (xi

′)(P ) = T I

J (xi)(P ). (1.1)

H apaÐthsh aut shmaÐnei ìti se kje shmeÐo P tou qwrìqronou h posìthta T ja ekfrzetaime thn Ðdia sunarthsiak morf tìso sto sÔsthma xi ìso kai sto sÔsthma x′i dhlad jadiathreÐtai h morf thc sunarthsiak c exrthshc (invariant form) kai mlista oi antÐstoiqecsunist¸sec sto Ðdio qwroqronikì shmeÐo èqoun thn Ðdia tim .

Sumperasmatik mporoÔme na poÔme ìti h kosmologÐa pou ja anaptÔxoume (kai pou eÐnai aut pou ufÐstatai s mera) eÐnai mia jewrÐa, h opoÐa basÐzetai en mèrei sth logik mac (dhlad stontrìpo pou skeptìmaste) kai en mèrei stic parathr seic mac kai thn ermhneÐa pou touc dÐnoume.Epanalambnoume ìti to kti eÐnai perissìtero apì to tÐpota kai autì eÐnai h kalÔterh dikaiologÐapou mporoÔme na broÔme. Apì thn llh prèpei na eÐmaste epifulaktikoÐ kai prosektikoÐ sthn

2Βλέπε και S. Weinberg ”Gravitation and Cosmology: Principles and Applications of the General Theory ofRelativity”, Willey 1972, p.409

3΄Οπου δεν αναϕέρουμε το σύνολο των δεικτών αναλυτικά αναϕερόμαστε γενικά μόνο στην τάξη του τανυστή.


apodoq sthn apìrriyh protsewn / ide¸n / ermhnei¸n oi opoÐec den `faÐnontai' mesa isqÔousec `logikèc '.

1.3.1 O Kosmikìc qrìnoc

Ektìc apì thn anuparxÐa perissìterwn tou enìc parathrht¸n h KosmologÐa èqei kai llh i-diaiterìthta. Melet èna fusikì sÔsthma, to sÔmpan, to opoÐo eÐnai to monadikì! Epiplèon oiparathrhtèc (emeÐc) eÐnai mèrh autoÔ tou sust matoc. Epomènwc oi ìpoioi qarakthrismoÐ tousÔmpantoc wc fusikì sÔsthma, aforoÔn tic parathr seic mac kai mìnon kai apoktoÔn antikeime-nikìthta mèsa apì thn Arq thc Sqetikìthtac thc KosmologÐac.

Oi kosmologikèc parathr seic mac deÐqnoun ìti uprqoun bajmwt kosmologik pedÐa ìpwch energeiak puknìthta kai h jermokrasÐa mèlanoc s¸matoc, ta opoÐa mei¸noun diark¸c thn tim touc. Ta pedÐa aut mporoÔme na ta qrhsimopoi soume wc eswterik rolìgia tou sÔmpantoc.Rolìi shmaÐnei qrìnoc epomènwc ta pedÐa aut mporoÔn na qrhsimopoihjoÔn prokeimènou na o-rÐsoun/ metr soun hlikÐa tou sÔmpantoc. H ènnoia tou qrìnou eÐnai basik sthn anjr¸pinhkatanìhsh thc exèlixhc twn fusik¸n susthmtwn.Kosmikìc qrìnocUprqei bajmwtì kosmologikì pedÐo, tou opoÐou oi timèc proèrqontai apì metr seic se parath-roÔmena fusik kosmologik pedÐa sto tupikì kosmikì sÔsthma suntetagmènwn. Tic timèc kjetètoiou pedÐou onomzoume kosmikì qrìno.

JewroÔme kat' arq n èna bajmwtì pedÐo S, to opoÐo metr ton kosmikì qrìno, t èstw.Epeid to pedÐo eÐnai bajmwtì prèpei na ikanopoieÐ thn Arq thc Sqetikìthtac thc KosmologÐac,apaitoÔme na èqei thn Ðdia èkfrash se ìla ta sust mata suntetagmènwn. Epiplèon epeid to SkajorÐzei to qrìno (se ìla ta sust mata suntetagmènwn) prèpei na eÐnai sunrthsh mìnon tout′, t se dÔo kosmik sust mata suntetagmènwn x′i kai xi antÐstoiqa. 'Ara èqoume:

S(t′) = S(t) (1.2)

apì ìpou prokÔptei:

t′ = t. (1.3)

SumperaÐnoume ìti wc apotèlesma thc Arq thc Sqetikìthtac thc KosmologÐac ìla ta ko-smologik sust mata suntetagmènwn, ta opoÐa eÐnai isodÔnama me to kosmikì sÔsthma sunte-tagmènwn - wc proc thn omda isometri¸n pou ja oristeÐ- qrhsimopoioÔn ton Ðdio kosmikì qrìnot. To sumpèrasma autì èqei sobarèc sunèpeiec tìso se parathrhsiakì ìso kai se gewmetrikìepÐpedo.

Se epÐpedo gewmetrÐac shmaÐnei ìti o qwrìqronoc pou jesmojeteÐ h Kosmologik Arq mporeÐna kalufjeÐ me mia monoparametrik oikogèneia mh temnìmenwn trisdistatwn uperepifanei¸n(aut h idiìthta eÐnai gnwst me to ìnoma fullopoÐhsh (foliation) kai mÐa pollaplìthtame aut n thn idiìthta onomzetai diaspsimh (decomposable) ). H parmetroc pou arijmeÐtic uperepifneiec eÐnai o kosmikìc qrìnoc kai h kje uperepifneia jewroÔme ìti perigrfeiton trisdistato q¸ro kje kosmik qronik stigm . To eÐdoc twn uperepifanei¸n (dhlad hgewmetrÐa touc) kaj¸c kai to pwc h mia uperepifneia susqetÐzetai me tic geitonikèc thc den èqeikajoristeÐ akìma. Autì ja gÐnei me thn apaÐthsh summetri¸n sto reustì twn parathrht¸n kaitou 3-q¸rou, isodÔnama me ton orismì thc metrik c tou tupikoÔ 3-q¸rou.


1.3.2 Arq Sqetikìthtac thc KosmologÐac Mèroc II

Prokeimènou na oloklhrwjeÐ h Arq Sqetikìthtac thc KosmologÐac prèpei na kajoristeÐ homda (Lie), h opoÐa ja kajorÐzei thn isodunamÐa twn kosmik¸n susthmtwn suntetagmènwn.SuneqÐzontac thn empeirÐa mac apì th Neut¸neia Fusik kai th JewrÐa thc Eidik c Sqetikì-thtac apaitoÔme h omda aut na orÐzetai apì tic isometrÐec thc metrik c, ìpwc aut ektimtaisto kosmikì sÔsthma suntetagmènwn pou knoume tic parathr seic mac. Dhlad sto kosmikìsÔsthma suntetagmènwn ja orÐsoume th metrik me bsh tic parathr seic mac kai met ja upo-logÐsoume thn omda isometri¸n (Lie), h opoÐa ja kajorÐsei kai th jewrÐa thc kosmologÐac pouja anaptÔxoume.

1.3.3 Oi summetrÐec thc metrik c

'Opwc exhg jhke sto prohgoÔmeno edfio h omda summetri¸n pou ja kajorÐsei thn isodunamÐatwn kosmik¸n susthmtwn suntetagmènwn ja jemeliwjeÐ apì th metrik tou tupikoÔ 3-q¸rou.Prin proqwr soume ja embajÔnoume thn prosèggis mac sthn ènnoia thc metrik c prokeimènouna gÐnoun antilhpt sthn èktash kai to bjoc pou apaiteÐtai ta ìsa apaiteÐ h Kosmologik Arq .

H ènnoia thc isometrÐac

JewroÔme èna q¸ro me distash n+n2 ìpou oi n diastseic aforoÔn èna q¸ro (pollaplìthta)me suntetagmènec x1, . . . , xn kai oi n2 diastseic aforoÔn mia pollaplìthta me suntetagmènec4

yij. JewroÔme n2 sunart seic twn xi, tic opoÐec sumbolÐzoume me gij me tic idiìthtec:a. Oi sunart seic gij eÐnai oi sunist¸sec enìc tanust txhc (0, 2)b. gij = gjig. detgij = 0.

Sto q¸ro xi, yij jewroÔme ton upìqwro pou orÐzetai me thn apaÐthsh gij = yij ìpou gij eÐnaio antalloÐwtoc tanust c txhc (2, 0), tou opoÐou oi sunist¸sec gij orÐzontai me thn apaÐthshgijgjk = δik. O upìqwroc autìc èqei distash n (ìsec kai oi eleÔjerec parmetroi x1, . . . , xn )kai ton sumbolÐzoume g. JewroÔme t¸ra èna metasqhmatismì suntetagmènwn:

xi → xi = f i(xj).

Epeid oi sunart seic gij eÐnai sunist¸sec tanust ja èqoume:

gij =∂f r

∂xi∂f s

∂xjgrs (1.4)

ìpou gij eÐnai h èkfrash thc metrik c sto sÔsthma suntetagmènwn xi. Oi sunart seic gij orÐzounmia nèa upopollaplìthta sto q¸ro xi, yij thn opoÐa sumbolÐzoume g. Sto sÔnolo ìlwn twnupoq¸rwn pou kajorÐzontai me ton anwtèrw trìpo , dhlad mèsw enìc sunìlou sunart sewngij me tic idiìthtec pou anafèrame orÐzoume mia sqèsh isodunamÐac gij ∼ gij me thn apaÐthsh naikanopoieÐtai h (1.4). Kje klsh aut c thc sqèshc isodunamÐac onomzoume metrik (metric)kai sumbolÐzoume [gij].

4Βλέπε Nail H Ibragimov “Transformation Groups Applied to Mathematical Physics” D Redel 1985 p. 53.Ο χώρος αυτός είναι ο χώρος δέσμης bundle space της δεύτερης εϕαπτόμενης δέσμης του χώρου xn.


Me ton orismì pou d¸same ìla ta sÔnola gij twn n2 sunart sewn pou ikanopoioÔn ticsunj kec pou jèsame kai sqetÐzontai me èna metasqhmatismì suntetagmènwn apoteloÔn ekfrseicthc Ðdiac metrik c. H apaÐthsh aut eÐnai isodÔnamh me thn prìtash metrik eÐnai o tanust ctou opoÐou oi gij eÐnai oi sunist¸sec sto sÔsthma suntetagmènwn xi.

H metrik ìmwc mporeÐ na qrhsimopoihjeÐ wc ergaleÐo me to opoÐo epilègoume eidik sust -mata suntetagmènwn. Prgmati èstw to sÔsthma suntetagmènwn xi sto opoÐo h metrik èqeisunist¸sec gij kai èstw [gij] h klsh twn metrik¸n pou orÐzoun oi gij. Sto sÔnolo [gij] jewroÔmeìla ta sust mata suntetagmènwn sta opoÐa h metrik èqei tic Ðdiec sunist¸sec, dhlad isqÔei:

gij =∂f r

∂xi∂f s

∂xjgrs. (1.5)

Oi lÔseic tic exÐswshc aut c exart¸ntai apì ènan arijmì eleÔjerwn paramètrwn, k èstw, kaiorÐzoun ta sust mata suntetagmènwn xi = f i(xj). MporeÐ na deiqteÐ ìti oi metasqhmatismoÐf i(xj) apoteloÔn omda, h opoÐa exarttai apì tic k eleÔjerec paramètrouc pou anafèrame.H omda aut eÐnai mia omda Lie distashc k. MporeÐ epÐshc na deiqteÐ 5 ìti h mègisth tim

tou k eÐnai n(n+1)2

kai autì sumbaÐnei mìnon ìtan h metrik eÐnai metrik tou q¸rou stajer ckampulìthtac.

Ti shmaÐnei ìmwc h apaÐthsh (1.5)? Prokeimènou na apant soume pollaplasizoume thn (1.5)me dxidxj kai èqoume:

gijdxidxj =

∂f r∂xi

∂f s∂xjdxidxj grs = grsdx

idxj. (1.6)

To aristerì mèloc eÐnai to m koc ds2 sto sÔsthma xi kai to dexÐ sto sÔsthma suntetagmènwnxi (giatÐ oi gij eÐnai koinèc sta dÔo sust mata suntetagmènwn). 'Ara h apaÐthsh (1.5) eÐnaiisodÔnamh me th sqèsh ìti h metasqhmatismìc pou orÐzei eÐnai mia isometrÐa (isometry). StosÔnolo ìlwn twn metasqhmatism¸n suntetagmènwn xi → xi = f i(xj) orÐzoume th sqèsh isodu-namÐac x ∼ xj me thn apaÐthsh h sunrthsh f i(xj) na eÐnai lÔsh thc (1.5). Ta sust matapou an koun sthn klsh aut onomzoume adraneiak sust mata (inertial coordinatesystems) thc metrik c gij.

Prokeimènou na gÐnoun ta anwtèrw antilhpt c jewr soume th metrik Lorentz sth JewrÐathc Eidik c Sqetikìthtac. H metrik Lorentz se opoiod pote sÔsthma suntetagmènwn prokÔ-ptei apì thn kanonik thc morf diag(−1, 1, 1, 1) me kpoio metasqhmatismì suntetagmènwn.JewroÔme t¸ra thn kanonik morf thc metrik c Lorentz kai anazhtoÔme ta adraneiak thc su-st mata (dhlad ta sust mata sta opoÐa h metrik èqei thn kanonik thc morf ). Apì ìlouctouc metasqhmatismoÔc periorizìmaste stouc grammikoÔc metasqhmatismoÔc, L èstw (uprqounkai lloi). Oi metasqhmatismoÐ L prosdiorÐzontai apì th lÔsh thc exÐswshc:

η = (L−1)tηL. (1.7)

h opoÐa eÐnai h gnwst sqèsh isometrÐac sth JewrÐa thc Eidik c Sqetikìthtac. H lÔsh thc

(1.7) dÐnei ìti oi metasqhmatismoÐ Lorentz exart¸ntai apì 4(4+1)2

= 10 paramètrouc pou eÐnai oigenn torec thc omdac tou Poncare. Ta dianÔsmata aut onomzoume dianÔsmata Killing.Ta adraneiak sust mata suntetagmènwn thc JewrÐac thc Eidik c Sqetikìthtac eÐnai ta gnwstLorentz sust mata suntetagmènwn, ta opoÐa sqetÐzontai me èna metasqhmatsmì Lorentz.

5Μία απόδειξη είναι η ακόλουθη. Η μετρική ικανοποιεί τη σχέση g[ab] = 0, η οποία είναι n(n−1)2 εξισώσεις. ΄Αρα

οι ελεύθεροι παράμετροι είναι n2 − n(n−1)2 = n(n+1)

2 .


1.3.4 H Kosmologik Arq kai o kajorismìc thc metrik c

Epistrèfoume sthn Kosmologik Arq kai jewroÔme sth jèsh tou fusikoÔ megèjouc T thmetrik gab h opoÐa, sÔmfwna me th jewrÐa thc Genik c Sqetikìthtac, perigrfei to barutikìpedÐo. Gia to pedÐo gij h (1.1) dÐnei th sqèsh:

gi′j′(xi′) = gij(x

i). (1.8)

SÔmfwna me ta ìsa anaptÔxame sto edfio 1.3.3 h sqèsh aut shmaÐnei ìti ta kosmologiksust mata suntetagmènwn eÐnai ekeÐna, ta opoÐa sqetÐzontai me to kosmikì sÔsthma suntetag-mènwn me mia isometrÐa. Oi metasqhmatismoÐ autoÐ èqoun genn torec ta dianÔsmata Killing thcmetrik c kai mporoÔn na upologistoÔn apì thn (1.8) en antikatast soume orÐsoume to gij stokosmikì sÔsthma suntetagmènwn. Se ìla ta kosmologik sust mata suntetagmènwn h metrik aut epekteÐnetai me bsh thn Arq thc Sqetikìthtac thc KosmologÐac.

H Kosmologik Arq kajorÐzei tic summetrÐec tou 3-q¸rou gia touc kosmikoÔc parathrhtèckai mìnon. Epeid ìpwc anafèrame oi summetrÐec eÐnai anexrthtec apì sugkekrimèno kosmologi-kì sÔsthma suntetagmènwn, oi summetrÐec autèc isqÔoun gia ìlouc touc kosmikoÔc parathrhtèc.

Oi upojèseic summetrÐac pou jewreÐ h Kosmologik Arq eÐnai oi akìloujec:

1. To monadiaÐo qronikì tetrnusma ua, to opoÐo orÐzei thn tetrataqÔthta twn kosmik¸nparathrht¸n, eÐnai èna sÔmmorfo dinusma Killing (conformal Killing vector) thc metrik cgij tou qwroqrìnou. To dinusma autì eÐnai parllhlo me to dinusma ∂t ìpou t eÐnai okosmologikìc qrìnoc.

2. Oi 3-uperepifaneiec pou orÐzontai me thn sqèsh t =stajerì eÐnai Lorentz kjetec stodinusma ua.

3. H gewmetrÐa twn 3-uperepifanei¸n eÐnai aut enìc q¸rou tri¸n diastsewn stajer c kam-pulìthtac - , isodÔnama, mègisthc summetrÐac.

Me bsh tic anwtèrw upojèseic eÐnai dunatìn na gryoume th metrik 6 tou qwroqrìnou poukajorÐzei h Kosmologik Arq . To kosmologikì montèlo tou qwroqrìnou autoÔ onomzoume totupikì kosmologikì montèlo (The standard cosmological model) kai eÐnai autì pouepÐ tou parìntoc faÐnetai na eÐnai to plèon apodektì. Prèpei na shmeiwjeÐ ìti eÐnai apl èna - kaimlista apì ta aploÔstera! - montèlo thc JewrÐac thc Genik c Sqetikìthtac. Kje lÔsh twnexis¸sewn tou Einstein eÐnai kai èna en dunmei kosmologikì montèlo. S mera pou arqÐzoume naèqoume plhj¸ra kosmologik¸n parathr sewn perissìterh èmfash dÐnetai sthn ermhneÐa twnparathr sewn sta plaÐsia kpoiou montèlou kai ìqi h diatÔpwsh genik¸n jewrhtik¸n melet¸n.Perimènoume de oi diaforopoi seic tou montèlou apì tic parathr seic na mac odhg soun sekalÔtera montèla.

6Για λεπτομέρειες βλέπε M. Tsamparlis (1998) “Conformal reduction of a spacetime metric” Class. QuantumGrav 15, 2901 - 2908.

Keflaio 2

Oi q¸roi stajer c kampulìthtac q¸roi mègisthc summetrÐac

Prin suneqÐsoume me thn anptuxh tou tupikoÔ kosmologikoÔ montèlou qreizetai na anafè-roume merik genik stoiqeÐa gia touc q¸rouc stajer c kampulìthtac touc q¸rouc mègisthcsummetrÐac.

2.1 Q¸roi stajer c kampulìthtac wc upìqwroi enìc

epÐpedou q¸rou

JewroÔme èna epÐpedo q¸ro Kn+1 distashc n+ 1 me Kartesianèc suntetagmènec xa, z ìpoua = 1, 2, . . . , n. Stic sunetetagmènec1 xa, z h metrik gAB (A,B = 1, . . . , n+ 1) grfetai:

ds2 = gABdxAdxB = gabdx

adxb +1

ε(dz)2 (2.1)

ìpou ε eÐnai mia mh mhdenik stajer kai gab eÐnai ènac n× n summetrikìc pÐnakac me stajeroÔcsuntelestèc.

Sto q¸ro Kn+1 jewroÔme mia uperepifneia (upìqwro) S distashc n thn opoÐa orÐzoume methn exÐswsh:

gabxaxb +

1

εz2 = 1. (2.2)

H exÐswsh aut qarakthrÐzetai apì thn idiìthta ìti apoteleÐtai apì shmeÐa, ta opoÐa apèqounthn Ðdia stajer apìstash apì thn arq tou sust matoc suntetagmènwn (aut eÐnai h basik upìjesh thc `sfairik c ' epifneiac). H stajer ε eÐnai anagkaÐa prokeimènou na kalufjoÔnìlec oi peript¸seic. H perÐptwsh ε > 0 antistoiqeÐ sthn EukleÐdeia sfaÐra kai h perÐptwshε = −1 kalÔptei thn perÐptwsh thc uperbolik c sfaÐrac, h opoÐa eÐnai pijan diìti h metrik den jewreÐtai ìti eÐnai EukleÐdeia. Tèloc mporoÔme na ermhneÔsoume oriak thn tim ε = 0 ìtikalÔptei thn perÐptwsh pou h sfaÐra ekfulÐzetai se mia epÐpedh epifneia.

Sthn epifneia S èqoume ìti (jumhjeÐte ìti ta stoiqeÐa tou pÐnaka gab eÐnai stajerèc):

2zdz + ε2gabxadxb = 0

1Αυτός είναι ο ορισμός των Καρτεσιανών συντεταγμένων για μια επίπεδη μετρική

13

14 KEFALAIO 2. QWROI STAJERHS KAMPULOTHTAS

apì ìpou èqoume:

dz2 = (dz)2 =ε2(gabx

adxb)2

z2=ε2(gabx

adxb)2

1 − εgcdxcxd. (2.3)

Antikajist¸ntac to dz2 sthn (2.1) brÐskoume ìti h metrik , sab èstw, h opoÐa epgetai sthnepifneia S apì th metrik gAB tou n+ 1 q¸rou Kn+1 eÐnai h akìloujh:

sabdxadxb = gabdx

adxb +ε(gabx

adxb)2

1 − εgabxaxb(2.4)

, isodÔnama:

sab = gab +ε

1 − εgcdxcxdgaex

egbfdxf . (2.5)

Epeid h metrik gab eÐnai h metrik tou epÐpedou n−q¸rou, eÐnai ènac sunduasmìc apì δ sunar-t seic.

En h metrik gab eÐnai h EukleÐdeia metrik èqoume:

sab = δab +ε

1 − ε(δcdxcxd)xadxb (2.6)

Se dianusmatik morf h (2.6) grfetai:

ds2 =1

ε

[dx2 +

(x · dx)2

1 − x2

]en ε > 0 (2.7)

ds2 =1

ε

[dx2 − (x · dx)2

1 + x2

]en ε < 0 (2.8)

ìpou èqoume upojèsei ìti 1 ≥ x2.En o n−q¸roc eÐnai o q¸roc Minkowski tìte gab = ηab ìpou ηab eÐnai h metrik Lorentz.

Se aut n thn perÐptwsh h metrik sthn epifneia S eÐnai:

sab = ηab +ε

1 − ε(ηcdxcxd)xadxb (2.9)

, se Kartesianèc suntetagmènec t, xµ stic opoÐec ηab = diag(−1, 1, 1, 1):

ds2 = −dt2 + dx2 +ε(x · dx− tdt)2

1 − ε(x2 − t2). (2.10)

'Askhsh 1. Upojèste ìti ε > 01. JewreÐste to metasqhmatismì:

t =1√ε

[εr′2

2cosh(

√εt′) +

(1 +

εr′2

2

)sinh(

√εt′)

]x = x′ exp(

√εt′)

ìpou r′2 = x′2 kai deÐxte ìti stic suntetagmènec (t′, x′) h metrik (2.10) gÐnetai:

ds2 = −dt′2 + e2√εt′dx′2. (2.11)

2.1. QWROI STAJERHS KAMPULOTHTAS WSUPOQWROI ENOS EPIPEDOUQWROU15

2. JewreÐste to deÔtero metasqhmatismì (t′,x′) → (t′′,x′′) o opoÐoc orÐzetai me tic sqèseic:

t′′ = t′ − 1

2√ε

ln[1 − εe2

√εt′x′2

]x′′ = x′e

√εt′

kai deÐxte ìti stic nèec suntetagmènec h metrik paÐrnei th morf (h metrik aut eÐnai gnwst wc metrik deSitter):

ds2 = −(1 − x′′2)dt′′2 + dx′′2 +ε(x′′ · dx′)2

1 − εx′′2 . (2.12)

2.1.1 To analloÐwto thc epifneiac S

AfoÔ h epifnia S orÐzetai mèsw thc apìstashc perimènoume ìti oi metasqhmatismoÐ sto n + 1q¸ro Kn+1 oi opoÐoi diathroÔn thn apìstash ja eÐnai ta dianÔsmata Killing thc metrik c gAB.Epiplèon epeid h epifneia S orÐzetai me thn apaÐthsh h apìstash twn shmeÐwn thc apì thn arq na eÐnai stajer , anamènoume ìti ta strobil (non-gradient) dianÔsmata Killing thc metrik cgAB ta opoÐa orÐzoun tic strofèc perÐ thn arq ja diathroÔn analloÐwth thn epifneia S.GnwrÐzoume ìti se èna epÐpedo q¸ro distashc n + 1 oi strofèc perÐ thn arq pargontai apì(n+1)(n+1−1)

2= n(n+1)

2dianÔsmata Killing, epomènwc aut eÐnai kai ta dianÔsmata Killing thc

epifneiac S.Arqik pargoume touc shmeiakoÔc metasqhmatismoÔc, oi opoÐoi pargontai apì aut ta

dianÔsmata Killing.O genikìc metasqhmatismìc ston n+ 1 q¸ro Kn+1 èqei th genik morf :(

z′

x′a

)=

(Rz

z Rzb

R az Ra

b

)(zxb

)(2.13)

= RAB

(zxb

)(2.14)

ìpou èqoume jèsei:

RAB =

(Rz

z Rzb

R az Ra

b

). (2.15)

Ta dianÔsmata Killing thc metrik c gAB gia touc shmeiakoÔc metasqhmatismoÔc, oi opoÐoi dia-throÔn thn epifneia S, upologÐzontai apì th sunj kh isometrÐac, h opoÐa se gl¸ssa pinkwnekfrzetai wc akoloÔjwc:

[RAB]t[gAC ][RB

D] = [gCD] (2.16)

kai apaitoÔme ìti sunist¸sec thc metrik c [gCD] na mhn allxoun, dhlad isqÔei:

[gAB] =

(ε−1 00 gab

). (2.17)

Antikajist¸ntac ta [RAB], [gAB] apì tic (2.15), (2.17) antÐstoiqa, èqoume:(

Rzz (R a

z )t

(Rzc)

t (Rac)

t

)(ε−1 00 gab

)(Rz

z Rzd

Rbz Rb

d

)=

(ε−1 00 gcd

)


apì ìpou prokÔptoun (lìgw thc summetrÐac thc [gAB]) oi akìloujec treic exis¸seic:

ε−1 (Rzz)

2 + (R az )t gabR

bz = ε−1 (2.18)

ε−1RzzR

zc + (R a

z )t gabRbz = 0 (2.19)

(Rac)

t gabRbd + ε−1 (Rz

c)tRz

d = gcd. (2.20)

H lÔsh tou sust matoc twn exis¸sewn (2.18)- (2.20) dÐnei ìlouc touc shmeiakoÔc metasqhma-tismoÔc sto q¸roKn+1, oi opoÐoi af noun thn uperepifneia S analloÐwth (dhlad apaikonÐzounèna shmeÐo thc S se èna llo shmeÐo thc S). Shmei¸noume ìti upeisèrqontai oi akìloujoi gnw-stoi:- Oi n2 posìthtec Ra

.c

- Oi 2n posìthtec Rz.c, R

a.z kai

- H 1 posìthta Rz.z.

Oi exis¸seic pou èqoume eÐnai:

1 (h (2.18) ) + n (oi (2.19) ) + n(n+1)2

(oi (2.20), oi opoÐec eÐnai summetrikèc stouc deÐktec c, d).Epomènwc o arijmìc twn eleÔjerwn paramètrwn eÐnai:

n2 + n+ 1 − 1 − n− n(n+ 1)

2=n(n+ 1)

2Autèc oi parmetroi katametroÔn touc metasqhmatismoÔc, oi opoÐoi pargontai apì ta antÐstoiqadianÔsmata Killing thc metrik c gab.

AnazhtoÔme eidikèc lÔseic tou sust matoc twn exis¸sewn (2.18)- (2.20). 'Opwc dh ana-

fèrame uprqoun n(n+1)2

grammik anexrthtec lÔsec (ìsa kai ta dianÔsmata Killing ta opoÐapargoun touc metasqhmatismoÔc).

a. Strofèc steraioÔ s¸matoc gÔrw apì thn arq

H lÔsh aut orÐzetai me tic sunj kec:

Ra.z = Rz

.a = 0, Rz.z = 1 (2.21)

(2.22)

Se aut n thn perÐptwsh oi exis¸seic (2.18), (2.19) ikanopoioÔntai tetrimmèna kai to sÔsthmakatal gei sth exÐswsh:

(Rac)

t gabRbd = gcd (2.23)

h opoÐa ekfrzei mia sunj kh isometrÐac gia thn n−metrikh gab. Se aut n thn perÐptwsh opÐnakac:

RAB =

(1 00 Ra

b

)kai o metasqhmatismìc ston n+ 1−q¸ro Kn+1 eÐnai:

z′ = z

x′a = Rabx

b.

Katal goume sto anamenìmeno sumpèrasma, dhlad ìti ta dianÔsmata Killing thc metrik cgab eÐnai oi strofèc (dhlad ta strobil dianÔsmata Killing) thc n + 1−metrik c gAB, ta opoÐaaf noun th suntetagmènh z− analloÐwth.


b. Yeudo-metaforèc

H lÔsh aut orÐzetai apì tic sunj kec:

Rbz = ab, Rz

b = −εgbcac, Rzz =

√1 − εgabaaab, R

ab = δab −Bεgaba

aab (2.24)

ìpou aa eÐnai èna tuqaÐo dinusma ston n−q¸ro kai B eÐnai mia parmetroc h opoÐa ja kajoristeÐapì thn apaÐthsh na ikanopoieÐtai to sÔsthma twn exis¸sewn (2.18) - to (2.20).

Prokeimènou to Rzz na eÐnai pragmatikì apaitoÔme thn akìloujh sunj kh sto dinusma aa:

εgabaaab ≤ 1. (2.25)

Antikajist¸ntac stic (2.18) - (2.20) briskoume ìti:

B =1 −

√1 − εgabaaab

εgabaaab. (2.26)

Epomènwc gia thn perÐptwsh lÔsh b. o pÐnakac RAB eÐnai:

RAB =

( √1 − εgabaaab −εgbcac

ab δab −Bεgabaaab

)(2.27)

ìpou to B dÐdetai sthn (2.26). Sungoume ìti o metasqhmatismìc ston q¸ro Kn+1 eÐnai:(z′

x′a

)=

( √1 − εgabaaab −εgbcac

ab δab −Bεgabaaab

)(zxb

)apì ìpou èqoume:

z′ = z√

1 − εgabaaab − εgbcxbac (2.28)

x′a = abz + (δab −Bεgacaaac)xb. (2.29)

O metasqhmatismìc (2.29) ìtan perioristeÐ sthn epifneia S gÐnetai:

x′a = xa + aa[√

1 − εgcdxcxd −Bεgbcabxc

]. (2.30)

Shmei¸noume ìti o metasqhmatismìc (2.30) metafèrei thn arq xa = 0 sto shmeÐo x′a = aa.Touc metasqhmatismoÔc sthc epifneiac S pou orÐzontai apì thn (2.30) onomzoume yeudo-metaforèc (quasi-translations). Shmei¸noume ìti gia autoÔc touc metasqhmatismìc èqoumekai èna metasqhmatismì thc suntetagmènhc z, o opoÐoc dÐnetai apì thn (2.28). Autì shmaÐnei ìtiautèc oi yeudometaforèc ston n+ 1−q¸ro Kn+1 den eÐnai strofèc perÐ ton xona z.

Oi yeudo-metaforèc exart¸ntai apì to dinusma aa, ra apì n paramètrouc, oi opoÐec eÐnai oisunist¸sec tou dianÔsmatoc aa. Sungoume ìti to sunolikì pl joc shmeiak¸n summetri¸n poujewr same eÐnai:

n(n+ 1)

2+ n =

(n+ 1)(n+ 2)

2− 1

dhlad ìsa kai ta dianÔsmata Killing thc n + 1−metrik c par èna. To dinusma Killing pouleÐpei eÐnai to ∂z, to opoÐo pargei shmeiakoÔc metasqhmatismoÔc, oi opoÐoi den eÐnai shmeiakoÐmetasqhmatismoÐ thc epifneiac S.


H pr¸th oikogèneia metasqhmatism¸n (h a.) jewroÔme ìti ekfrzei thn isotropÐa thc epif-neiac. Ktw apì th drsh aut¸n twn metasqhmatism¸n h epifneia kineÐtai wc èna stereì s¸magÔrw apì thn arq twn axìnwn tou n + 1−q¸rou. H deÔterh oikogèneia metasqhmatism¸n (hb.) jewroÔme ìti ekfrzei thn omogèneia tou q¸rou lìgw thc idiìthtac x′a = aa. Tic dÔo autècidiìthtec ekfrzoume me th frsh h epifneia S eÐnai omogen c kai isìtroph ston n + 1−q¸roKn+1.

2.1.2 Upologismìc twn dianusmtwn Killing thc metrik c sab

Prokeimènou na upologÐsoume ta dianÔsmata Killing ta opoÐa af noun analloÐwth thn epifneiaS jewroÔme thn apeirost morf twn shmeiak¸n metasqhmatism¸n kai grfoume:

x′a = xa + εξa +O(ε2). (2.31)

ìpou ξa eÐnai èna dianusmatikì pedÐo sthn pollaplìthta pou orÐzei h epifneia S kai sugkrÐnoumeme th sqèsh pou ekfrzei to x′a.

Ta dianÔsmata Killing thc isotropÐac (ξaI )

Gia to pr¸to sÔnolo twn metasqhmatism¸n (thn isotropÐa) èqoume:

x′a = Rabx

b

Upojètoume ìti o metasqhmatismìc autìc eÐnai apeirostìc kai grfoume:

Ra.b = δab + εΩa

.b (2.32)

ìpou Ωa.b eÐnai ènac pÐnakac me stajeroÔc suntelestèc. Tìte:

x′a = Ra.bx

b = (δab + εΩa.b)x

b = xa + εΩa.bx

b. (2.33)

SugkrÐnontac tic (2.31) kai (2.33) brÐskoume:

ξaI = Ωa.bx

b. (2.34)

Prokeimènou na upologÐsoume ton pÐnaka Ωa.b jewroÔme thn exÐswsh Killing gia th metrik

sab sthn epifneia S kai grfoume:

sabξbI,c + scbξ

bI,a = 0.

Antikajist¸ntac to ξaI apì thn (2.34) prokÔptei ìti o pÐnakac Ωa.b prèpei na eÐnai lÔsh thc

exÐswshc:sabΩ

b..a + sbaΩ

a..b = 0 ⇔ Ωab = −Ωba. (2.35)

ra o pÐnakac Ωab eÐnai ènac antisummetrikìc pÐnakac me stajeroÔc suntelestèc. Profan¸c upr-

qoun n(n−1)2

to pl joc tètoia dianÔsmata Killing ìsa kai o arijmìc twn anexrthtwn stoiqeÐwntou pÐnaka Ωab.


Ta dianÔsmata Killing thc omogèneiac (ξaH)

Sth sunèqeia upologÐzoume ta dianÔsmata Killing tou deÔterou sunìlou twn metasqhmatism¸n(thc omogèneiac). Sthn perÐptwsh aut èqoume brei ìti:

x′a = xa + aa[√


]ìpou B =

1−√

1−εgabaaab

εgabaaab. Epeid o metasqhmatismìc eÐnai apeirostìc jètoume aa = εAa ìpou

Aa eÐnai èna dinusma sthn pollaplìthta pou orÐzei h epifneia S, to opoÐo orÐzei to shmeiakìmetasqhmatismì sthn perioq thc arq c twn suntetagmènwn, kai èqoume:

B =1 −

√1 − εgabaaab

εgabaaab=

1

ε+O(ε2).

Apì ed¸ prokÔptei:

x′a = xa + εAa√

1 − εgcdxcxd +O(ε2).

SugkrÐnontac me thn (2.31) brÐskoume:

ξaH = Aa√

1 − εgcdxcxd. (2.36)

Profan¸c uprqoun n anexrthta dianusmatik pedÐa ξaH ìsa kai oi sunist¸sec tou dianÔsma-toc Aa. SumperaÐnoume ìti sunolik o arijmìc twn dianusmtwn Killing thc metrik c sab thcepifneiac S eÐnai:

n(n− 1)

2+ n =

n(n+ 1)

2.

Autìc eÐnai o mègistoc arijmìc dianusmtwn pou mporeÐ na èqei mia metrik enìc q¸rou n dista-shc. 'Otan ε = 0 ta dianÔsmata aut sumpÐptoun me ta dianÔsmata Killing tou epÐpedou q¸roudistashc n.

2.1.3 H gewmetrÐa thc epifneiac S

Sto parìn edfio upologÐzoume tic diforec gewmetrikèc posìthtec thc epifneiac S. Prokei-mènou na upologÐsoume tic gewdesiakèc thc metrik c sab thc epifneiac S upologÐzoume toucsuntelestèc sunoq c thc metrik c gAB me deÐktec Γa

bc. BrÐskoume:

Γabc = εxagbc. (2.37)

Epomènwc h exÐswsh twn gewdesiak¸n sthn epifneia S eÐnai:

d2xa

dτ 2+ εxa = 0. (2.38)

H lÔsh aut c thc exÐswshc eÐnai ènac grammikìc sunduasmìc twn sunart sewn sin t√ε kai cos t

√ε

gia ε > 0 kai sinh t√−ε kai cosh t

√−ε gia ε < 0. H pr¸th lÔsh eÐnai EukleÐdeioi kÔkloi kai h

deÔterh uperbolikoÐ kÔkloi ston EukleÐdeio q¸ro.


Sth sunèqeia upologÐzoume thn kampulìthta thc epifneiac S. UpologÐzoume gia tic suni-st¸sec tou tanust kampulìthtac (qrhsimopoi¸ntac th metrik tou n+ 1 q¸rou):

Rabcd = ε(sacsbd − sadsbc). (2.39)

O tanust c Ricci thc metrik c thc epifneiac S eÐnai:

Rab = (n− 1)εsab (2.40)

kai bajmwt kampulìthta Ricci:R = n(n− 1)ε. (2.41)

'Askhsh 2. DeÐxte ìti o tanust c Weyl thc metrik c thc epifneiac S eÐnai mhdèn (n > 3).

O mhdenismìc tou tanust Weyl shmaÐnei ìti h pollaplìthta pou orÐzei h epifneia S eÐnaisÔmmorfa epÐpedh2 (conformally flat), opìte h metrik sab grfetai sth morf :

sab = 2ϕ(xc)gab (2.42)

ìpou gab eÐnai h metrik tou epÐpedou n−distatou q¸rou kai ϕ(xc) eÐnai o sÔmmorfoc pargontac(conformal factor). To apotèlesma autì mac epitrèpei na upologÐsoume t sÔmmorfh lgebraenìc q¸rou stajer c kampulìthtac apì th sÔmmorfh lgebra tou epÐpedou q¸rou me bsh thnprìtash ìti:

SÔmmorfoi q¸roi èqoun thn Ðdia sÔmorfh lgebra (ìqi ìmwc ta Ðdia dianÔsmataKilling kai omojetik dianÔsmata).

SuneqÐzoume me thn gewmetrik dom twn q¸rwn gia ε >, ε < 0.

2.1.4 H perÐptwsh ε > 0

Aut eÐnai h EukleÐdeia perÐptwsh me metrik (2.7). To stoiqeÐo ìgkou tou n−distatou q¸roupou antistoiqeÐ se aut n th metrik eÐnai:

Vn =

∫1≥x2

√gdx1...dxn = 2ε−

n2

∫1≥x2

dx1...dxn√1 − x2

(2.43)

UpologÐzoume:

Vn =2π

n+12

Γ(n+12

)ε−n2 (2.44)

ìpou Γ eÐnai h sunrthsh Gmma.Gia n = 1 brÐskoume thn perÐmetro tou kÔklou aktÐnac

√ε. Prgmati èqoume

L = V1 =2π

1+12

Γ(1+12

)ε− 12 =

2π

Γ(1)ε−

12 =

2π√ε. (2.45)

2Αυτή είναι μια γενική ιδιότητα των χώρων σταθερής καμπυλότητας.


Gia n = 2 brÐskoume thn epifneia thc sfaÐrac aktÐnac√ε. Se aut n thn perÐptwsh èqoume:

S = V2 =2π

2+12

Γ(2+12

)ε− 22 =

2π3/2

Γ(32)ε−1 = 4πε−1. (2.46)

Gia n = 3 brÐskoume ton ìgko thc sfaÐrac aktÐnac√ε :

V = V3 =2π

3+12

Γ(3+12

)ε− 32 =

2π2

Γ(2)ε−

32 = 2π2ε−

32 . (2.47)

UpologÐzoume t¸ra thn perÐmetro enìc q¸rou distashc n qrhsimopoi¸ntac tic gewdesiakèc.H gewdesiak c pou pern apì to shmeÐo x = 0 eÐnai:

x = E sin(τ√ε)

ìpou E eÐnai èna dinusma to opoÐo orÐzei th dieÔjunsh thc gewdesiak c se kje shmeÐo (isodÔ-nama se kje tim thc paramètrou τ). Antikajist¸ntac x sthn (2.7) brÐskoume E2 = 1, dhlad to E eÐnai èna monadiaÐo dinusma. IsodÔnama h parmetroc τ eÐnai mia afinik parmetroc.

Ac upojèsoume ìti xekinme apì to shmeÐo x = 0 (èstw to Bìreio Pìlo) tou q¸rou me tim paramètrou τ = 0. 'Otan ja xanagÐnei to x = 0 (èstw o Nìtioc Pìloc) èqoume mia apìstash(dhlad metabol ) thc paramètrou τ = π√

ε. H gewdaisiak gurÐzei sto arqikì shmeÐo x = 0

ìtan τ = 2 π√ε. Epomènwc prèpei na metakinhjoÔme kat gewdaisiak apìstash 2π√

εprokeimènou

na pragmatopoi soume mia kleist troqi kinoÔmenoi diark¸c epÐ thc gewdesiak c. H metabol aut eÐnai h perÐmetroc tou q¸rou kai isqÔei anexrthta apì th distash tou q¸rou me mìnhproôpìjesh ìti h kampulìthta eÐnai jetik . To apotèlesma autì qarakthrÐzoume me th frsh` o q¸roc eÐnai peperasmènoc ' (diìti to 2π√

εeÐnai peperasmèno ) kai mh fragmènoc (unbounded)

diìti 2π√ε→ ∞ ìtan ε→ 0. To apotèlesma autì eÐnai sumbatì me thn (2.45).

2.1.5 H perÐptwsh ε < 0

Se aut n th perÐptwsh oi gewdesiakèc apì thn arq eÐnai:

x = E sinh(τ√−ε)

ìpou pli to E eÐnai èna monadiaÐo dinusma(E2 = 1) kai τ eÐnai mia afinik parmetroc. Seaut n thn perÐptwsh h sunrthsh sinh den eÐnai periodik epomènwc eÐnai dunatìn kinoÔmenoikat m koc miac gewdesiak c, h opoÐa dièrqetai apì thn arq kat peirh apìstash, qwrÐc naepistrèyoume sto shmeÐo ekkÐnhshc. To gegonìc autì ermhneÔoume me th frsh `o q¸roc eÐnaipeiroc '. Gia n = 2 h epifneia S eÐnai o q¸roc Gauss - Bo lyai - Lobachevsky (oi gewdesiakècden tèmnontai sto peiro, to opoÐo eÐnai h sunj kh h opoÐa orÐzei thn parallhlÐa).

Keflaio 3

To tupikì kosmologikì montèlo

3.1 H metrik twn Friedman Robertson Walker

JewroÔme tic akìloujec suntetagmènec sto qwrìqrono:a. Qronik suntetagmènh: o kosmikìc qrìnoc tb. Qwrikèc suntetagmènec: Treic suntetagmènec stic trisdistatec uperepifneiec oi opoÐecfullopoioÔn to qwrìqrono. H metrik twn uperepifanei¸n aut¸n eÐnai aut enìc trisdistatouq¸rou stajer c kampulìthtac kai eÐnai gnwst (ektìc apì thn tim thc kampulìthtac).

Epeid oi trisdistatec uperepifneiec kajorÐzontai apì mÐa tim tou kosmikoÔ qrìnou, dhla-d èqoun exÐswsh sto sÔsthma suntetagmènwn pou jewr same t =stajerì, h metrik grfetai:

ds2 = −dt2 + a2(t)gµνdxµdxν (3.1)

ìpou h 3-metrik gµν eÐnai h metrik enìc q¸rou Riemman stajer c kampulìthtac1. H sunrthsha(t) eÐnai mÐa gnwsth parmetroc thc metrik c ds2 pou prèpei na upologisteÐ apì tic exis¸seicpedÐou tou Einstein. Onomzetai o pargontac klÐmakac (scale factor) tou kosmologikoÔmontèlou. ParathroÔme ìti h Kosmologik Arq den kajorÐzei monos manta to montèlo touqwroqrìnou all kajorÐzei mia oikogèneia montèlwn, ta opoÐa parametrÐzontai apì ton pargontaklÐmakac. Aut h eleujerÐa tou montèlou mac epitrèpei na epilèxoume to katllhlo montèlo mebsh tic parathr seic llec epiprìsjetec upojèseic. H metrik (3.1) èqei onomasteÐ metrik twn Friedmann Robertson Walker (sta epìmena ja anafèretai en suntomÐa FRW) giatÐprotjhke (kai exelÐqthke) arqik apì autoÔc. Sunart sei tou pargonta klÐmakac orÐzoumeèna nèo qrìno η, ton opoÐo onomzoume sÔmmorfo qrìno (conformal time) me th sqèsh:

η =

∫a−1dt. (3.2)

Sunart sei tou sÔmorfou qrìnou h metrik grfetai:

ds2 = a2(−dη2 + gµνdxµdxν). (3.3)

Mènei akìma na kajorÐzoume thn 3-metrik gµν . 'Opwc anafèrame sto keflaio 2 oi q¸roistajer c kampulìthtac opoiasd pote peperasmènhc distashc diakrÐnontai anloga me th baj-mwt touc kampulìthta se trÐa eÐdh.

1Δηλαδή η κανονική μορϕή της χωρικής μετρικής είναι (+,+,+).

23

24 KEFALAIO 3. TO TUPIKO KOSMOLOGIKO MONTELO

a. Q¸roi me jetik bajmwt kampulìthta (ìpwc p.q. h sfairik epifneia), oi opoÐoi onomzon-tai parabolikoÐ (parabolic)b. Q¸roi me mhdenik bajmwt kampulìthta (ìpwc p.q. h epÐpedh epifneia), oi opoÐoi onom-zontai epÐpedoi (flat)g. Q¸roi me arnhtik bajmwt kampulìthta (ìpwc p.q. to fÔllo uperboloeidoÔc), oi opoÐoionomzontai uperbolikoÐ (hyperbolic) .

En sumbolÐsoume me ε thn tim thc stajer c kampulìthtac, tìte se sfairikèc suntetagmènecr, θ, ϕ h 3-metrik gµν èqei th morf 2 (blèpe (2.5) ):

gµνdxµdxν =

dr2

1 − εr2+ f 2(r)dΩ2 (3.4)

ìpou:dΩ2 = dθ2 + sin2 θdϕ2 (3.5)

kai

f 2(r) =

sin r gia ε = 1r gia ε = 0

sinh r gia ε = −1

. (3.6)

Epomènwc h metrik twn FRW stic suntetagmènec t, r, θ, ϕ eÐnai:

ds2 = −dt2 + a(t)

(dr2

1 − ε4r2

+ f 2(r)(dθ2 + sin2 θdϕ2)

). (3.7)

Ektìc apì sfairikèc suntetagmènec qrhsimopoioÔntai kai oi Kartesianèc suntetagmènec t, x, y, zìpou oi qwrikèc suntetagmènec x, y, z stic trisdistatec uperepifneiec sundèontai me ticr, θ, ϕ me tic sun jeic sqèseic metasqhmatismoÔ twn sfairik¸n suntetagmènwn. Stic Karte-sianèc suntetagmènec h metrik FRW eÐnai:

ds2 = −dt2 + a2(t)1

(1 + ε4x2)2

(dx2 + dy2 + dz2) (3.8)

ìpou x2 = x2 + y2 + y2 kai ε = ±1.Prokeimènou h metrik twn FRW na sundejeÐ me thn parat rhsh (dhlad th Fusik ) prèpei

na sundejoÔn oi suntetagmènec stic opoÐec eÐnai ekpefrasmènh me parathr sima megèjh. HKosmologik arq oloklhr¸netai me thn paradoq :

Oi suntetagmènec t, r, θ, ϕ stic opoÐec ekfrzetai h metrik (3.7) sumpÐptounme tic antÐstoiqec suntetagmènec tou tupikou kosmikoÔ sust matoc suntetagmènwn(to opoÐo, ìpwc anafèrame, kajorÐzetai parathrhsiak!).

H metrik twn FRW eÐnai apotèlesma twn upojèsewn thc Kosmologik c Arq c kai eÐnai hmetrik pou qrhsimopoioÔme prokeimènou na anaptÔxoume thn KosmologÐa. Perièqei dÔo para-mètrouc ton pargonta klÐmakac kai thn stajer ε pou eÐnai h kampulìthta tou 3-q¸rou. ToantikeÐmeno thc KosmologÐac eÐnai na kajorÐsei tic timèc aut¸n twn paramètrwn mèsw twn ko-smologik¸n parathr sewn kai kat sunèpeia na perigryei sta plaÐsia thc JewrÐac thc Genik cSqetikìthtac to sÔmpan wc èna dunamikì sÔsthma.

2Οι συντεταγμένες r, θ, ϕ δεν είναι αναγκαία οι ίδιες με τις τυπικές κοσμικές συντεταγμένες στον 3-χώρο πουθα ορίσουμε στα επόμενα, ενώ η συντεταγμένη t είναι ο κοσμικός χρόνος ή μια συνάρτηση του κοσμικού χρόνου

3.2. OI EPIPTWSEIS THS KOSMOLOGIKHS ARQHS 25

3.2 Oi epipt¸seic thc Kosmologik c Arq c

3.2.1 H Kosmologik Arq kai h Arq thc IsodunamÐac

H metrik eÐnai èna gewmetrikì stoiqeÐo, to opoÐo apokt dunamikì perieqìmeno sth JewrÐathc Genik c Sqetikìthtac mìnon mèsa apì thn upìjesh thc Arq c thc isodunamÐac, dhlad thnupìjesh ìti

H kÐnhsh enìc ulikoÔ shmeÐou sto qwrìqrono ktw apì thn epÐdrash thc barÔ-thtac mìnon (eleÔjerh pt¸sh) gÐnetai kat m koc gewdesiak c.

En jewr soume to qronikì dinusma ∂t, to opoÐo sto sÔsthma suntetagmènwn t, r, θ, ϕ èqeisunist¸sec3 δa0 tìte upologÐzoume ìti h tetraepitqunsh:

ua = ua;bub = ua,bu

b + Γabcu

buc = Γa00. (3.9)

All gia th metrik twn FRW stic suntetagmènec t, r, θ, ϕ to Γa00 = 0 epomènwc ua = 0.

Autì shmaÐnei ìti oi parathrhtèc me tetrataqÔthta ua h opoÐa sto kosmikì sÔsthma suntetag-mènwn èqei sunist¸sec δa0 , kinoÔntai upì thn epÐdrash tou pedÐou barÔthtac kat m koc twnt−gewdaisiak¸n. Alla oi t−gewdaisiakèc orÐzontai me th sqèsh r = c1, θ = c2, ϕ = c3 ìpouc1, c2, c3 eÐnai stajerèc. Epeid sÔmfwna me thn Kosmologik Arq sto kosmologikì sÔsth-ma suntetagmènwn oi tupikoÐ galaxÐec èqoun stajerèc qwrikèc suntetagmènec, sungoume ìtioi parathrhtèc me tetrataqÔthta ua eÐnai oi sunkinoÔmenoi parathrhtèc. To sumpèrasma autìeÐnai basikì giatÐ h kosmologÐa pou anaptÔssoume sthrÐzetai mìnon se ènan parathrht (emc)o opoÐoc brÐsketai se eleÔjerh pt¸sh sto kosmikì pedÐo barÔthtac kai apì jewrhtik poyh hjewrÐa pou ja qrhsimopoi soume eÐnai h jewrÐa thc Genik c Sqetikìthtac gia thn opoÐa h Arq thc isodunamÐac eÐnai jemeliak upìjesh.

3.2.2 Oi kinhmatikèc kai oi dunamikèc epipt¸seic thc Kosmo-

logik c Arq c

Oi kinhmatikèc epipt¸seic thc Kosmologik c Arq c aforoÔn to reustì twn parathrht¸n en¸oi dunamikèc ton tanust enèrgeiac orm c, o opoÐoc antistoiqeÐ sthn Ôlh (perilambanomènwntwn dunamik¸n pedÐwn) pou dhmiourgoÔn to pedÐo barÔthtac. Pèra apì tic epipt¸seic autècuprqoun kai oi gewmetrikèc epipt¸seic, oi opoÐec aforoÔn ta difora gewmetrik antikeÐmenapou upologÐzontai apì th metrik kai upeisèrqontai stic kinhmatikèc kai dunamikèc exis¸seic.TonÐzoume ìti oi gewmetrikèc epipt¸seic eÐnai pèra kai pnw apì thn kinhmatik kai th dunamik kai aforoÔn to upìbajro sto opoÐo ja anaptuqjeÐ h kinhmatik kai h dunamik thc jewrÐac. Staepìmena exetzoume thn kje epÐptwsh qwrist.

3.2.3 Gewmetrikèc epipt¸seic

'Opwc tonÐsthke oi gewmetrikèc epipt¸seic aforoÔn ta difora gewmetrik antikeÐmena pou upo-logÐzontai apì th metrik . Ta kÔria antikeÐmena eÐnai:a. Oi suntelestèc sunoq c Γa

bc

3Για ευκολία αριθμούμε από εδώ και στο εξής τις συντεταγμένες t, r, θ, ϕ με τους αριθμούς 0, 1, 2, 3 αντίστοιχα.


b. O tanust c Riemann kai oi sustolèc tou Rab (tanust c tou Ricci) kai R (tanust c bajmwt ckampulìthtac).g. O tanust c tou Einstein Gab = Rab − 1

2Rgab.

O upologismìc twn megej¸n aut¸n eÐnai tupikìc kai gÐnetai me ta ìsa anaptÔxame sta proh-goÔmena. DÐnoume ta apotelèsmata wc skhsh.

'Askhsh 3. DeÐxte ìti oi mh mhdenizìmenec sunist¸sec twn suntelest¸n sunoq c, tou tanust Riemann, tou tanust tou Ricci, tou tanust Einstein kai thc bajmwt c kampulìthtac RdÐnontai apì tic akìloujec sqèseic:a. Sfairikèc suntetagmènec.Metrik

ds2 = −dt2 + a2 (t)

(dr2

1 − εr2+ f 2(r)dΩ2

)(3.10)

Suntelestèc sunoq c : Γcab

Γ0µν = aagµν ,Γ

µν0 =

a

aδµν

Γ111 =

Kr

1 −Kr2,Γ1

22 = −r(1 −Kr).Γ133 = −r(1 −Kr2) sin2 θ

Γ233 = − sin θ cos θ,Γ2

12 =1

r

Γ313 =

1

r,Γ3

23 = cot θ (3.11)

Tanust c kampulìthtac: Rijkl (NA ELEQTEI)

R0µ0µ = −aa (3.12)

Rµν = a2a2 (3.13)

Tanust c Ricci: Rij

R00 = −3(H2 + H) (3.14)

Rµν = a2(

3H2 + H +2K

a2

)gµν (3.15)

ìpou

H =a

a. (3.16)

eÐnai o pargontac Hubble.Bajmwt kampulìthta (Ricciscalar): R

R = 6

(2H2 + H +

K

a2

). (3.17)

Tanust c Einstein: Gij

G00 = −3

(H2 +

K

a2

)(3.18)

Gµν = −

(3H2 + 2H +

K

a2

)δµν . (3.19)


b. Kartesianèc suntetagmènec NA TO DIORJWSWMetrik

ds2 = −dt2 + a2(t)1

(1 + ε4x2)2

(dx2 + dy2 + dz2) (3.20)

Suntelestèc sunoq c : Γcab

Γata =

a

a(3.21)

Γtaa = aa (3.22)

Tanust c kampulìthtac: Rijkl

Rtata = −aa (3.23)

Rabab = a2a2 (3.24)

Tanust c Ricci: Rij

R00 = −3a

a(3.25)

Rµν =

[a

a+ 2

a2 + ε

a2

]gµν (3.26)

Bajmwt kampulìthta: (Ricci scalar)

R = 6

[a

a+a2 + ε

a2

]. (3.27)

Tanust c Einstein: Gij

G00 = 3a2 + ε

a2(3.28)

Gµν = −[2a

a+a2 + ε

a2

]gµν (3.29)

H teleÐa pnw apì èna sÔmbolo shmaÐnei parag gish4 wc proc t kai ε = 0,±1 eÐnai h stajer kampulìthta twn trisdistatwn uperepifanei¸n pou fullopoioÔn to qwrìqrono.

3.2.4 H idiaiterìthta twn sfairik¸n suntetagmènwn

Oi sfairikèc suntetagmènec èqoun idiaÐtero rìlo sthn KosmologÐa giatÐ ìlec oi parathr seicaforoÔn optikèc parathr seic aktinobolÐac apì sugkekrimènh kateÔjunsh. Epomènwc oi gwnÐecθ, ϕ, oi opoÐec kajorÐzoun thn kateÔjunsh pou lambnetai mia fwtein aktÐna apì èna fwteinì an-tikeÐmeno, eÐnai aparaÐthtec metablhtèc sthn parat rhsh. Epeid èqoume upojèsei ìti o 3-q¸roc

4Επειδή όσα Γ περιέχουν δείκτη t μηδενίζονται η μερική παραγώγιση ως προς t είναι και συναλλοίωτη παραγώ-γιση.


eÐnai q¸roc stajer c kampulìthtac, ra isìtropoc kai omogen c, tautÐzoume ta dianÔsmata ∂θ, ∂ϕme dÔo apì ta dianÔsmata Killing pou orÐzoun thn isotropÐa tou q¸rou5.

To trÐto dinusma Killing pou apomènei, to tautÐzoume me th suntetagmènh r, dhlad to ∂r,kai jewroÔme ìti orÐzei th mètrhsh twn qwrik¸n apostsewn kat m koc thc gewdesiak c pouorÐzoun oi dieujÔnseic6 θ, ϕ. Praktik autì shmaÐnei ìti h apìstash enìc shmeÐou P kat m kocthc r−gramm c suntetagmènwn gia mia genik metrik gab eÐnai:

dproper(t) =

∫ r

0

√grr dr. (3.30)

H apìstash aut jewroÔme ìti eÐnai h idioapìstash (proper distance) tou P , dhlad aut poumetr ènac parathrht c en kinhjeÐ kat m koc thc r−gramm c suntetagmènwn apì thn arq tou sust matoc suntetagmènwn sto shmeÐo P . H antÐstoiqh taqÔthta kÐnhshc brÐsketai apì thnparat rhsh ìti h kÐnhsh kat m koc thc r−gramm c suntetagmènwn shmaÐnei ìti ϕ = stajerì,θ = stajerì ra h taqÔthta

vr =∂dproper(t)

∂t=

∫ r

0

∂√grr

∂tdr. (3.31)

Lìgw thc adunamÐac mac na kinhjoÔme se kosmologik klÐmaka den mporoÔme na pistopoi -soume ìti prgmati h upìjesh aut eÐnai swst (dhlad sumfwneÐ me thn parat rhsh / mètrhsh).To mìno pou mac apomènei eÐnai na thn pistèyoume kai na prospaj soume na thn epalhjeÔsou-me èmmesa apì llec ektim seic qwrik c kosmik c apìstashc metroÔmenwn megej¸n oi opoÐecsthrÐzontai se difora fusik pedÐa.

Sth merik twn FRW h grr = a2(t)1−εr2

opìte:

vr =

∫ r

0

∂√grr

∂tdr =

∫ r

0

a√1 − εr2

dr . (3.32)

3.2.5 Kinhmatikèc epipt¸seic

H Ôparxh thc tetrataqÔthtac mac epitrèpei na orÐsoume ton probolikì telest hab me th sqèsh7

(c = 1):hab = gab + uaub. (3.33)

O tanust c hab probllei kjeta sthn tetrataqÔthta. Me th qr sh tou probolikoÔ tanust orÐzoume thn 1+3 dispash thc tanustik c lgebrac pnw sto qwrìqrono sÔmfwna me ta ìsaanafèrame sto edfio .... Sto kosmikì sÔsthma suntetagmènwn t, xµ h tetrataqÔthta twn ko-smik¸n parathrht¸n (oi parathrhtèc autoÐ onomzontai kai sunkinoÔmenoi) èqoun suntetagmènhmìnon sth dieÔjunsh t, ra èqoume:

ua = δta. (3.34)

5Θυμηθείτε ότι σε ένα χώρο σταθερής καμπυλότητας ή, ισοδύναμα, χώρο μέγιστης συμμετρίας με διάσταση nυπάρχουν n(n−1)

2 διανύσματα Killing, τα οποία αϕορούν την ισοτροπία του χώρου. Επομένως σε ένα χώρο με n = 3έχουμε τρία διανύσματα Killing τα οποία αϕορούν την ισοτροπία του χώρου.

6Στη σϕαίρα με ακτίνα r = 1 οι θ, ϕ ορίζουν ένα σημείο από το οποίο διέρχεται η γεωδαισιακή, η οποία συμπίπτειμε την r−γραμμή συντεταγμένων.

7Για c = 1 ο προβολικός τανυστής είναι hab = gab +1c2uaub.


O kajorismìc thc tetrataqÔthtac sunepgetai ìti to kosmologikì reustì twn parathrht¸n(dhlad twn sÔnolo twn parathrht¸n me touc opoÐouc ja melet soume thn kosmologÐa, oi opoÐ-oi eÐnai mia eidik kathgorÐa parathrht¸n sto sÔmpan / qwrìqrono pou orÐzetai me thn omdaisometrÐac thc Arq c thc Sqetikìthtac thc KosmologÐac) den èqei sqetik kÐnhsh. Kinhmatikautì ermhneÔetai me thn apaÐthsh oi qwrikèc suntetagmènec enìc kosmologikoÔ gegonìtoc ( kosmikoÔ antikeimènou ìpwc p.q. enìc galaxÐa) na eÐnai stajerèc sto kosmikì sÔsthma sunte-tagmènwn. Dhlad ènac sugkekrimènoc galaxÐac èqei mia sugkekrimènh kai monadik jèsh sthnournia sfaÐra pou parathroÔme. Mia eikìna pou èqei protajeÐ gia autì to gegonìc eÐnai ìti hournia sfaÐra eÐnai èna uper-mpalìni trei¸n diastsewn sto opoÐo èqoun sqediasteÐ oi kampÔlecθ = θ1, ϕ = ϕ1, r = r1 opìte kje kosmologikì s¸ma brÐsketai sthn tom trei¸n tètoiwn kampu-l¸n. 'Opwc to uper-mpalìni metablletai dhlad fousk¸nei xefousk¸nei oi kampÔlec autècexakoloujoÔn na tèmnontai sto Ðdio shmeÐo pou eÐnai topojethmènoc o sugkekrimènoc galaxÐac.

Me bsh ta anwtèrw upologÐzoume8:

hab = a2(t)1

(1 + ε4x2)2

δµaδνb . (3.35)

SÔmfwna me thn Kosmologik Arq to reustì twn parathrht¸n tou kosmikoÔ sust matocsuntetagmènwn èqei tetrataqÔthta, h opoÐa eÐnai èna bajmwtì sÔmorfo9 dinusma Killing thcmetrik c. H apaÐthsh aut majhmatik ekfrzetai me thn exÐswsh:

ua;b = 2ψgab (3.36)

ìpou ψ(xa) eÐnai o sÔmmorfoc pargontac. Sustèllontac me gab kai knontac qr sh thc sqèshcgabg

ab = 4 brÐskoume:

ψ =1

2ua;a. (3.37)

Epiplèon sustèllontac thn (3.36) me ub èqoume:

ua;bub = 2ψua = 0 (3.38)

ìpou èqoume qrhsimopoi sei ìti ψ = 0 llwc h tetrataqÔthta eÐnai èna dinusma Killing gegonìcpou apokleÐei h Kosmologik Arq . All to dinusma ua = ua;bu

b eÐnai h tetraepitqunsh twnkosmik¸n parathrht¸n. To gegonìc autì eÐnai sunepèc me thn apaÐthsh ìti oi kosmikèc grammèctwn kosmik¸n parathrht¸n eÐnai gewdesiakèc.

H pargwgoc thc tetrataqÔthtac ua;b analÔetai se 1+3 anlush sÔmfwna me thn sqèsh:

ua;b = ωab + σab +1

3θhab − uaub (3.39)

ìpou ωab eÐnai h strèyh, σab eÐnai h ditmhsh kai θ eÐnai h diastol . SugkrÐnontac me thn exÐswsh(3.36) èqoume ìti sto tupikì Kosmologikì montlelo twn FRW:

ωab = σab = 0, ua = 0 (3.40)

8Ο υπολογισμός έχει ως ακολούθως

hab = gab + uaub = gab + δ0aδ0b = −δ0aδ0b + a2(t)dΩ2 + δ0aδ

0b . = a2(t)dΩ2

.9Σύμορϕο σημαίνει u(a;b) = 2ψgab. Βαθμωτό σημαίνει u[a;b] = 0.


ra:

ua;b =1

3θhab. (3.41)

Prokeimènou na broÔme th sqèsh metaxÔ tou sÔmmorfou pargonta ψ kai thc diastol c θgrfoume:

2ψgab =1

3θhab

opìte sustèllontac me hab prokÔptei:

2ψgabhab =

1

3θhabh

ab.

All:gabh

ab = (hab − uaub)hab = habh

ab

kai telik:

ψ =1

6θ. (3.42)

Sta epìmena ja qrhsimopoioÔme to θ kai ìqi to ψ giatÐ to teleutaÐo den eÐnai fusikì mègejocall gewmetrikì.

3.2.6 Dunamikèc epipt¸seic

Apì th metrik upologÐzoume ton tanust tou Einstein kai mèsw twn exis¸sewn pedÐou thcjewrÐac:

Gab = kTab (3.43)

ìpou k = 8πG eÐnai h stajer thc barÔthtac tou Einatein kai G h pagkìsmia stajer barÔthtactou NeÔtwna, upologÐzoume ton tanust enèrgeiac orm c, o opoÐoc antistoiqeÐ sthn Ôlh kai ìla tadunamik pedÐa pou dhmiourgoÔn to pedÐo barÔthtac. ParathroÔme ìti o orismìc thc metrik c megewmetrikèc apait seic mac kajorÐzei, mèsw twn exis¸sewn Einstein, th morf thc Ôlhc. Dhlad sth JewrÐa thc Genik c Sqetikìthtac h GewmetrÐa fteiqnei Fusik . Antikajist¸ntac to Gab

apì thn (3.28), (3.29) brÐskoume:

kTab = 3a2 + ε

a2uaub −

[2a

a+a2 + ε

a2

]hab. (3.44)

SumperaÐnoume ìti h Ôlh pou mac epilègei h Kosmologik Arq gia touc kosmikoÔc parathrh-tèc (ìqi giac llouc!) se sunduasmì me tic exis¸seic pedÐou tou Einstein gia to tupikì montèlotou sÔmpantoc eÐnai to idanikì reustì. Apì diaisjhtik poyh h apaÐthsh aut eÐnai apodekt mia kai deÐqnei ìti h kÐnhsh twn galaxi¸n den apaiteÐ metafor orm c apì ton ènan ston llo(isodÔnama den èqoume ro jermìthtac apì to ènan galaxÐa ston llo) kai èqoume apokleÐseithn qwrik anisotropÐa sthn kÐnhsh. En µ eÐnai h puknìthta enèrgeiac kai p h isìtroph pÐeshthc Ôlhc kai twn pedÐwn pou dhmiourgoÔn to pedÐo barÔthtac (gia touc kosmikoÔc parathrhtèc!)tìte èqoume:

kTab = µuaub + phab. (3.45)


Antikajist¸ntac sthn (3.44) prokÔptoun exis¸seic pedÐou (gia touc kosmikoÔc parathrhtèc!):

3a2 + ε

a2= kµ (3.46)

2a

a+a2 + ε

a2= −kp (3.47)

oi opoÐec sundèoun tic dunamikèc metablhtèc µ, p me tic kinhmatikèc / gewmetrikèc metablhtèca(t), ε.

JewroÔme thn posìthta

H =a

a(3.48)

thn opoÐa onomzoume parmetro Hubble (Hubble parameter). Prokeimènou na doÔme thnkinhmatik ermhneÐa thc paramètrou Hubble parathroÔme ìti:

θ = ua;a = ua,a + Γabcu

buc = Γ000 = 3

a

a= 3H. (3.49)

Sunart sei thc paramètrou Hubble h pr¸th exÐswsh pedÐou grfetai:

3(H2 +

ε

a2

)= kµ. (3.50)

H exÐswsh aut eÐnai gnwst wc exÐswsh Friedman. 'Oson afor th deÔterh exÐswsh èqoume:

2H + 3H2 +ε

a2= −8kp. (3.51)

Apì tic exis¸seic pedÐou prokÔptei ìti:

p(t), µ(t).

dhlad h puknìthta enèrgeiac kai h isotropik pÐesh eÐnai sunart seic tou kosmikoÔ qrìnou mì-non kai eÐnai stajerèc se kje trisdistath uperepifneia. To gegonìc autì to qarakthrÐzoumeme th frsh

H puknìthta enèrgeiac kai h isotropik pÐesh enswmat¸noun (inherit) tic sum-metrÐec thc metrik c

apotèlesma sumbatì me thn apaÐthsh (1.2) thc Kosmologik c Arq c.

3.2.7 H Ôlh sto sÔmpan

Eisgoume tic posìthtec:

ΩM =kµ

3H2(3.52)

ΩK = − ε

(aH)2(3.53)


tic opoÐec onomzoume parmetroc puknìthtac Ôlhc (matter density parameter) kaiparmetroc puknìthtac kampulìthtac (curvature density parameter). Sunart -sei twn paramètrwn puknìthtac h exÐswsh Friedmann grfetai:

ΩM + ΩK = 1 (3.54)

Sth shmerin KosmologÐa deqìmaste ìti h Ôlh apoteleÐtai apì treic morfèc:a. Sqetikistik swmatÐdiab. Mh sqetikistik Ôlhg. Skotein enèrgeia (dark energy).Gia kje morf Ôlhc eisgoume antÐstoiqec paramètrouc puknìthtac wc akoloÔjwc:

Ω(0)r =

kµ(0)r

3H20

(3.55)

Ω(0)m =

kµ(0)m

3H20

(3.56)

Ω(0)DE =

kµ(0)DE

3H20

(3.57)

ìpou o deÐkthc (0) na sumbolÐzei to parìn. H parmetroc Hubble H0 sun jwc grfetai sthmorf :

H0 = 100h Kmsec−1Mpc−1 = 2.1332h× 10−42GeV (3.58)

ìpou1Mpc = 3.08568 × 1022 cm = 3.26156 × 106 ly. (3.59)

H stajer h ekfrzei thn abebaiìthta sthn tim tou H0. Oi parathr seic apì to prìgrammaHubble Key Prooject dÐnoun thn tim

h = 0.72 ± 0.08. (3.60)

OrÐzoume to qrìno Hubble (Hubble time) me th sqèsh:

tH = 1/H0 = 9.78 × 109 h−1èth. (3.61)

O qrìnoc Hubble dÐnei mia proseggistik ektÐmhsh thc hlikÐac tou sÔmpantoc. EpÐshc orÐzoumethn aktÐna Hubble (Hubble radius) me th sqèsh:

DH =c

H0

= 2.998h−1 Mpc (3.62)

h opoÐa kajorÐzei kat prosèggish th megalÔterh klÐmaka pou mporoÔme na parathr soume t¸ra.Me bsh th shmerin tim thc paramètrou Hubble orÐzoume kai thn krÐsimh puknìthta

(critical density) me th sqèsh:

µ0c =

3H20

k= 1.88h2 × 10−29 gcm−3. (3.63)

H krÐsimh puknìthta antistoiqeÐ sth mèsh kosmologik puknìthta tou sÔmpantoc s mera. HkrÐsimh puknìthta eÐnai polÔ mikrìterh apì tic puknìthtec twn topik¸n domik¸n mondwn tousÔmpantoc. Gia pardeigma h mèsh puknìthta thc Ghc eÐnai µΓ ≈ 5 gcm−3.


3.2.8 Nìmoi diat rhshc

Oi nìmoi diat rhshc eÐnai apotèlesma thc 1+3 anlushc thc tautìthtac T ab;a = 0. Oi nìmoi

diat rhshc ekfrzoun tic exis¸seic pedÐou sunart sei twn parag¸gwn thc puknìthtac enèrgeiackai thc isotropik c pÐeshc. Apì th genik sqèsh (blèpe Shmei¸seic Genik c Sqetikìthtac)gia èna idanikì reustì me puknìthta enèrgeiac µ kai isotropik pÐesh p èqoume tic akìloujecexis¸seic diat rhshc10:

ua = habp,bµ+ p

(3.64)

µ = −(µ+ p)θ (3.65)

ìpou θ = ua;a eÐnai h diastol thc tetrataqÔthtac ua.

3.2.9 Exis¸seic didoshc

Epeid h mình kinhmatik posìthta sto kosmologikì montèlo twn FRW eÐnai h diastol θ apìto sÔnolo twn exis¸sewn didoshc (blèpe ....) epibi¸nei mìnon aut pou afor thn parmetroθ. H exÐswsh aut onomzetai exÐswsh tou Raychaudhuri (Raychaudhuri equation).Genik h exÐswsh Raychaudhuri eÐnai h akìloujh:

θ +1

3θ2 = −Rabu

aub − σ2 + ω2 (3.66)

kai sthn perÐptwsh twn kosmik¸n parathrht¸n (gia touc opoÐouc isqÔei σ2 = ω2 = 0 )gÐnetai:

θ +1

3θ2 = −R00 ⇒

θ +1

3θ2 = 3

a

a. (3.67)

H exÐswsh Raychaudhuri den eÐnai mia nèa exÐswsh all perièqetai stic exis¸seic pedÐou. Giana to apodeÐxoume autì ergazìmaste wc akoloÔjwc:

H =aa− a2

a2=a

a−H2 = −1

2

(kp+ 3H2 +

ε

a2

)= −1

2

(kp+ kµ− 2ε

a2

)= −1

2

(kp+

1

3θ2 +

ε

a2

)(3.68)

θ = 3H = −3

2

(kp+

1

3θ2 +

ε

a2

)= −3

2

(kp+

ε

a2

)− 1

2θ2 ⇒

θ +1

3θ2 = −3

2

(kp+

ε

a2

)− 1

2θ2 +

1

3θ2 = −3

2

(kp+

ε

a2

)− 1

6θ2 = −3

2

(kp+H2 +

ε

a2

).

(3.69)

Antikajist¸ntac to aristerì mèloc brÐskoume thn exÐswsh pedÐou (3.47).

10Οι εξισώσεις αυτές ισχύουν για ιδανικό ρευστό ανεξάρτητα από τη μετρική, δηλαδή δεν είναι χαρακτηριστικέςτου τυπικού κοσμολογικού μοντέλου των FRW .


H sqèsh (3.67) mac odhgeÐ na eisgoume mia nèa parmetro sth melèth tou tupikoÔ kosmo-logikoÔ montèlou thn opoÐa onomzoume parmetroc epibrdunshc (decelerating para-meter) kai orÐzoume me th sqèsh:

q = − aa

1

H2= − aa

a2. (3.70)

Knontac qr sh thc exÐswshc pedÐou (3.47) brÐskoume ìti h parmetroc epibrdunshc dÐnetaiapì th sqèsh:

q = − aa

1

H2=

1

2+

1

2H2

(kp+

ε

a2

). (3.71)

Apì th sqèsh orismoÔ thc paramètrou q èqoume ìti:

a

a= −qH2. (3.72)

Antikajist¸ntac sthn (3.67) kai knontac qr sh thc θ = 3H h exÐswsh Raychaudhuri (kaiisodÔnama h deÔterh exÐswsh pedÐou) grfetai:

θ +1

3(1 + q)θ2 = 0. (3.73)

'Askhsh 4. JewreÐste th metrik

ds2 = −dt2 + a (t)2(dxidxi

)kai tic paramètrouc

H =a

aq = − aa

a2= −1 − H

H2.

JewreÐste sth sunèqeia to metasqhmatismì thc qronik c suntetagmènhc:

dt = N (τ) dτ

kai deÐxte ìti h metrik grfetai:

ds2 = −N2 (t) dτ 2 + a (τ)2(dxidxi

)UpologÐste tic paramètrouc H, q stic nèec suntetagmènec.LÔsh 'Eqoume:

da

dt=da

dτ

dτ

dt=a′

N, a′ =

da

dτra:

H =1

a

da

dt=

1

N

a′

a, a′ =

da

dτ.

Gia thn parmetro q èqoume:

q = − aaa2

= −1 − H

H2.

All:dH

dt=dH

dτ

dτ

dt=

1

N

dH

dτ=

1

NH ′.

Antikajist¸ntac upologÐzoume:

q = −1 − 1

N

H ′

H2.

3.3. H SUNALLOIWTH MORFH TOU JEWRHMATOS TOU GAUSS 35

Me bsh ta anwtèrw katal goume sto akìloujo:

Sumpèrasma:

Oi exis¸seic pedÐou tou tupikoÔ kosmologikoÔ montèlou ( montèlou twn FRW)me stajer kampulìthta ε eÐnai opoiod pote zeugri apì tic treic exis¸seic (3.46),(3.47), (3.65) (3.50), (3.73),(3.65). Shmei¸noume ìti oi exis¸seic pedÐou exart¸ntaiapì thn stajer kampulìthta ε en¸ h exÐswsh diat rhshc den exarttai. Oi gnw-stec metablhtèc tou montèlou eÐnai oi sunart seic a(t), p(t) kai µ(t). ParathroÔmeìti èqoume mìno dÔo exis¸seic epomènwc qreizetai na èqoume akìma mÐa exÐswsh pro-keimènou na mporèsoume na lÔsoume to dunamikì sÔsthma. Thn epiplèon exÐswshonomzoume exÐswsh katstashc (equation of state) kai thn eisgoume me epi-plèon fusikèc gewmetrikèc upojèseic akìma kai me upojèseic ergasÐac. SthnperÐptwsh tou epÐpedou kosmologikoÔ montèlou (dhlad ε = 0) oi exis¸seic pedÐouaplopoioÔntai stic ktwji:

3H2 = kµ (3.74)

2a

a+H2 = −kp (3.75)

en¸ h exÐswsh diat rhshc paramènei h Ðdia. Epomènwc oi exis¸seic pou èqoume gia toepÐpedo FRW montèlo eÐnai opoiesd pote dÔo exis¸seic apì tic treic exis¸seic (3.74),(3.75), (3.65).

3.3 H sunalloÐwth morf tou jewr matoc tou Gauss

L mma 1. DeÐxte ìti gia èna tuqaÐo pÐnaka A (x) isqÔei ìti

∂

∂xiln [det (A (x))] = Tr

A−1 ∂A

∂xi

. (3.76)

Apìdeixh'Eqoume th sqèsh:

det (A) =∑

aijbji

ìpou Bji eÐnai o cofactor tou aij. Tìte

∂ (detA)

∂aij= bji

kai

d (detA) =∑i,j

∂ (detA)

∂aijdaij =

∑i,j

bjidaij = Tr Adj (A) dA

opìte

∂

∂xiln [det (A (x))] =

(det (A));idet (A)

=1

det (A)Tr Adj (A) dA = Tr

A−1A,i

.


Pardeigma 1. DeÐxte ìti:

Γaab =

1√|g|

(√|g|

),b. (3.77)

ApìdeixhApì ton orismì tou Γa

ab èqoume:

Γaab =

1

2gacgac,b

kai me qr sh tou lÔmatoc èqoume p¸c:(GIATI ??? KALUTERA!!)

Γaab =

1

2

(ln√|g|

)b

=1√|g|

(√|g|

),b.

'Askhsh 5. Knontac qr sh tou paradeÐgmatoc 1 deÐxte ìti gia th sunalloÐwth pargwgo ènocdianusmatikoÔ pedÐou isqÔei h sqèsh:

Xa;a =

1√|g|

(√|g|Xa

),a. (3.78)

Sungete ìti en to Xa mhdenÐzetai se mÐa epifnia S tìte:∮S

Xa;a

√|g|dxn = 0. (3.79)

Aut eÐnai h genik morf tou jewr matoc tou Gauss.

'Askhsh 6. DeÐxte ìti gia ènan tanust deÔterhc txhc isqÔei h akìloujh sqèsh:

T ab;b =

1√|g|

(√|g|T ab

),b

+ ΓabcT

bc.

Jewr ste t¸ra ìti T ab = F ab, ìpou F ab eÐnai ènac antisummetrikìc tanust c, dhlad F (ab) = 0kai deÐxte ìti:

F ab;b =

1√|g|

(√|g|F ab

),b.

3.4 Diat rhsh tou arijmoÔ twn galaxi¸n

JewroÔme se kpoia kosmik qronik stigm t ìti h puknìthta twn galaxi¸n eÐnai nG, ìpou methn ènnoia puknìthta ennooÔme arijmì galaxi¸n an monda ìgkou tou 3-q¸rou thn kosmik idiostigm t. Epeid èqoume upojèsei ìti oi galaxÐec kinoÔntai me thn tetrataqÔthta twnkosmik¸n parathrht¸n ua orÐzoume to reÔma tou arijmoÔ twn galaxi¸n me th sqèsh:

JaG = nGu

a. (3.80)

H puknìthta nG(t) eÐnai sunrthsh tou kosmikoÔ qrìnou kai mìnon. JewroÔme ìti den èqoumedhmiourgÐa katastrof galaxi¸n to opoÐo shmaÐnei ìti to reÔma Ja

G diathreÐtai dhlad :

J aG ;a = 0. (3.81)

3.4. DIATHRHSH TOU ARIJMOU TWN GALAXIWN 37

Apì th sqèsh (3.78) pou dÐnei thn apìklish enìc dianusmatikoÔ pedÐou èqoume:

JaG..;a =

1√|g|

∂

∂xa

(√|g|nGu

a)

=1√|g|

∂

∂xa

(√|g|nGδ

at

)=

1√|g|

∂

∂xa

(√|g|nG

)δat =

1√|g|

∂

∂t

(√|g|nG

)apì ìpou prokÔptei

∂

∂t

(√|g|nG

)= 0.

Apì th metrik FRW se sfairikèc suntetagmènec upologÐzoume thn orÐzousa:

g = −a6 r4 sin2 θ

1 −Kr2

kai antikajist¸ntac sthn teleutaÐa sqèsh brÐskoume ìti h posìthta:

nGa3 = stajerì. (3.82)

Prokeimènou na d¸soume fusik ermhneÐa se aut n thn posìthta anatrèqoume sth sqèsh (2.47),h opoÐa dÐnei ton ìgko suntagmènwn (coordinate volume) se èna q¸ro stajer c kampulìth-tac me EukleÐdeio qarakt ra. SÔmfwna me thn Kosmologik Arq tètoioi eÐnai kai oi q¸roit = stajerì. Epomènwc h posìthta nGa

3 eÐnai o arijmìc twn galaxi¸n an monda ìgkou sunte-tagmènwn kai paramènei stajerìc sto sunkinoÔmeno sÔsthma suntetagmènwn (kai mìnon!). Pro-keimènou na gÐnei to mègejoc autì katanohtì jewroÔme touc galaxÐec se difora shmeÐa stoq¸ro t = stajerì stic antÐstoiqec suntetagmènec r, θ, ϕ. Epeid kje sunkinoÔmenoc parathrh-t c èqei stajerèc suntetagmènec r, θ, ϕ ìpwc o 3-q¸roc exelÐssetai sto qrìno oi suntetagmènectwn galaxi¸n paramènoun oi Ðdiec kai epomènwc h puknìtht touc me bsh tic suntetagmènec touceÐnai stajer , parìlo pou h epifneia eÐnai dunatìn na sustèlletai na diastèlletai.

Keflaio 4

EktÐmhsh kosmologik¸n apostsewn

4.1 Mèjodoi ektÐmhshc kosmologik¸n apostsewn

Profan¸c h ektÐmhsh (giatÐ h mètrhsh eÐnai adÔnath!) kosmologik¸n apostsewn den mporeÐ naeÐnai mia akrib c kai monos manth mejodologÐa. Gia pardeigma sthn KosmologÐa oi galaxÐecjewroÔntai shmeÐa en¸ sthn pragmatikìthta ekteÐnontai se polÔ meglec apostseic. Autìsunepgetai ìti oi mejodologÐec pou ja anaptÔxoume ja èqoun meglh abebaiìthta ( akrÐbeia) sesÔgkrish me ta g ina dedomèna. 'Omwc to gegonìc autì den prèpei na mac odhg sei se mhdenismìthc axiopistÐac twn mejìdwn aut¸n mia kai ta kosmologik dedomèna aforoÔn fainìmena seklÐmaka mh sugkrÐsimh me th g inh. En gènei prèpei na èqoume pnta kat nou ìti h KosmologÐaden eÐnai mia tupik epist mh thc opoÐac to apotèlesma pistopoieÐtai eÐnai pistopoi simo, allafor ektim seic gia polÔ megla sust mata me meglh pijanìthta ljouc (ìqi sflmatoc!), oiopoÐec gÐnontai me th qr sh thc episthmonik c mejìdou.

4.1.1 H erujr metatìpish

H erujr metatìpish eÐnai h pr¸th mejodologÐa pou mac epitrèpei thn ektÐmhsh twn apostsewntwn galaxi¸n. BasÐzetai sthn parat rhsh ìti to fsma thc aktinobolÐac pou lambnoume apìèna galaxÐa eÐnai metatopismèno (wc sÔnolo, dhlad oi apostseic metaxÔ twn fasmatik¸n gram-m¸n den metabllontai isodÔnama den èqoume dhmiourgÐa ap¸leia suqnot twn - qrwmtwn)wc proc to fsma Ðdiou fwtìc pou lambnoume apì akÐnhth phg sto ergast rio. Apì thn Eidik Sqetikìthta, all kai apì thn Neut¸neia Fusik , aut h metatìpish tou fsmatoc sumbaÐneiìtan uprqei sqetik kÐnhsh phg c kai dèkth kai eÐnai gnwstì wc fainìmeno Doppler. Epomè-nwc eÐnai logikì na apod¸sei kpoioc th metatìpish tou fsmatoc enìc galaxÐa sthn kÐnhsh(apomkrunsh plhsÐasma) tou sugkekrimènou galaxÐa proc emc. Epeid h aktinobolÐa apì togalaxÐa metafèretai me hlektromagnhtik kÔmata jewroÔme ìti h kÐnhsh twn fwtonÐwn gÐnetaikat m koc mhdenik¸n gewdesiak¸n. EpÐshc epeid oi aktinobolÐa lambnetai apì sugkekrimènhdieÔjunsh ston ouranì (jumÐzoume ìti oi tupikoÐ galaxÐec eÐnai stajer sundedemènoi sthn ou-rnia sfaÐra) oi suntetagmènec ϕ, θ eÐnai stajerèc epomènwc h kÐnhsh ston 3-q¸ro gÐnetai katm koc miac r−gramm c suntetagmènwn. Majhmatik ta anwtèrw ekfrzontai me thn apaÐthsh:

0 = dτ 2 = −dt2 + a2(t)dr2

1 − εr2(4.1)

39

40 KEFALAIO 4. EKTIMHSH KOSMOLOGIKWN APOSTASEWN

apì ìpou èqoume:dt

a(t)= f(r)dr (4.2)

me

f(r) =1√

1 − εr2. (4.3)

ParathroÔme ìti to aristerì mèloc eÐnai sunrthsh mìnon tou t kai to dexÐ mìnon tou r. AutìermhneÔetai wc akoloÔjwc. Epeid h jèsh twn galaxi¸n ston ouranì eÐnai stajer h dieÔjunshapì thn opoÐa lambnoume thn aktinobolÐa eÐnai stajer ra afor thn r− gramm suntetagmè-nwn apì thn dieÔjunsh tou galaxÐa. 'Omwc sto qwrìqrono oi galaxÐec kinoÔntai mazÐ me toucsunkinoÔmenouc parathrhtèc pnta sto Ðdio t−epÐpedo kai oi qwrikèc touc apostseic metabl-lontai. H eikìna aut èqei dojeÐ petuqhmèna me èna mpalìni sthn epifneia tou opoÐou eÐnaistajer kollhmènoi oi galaxÐec, to opoÐo diastèlletai sustèlletai opìte oi suntetagmènecϕ, θ twn galaxi¸n paramènoun oi Ðdiec en¸ metablletai mìnon h suntetagmènh r.

4.1.2 To fainìmeno Doppler sth Genik Sqetikìthta

Prokeimènou na suzht soume to fainìmeno Doppler sth Genik Sqetikìthta jumìmaste merikstoiqeÐa apì thn Eidik Sqetikìthta, ta opoÐa metafèroume sth Genik Sqetikìthta. Ta fwtìniaqarakthrÐzontai apì mhdenik tetranÔsmata ka me sunist¸sec:

ka =

(hνc

hνce

)Σ

(4.4)

ìpou e eÐnai to monadiaÐo kat thn didosh tou fwtonÐou ston 3-q¸ro enìc sust matoc Σ kai νh suqnìtht tou. Epeid jewroÔme c = 1 h sqèsh aut grfetai:

ka = hν

(1e

)Σ

. (4.5)

'Oson afor thn tetrataqÔthta ua enìc parathrht èqoume (sto Ðdio Σ!) thn anlush (c = 1):

ua =

(γγv

)Σ

ìpou v eÐnai h 3-taqÔthta tou parathrht sto Σ. En to Σ eÐnai Lorentz Kartesianì sÔsthmaopìte h metrik Lorentz ηab = diag(−1, 1, 1, 1), tìte èqoume:

ηabuakb = hvγ(−1 + v · e)

Jewr¸ntac dÔo parathrhtèc me tetrataqÔthtec ua1, ua2, oi opoÐoi parathroÔn to Ðdio fwtìnio,

èqoume th sqèsh:ηabu

a1k

b

ηabua2kb

=γ1(−1 + v1·e)

γ2(−1 + v2·e). (4.6)

ParathroÔme ìti to dexÐ mèloc exarttai mìnon apì tic taqÔthtec twn parathrht¸n sto Σ. Oiposìthtec ηabua1k

b kai ηabua2kb eÐnai analloÐwtec epomènwc mporoÔn na upologistoÔn se opoiod -

pote sÔsthma mac boleÔei. JewroÔme ta idiosust mata Σ1, Σ2 twn parathrht¸n 1,2 sta opoÐav1 = 0,v2 = 0 antÐstoiqa kai èqoume:

ηabua1k

b = hν1, ηabua2k

b = hν2 (4.7)

4.1. MEJODOI EKTIMHSHS KOSMOLOGIKWN APOSTASEWN 41

ìpou ν1, ν2 eÐnai oi suqnìthtec tou fwtonÐou ìpwc thn metroÔn oi parathrhtèc 1,2 antÐstoi-qa. Telik èqoume thn akìloujh sqèsh, h opoÐa sundèei tic metroÔmenec suqnìthtec apì toucparathrhtèc me tic taqÔthtec twn parathrht¸n sto Σ:

ν1ν2

=γ1(−1 + v1·e)

γ2(−1 + v2·e). (4.8)

H sqèsh aut ekfrzei to fainìmeno Doppler sthn Eidik Sqetikìthta. To anwtèrw senriometafèretai autoÔsio sth JewrÐa thc Genik c Sqetikìthtac en antikatastajeÐ h metrik touLorentz me th metrik tou qwrìqronou. Kat sunèpeia jewroÔme ìti to fainìmeno Doppler sthGenik Sqetikìthta perigrfetai apì thn exÐswsh:

ν1ν2

=gabu

a1k

b

gabua2kb

(4.9)

ìpou ta difora sÔmbola èqoun thn Ðdia shmasÐa.Ac efarmìsoume th sqèsh aut stic kosmologikèc parathr seic. JewroÔme èna galaxÐa, o

opoÐoc (met apì katllhlh strof twn axìnwn, h opoÐa den allzei oÔte periorÐzei tÐpota afoÔo 3-q¸roc eÐnai isìtropoc) brÐsketai sthn r dieÔjunsh, epomènwc h aktinobolÐa pou ekpèmpeilambnetai kat m koc thc gewdesiak c sth dieÔjunsh aut , sÔmfwna me ta ìsa anafèrame piopnw. Sto kosmikì sÔsthma suntetagmènwn tìso o galaxÐac ìso kai o parathrht c sth GhkinoÔntai ètsi ¸ste na mhdenÐzetai h sqetik touc taqÔthta (sunkinoÔmenoi parathrhtèc, rakoinì idiosÔsthma). Autì shmaÐnei ìti èqoun koinì idiìqrono, τ èstw, en¸ o kosmikìc qrìnoctou Σ eÐnai h suntetagmènh t. Epeid oi parathrhtèc eÐnai sunkinoÔmenoi to τ = γ(t) ìpou γ(t)eÐnai mia tuqaÐa sunrthsh. OrÐzoume thn γ(t) me thn apaÐthsh:

γ(t) =1

a(t)t. (4.10)

Me ton orismì autìn den eisgoume mia akìmh gnwsth sunrthsh1. Stic suntetagmènec τ, r, θ, ϕh metrik gÐnetai:

ds2 = a2(τ)(−dτ 2 + dr2 + f 2(r)dΩ2

)(4.12)

ìpou h sunrthsh f 2(r) dÐnetai sth sqèsh (3.6).Apì th sqèsh (4.10) èqoume:

τ =1

a(t)t. (4.13)

H sqèsh (4.13)deÐqnei pwc metasqhmatÐzetai o idiìqronoc (τ) twn sunkinoÔmenwn parathrht¸n meton kosmologikì qrìno (t), o opoÐoc eÐnai anexrthtoc parhthrht kai metriètai apì èna kosmikìbajmwtì pedÐo ìpwc perigryame sto edfio 1.3.1.

1Μια δικαιολόγηση του ορισμού είναι η ακόλουθη. Προκειμένου να ορίσουμε τη συνάρτηση γ(t) θεωρούμε δύοϕωτόνια, τα οποία εκπέμπονται από τον παρατηρητή ui1 τη χρονική στιγμή t1 από τη θέση με συντεταγμένες (στοΣ) r1, θ1, ϕ1 και τη χρονική στιγμή t1 + δt1 από τη ίδια θέση και λαμβάνονται από τον παρατηρητή ui2 τις χρονικέςστιγμές t2 και t2 + δt2 στη θέση r2, θ2, ϕ2 αντίστοιχα. Επειδή η διάδοση των ϕωτονίων στο Σ είναι κατά μήκοςτης μηδενικής γεωδαισιακής έχουμε:

−dt2 + a2(t)dr2

1− ϵr2= 0 ⇒ dt

a(t)=

dr√1− ϵr2

. (4.11)

ΝΑ ΣΥΜΠΛΗΡΩΘΕΙ.


Prokeimènou na upologÐsoume to metasqhmatismì thc idioqronik c dirkeiac metaxÔ dÔo sun-kinoÔmenwn parathrht¸n ergazìmaste wc akoloÔjwc. JewroÔme èna mètwpo kÔmatoc (gegonìcA) to opoÐo ekpèmpetai apì ton parathrht 1, èstw, th qronik idiostigm tou τA kai èstw ìti okosmologikìc qrìnoc tou gegonìtoc A eÐnai tA. 'Estw ìti to epìmeno mètwpo kÔmatoc (gegonìcB) ekpempetai th qronik idiostigm τB kai èstw t1 + δt1 o kosmologikìc qrìnoc tou gegonìtocB. Tìte èqoume:

τA =1

a(t1)t1, τB =

1

a(t1 + δt1)t1 + δt1 ⇒

τB − τA =1

a(t1 + δt1)(t1 + δt1) −

1

a(t1)t1.

All h diafor qrìnou δt1 gia diadoqik mètwpa kÔmatoc eÐnai polÔ mikr (thc txhc twn 10−14 s)epomènwc mporoÔme na jewr soume ìti a(t1 + δt1) = a(t1). Tìte h teleutaÐa sqèsh dÐnei:

τB − τA =1

a(t1)δt1 (4.14)

Mia mesh efarmog thc sqèshc aut c eÐnai h akìloujh. 'Estw ìti ta gegonìta A, B ta parathreÐkai ènac lloc kosmikìc parathrht c 2 tic qronikèc idiostigmèc tou t2 kai t2 + δt2. Tìte èqoumeth sqèsh:

1

a(t1)δt1 =

1

a(t2)δt2 (4.15)

h opoÐa deÐqnei pwc metasqhmatÐzontai polÔ mikr (se sqèsh me tic metabolèc tou pargontaklÐmakac) qronik idiodiast mata metaxÔ sunkinoÔmenwn parathrht¸n.

Metasqhmatismìc periìdou kai suqnìthtac hlektromagnhtikoÔ kÔmatoc

JewroÔme fwtìnio suqnìthtac ν sto kosmologikì sÔsthma suntetagmènwn, to opoÐo ekpèmpetaiapì ton parathrht ui1 th qronik idiostigm tou t1 kai lambnetai apì ton parathrht ui2 thqronik idiostigm tou t2. En T eÐnai h perÐodoc tou fwtonÐou sto kosmologikì sÔsthmasuntetagmènwn, tìte èqoume2 :

ν =2π

T=

2π

T1a(t1) = ν1a(t1) (4.16)

ìpou ν1 eÐnai h suqnìthta tou fwtonÐou sto idiosÔsthma tou parathrht ui1 . Me ìmoio trìpo ( diaforetik) deÐqnoume ìti gia th suqnìthta ν2 tou fwtonÐou sto idiosÔsthma tou parathrht ui2 isqÔei ν = ν2a(t2). Epomènwc èqoume th sqèsh3:

ν1a(t1) = ν2a(t2) ⇒ν1ν2

=a(t2)

a(t1). (4.17)

Sthn prxh èqei epikrat sei na qrhsimopoieÐtai to mègejoc:

z =λ2 − λ1λ1

(4.18)

2Γιατί από την (4.14) έχουμε T1 = 1a(t1 ).

3Οι παρατηρητές είναι συνκινούμενοι άρα έχουν ίδιες ιδιοχρονικές στιγμές.

4.2. KOSMOLOGIKES APOSTASEIS 43

ìpou λ2, λ1 eÐnai ta antÐstoiqa m kh kÔmatoc tou fwtonÐou (dhlad thc aktinobolÐac pou para-threÐtai). Epeid prìkeitai gia hlektromagnhtik aktinobolÐa to λ = c

νopìte upologÐzoume:

z =a(t2)

a(t1)− 1. (4.19)

Thn posìthta z onomzoume parmetroc erujrc metatìpishc (red-shift parameter)kai apoteleÐ ènan deÐkth thc apìstashc enìc galaxÐa apì thn arq tou kosmikoÔ sust matocsuntetagmènwn, thn opoÐa jewroÔme ìti brÐsketai sto kèntro tou galaxÐa mac. En z > 0 tìtelème ìti èqoume erujr metatìpish kai en z < 0 tìte lème ìti èqoume metatìpish sto mplè(blue-shit).

4.2 Kosmologikèc apostseic

Prokeimènou na suzht soume tic diforec kosmologikèc apostseic kai th mejodologÐa ektÐmh-s c touc grfoume th 3-metrik gµν tou tupikoÔ kosmologikoÔ montèlou se katllhlh morf .Sth sqèsh (3.4) èqoume deÐxei ìti

gµνdxµdxν = dσ2 =

dr2

1 − εr2+ f 2(r)dΩ2

ìpou

f 2(r) =

sin r gia ε = 1r gia ε = 0

sinh r gia ε = −1

.

Eisgoume th nèa suntetagmènh χ me th sqèsh:

r =

sinχ gia ε = 1χ gia ε = 0

sinhχ gia ε = −1

(4.20)

opìte h 3-metrik grfetai:dσ2 = dχ2 + (fε(χ))2 dΩ2. (4.21)

ìpou h sunrthsh

fε(χ) =

sinχ gia ε = 1χ gia ε = 0

sinhχgia ε = −1

. (4.22)

H sunrthsh fε(χ) mporeÐ na grafeÐ me mia sqèsh wc akoloÔjwc

fε(χ) =1√−ε

sinh(√−εχ) (4.23)

ìpou h perÐptwsh tou epÐpedou sÔmpantoc (dhlad ε = 0) lambnetai me to ìrio ε→ 0.Sthn KosmologÐa diakrÐnoume treÐc apostseic (a) Thn sunkinoÔmenh apìstash (b) Thn

apìstash aktinobolÐac kai (g) Thn apìstash thc gwniak c diamètrou.


4.2.1 H sunkinoÔmenh apìstash

JewroÔme mia phg ekpomp c hlektromagnhtik c aktinobolÐac me suntetagmènec θ, ϕ sthn our-nia sfaÐra kai se apìstash χ1, h opoÐa ekpèmpei th kosmologik qronik stigm t1. H aktinobolÐathc phg c kineÐtai kat m koc thc χ−gramm c suntetagmènwn kai parathreÐtai thn kosmologi-k qronik stigm t0 (dhlad t¸ra). H erujr metatìpish tou fsmatoc thc phg c, h opoÐaantistoiqeÐ sthn apìstash thc phg c èstw ìti eÐnai z en¸ h antÐstoiqh erujr metatìpish toufsmatoc sthn parat rhsh (pou eÐnai to shmeÐo me χ = 0) eÐnai z = 0. H exÐswsh thc thc troqictou fwtonÐou eÐnai (jewroÔme c = 1) (blèpe (4.1):

−c2dt2 + a2(t)dχ2 = 0 ⇒ dχ =c

a(t)dt. (4.24)

H sunkinoÔmenh apìstash (comoving distance) orÐzetai me th sqèsh:

dc ≡ χ1 =

∫ χ1

0

dχ = −∫ t1

t0

c

a(t)dt (4.25)

dhlad eÐnai Ðsh me th metabol thc suntetagmènhc χ metaxÔ twn gegonìtwn thc ekpomp c kaithc l yhc (to meÐon giatÐ t0 > t1). ParathroÔme ìti isqÔei:

dc = cη (4.26)

ìpou η eÐnai o sÔmorfoc qrìnoc.Apì th sqèsh (4.19) èqoume (t kosmologikìc qrìnoc):

z =a0a(t)

− 1 ⇒ z =a0a2(t)

a(t) =a0a(t)

H(t) =a0H0

a(t)

H(t)

H0

=a0H0

a(t)E(z)

ìpou èqoume jèsei:

E(z) =H(z)

H0

. (4.27)

'Ara:

z =a0H0

a(t)E(z) ⇒ c

a(t)dt =

c

a0H0

1

E(z)dz

AntikajistoÔme to cdta(t)

sthn (4.24) kai èqoume thn sunkinoÔmenh apìstash sunart sei thc para-mètrou z:

dc ≡c

a0H0

∫ z

0

1

E(z)dz. (4.28)

AnaptÔssoume th sunrthsh∫ z

01

E(z)dz gÔrw apì to shmeÐo z = 0 (th l yh) kai èqoume4:∫ z

0

1

E(z)dz = z − E ′(0)

2z2 +

1

62(E ′(0))2 − E ′′(0)z3 +O(z4)

ìpou o tìnoc upodhl¸nei pargwgo wc proc z. Antikajist¸ntac blèpoume ìti gia z ≪ 1 hsunkinoÔmenh apìstash dÐnetai apì th th sqèsh:

dc ≃c

a0H0

z gia z << 1. (4.29)

4Αποδείξτε το.


Apì th sqèsh (4.9) jewr¸ntac thn ekpomp wc to gegonìc 2 kai thn parat rhsh wc togegonìc 1 èqoume se ènan profan sumbolismì (eisgoume pli to c):

ν0ν

=−1

γ(−1 + v·e/c)(4.30)

ìpou èqoume jèsei thn taqÔthta sthn parat rhsh (sth jèsh r0) Ðsh me mhdèn. H ekpomp gÐnetaikat m koc thc χ− gramm c suntetagmènwn, dhlad thc eiserqìmenhc aktinik c dieÔjunshc,ra v·e = −vr ìpou vr eÐnai h aktinik taqÔthta. Sunart sei tou m kouc kÔmatoc èqoume taakìlouja:

c = ν0λ0 = νλ⇒ λ0λ

=ν

ν0= γ(1 + vr/c). (4.31)

AnaptÔssontac to γ gÔrw apì thn taqÔthta mhdèn èqoume:

γ = 1 +1

2β2 +

3

8β4 + . . . (4.32)

opìte prokÔptei:

λ0λ

=

(1 +

1

2β2 + ...

)(1 + vr) =1 + vr.

Apì ton orismì tou z èqoume:

z =λ0λ

− 1 = vr/c (4.33)

dhlad gia mikrèc taqÔthtec h erujr metatìpish eÐnai anlogh thc aktinik c taqÔthtac. Sun-diasmìc me th sqèsh (4.29) dÐnei:

vr = a0H0dc. (4.34)

H sqèsh aut deÐqnei ìti h taqÔthta apomkrunshc thc phg c eÐnai anlogh thc sunkinoÔmenhcapìstashc me stajer analogÐac thn posìthta a0H0. H apìstash kat m koc thc suntetagmènhcr, dhlad h fusik apìstash thc phg c thn kosmologik qronik stigm t, h opoÐa antistoiqeÐse erujr metatìpish z, dÐnetai apì th sqèsh:

r = a0dc(z). (4.35)

Gia mikrèc taqÔthtec apì th sqèsh (4.29) prokÔptei:

r ≃ c

H0

z = vr/H0 (4.36)

dhlad h fusik apìstash thc phg c eÐnai anlogh thc taqÔthtac apomkrunshc me stajeranalogÐac thn shmerin tim thc paramètrou Hubble. H sqèsh aut isqÔei gia mikr z ≪ 1 kaieÐnai gnwst wc nìmoc tou Hubble (Hubble’s law). Gia z ≥ 1 h sqèsh metaxÔ apìstashckai taqÔthtac apomkrunshc den eÐnai grammik .


4.2.2 H apìstash parllaxhc

JewroÔme sthn ournia sfaÐra èna tupikì galaxÐa, tou opoÐou oi suntetagmènec sto kosmologikìsÔsthma suntetagmènwn , x èstw, eÐnai oi stajerèc r1, θ1, ϕ1. JewroÔme epÐshc to sÔsthmasuntetagmènwn, x′ èstw, me arq sth fwtein phg , to opoÐo eÐnai isodÔnamo me to tupikìkosmologikì sÔsthma. Ta dÔo sust mata suntetagmènwn sundèontai me èna metasqhmatismìhmi-metatìpishc, o opoÐoc èstw ìti orÐzetai me to dinusma aa sto x sÔsthma suntetagmènwn.To dinusma aa eÐnai to dinusma jèshc thc arq c tou x′ sto x. O metasqhmatismìc autìcsundèei ta dianÔsmata jèshc x′a, xa enìc shmeÐou sthn ournia sfaÐra wc akoloÔjwc:

x′a = xa + aa[√


](4.37)

ìpou B =1−√

1−εgabaaab

εgabaaabkai gab eÐnai oi sunist¸sec thc metrik c sto sÔsthma suntetagmènwn

x. Epeid to sÔsthma suntetagmènwn r, θ.ϕ eÐnai orjokanonikì (all ìqi olìnomo!) èqoumegab = δab. Se dianusmatikì formalismì h sqèsh (4.37) grfetai:

x = x′ + a[√

1 − εx′2 −Bε(x′ · a)]

me

B =1 −

√1 − εa2

εa2

:

x = x′ + a

[√1 − εx′2 −

1 −

√1 − εa2

x′ · aa2

]. (4.38)

ìpou ta eswterik ginìmena eÐnai EukleÐdeia eswterik ginìmena.'Estw x1 to dinusma jèshc tou sust matoc suntetagmènwn x′ sto sÔsthma suntagmè-

nwn x. 'Opwc èqoume deÐxei, ta sust mata suntetagmènwn x′ kai x sqetÐzontai me ènametasqhmatismì hmi-metatìpishc me dinusma a = x1. H (4.38) dÐnei:

x = x′ + x1

[√1 − εx′2 −

1 −

√1 − εx2

1

x′ · x1

x21

]. (4.39)

JewroÔme fwtein aktÐna h opoÐa ekpèmpetai apì thn phg th qronik idiostigm (thc phg c!)t′0 = 0 kai se kateÔjunsh h opoÐa sto idiosÔsthma thc phg c x′ dÐnetai apì to monadiaÐo dinu-sma n, me suntetagmènec θ′0, ϕ

′0. Epeid to fwc kineÐtai kat m koc miac mhdenik c gewdaisiak c

kai epeid sto sunkinoÔmeno sÔsthma suntetagmènwn h phg brÐsketai se èna stajerì shmeÐosthn ournia sfaÐra, jewroÔme ìti gia opoiod pote shmeÐo kat m koc thc mhdenik c gewdaisia-k c isqÔei θ′ = θ′0, ϕ

′ = ϕ′0. Tìte to dinusma jèshc twn shmeÐwn thc mhdenik c gewdaisiak c

eÐnai5:x′(ρ) = n ρ (4.40)

ìpou ρ eÐnai mia jetik parmetroc h opoÐa aparijmeÐ shmeÐa kat m koc thc gewdaisiak c (meρ = 0 sthn phg , pou sumpÐptei me thn arq tou sust matoc suntetagmènwn x′). To dinusma

5Η μηδενοκή γαιοδεσιακή θεωρούμε ότι είναι ευθεία. Αυτό πρέπει να αποδειχτεί δεν είναι προϕανές. Στα επόμεναθεωρούμε το n ανεξάρτητο του ρ το οποίο δεν ισχύει γενικά, εκτός στην περίπτωση που η γαιδεσιακή θεωρείταιότι είναι ευθεία.


jèshc thc fwtein c aktÐnac sto sÔsthma suntetagmènwn x upologÐzetai apì to metasqhmatismì(4.39), epomènwc èqoume:

x = n ρ+ x1

[√1 − ερ2 −

1 −

√1 − εx2

1

ρn · x1

x21

]. (4.41)

Metablloume thn kateÔjunsh twn axìnwn tou sust matoc x ètsi ¸ste |x1| = r1 ìpou reÐnai h aktinik suntetagmènh. Autì eÐnai pnta dunatìn lìgw tou ìti o 3-q¸roc eÐnai isìtropoc.Tìte h teleutaÐa sqèsh dÐnei:

x(ρ) = n ρ+ x1

[√1 − ερ2 −

1 −

√1 − εr21

ρn · x1

r21

]. (4.42)

To efaptìmeno dinusma sth fwtein aktÐna x(ρ) to opoÐo dÐnei th dieÔjunsh thc aktÐnac sekje shmeÐo sto q¸ro eÐnai:

dx(ρ)

dρ= n + x1

[−ερ√

1 − ερ2−

1 −√

1 − εr21

n · x1

r21

]. (4.43)

Upojètoume6 ìti kont sthn arq tou sust matoc suntetagmènwn x (shmeÐo parat rhshc) htim thc paramètrou ρ = r1. UpologÐzoume apì thn (4.42):

x(r1) = nr1 + x1r1

[√1 − εr21 −

1 −

√1 − εr21

n · x1

]= (n− (n · x1)x1)r1 + x1r1

[√1 − εr21 +

√1 − εr21n · x1

]= n⊥r1 + x1r1

[√1 − εr21 +

√1 − εr21n · x1

](4.44)

ìpou x1 = r1x1 kai n⊥ eÐnai h kjeth probol tou n kat m koc thc dieÔjunshc x1. 'Omoiah dieÔjunsh thc fwtein c aktÐnac kont sthn arq tou sust matoc suntetagmènwn x eÐnai(blèpe (4.43) ):

dx(ρ)

dρ|ρ=r1 = n + x1

[−εr21√

1 − εr12−

1 −√

1 − εr21

n · x1

]= n− (n · x1)x1 + x1

[−εr21√

1 − εr12+√

1 − εr21n · x1

]= n⊥ +

x1√1 − εr12

[−εr21 + (1 − εr21)n · x1

]= n⊥ +

x1√1 − εr12

[−εr21(1 + n · x1) + n · x1

]. (4.45)

H gwnÐa metaxÔ thc dieÔjunshc parat rhshc (direction of sight) (dhlad thc dieÔjunshc sthnphg (!)pou ja eÐqe to fwc sthn arq tou sust matoc suntetagmènwn x en o q¸roc tan

6Η υπόθεση αυτή είναι σημαντική!


EukleÐdeioc) −x1 kai thc dieÔjunshc thc pragmatik c ekpomp c thc fwtein c aktÐnac sthn phg n metr thn parllaxh thc fwtein c phg c sto sÔsthma suntetagmènwn x. Upojètoume ìtiaut h gwnÐa, θ èstw, eÐnai polÔ mikr opìte mporoÔme na proseggÐsoume to sin θ ≃ θ, cos θ ≃ 1.Autì sunepgetai:

n · x1 = −(n · −x1) = − cos θ ≃ −1.

Antikajist¸ntac sthn (4.44) kai (4.45) antÐstoiqa brÐskoume:

x(r1) ≃ n⊥r1 + x1r1

[√1 − εr21 −

√1 − εr21

]≃ n⊥r1 (4.46)

dx(ρ)

dρ|ρ=r1 ≃ n⊥ +

x1√1 − εr12

[−εr21(1 − 1) − 1

]≃ n⊥ − x1√

1 − εr12. (4.47)

H dieÔjunsh thc fwtein c aktÐnac kont sthn arq tou x dÐnetai apì monadiaÐo dinusma kat

m koc tou dianÔsmatocdx(ρ)

dρ|ρ=r1 . UpologÐzoume:∣∣∣∣dx(ρ)

dρ|ρ=r1

∣∣∣∣2 ≃ n2⊥ +

1

1 − εr12[−εr21(1 + n · x1) + n · x1

]2 − 2n⊥ · x1√1 − εr12

[−εr21(1 + n · x1) + n · x1

]= n2

⊥ +1

1 − εr12[−εr21(1 + n · x1) + n · x1

]2.

Sthn prosèggis mac n2⊥ ≃ θ2 ≃ 0 opìte èqoume:∣∣∣∣dx(ρ)

dρ|ρ=r1

∣∣∣∣ ≃ 1√1 − εr12

. (4.48)

Telik to monadiaÐo dinusma, u èstw, pou kajorÐzei th dieÔjunsh thc fwtein c aktÐnac kontsthn arq tou x eÐnai:

u ≃ −√

1 − εr12[n⊥ − x1√

1 − εr12

]= x1 −

√1 − εr12n⊥. (4.49)

H parllaxh θ thc fwtein c phg c sto sÔsthma suntetagmènwn x dÐnetai apì th diaformetaxÔ twn dÔo dieujÔnsewn u (th fainìmenh dieÔjunsh thc fwtein c aktÐnac (the apparentdirection of sight) ) kai x1 (thn alhj dieÔjunsh thc fwtein c aktÐnac (the true direction ofsight) ). UpologÐzoume:

u− x1 ≃ −√

1 − εr12n⊥ ⇒ (4.50)

|u− x1| ≃√

1 − εr12|n⊥|. (4.51)

Th qronik stigm thc mètrhshc thc fwtein c aktÐnac sto sÔsthma suntetagmènwn x, t0èstw, h fwtein aktÐna eÐnai sth jèsh x(r1). Sto sÔsthma suntetagmènwn x jewroÔme tatautìqrona gegonìta (simultaneous events) ( t0, 0) kai (t0,x(r1)), ta opoÐa brÐskontai stonidiìqwro tou parathrht me idiosÔsthma x thn idiostigm t0. H qwrik apìstash aut¸n twngegonìtwn eÐnai h idio-apìstash (proper distance) (giatÐ eÐnai tautìqrona) kai èqoume (blèpe(4.46) ):

b =√ds2 = R(t0)|x(r1)| ≃ R(t0)r1|n⊥|. (4.52)


Thn apìstash b onomzoume impact parameter thc fwtein c aktÐnac. Apì thn (4.52) prokÔ-ptei:

|n⊥| ≃b

R(t0)r1. (4.53)

Antikajist¸ntac sthn (4.51) brÐskoume:

|u− x1| ≃√

1 − εr12b

R(t0)r1. (4.54)

All to u− x1 eÐnai to dinusma pou dÐnei thn apìklish metaxÔ thc alhjoÔc kai thc fainì-menhc kateÔjunshc thc fwtein c aktÐnac kont sthn arq tou sust matoc suntetagmènwn x(blèpe Sq ma ....), kai eÐnai to mètro thc gwnÐac parllaxhc thc fwtein c aktÐnac sto sÔsthmasuntetagmènwn x. Sunep¸c jètoume θ ≃ |u− x1| kai telik èqoume th sqèsh7 :

θn ≃√

1 − εr12b

R(t0)r1. (4.55)

Sthn EukleÐdeia gewmetrÐa mia phg se apìstash d me impact parameter b èqei gwnÐa pa-rllaxhc (upojètoume ìti h gwnÐa eÐnai mikr opìte proseggÐzoume to sin me th gwnÐa):

θn =b

dgia θn → 0, b→ 0. (4.56)

H sqèsh (4.56) mac odhgeÐ na orÐsoume thn apìstash parllaxhc (parallax distance)miac fwtein c phg c sto kosmologikì sÔsthma suntetagmènwn x me th sqèsh:

dP = R(t0)r1√

1 − εr12. (4.57)

Shmei¸noume ìti se èna montèlo sÔmpantoc me jetik kampulìthta, dhlad ε = +1, se apì-stash r1 = 1 èqoume apeirh apìstash parllaxhc kai gia fwteinèc phgèc se fusikèc apostseicr1 < 1, apìstash parllaxhc mei¸netai.

4.2.3 H apìstash lamprìthtac

Xekinme me dÔo basikoÔc orismoÔc.a. Apìluth lamprìthta miac kosmik c fwtein c phg c (absolute luminosity) (sÔm-

bolo L) eÐnai h olik isqÔc pou aktinoboleÐ mia fwtein phg sto sÔmpan. JewroÔme ìti hekpomp eÐnai isìtroph ra h fwtein isqÔc pou ekpèmpetai se mia stere gwnÐa dΩ eÐnai L/dΩ.

b. Fainìmenh lamprìthta miac kosmik c fwtein c phg c (apparent luminosity) (sÔm-bolo l) eÐnai h olik isqÔc pou lambnetai an stere gwnÐa apì thn kosmik phg ston kajrèpthtou thleskopÐou. Sth Neut¸neia Fusik ìpou o q¸roc eÐnai EukleÐdeioc, h fainìmenh lamprì-thta lE miac fwtein c phg c apìluthc lamprìthtac L h opoÐa brÐsketai se EukleÐdeia apìstashdE apì to epÐpedo tou kajrèpth tou mikroskopÐou dÐnetai apì th sqèsh:

l =L

4πd2E. (4.58)

7΄Οπως δείξαμε |n⊥| = sinθ ≃ θ.


Apì th sqèsh aut orÐzoume thn (EukleÐdeia) apìstash lamprìthtac (luminocity dista-nce) :

dE =

√LE

4πlE(4.59)

Ja upologÐsoume thn apìstash lamprìthtac sto tupikì kosmologikì montèlo twn FRW.JewroÔme mia fwtein phg kat m koc thc dieÔjunshc parat rhshc (line of sight) me dinusmajèshc x1 sto sÔsthma suntetagmènwn x. 'Estw b h dimetroc tou kajrèpth tou thleskopÐouìpou lambnetai h fwtein aktinobolÐa apì th fwtein phg . Tìte b eÐnai o pargontac kroÔshc(impact parameter) pou jewr same me sthn (4.52).

'Estw ìti h apìluth lamprìthta thc phg c sto idiosÔsthma thc eÐnai Ls. Autì shmaÐnei ìtih phg ekpèmpei fwtein isqÔ isìtropa me rujmì Nhν1

δt1ìpou N eÐnai o arijmìc twn fwtonÐwn

th qronik idiostigm t1 gia qronikì disthma δt1 me suqnìthta ν1 ìpou ìla ta megèjh eÐnaisto idiosÔsthma thc phg c. Ta fwtìnia pou lambnontai ston kajrèpth tou thleskopÐou èqounsuqnìthta ν0 kai lambnontai th qronik idiostigm t0 gia qronikì disthma (sto idiosÔsthmatou thleskopÐou !) δt0. 'Eqoume gia thn apìluth lamprìthta thc phg c sta dÔo sust matasuntetagmènwn (dhlad to idiosÔsthma thc phg c kai to idiosÔsthma tou thleskopÐou):

Ls =Nhν1δt1

, Lt =Nhν0δt0

.

H sqèsh metaxÔ twn suqnot twn ν1, ν2 kai twn qronik¸n periìdwn δt1, δt0 dÐnetai apì tic sqè-seic (4.17) kai (4.15) antÐstoiqa. Epomènwc h apìluth lamprìthta Lt thc phg c sto idiosÔsthmatou kajrèpth eÐnai:

Lt =Nhν0Nhν1δt0

δt1δt0

Ls =a2(t1)

a2(t0)Ls. (4.60)

H isqÔc tou ekpempìmenou fwtìc an stere gwnÐa eÐnai Ls

4π, epomènwc sto idiosÔsthma tou

kajrèpth èqoume an monda sterec gwnÐac:

1

4πLt =

1

4π

a2(t1)

a2(t0)Ls. (4.61)

Oi fwteinèc aktÐnec pou lambnontai ston kajrèpth tou thleskopÐou brÐskontai sto eswte-rikì enìc orjog¸niou k¸nou me embadìn epifneiac bshc πb2 en¸ sto idiosÔsthma thc phg cekpèmpontai sto eswterikì mis c sterec gwnÐac, h opoÐa orÐzetai apì to m koc thc kjethcprobol c n⊥| (blèpe Sq ma ...). H stere gwnÐa8 isoÔtai me π|n⊥|2 diìti to dinusma n eÐnaimonadiaÐo kai n⊥ eÐnai h kjeth probol tou dianÔsmatoc n sth dieÔjunsh parat rhshc (lineof sight) x1. Epomènwc h isqÔc pou lambnetai ston kajrèpth tou thleskopÐou eÐnai Ðsh me14πLtπ|n⊥|2. 'Ara:

1

4πLtπ|n⊥|2 =

1

4

a2(t1)

a2(t0)Ls|n⊥|2 =

1

4

a2(t1)

a2(t0)Ls

b2

a2(t0)r21

8Η στερεά γωνία Ω την οποία υποτείνει μια επιϕάνεια S ορίζεται ως η το εμβαδόν που ορίζει η προβολή τηςεπιϕάνειας S στην μοναδιαία σϕαίρα με κέντρο την κορυϕή της γωνίας. Για μια κυκλική επιϕάνεια S έχουμε ότι η

στερεά γωνία είναι∫Ssin θ cosϕdϕ. Για κώνο μισής γωνίας θ0 << 1 το ολοκλήρωμα είναι

∫ θ00

∫ 2π

0sin θdθdϕ ≃ 2π∫ θ0

0θdθ = 2π 1

2θ20 = πθ20. Εδώ έχουμε ότι sin θ0 = |n⊥| (γιατί η υποτείνουσα του τριγώνου είναι ίση με 1), επομένως

η στερεά γωνία ισούται με π|n⊥|2.


ìpou èqoume antikatast sei to |n⊥|2 apì th sqèsh (4.53) kai b eÐnai o impact parameter pouupologÐsame sthn (4.52) (oi aktÐnec sthn apìstash tou pargonta kroÔshc impact parameterglÔfoun to epÐpedo tou kajrèpth tou thleskopÐou. H fainìmenh lamprìthta lt thc fwtein cphg c eÐnai h isqÔc pou lambnetai ston kajrèpth tou thleskopÐou sto idiosÔsthma tou thle-skopÐou an monda epifneiac tou kajrèpth. H epifneia tou kajrèpth eÐnai πb2 , epomènwc:

lt =1

πb21

4πLtπ|n⊥|2 =

1

4π

a2(t1)

a2(t0)

1

a2(t0)r21Ls. (4.62)

Me bsh thn EukleÐdeia apìstash lamprìthtac dE orÐzoume thn (kosmologik ) apìstashlamprìthtac (luminosity distance) dL me thn apaÐthsh9

dL =

√Ls

4πlt= a2(t0)

r1a(t1)

. (4.63)

4.2.4 H apìstash gwniak c diamètrou

Sth Neut¸neia Fusik h gwniak dimetroc (angular diameter) (sÔmbolo δ) gia mia phg me dimetro D se apìstash d ≫ D orÐzetai wc to phlÐko:

δ =D

d, δ ≪ 1.

UpologÐzoume thn (kosmologik ) gwniak dimetro miac fwtein c phg c sta plaÐsia toutupikoÔ kosmologikoÔ montèlou (dhlad tou FRW).

JewroÔme mia fwtein phg me dimetro sto idiosÔsthm thc (proper diameter) D (that isdiameter in the cosmological frame) h opoÐa brÐsketai se apìstash r1 kai aktinoboleÐ th qronik idiostigm thc t1. H fwtein aktinobolÐa parathreÐtai sth jèsh r = 0 th qronik idiostigm (stoidiosÔsthma) tou thleskopÐou t0. Oi fwteinèc aktÐnec sto idiosÔsthma thc phg c ekpèmpontaiisìtropa kai diadÐdontai sto q¸ro mèqri pou estizoun ston kajrèpth tou thleskopÐou sth jèshr = 0. QwrÐc periorismì thc genikìthtac strèfoume to thleskìpio ¸ste o xonac tou thlesko-pÐou (line of sight) na eÐnai sto kèntro thc fwtein c phg c. JewroÔme ìti aut h dieÔjunshantistoiqeÐ sth gwnÐa θ = 0 opìte h gwniak dimetroc thc phg c orÐzetai apì tic timèc −δ/2kai δ/2. Tìte h dimetroc thc phg c sto idiosÔsthm thc (proper diameter) eÐnai:

D =

∫ δ/2

−δ/2

√gθθdθ =

∫ δ/2

−δ/2

a(t1)r1dθ = a(t1)r1δ.

Epomènwc h (kosmologik ) gwniak dimetroc eÐnai:

δ =D

a(t1)r1. (4.64)

OrÐzoume thn apìstash gwniak c diamètrou (angular diameter distance) dA miacfwtein c phg c me (kosmologik ) gwniak dimetro δ me th sqèsh:

dA =D

δ= a(t1)r1. (4.65)

9Το αποτέλεσμα αυτό μπορεί να παραχθεί και με διαϕορετική μεθοδολογία. Βλέπε H.P. Robetson, Z. Astrophys.15,69 (1937) and Z. f. Astrophys., 15, 69 (1938).


Shmei¸noume ìti ìtan to r1 auxnei o pargontac klÐmakac a(t1) mei¸netai epomènwc semerik montèla (dhlad lÔseic a(t1))) h apìstash gwniak c diamètrou dA eÐnai dunatìn na èqeièna mègisto, gia antikeÐmena se polÔ meglec apostseic ta opoÐa èqoun gwniakèc diamètrouc oiopoÐec mei¸nontai ìso auxnei h apìstash lamprìthtac (4.53)

4.2.5 H apìstash kosmik c kÐnhshc

JewroÔme fwtein phg h opoÐa sto idiosÔsthma tou thleskopÐou parathreÐtai ìti sto kosmikìsÔsthma suntetagmènwn èqei taqÔthta V⊥ kjeth sth dieÔjunsh parat rhshc (line of sight).Tìte gia to qronikì disthma t1 kai t1 + ∆t1 sto idiosÔsthma thc phg c to fwc kineÐtai stokosmikì sÔsthma suntetagmènwn apìstash ∆D = V⊥∆t1. H qronik dirkeia thc kÐnhshc thcphg c sto idiosÔsthma tou thleskopÐou dÐnetai apì th sqèsh (4.15). Epomènwc, upojètontacìti h parat rhsh thc fwtein c phg c sto idiosÔsthma tou thleskopÐou gÐnetai metaxÔ twn antÐ-stoiqwn qronik¸n idiostigm¸n t0 kai t0 + ∆t0 sto idiosÔsthma tou thleskopÐou èqoume:

∆D = V⊥R(t1)

R(t0)∆t0.

H kÐnhsh thc fwtein c phg c orÐzei sthn ournia sfaÐra mia troqi me gwniak dimetro ∆δh opoÐa dÐnetai apì thn (4.64). 'Ara:

∆δ =∆D

a(t1)r1∆t0. (4.66)

Se analogÐa me ton orismì thc apìstashc gwniak c diamètrou orÐzoume thn apìstash ko-smik c kÐnhshc (angular diameter distance) dM me th sqèsh:

dM =V⊥µ

(4.67)

ìpou to µ metr thn kosmik kÐnhsh kai orÐzetai wc akoloÔjwc:

µ =∆δ

∆t0. (4.68)

Apì thn (4.66) èqoume:dM = a(t1)r1. (4.69)

'Askhsh 7. Qrhsimopoi¸ntac th sqèsh z = a(t0)a(t1)

− 1 h opoÐa sundèei thn erujr metatìpish

me to phlÐko tou pargonta klÐmakac, deÐxte ìti oi diforec (kosmologikèc) apostseic poujewr same sqetÐzontai me tic akìloujec sqèseic:

dAdL

=

(R(t1)

R(t0)

)2

= (1 + z)−2 (4.70)

dMdL

=R(t1)

R(t0)= (1 + z)−1. (4.71)

4.3. OI DIAFORES LAMPROTHTES 53

Apì thn skhsh 7 sungoume ìti en kpoioc eÐnai se jèsh na metr sei thn erujr metatìpishz miac fwtein c phg c me akrÐbeia, den uprqei lìgoc na metr sei tic apostseic dL, dA kai dMektìc en epijumeÐ na epibebai¸sei th metrik FRW thn kosmologik proèleush thc erujrcmetatìpishc. Se antÐjesh h mètrhsh thc apìstashc parllaxhc dP ja mporoÔse (kat' arq n) nad¸sei perissìterh plhroforÐa apì aut n pou mporeÐ na d¸sei h mètrhsh tou dL kai tou z, allmèqri s mera eÐnai dunatìn na metr soume thn parllaxh mìno gia fwteinèc phgèc, oi opoÐec eÐnaipolÔ kont, dhlad gia fwteinèc phgèc me z ≪ 1 kai r1 ≪ 1. Se autèc ìmwc tic apostseicìlec oi parathr simec apostseic pou jewr same eÐnai praktik Ðsec metaxÔ touc kai Ðsec me thnapìstash kosmik c kÐnhshc:

dA ≃ dL ≃ dM ≃ dP ≃ dprop(t0) ≃ R(t0)r1.

H diaforopoÐhsh metaxÔ twn kosmologik¸n apostsewn eÐnai shmantik mìnon gia fwteinècphgèc oi opoÐec apèqoun perissìtero apì 109 èth fwtìc.

4.3 Oi diforec lamprìthtec

Oi perissìteroi metrhtèc aktinobolÐac èqoun euaisjhsÐa mìnon se èna tm ma tou hlektroma-gnhtikoÔ fsmatoc. H apìluth kai h fainìmenh lamprìthta pou jewr same sto edfio 4.2aforoÔse ìlo to hlektromagnhtikì fsma kai tic onomzoume bolometrikèc lamprìthtec(bolometric luminosities).

Anloga me thn euaisjhsÐa sto antÐstoiqo tm ma tou fsmatoc diakrÐnoume thn uperi¸dh(ultraviolet), thn mple (blue), thn fwtografik (photograpic), thn optik (visual) kai thn u-pèrujrh (infrared) lamprìthta. En h fwtein phg ekpèmpei sto idiosÔsthma thc se ìlec ticsuqnìthtec tic mikrìterec apì mia suqnìthta ν1 isqÔ aktinobolÐac L(ν1), tìte sÔmfwna me thn(4.62) h fainìmenh lamprìthta thc phg c sto idiosÔsthma tou thleskopÐou eÐnai:

l(ν0) =1

4π

a2(t1)

a4(t0)r21L[ν0a(t0)/a(t1)].

H isqÔc pou aktinoboleÐ èna mèlan s¸ma jermokrasÐac T1 se suqnìthtec mikrìterec apì miasuqnìthta ν dÐnetai apì ton tÔpo tou Plank:

L(ν) =15L

π4ν

(hν

kT1

)4(exp

(hν

kT1

)− 1

)−1

ìpou L(ν)∆(ν) eÐnai h bolometrik apìluth lamprìthta sto tm ma tou fsmatoc ∆(ν), k eÐnaih stajer Boltzmann kai h eÐnai h stajer tou Plank. Sungoume ìti h fainìmenh lamprìthtal(ν0)∆(ν0) thc fwtein c phg c gia suqnìthtec mikrìterec thc ν0 = νR(t1)/R(t0) sto idiosÔsthmatou thleskopÐou eÐnai:

l(ν0) =15l

π4ν0

(hν0kT0

)4 (exp

(hν0kT0

)− 1

)−1

ìpou l eÐnai to fainìmeno bolometrikì mègejoc thc fwtein c phg c kai T0 eÐnai h anhgmènhjermokrasÐa sto idiosÔsthma tou thleskopÐou h opoÐa dÐnetai apì th sqèsh:

T0 =a(t1)

a(t0)T1.


Shmei¸noume ìti en gnwrÐzoume tic jermokrasÐec T1, T0 thc phg c sto idiosÔsthm thckai sto idiosÔsthma tou thleskopÐou tìte eÐnai eÔkolo na upologÐsoume thn apìluth kai thnfainìmenh bolometrik lamprìthta thc phg c en gnwrÐzoume epiplèon thn perioq tou fsmatoc∆ν1, dhlad na gnwrÐzoume tic posìthtec L(ν1)∆ν1 kai l(ν1)∆ν1.

4.4 Oi mondec twn astronomik¸n apostsewn

H basik astronomik monda eÐnai h mèsh apìstash Ghc - HlÐou kai onomzetai astronomik monda (astronomical unit) kai sumbolÐzetai a.u.. IsqÔei ìti:

1a.u. = 1.49598 × 108Km (4.72)

Jewr¸ntac thn troqi thc Ghc kuklik , h probol tou dianÔsmatoc jèshc thc Ghc wc proc ton'Hlio se èna epÐpedo kjeto sthn dieÔjunsh pou kajorÐzei èna stajerì astèra sthn ourniasfaÐra kat th dirkeia tou ètouc paÐrnei th mègisth tim pou eÐnai 1 a.u.. Se antistoiqÐa oastèrac sthn ournia sfaÐra diagrfei mia èlleiyh kat th dirkeia tou qrìnou thc opoÐac omègistoc xonac π dÐnetai apì th sqèsh (Sqhma na gÐnei):

π ( se rad) =1

dP( se pc) (4.73)

ìpou dP eÐnai h apìstash parllaxhc tou astèra. Thn posìthta π onomzoume trigwnome-trik parllaxh (trigonometric parallax). H monda apìstashc parsec (sumbolÐzetai ensuntomÐa pc) orÐzetai wc h apìstash parllaxhc gia thn opoÐa h trigwnometrik parllaxh eÐnaiÐsh me 1

′′. Epeid 1 rad = 206264, 8 seconds èqoume:

1pc = 206264, 8 a.u. = 3, 0856 × 1013Km = 3, 2615 ly. (4.74)

Epomènwc h (4.73) grfetai:

π (seseconds) =1

dP( se pc). (4.75)

Parìlo pou mìno oi plhsièsteroi astèrec èqoun metr simh trigwnometrik parllaxh, èqeiepikrat sei ìlec oi apostseic ektìc tou hliakoÔ sust matoc na dÐnontai se parsec kai oi apo-stseic, anexrthta apì ton trìpo pou metrioÔntai, na ekfrzontai se isodÔnamh trigwnometrik parllaxh.

4.5 Oi mondec twn bolometrik¸n lamprot twn

H fainìmenh bolometrik fwteinìthta èqei epikrat sei gia istorikoÔc lìgouc na ekfrzetai su-nart sei enìc megèjouc to opoÐo onomzoume fainìmeno bolometrikì mègejoc (apparentbolometric magnitude) , to sumbolÐzoume mbol, kai to orÐzoume me th sqèsh:

l = 10−2m/5 × 2, 52 × 10−5erg/cm2sec. (4.76)

To apìluto bolometrikì mègejoc (absolute bolometric magnitude) M orÐzetai wcto fainìmeno bolometrikì mègejoc pou ja eÐqe mia fwtein phg se apìstash 10 pc. 'Ara èqoume:

L = 10−2M/5 × 3, 02 × 1035erg/sec. (4.77)

Th diafor m−M onomzoume bolometrikì mètro apìstashc (distance modulus).

4.5. OI MONADES TWN BOLOMETRIKWN LAMPROTHTWN 55

'Askhsh 8. DeÐxte ìti h apìstash aktinobolÐac dL sundèetai me to mètro apìstashc me thsqèsh

dL = 101+(m−M)/5 pc. (4.78)

Gia tic diforec perioqèc tou fsmatoc orÐzoume ta antÐstoiqa fainìmena bolometrik megèjhmU ,mB, ktl me sqèseic anlogec thc (4.76) all me diaforetikoÔc suntelestèc. Oi suntelestècautoÐ epilègontai ètsi ¸ste ìla ta fainìmena megèjh na eÐnai ta Ðdia gia astèrec tou fasmatikoÔtÔpou Ao megèjouc metaxÔ pèmptou kai èktou. Ta antÐstoiqa apìluta megèjh orÐzontai me thnapaÐthsh ta mètra apìstashc gia ìlec tic perioqèc tou fsmatoc p.q. mU −MU ,mB −MB, ktlna eÐnai Ðsa me to bolometrikì mètro apìstashc m−M .

Th diafor mB −mV = MB −MV onomzoume deÐkth qr¸matoc (colour index). Oiastèrec me arnhtikì deÐkth qr¸matoc eÐnai pio mple apì astèrec me jetikì deÐkth qr¸matoc.

'Askhsh 9. Gia ton lio èqome ta akìlouja apìluta megèjh:

Bolometrikì mègejocM = +4, 72, MU = 5, 51, MB = 5, 41, MV = 4, 79

kai ta akìlouja fainìmena megèjh:

Bolometrikì mègejoc m = −26, 85, mU = −26, 06, mB = −26, 16, mV = −26, 78

DeÐxte ìti to bolometrikì mètro apìstashc eÐnai −41, 57 kai o deÐkthc qr¸matoc 0, 62.


4.6 Comoving coordinates

While general relativity allows one to formulate the laws of physics using arbitrary coordinates,some coordinate choices are more natural (e.g. they are easier to work with). Comovingcoordinates are an example of such a natural coordinate choice. They assign constant spatialcoordinate values to observers who perceive the universe as isotropic. Such observers are called”comoving” observers because they move along with the Hubble flow.A comoving observer is the only observer that will perceive the universe, including the cosmicmicrowave background radiation, to be isotropic. Non-comoving observers will see regions ofthe sky systematically blue-shifted or red-shifted. Thus isotropy, particularly isotropy of thecosmic microwave background radiation, defines a special local frame of reference called thecomoving frame. The velocity of an observer relative to the local comoving frame is called thepeculiar velocity of the observer.Most large lumps of matter, such as galaxies, are nearly comoving, i.e., their peculiar velocities(due to gravitational attraction) are low.The comoving time coordinate is the elapsed time since the Big Bang according to a clock ofa comoving observer and is a measure of cosmological time. The comoving spatial coordinatestell us where an event occurs while cosmological time tells us when an event occurs. Together,they form a complete coordinate system, giving us both the location and time of an event.Space in comoving coordinates is usually referred to as being ”static”, as most bodies onthe scale of galaxies or larger are approximately comoving, and comoving bodies have static,unchanging comoving coordinates. So for a given pair of comoving galaxies, while the properdistance between them would have been smaller in the past and will become larger in the futuredue to the expansion of space, the comoving distance between them remains constant at alltimes.The expanding Universe has an increasing scale factor which explains how constant comovingdistances are reconciled with proper distances that increase with time.See also: metric expansion of space.

Comoving distance and proper distance

Comoving distance is the distance between two points measured along a path defined at thepresent cosmological time. For objects moving with the Hubble flow, it is deemed to remainconstant in time. The comoving distance from an observer to a distant object (e.g. galaxy)can be computed by the following formula: \chi = \int t eˆt c \; \mboxd t’ \over a(t’)where a(t’) is the scale factor, te is the time of emission of the photons detected by the observer,t is the present time, and c is the speed of light in vacuum.Despite being an integral over time, this does give the distance that would be measured bya hypothetical tape measure at fixed time t, i.e. the ”proper distance” as defined below,divided by the scale factor a(t) at that time. For a derivation see (Davis and Lineweaver, 2003)”standard relativistic definitions”.AnotherIn standard cosmology, comoving distance and proper distance are two closely related distancemeasures used by cosmologists to define distances between objects. Proper distance roughlycorresponds to where a distant object would be at a specific moment of cosmological time,measured using a long series of rulers stretched out from our position to the object’s positionat that time, and which can change over time due to the expansion of the universe. Comoving

4.7. USES OF THE PROPER DISTANCE 57

distance factors out the expansion of the universe, giving a distance that doesn’t change overtime, though it is defined to be equal to the proper distance at the present time.

Definitions * Many textbooks use the symbol \! \chi for the comoving distance. Howe-ver, this \! \chi must be distinguished from the coordinate distance r in the commonly-usedcomoving coordinate system for a FLRW universe where the metric takes the form \! dsˆ2 =-cˆ2 d\tauˆ2 = - cˆ2 dtˆ2 + a(t)ˆ2 ( \fracdrˆ21 - krˆ2 + rˆ2 (d\thetaˆ2 + sinˆ2 \thetad\phiˆ2 )). In this case the comoving coordinate distance \! r is related to \! \chi by \! \chi =r if k=0 (a spatially flat universe), by \! \chi = sinˆ-1 r if k=1 (a positively-curved ’spherical’universe), and by \! \chi = sinhˆ-1 r if k=-1 (a negatively-curved ’hyperbolic’ universe).[1]* Most textbooks and research papers define the comoving distance between comoving observersto be a fixed unchanging quantity independent of time, while calling the dynamic, changingdistance between them proper distance. On this usage, comoving and proper distances are nu-merically equal at the current age of the universe, but will differ in the past and in the future;if the comoving distance to a galaxy is denoted χ, the proper distance d(t) at an arbitrary timet is simply given by d(t) = a(t)χ where a(t) is the scale factor. (e.g. Davis and Lineweaver,2003) The proper distance d(t) between two galaxies at time t is just the distance that wouldbe measured by rulers between them at that time.[2]

4.7 Uses of the proper distance

Cosmological time is identical to locally measured time for an observer at a fixed comovingspatial position, that is, in the local comoving frame. Proper distance is also equal to thelocally measured distance in the comoving frame for nearby objects. To measure the properdistance between two distant objects, one imagines that one has many comoving observers ina straight line between the two objects, so that all of the observers are close to each other,and form a chain between the two distant objects. All of these observers must have the samecosmological time. Each observer measures their distance to the nearest observer in the chain,and the length of the chain, the sum of distances between nearby observers, is the total properdistance.[3]It is important to the definition of both comoving distance and proper distance in the cosmolo-gical sense (as opposed to proper length in special relativity) that all observers have the samecosmological age. For instance, if one measured the distance along a straight line or spacelikegeodesic between the two points, observers situated between the two points would have diffe-rent cosmological ages when the geodesic path crossed their own world lines, so in calculatingthe distance along this geodesic one would not be correctly measuring comoving distance orcosmological proper distance. Comoving and proper distances are not the same concept ofdistance as the concept of distance in special relativity. This can be seen by considering thehypothetical case of a universe empty of mass, where both sorts of distance can be measured.When the density of mass in the FLRW metric is set to zero (an empty ’Milne universe’), thenthe cosmological coordinate system used to write this metric becomes a non-inertial coordinatesystem in the flat Minkowski spacetime of special relativity, one where surfaces of constanttime-coordinate appear as hyperbolas when drawn in a Minkowski diagram from the perspe-ctive of an inertial frame of reference.[4] In this case, for two events which are simultaneousaccording the cosmological time coordinate, the value of the cosmological proper distance is


not equal to the value of the proper length between these same events,(Wright) which wouldjust be the distance along a straight line between the events in a Minkowski diagram (and astraight line is a geodesic in flat Minkowski spacetime), or the coordinate distance between theevents in the inertial frame where they are simultaneous.If one divides a change in proper distance by the interval of cosmological time where the changewas measured (or takes the derivative of proper distance with respect to cosmological time) andcalls this a ”velocity”, then the resulting ”velocities” of galaxies or quasars can be above thespeed of light, c. This apparent superluminal expansion is not in conflict with special or generalrelativity, and is a consequence of the particular definitions used in cosmology. Even light itselfdoes not have a ”velocity” of c in this sense; the total velocity of any object can be expressed asthe sum v tot = v rec + v pec v rec is the recession velocity due to the expansion ofthe universe (the velocity given by Hubble’s law) and v pec is the ”peculiar velocity” measu-red by local observers (with v rec = dota(t)chi(t) and v pec = a(t)dotchi(t), the dotsindicating a first derivative), so for light v pec is equal to c (-c if the light is emitted towardsour position at the origin and +c if emitted away from us) but the total velocity v tot isgenerally different than c.(Davis and Lineweaver 2003, p. 19) Even in special relativity thecoordinate speed of light is only guaranteed to be c in an inertial frame, in a non-inertial framethe coordinate speed may be different than c;[5] in general relativity no coordinate system ona large region of curved spacetime is ”inertial”, but in the local neighborhood of any point incurved spacetime we can define a ”local inertial frame” and the local speed of light will be cin this frame,[6] with massive objects such as stars and galaxies always having a local speedsmaller than c. The cosmological definitions used to define the velocities of distant objects arecoordinate dependent - there is no general coordinate independent definition of velocity betweendistant objects in general relativity (Baez and Bunn, 2006). The issue of how to best describeand popularize the apparent superluminal expansion of the universe has caused a minor amountof controversy. One viewpoint is presented in (Davis and Lineweaver, 2003). Proper distancevs. comoving distance from small galaxies to galaxy clusters

Within small distances and short trips, the expansion of the universe during the trip can beignored. This is because the travel time between any two points for a non-relativistic movingparticle will just be the proper distance (i.e. the comoving distance measured using the scalefactor of the universe at the time of the trip rather than the scale factor ”now”) between thosepoints divided by the velocity of the particle. If the particle is moving at a relativistic velocity,the usual relativistic corrections for time dilation must be made.

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