Barutik Aktinobola sthnUpologistik SqetikìthtaEpiblèpwn Kajhght :N. StergioÔla

Ptuqiak Ergasa tou foitht Mlea Qr stou

2005Aristotèleio Panepist mio Jessalonkh Sqol Jetik¸n Episthm¸n-Tm ma Fusik 20 Septembrou 2005

1Euqariste Ja jela na euqarist sw jerm ton Epkouro Kajhght tou Aristoteleou Panepisthmou,kai epiblèponta kajhght th diplwmatik mou ergasa , k. StergioÔla Nko, gia thn upìdeixhtou jèmato kai gia th suneq kai polÔtimh bo jeia tou se kje mou duskola kai apora katth dirkeia th melèth aut th ergasa . Na euqarist sw epsh tou flou mou giathn yuqologik kai episthmonik tou bo jeia. Tèlo jèlw na euqarist sw thn oikogèneiamou pou stjhke dpla mou kai me st rixe ìqi mìno kat th dirkeia epexergasa aut th diplwmatik ergasa , all kai se kje stigm th zw mou.

2PerlhyhSkopì th diplwmatik aut ergasa tan h melèth th suneisfor twn peristrofik¸nìrwn th metrik tou exwterikoÔ qwroqrìnou enì peristrefìmenou sqetikistikoÔ astèra,sthn ekpempìmenh barutik aktinobola. Xekinme me th melèth twn barutik¸n kumtwn stoneppedo qwroqrìno, mèsw twn grammikopoihmènwn exis¸sewn pedou tou Einstein. Sto epì-meno stdio meletme ti diataraqè th metrik Schwarzschild enì sfairik summetrikoÔqwroqrìnou. Qwrzonta ti diataraqè se rtie kai perittè , katal goume se dÔo kumatikè exis¸sei , thn exswsh twn Regge-Wheeler gia perittè diataraqè kai thn exswsh tou Zerilligia rtie diataraqè . To epìmeno b ma enai pli h exagwg twn exis¸sewn Regge-Wheelerkai Zerilli, katal gonta sto sumpèrasma ìti enai dunat h qr sh tètoiwn sunart sewn pouna enai anallowte ktw apì metsqhmatismoÔ bajmda , ¸ste na upakoÔoun ti exis¸sei twn Regge-Wheeler kai Zerilli se opoiod pote sÔsthma suntetagmènwn. Sto telikì stdio me-letme thn efarmog ma gia peristrefìmenou astère , h metrik twn opown apoklenei apìth sfairik summetra, qrhsimopoi¸nta ma mèjodo sÔmfwna me thn opoa, mporoÔme na qwr-soume ma genik mh sfairik metrik se èna sfairikì kommti kai se èna mh sfairikì kommti,oÔtw ¸ste na mporèsoume na melet soume to prìblhma ma me th bo jeia th prohgoÔmenh jewra pou edame gia sfairikoÔ qwroqrìnou . Prosdiorzoume ètsi tou analutikoÔ majh-matikoÔ tÔpou twn suneisfor¸n twn peristrof¸n (stsimo sflma) kai tou efarmìzoumegia diforou astère .

3Abstract

Our goal in this project, was to study the contributions of the rotating parts of the metricoutside a rotating relativistic star,on the emmited gravitational waves. We begin by studingthe gravitational waves on flat spacetime, through the Einstein’s linearized field equations. Onthe next chapter we investigate the perturbations of the Schwarzschild metric of a sphericallysymetric spacetime. Separating the perturbations in even and odd perturbations, we find twowave equations, the Regge-Wheeler equation for odd perturbations and the Zerilli equationfor even perturbtions. On the next chapter we find a way to build gauge-invariant functionsthat obey the Regge-Wheeler and Zerilli equations in any coordinate frame. On the lastchapter we study our problem for rotating stars, whose metric slightly differs from sphericallysymmetry, using a method that, allows us to separate a non spherical metric into a sphericalpart and a non spherical one, so as to study our problem with the aid of the theory developedearlier for spherical spacetimes. So we define the analytical mathematical relations of therotations’ contributions (stationary error) and we use them for various stars.

Perieqìmena1 Eisagwg sta Barutik KÔmata 51.1 Grammikopohsh twn exis¸sewn Eintein . . . . . . . . . . . . . . . . . . . . . . 51.2 Kumatik LÔsh twn Exis¸sewn Einstein . . . . . . . . . . . . . . . . . . . . . 101.3 Shmasa th Bajmda TT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Pìlwsh twn Barutik¸n Kumtwn . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Paragwg twn Barutik¸n Kumtwn . . . . . . . . . . . . . . . . . . . . . . . . 182 Diataraqè tou qwroqrìnou Schwarzschild kai barutik kÔmata 222.1 Grammikè Diataraqè tou Qwroqrìnou Schwarzschild . . . . . . . . . . . . . . 222.2 Anlush se Sfairikè Armonikè . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Anlush Suqnot twn:Eidkeush gia M = 0 . . . . . . . . . . . . . . . . . . . . 262.4 Metasqhmatismo Bajmda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Aktinikè Exis¸sei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.1 Perittè Diataraqè . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.2 'Artie Diataraqè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Enallaktik 'Ekfrash se Sfairikè Armonikè . . . . . . . . . . . . . . . . . . 333 Anallowte Bajmda tou Qwroqrìnou Schwarzschild 383.1 Arq Metabol¸n sth Jewra Diataraq¸n . . . . . . . . . . . . . . . . . . . . 393.2 Diataraqè twn Ken¸n kai Statik¸n Metrik¸n . . . . . . . . . . . . . . . . . . 413.3 Perittè Diataraqè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 'Artie Diataraqè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Eustjeia Melan¸n Op¸n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Barutik Aktinobola apì Peristrefìmenou Astère 544.1 Exagwg Kumatomorf¸n se mh-sfairikoÔ Qwroqrìnou . . . . . . . . . . . . . 544.2 H Metrik twn Hartle-Thorne . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 EÔresh twn sunart sewn twn diataraq¸n . . . . . . . . . . . . . . . . . . . . . 604.4 Poluwnumikì anptugma tou statikoÔ sflmato . . . . . . . . . . . . . . . . . 634.5 SÔgkrish th kampÔlh stsimou sflmato gia diaforetikoÔ astère . . . . 654.6 Sumpersmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Keflaio 1Eisagwg sta Barutik KÔmataMèqri ti arqè tou 20oÔ ai¸na, h mình phg parat rhsh tou sÔmpanto tan h hlektro-magnhtik aktinobola, se kje th morf . H anptuxh th Genik Jewra th Sqetikìthta ìmw noixe nèou orzonte gia th melèth kai parat rhsh tou sÔmpanto . 'Ena apì tou nèou autoÔ kldou apoteloÔn kai ta barutik kÔmata, h melèth twn opown an kai xeknhselgo met th dhmoseush th Genik Jewra th Sqetikìthtas(1915) me to prwtoporiakìèrgo tou Albert Einstein , kmase met th dekaeta tou 1970, me ti ergase twn Regge kaiWheeler , Zerilli , Mathews , Moncrief , k.a.H ergasa qwrzetai se tèssera keflaia. Sto pr¸to keflaio gnetai ma eisagwg sthjewra twn barutik¸n kumtwn, th didosh tou sto kenì, thn pìlwsh tou kaj¸ kai thnparagwg tou apì astrik s¸mata. Sto deÔtero keflaio meletoÔntai oi diataraqè touqwroqrìnou Schwarzschild kai ta barutik kÔmata pou autè prokaloÔn. Sto trto kef-laio melettai to dio prìblhma, apì ma diaforetik ìmw skopi, basismènh sthn kataskeu posot twn pou enai anallowte se apeirostoÔ metasqhmatismoÔ suntetagmènwn. Tèlo efarmìzoume thn ìlh jewra pou melet same sta prohgoÔmena keflaia sthn perptwsh peri-strefìmenwn astèrwn.Sta paraktw ja qrhsimopoihje h qwroeid morf twn pros mwn, (−,+, +,+), ìpou oiellhniko dekte ja parnoun timè apì 0 èw 3, en¸ oi latiniko apì 1 èw 3. H sunallowthpargwgo sumbolzetai me to ellhnikì erwthmatikì (;) kai merikè forè me to sÔmbolo touandelta, ∇, en¸ h sun jh me èna kìmma(,) me to sÔmbolo ∂. Oi tanustè ja grfontai mepio skoÔra grmmata (p.q. T) kai ta tra-dianÔsmata me to bèlo apì epnw (p.q. ~a). Tèlo ja doulèyoume sto gewmetrikopoimèno sÔsthma mondwn, ìpou G = c = 1 .1.1 Grammikopohsh twn exis¸sewn EinteinH melèth twn barutik¸n kumtwn xekin apì thn prwtoporiak doulei tou Einstein kaibaszetai sth grammikopoihmènh morf twn exis¸sewn pedou tou Einstein , oi opoe enai

Gµν ≡ Rµν −1

2gµνR = 8πTµν (1.1)

1.1 Grammikopohsh twn exis¸sewn Eintein 6ìpou Gµν o tanust tou Einstein ,Rµν kai R o tanust kai to bajmwtì tou Ricci antstoi-qa, gµν h metrik kai Tµν o tanust enèrgeia orm . Se aut thn prosèggish ja apodeiqteìti h grammikopoihmènh jewra odhge se lÔsei twn exis¸sewn (1.1) pou sumperifèrontai w kÔmata(barutik kÔmata). Prèpei ìmw na toniste(ìpw ja deiqte paraktw) ìti h ènnoiatwn barutik¸n kumtwn w lÔsei twn exis¸sewn Einstein enai ègkurh mìno ktw apì merikè exidanikeumène upojèsei :ènan kenì kai asumptwtik eppedo qwroqrìno kai ma grammikopoih-mènh morf tou barutikoÔ pedou. Aut enai exllou kai h prosèggish th grammikopoihmènh jewra . Se perptwsh ìmw pou de plhroÔntai oi upojèsei autè de shmanei ìti den enaidunatì o orismì twn barutik¸n kumtwn [16. Apl tìte h melèth gnetai pio perpokh kaidÔskolh(Tètoia perptwsh ja sunantsoume sto 2o keflaio).H ìlh melèth xekin jewr¸nta ma metrik , gµν , h opoa diafèrei elqista apì th metrik Minkowski , se suntetagmène Minkowski (t, x, y, z) , ètsi ¸ste

gµν = ηµν + hµν (1.2)ìpouηµν = diag(−1, 1, 1, 1). (1.3)To ìti briskìmaste sth grammikopoihmènh perioq shmanei ìti

|hµν | ≪ 1.Sunep¸ ìsoi ìroi emfanzontai deutèra kai anwtèra txew ja agnooÔntai. Apì th morf th (1.2) fanetai ìti kai o tanust hµν enai diag¸nio .H antallowth morf tou hµν enai [11hµν ≡ ηµαηνβhαβ. (1.4)IsqÔei ìti

(ηαβ + hαβ)(ηµν − hµν) = δνα (1.5)Sugkrnonta thn teleutaa exswsh me thn gαβgβν = δν

α parnoume ìtigµν = ηµν − hµν (1.6)Prin grafoÔn oi grammikopoihmène exis¸sei pedou prèpei na ekfrastoÔn ta sÔmbola

Christoffel sth grammikopoihmènh tou morf . AfoÔ ta ηµν enai stajer,Γµ

αβ =1

2gµν(gνα,β + gβν,α − gαβ,ν)

=1

2(ηµν − hµν)(hνα,β + hβν,α − hαβ,ν)

=1

2ηµν(hνα,β + hβν,α − hαβ,ν)

=1

2(hµ

α,β + hµβ,α − hµ

αβ,) (1.7)

1.1 Grammikopohsh twn exis¸sewn Eintein 7afoÔ ηµν,α = 0 kai hµν(hνα,β + hβν,α − hαβ,ν) enai ìro deÔtèra txew . ParathroÔme ìtito anèbasma kai katèbasma twn deikt¸n gnetai me to metrikì tanust ηµν ki ìqi me ton gµν .Autì enai to apotèlesma th grammik prosèggish . Par' ìlh ìmw th grammik morf twnexis¸sewn o qwroqrìno paramènei kampulwmèno .Me bsh ta parapnw o tanust tou Riemann dnetai apì th sqèshRαµβν ≡ 1

2(gαν,µβ − gαβ,µν + gµβ,αν − gµν,αβ)

≡ 1

2(hαν,µβ − hαβ,µν + hµβ,αν − hµν,αβ) (1.8)Oi tautìthte tou Bianchi

∇αRδǫβγ + ∇γRδǫαβ + ∇βRδǫγα ≡ 0 (1.9)gnontai∂αRδǫβγ + ∂γRδǫαβ + ∂βRδǫγα ≡ 0 (1.10)kai ikanopoioÔntai apì thn (1.8).O tanust tou Ricci enai1

Rµν = ηαβRαµβν =1

2ηαβ(hν

β,µβ − hββ,µν + hα

µ,αν − h αµν,α )

=1

2(hα

ν,µα − h αµν,α + hα

µ,αν − h,µν) (1.11)ìpouhβ

β = ηβαhβα ≡ h (1.12)O ìro hαµν,α grfetai ki w ¤hµν ,ìpou ¤ enai o telest tou d’Alembert

¤ = ηαβ∂α∂β

= ∂α∂α

=∂2

∂t2−∇2

=∂2

∂t2− (

∂2

∂x2+

∂2

∂y2+

∂2

∂z2) (1.13)'Etsi h (1.11) gnetai

Rµν =1

2(hν,µα + hµ,αν − ¤hµν − ¤h,µν) (1.14)1Sthn dia sqèsh ja katal game an xekinoÔsame apì th sqèsh

Rµν = Γαµν,α − Γ

αµα,ν + Γ

αβαΓ

βµν − Γ

αβνΓ

βµα

1.1 Grammikopohsh twn exis¸sewn Eintein 8To bajmwtì tou Ricci

R = ηµνRµν =1

2ηµν(hα

ν,µα + hαµ,αν − ¤hµν − ¤h,µν)

=1

2(hµα

,µα + hνα,να − ¤hν

ν − h ν,ν ) (1.15)Oi dekte µ, α, ν enia bwbo dekte .Epsh me bsh thn (1.13) isqÔei ìti h ν

,ν = ¤h. Jètonta µ = ν = β h (1.15) gnetai

R =1

2(2hβα

,βα − 2¤h)

= hβα,βα − ¤h (1.16)'Ara o tanusth tou Einstein ja enai

Gµν = Rµν −1

2ηµνR

=1

2[hα

ν,µα + hαµ,αν − ¤hµν − h,µν − ηµν(h

αβ,αβ − ¤h)] (1.17)kai h exswsh tou Einstein

hαν,µα + hα

µ,αν − ¤hµν − h,µν − ηµνhαβ

,αβ + ηµν¤h = 16πTµν (1.18)Apì th morf th (1.18) den fanetai ìti h exswsh aut apotele kumatik exswsh. MporoÔmena qrhsimopoi soume nèo sumbolismì pou knei qr sh twn tanust¸n mhdenikoÔ qnou (trace-free), oi opooi orzontai w

hµν ≡ hµν −1

2ηµνh (1.19)ìpou o telest paÔla mpore na efarmoste se kje summetrikì tanust , ìpw gia par-deigma, Rµν = Gµν kai hµν = hµν . Me th bo jeia autoÔ tou sumbolismoÔ oi exis¸sei tou

Einstein (1.18) gnontaih

α

µα,ν + hα

να,µ − ηµνh,αβ

αβ − ¤hµν = 16πTµν (1.20)Sto shmeo autì ja qrhsimopoi soume thn eleujera pou ma dnei h Genik Sqetikìthta sthnepilog twn suntetagmènwn.To sÔsthma suntetagmènwn sto opoo grfthkan oi exis¸sei (1.2) kai (1.3) lègetai sqedìn-Lorentz. Uprqoun dÔo edh metasqhmatism¸n pou odhgoÔnapì èna sÔsthma sqedìn-Lorentz se èna llo: oi metasqhmatismo Lorentz upobjrou kai oimetasqhmatismo bajmda .Oi metasqhmatism bajmda [10 enai èna edo allag suntatagmènwn th morf xα′

= xα + ξα (1.21)apì èna sÔsthma xα se èna llo xα′ me th bo jeia enì aujaretou dianÔsmato ξα, touopoou oi sunist¸se enai sunart sei th jèsh . To ξα enai mikrì ètsi ¸ste kai |ξα,β| ≪ 1.

1.1 Grammikopohsh twn exis¸sewn Eintein 9Tìte o metrikì tanust th (1.2) metasqhmatzetai apo to èna sÔsthma xα sto llosÔsthma xα′ w :g′

µν(x′) =

∂xα

∂x′µ∂xβ

∂x′ν gαβ(x) (1.22)Apì thn (1.21) prokÔptei∂xα

∂x′β = δαβ − ξα

,β (1.23)kai h (1.22) me th bo jeia th parapnw sqèsh parnei th morf g′

µν = (δαµ − ξα

,µ)(δβν − ξβ

,ν)(ηαβ + hαβ

= ηµν + hµν − δαν ηαβξβ

,ν − δβν ηαβξα

,µ + O[ξ2, ξ · h]

= ηµν + hµν − ξµ,ν − ξµ,ν (1.24)ìpou gia na èqei o metrikì tanust th morf th (1.2) sto nèo sÔsthma prèpeih′

µν = hµν − ξµ,ν − ξν,µ (1.25)Epsh parathroÔme ìti |h′µν ≪ 1|. 'Ara to nèo sÔsthma enai pli sÔsthma sqedìn-Lorentz.H onomasa metasqhmatismì bajmda dìjhke apo thn omoiìthta th (1.25) me tou metasqh-matismoÔ bajmda tou HlektromagnhtismoÔ.An t¸ra upologsoume apì thn (1.8) ton tanust tou Riemann sto nèo sÔsthma suntetg-mènwn, qrhsimopoi¸nta thn (1.25), brskoume polÔ eÔkola ìti

Rpaliìαβµν = Rnèo

αβµν (1.26)Dhlad o tanust tou Riemann enai anallowto mègejo ktw apì metasqhmatismoÔ baj-mda . Omow kai o tanust tou Einstein kai o tanust enèrgeia -orm enai anallowtoiktw apì tou metasqhmatismoÔ autoÔ . 'Etsi an enai gnwst ma lÔsh hµν twn grmmiko-poihmènwn exis¸sewn (1.20) gia èna dosmèno Tµν , me èna metasqhmatismì bajmda mpore nabreje ma llh lÔsh pou perigrfei thn dia fusik katstash (ìla ta parathr sima megèjhparamènoun anallowta).Parathr¸nta thn exswsh (1.20) gnetai fanerì ìti mpore na aplopoihje arket anh

µν

,ν = 0 (1.27)H parapnw sqèsh ekfrzei tèsseri exis¸sei ,ìse kai oi eleÔjere sunart sei bajmda ξα. Enai pnta dunatì na epilege ma bajmda (dhlad èna ξα ) ¸ste na isqÔei h (1.27). Jaanaferìmaste ètsi sthn (1.27) w sunj kh bajmda Lorentz. H Ôparxh th apodeiknÔetai w ex . 'Estw kpoio aujareto h

palaiìµν 6= 0. Se èna metasqhmatismì bajmda (1.25) to hµνallzei w

hnèoµν = h

palaiìµν − ξµ,ν − ξν,µ + ηµνξ

α,α (1.28)kai

hnèoµν

,ν= h

palaiìµν

,ν− ξµ ν

,ν = hpalaiìµν

,ν− ¤ξµ (1.29)

1.2 Kumatik LÔsh twn Exis¸sewn Einstein 10Gia na enai to hnèoµν = 0 to ξµ prosdiorzetai apì thn exswsh

¤ξµ = ξµ ν,ν = h

palaiìµν

,ν(1.30)H exswsh (1.30) enai ma kumatik exswsh, mh-omogen , h opoa èqei pnta lÔsh gia kje

hµν

,ν . 'Ara uprqei pnta kpoio ξµ, gia to opoo ja lambnoume th bajmda Lorentz gia kpoioaujareto hµν . Tèlo ìson afor th bajmda Lorentz axzei na shmeiwje ìti aut den enaipl rw kajorismènh,giat kje metasqhmatismì ¤ξα = 0 af nei anephrèastei th sunj khbajmda hµν

,ν = 0. Sthn ousa h bajmda Lorentz apotele ma klsh bajmdwn.Sth bajmda Lorentz, oi exis¸sei tou Einstein (1.20) grfontai w ¤hµν = 16πTµν (1.31)Oi exis¸sei pedou sto kenì, ìpou Tµν = 0 gnontai

¤hµν = 0 (1.32)Parnonta to qno th (1.32) èqoumeηµν

¤hµν = ¤(ηµνhµν)

= ¤(h − 2h) = −¤h = 0 (1.33)lìgw th (1.19). An sundisoume t¸ra ti (1.31), (1.32) kai (1.19) parnoume¤hµν = 0 (1.34)Deqthke ètsi ìti, se ma bajmda Lorentz, oi diataraqè tou qwroqrìnou metaddontai se autìnsan kÔmata.1.2 Kumatik LÔsh twn Exis¸sewn EinsteinDeqthke ìti ta hµν upakoÔoun sthn kumatik exswsh (1.32), se ma sugkekrimènh bajmda.H aploÔsterh lÔsh th exswsh (1.32) enai èna eppedo kÔma th morf

hµν = Aµνeikαxα (1.35)ìpou o Aµν enai èna summetrikì tanust ki ekfrzei to plto tou kÔmato kai kα ènatetradinusma, ti idiìthte tou opoou ja doÔme t¸ra.H (1.32) grfetai kai w

ηαβhµν,αβ = 0 (1.36)Paragwgzonta dÔo forè thn exswsh (1.35) kai me th bo jeia th (1.36) parnoumeηαβhµν,αβ = −ηαβkαkβhµν (1.37)

1.2 Kumatik LÔsh twn Exis¸sewn Einstein 11Gia na epalhjeÔetai h (1.37) prèpeiηαβkαkβ = kαkα = 0 (1.38)Dhlad ,sÔmfwna me thn teleutaa sqèsh to kα enai èna fwtoeidè dinusma. Ef' ìson h (1.35)paristnei èna eppedo kÔma, to hµν ja lambnei ma stajer tim sthn uperepifneiakαxα = k0t + ~k · ~x (1.39)Th sunist¸sa k0 th lème kai suqnìthta kÔmato kai sumbolzetai me to ω, ki ètsi to kα →

ω,~k. 'Estw twra èna fwtìnio pou kinetai kat th dieÔjunsh tou fwtoeidoÔ tetradianÔ-smato kα, sthn kampÔlhxµ(λ) = kµ(λ) + lµ (1.40)ìpou λ parmetro kai lµ stajerì dinusma th arqik jèsh tou fwtonouλ = 0. Apì thn(1.38)

kµxµ = kµl

µ = stajerì (1.41)Sugkrnonta aut th sqèsh me thn (1.39) bganei to sumpèrasma ìti to fwtìnio kinetai me tobarutikì kÔma èqonta thn dia fsh. Me lla lìgia to kÔma diaddetai me thn taqÔthta toufwtì sthn dieÔjunsh ~k = (kx,ky ,kz)

ω. Apì ta parapnw, ef' ìson to kα enai fwtoeidè ,

ω2 = |~k|2 (1.42)H (1.42) apotele thn exswsh diaspor tou kÔmato , apì thn opoa prokÔptei ìti h taqÔthtafsh kai h taqÔthta omdo tou kÔmato enai se me 1 [18.Apì th sunj kh tou Lorentz,hµν

,ν = 0, prokÔptei gia to kÔma (1.35)h

µν

,ν = ikνhµν

= 0 ⇒ kνAµν = 0 (1.43)Dhlad oi tèsseri exis¸sei (1.43) periorzoun to Aµν sto na enai kjeto sto kα. Epomènw apì ti dèka anexrthte sunist¸se tou Aµν , ma mènoun mìno èxi anexrthte sunist¸se ex'aita twn sunjhk¸n (1.43). 'Omw den èqoume exantl sei ìla ta perij¸ria th bajmda ma .Uprqei h dunatìthta, ìpw anafèrjhke sthn prohgoÔmenh pargrafo, na gnei qr sh enì aujaretou dianÔsmato , lÔsh th exswsh ¤ξα = 0 qwr na ephreaste h bajmda. 'Estw hlÔsh th parapnw kumatik exswsh

ξα = Bαeikµxµ (1.44)ìpou Bα ma stajer kai kµ fwtoeidè dinusma. To ξα pargei ma metabol tou hµν poudnetai apì thn exswsh (1.28), h opoa se sunergasa me ti (1.35) kai (1.44) dneiAnèoµν = Apaliìµν − iBµkν − iBνkµ + iηµνB

αkα (1.45)Epilègonta katllhla to Bα mporoÔme na proume dÔo periorismoÔ akìma gia to Anèoαβ

1.3 Shmasa th Bajmda TT 12(i) Aα

α = 0 (1.46)(ii) AµνUν = 0 (1.47)ìpou U kpoia stajer tetrataqÔthta,dhlad opoiod pote stajerì qronoeidè monadiao di-nusma.Oi dÔo pnw periorismo apoteloÔn tèsseri epiplèon exis¸sei kai ìqi pènte ìpw fanetaiex' arq . Autì giat ma apì ti exis¸sei th (1.47), h kµ(AµνU

ν) ikanopoietai dh apì thn(1.43). To sÔnolo twn oqt¸ anexrthtwn exis¸sewn (1.43), (1.46) kai (1.47) af noun tontanust tou pltou Aµν me dÔo anexrthte sunist¸se kai onomzetai sunj ke egkrsia -mhdenikoÔ qnou bajmda TT (Transverse-Traceless). ApodeiknÔetai eÔkola ìti sth bajmdaTT, ex' aita th (1.46), enai hTT

µν = hTTµν .'Estw t¸ra ìti metaferìmaste se èna sÔsthma anafor Lorentz, ¸ste to dinusma Upou bassame th bajmda TT, na enai to qronikì dinusma bsh U ν = δν

0 = (1, 0, 0, 0). Apìthn exswsh (1.47) prokÔptei ìti Aµ0 = 0 gia kje µ. MporoÔme na prosanatolsoume tou xone ¸ste to kÔma na diaddetai kat ton xona z, dhlad to kα na enai to (ω, 0, 0, ω). Tìteapì thn (1.43) èpetai ìti(kai mèsw th 1.47) Aµz = 0 gia kje µ(Apì ed¸ bganei h onomasaegkrsia gia th bajmda:to Aµν tèmnei th dieÔjunsh didosh ~ez). Apì ta parapnw oimìne mh-mhdenikè sunist¸se tou Aµν enai oi Axx, Ayy kai Axy = Ayx. Apì thn (1.46) tèlo parnoume ìti Axx = −Ayy [18. Epomènw se auto to eidikì sÔsthma anafor isqÔeiA =

0 0 0 00 Axx Axy 00 Axy −Axx 00 0 0 0

(1.48)'Olh h parapnw doulei ègine gia eppeda kÔmata. Mpore ìmw na deiqte kti antstoiqogia aujareta kÔmata, sth grammikopoihmènh pnta jewra. Kje barutikì kÔma mpore, sÔm-fwna me thn anlush Fourier, na analuje se ma upèrjesh eppedwn barutik¸n kumtwn. Giakje eppedo kÔma mporoÔme na eisgoume th bajmda TT. ParathroÔme ìti oi trei sunj ke th bajmda (1.43), (1.46) kai (1.47) enai grammikè w pro to hµν . 'Ara kai to aujaretokÔma ja ti ikanopoie. Ftnoume ètsi sto je¸rhma: Dialègoume èna sugkekrimèno sÔsthmaanafor Lorentz (sugkekrimènh tetrataqÔthta U ν )th grammiokpoihmènh jewra . Se autìto sÔstma ( uα = δα

0 ), exetste èna sugkekrimèno barutikì kÔma aujareth morf . Ja mpo-rete pnta na brete ma bajmda sthn opoa to hµν na ikanopoie ti sunj ke (1.43), (1.46)kai (1.47) .1.3 Shmasa th Bajmda TTMèqri stigm h eisagwg th bajmda TT fnhke lgo aujareth, qwr idiatero endia-fèron. 'Opw ja fane ìmw paraktw, h qr sh th bajmda TT eisgei kpoia shmantik

1.3 Shmasa th Bajmda TT 13pleonekt mata sth melèth twn barutik¸n kumtwn. To pio shmantikì apotèlesma pou dnei hbajmda TT enai ìti oi mìne mh-mhdenikè sunist¸se tou tanust tou Riemann enai oiRj0i0 = R0j0i = −Rj00i = −R0ji0Efoson gia th sunist¸sa Rj0i0 isqÔei

Rj0i0 = −1

2hTT

ji,00 (1.49)apì th qr sh th bajmda TT bganei to sumpèrasma ìti, èna taxideÔon barutikì kÔma meperiodik qronik sumperifor th morf hTTji ∼ exp(iωt) mpore na sqetiste me ma topik talntwsh tou qwroqrìnou, afoÔ

hTTji,00 ∼ −ω2exp(iωt) ∼ Rj0i0 kai Rj0i0 =

1

2ω2hTT

jiA doÔme t¸ra thn epdrash th didosh twn barutik¸n kumtwn se swmatdia. Gia to sko-pì autì jewroÔme èna eleÔjero swmatdio, se ma perioq tou qwroqrìnou ìpou arqik denuprqoun barutik kÔmata, pou sunant èna barutikì kÔma. Dialègoume èna topikì sÔsthmaLorentz2 sto opoo arqik to swmatdio na hreme kai sth sunèqeia epilègoume th bajmda TTw pro autì to sÔsthma. Autì shmanei pw h tetrataqÔthta th exswsh (1.47) enai harqik tetrataqÔthta tou swmatidou Uα = δα

0 . AfoÔ to swmatdio enai eleÔjero ikanopoiethn exswsh twn gewdaisiak¸ndUα

dt+ Γα

µνUµUν = 0 (1.50)Kaj¸ to swmatdio hreme arqik, h arqik epitqunsh tou ìtan sunant to kÔma, enai

(dUα

dt

)

0

= −Γα00 = −1

2ηαβ(hβ0,0 + h0β,0 − h00,β (1.51)'Omw sth bajmda TT pou briskìmaste deqthke ìti oi mìne anexrthte sunist¸se A

TTenai oi hTTxx =-hTT

yy kai hTTxy = hTT

yx . 'Ara h (1.51) dnei Γα00 ki epomènw h arqik epitqunshenai mhdèn. Autì shmanei ìti amèsw met to swmatdio ja suneqsei na hreme, all kai hepitqunsh tou ja paramenei mhdenik . 'Ara to swmatdio ja hreme suneq¸ anexrthta apìto kÔma. Par' ìla aut h hrema aut de shmanei ìti to swmatdio de kinetai kajìlou. Apl¸ to swmatdio paramènei se ma jèsh èqonta stajerè suntetagmène . AnakalÔfjhke pro to2 'Ena topikì sÔsthma Lorentz enai èna sÔsthma anafor sto opoo h metrik parnei th morf

gαβ ≡ eα · eβ = ηαβsto gegonì P pou apotele thn arq tou sust mato . Epsh gia th metrik isqÔei∂gαβ

∂xµ= 0 ⇒ Γ

αβγ = 0 sto P

1.3 Shmasa th Bajmda TT 14parìn ìti h epilog th bajmida TTdhlad h eidik prosarmog sti paramorf¸sei twnsuntetagmènwn ma odhge sthn eÔresh enì sust mato suntetagmènwn, to opoo paramèneisundedemèno me to swmatdio. 'Etsi to sugkekrimèno sÔsthma Lorentz den enai tètoio sÔsthmamìno se èna gegonì th gewdaisiak kosmik gramm tou swmatidou, all se ìlh thnkosmik gramm tou swmatidou. Se autì to sÔsthma twn suntatagmènwn enaids2 = −dτ 2 + δijdxidxj + O(|xi|2)dxαdxβ (1.52)'Estw èna deÔtero swmatdio Bi sth geitoni tou arqikoÔ ma swmatidou (to opoo ja ono-

Sq ma 1.1: Anaparstash th metabol tou dianÔsmato sÔndesh ηj metaxÔ dÔo swmatidwn,pou brskontai pnw se gewdaisiakè troqiè , pou prokaletai apì thn èleush tou barutikÔkÔmato .maste A). Jewroume epsh ìti h arq tou sust mato suntetagmènwn ma tautzetai me toarqikì ma swmatdio A. H fixh tou barutikoÔ kÔmato ja diatarxei th gewdaisiak knhshtwn dÔo swmatidwn, ìpw fanetai sto sq ma 1.1(H eikìna enao apì thn ergasa [16, selda7) kai ja d¸sei ma mh-mhdenik suneisfor sthn exswsh th gewdaisiak apìklish :UγUβηα

;βγ = −RαβγδU

βηγU δ (1.53)

1.3 Shmasa th Bajmda TT 15 isodÔnamaUγUβ

(D2ηα

Dτ 2

)≡ UγUβ

(d2ηα

dτ 2+ Γα

βγ

dηα

dτ

dηβ

dτ

)= −Rα

βγδUβηγU δ (1.54)ìpou ηα enai to tetradinusma sÔndesh twn dÔo swmatidwn, D

Dτh olik pargwgo kai τo idio-qrìno . Ef' ìson to swmatdio A brsketai sthn arq twn suntetagmènwn oi qwrikè sunist¸se tou ηα enai ηj ≡ xj

B − xjA = xj

B kai h exswsh (1.54) gnetaiD2xj

B

Dτ 2= −Rj

0i0xiB = −Rj0i0x

iB (1.55)AfoÔ emaste se èna sÔsthma Lorentz kont sth geitoni tou swmatidou A ta sÔmbola Chri-

stoffel exafanzontai, ìpw kai oi pargwgoi tou dΓµαβ

dτkai h sunallowth pargwgo gnetaih sun jh pargwgo . 'Etsi h (1.55) gnetai

d2xjB

dτ 2= −Rj0i0x

iB (1.56)'Opw anafèrame to sÔsthma pou dialèxame taxideÔei maz me to swmatdio ki epomènw o idio-qrìno tautzetai me to suntetagmèno qrìno t. Epsh sth bajmda TT isqÔei, ìpw deqthkenwrtera, ìti RTT

j0i0 = Rj0i0 kai h (1.56) gnetai telikd2xj

B

dτ 2= −RTT

j0i0xiB =

1

2

(∂2hTT

ji

∂t2xi

B

) (1.57)Me arqik sunj kh na brskontai ta dÔo swmatdia se hrema metaxÔ tou prin thn èleush toukÔmato (xiB = xi

B(0), oloklhr¸nonta thn (1.57) parnoume th lÔshxi

B(t) = xiB(0)[δij +

1

2hTT

ji ]sth jèsh tou A (1.58)H parapnw exswsh dnei ti talant¸sei pou ektele to swmtio B, ìpw ti blèpei to swmtioA kajw pern to kÔma. Apì thn exswsh aut fanetai ìti h idiapìstash, pou enai anloghtou ηj kai enai mègejo anexrthto tou sust mato suntatagmènwn, metablletai qronik,en¸ h suntatagmènh apìstash, mègejo exart¸meno apì to sÔsthma, paramènei stajer .An èqoume èna eppedo kÔma kai to tetradinusma sÔndesh enai parllhlo sto tetradi-nusma didosh kα tìte den èqoume talntwsh afoÔhTT

ji xiB(0) ∼ hTT

ji ki = 0Katal goume ètsi sto sumpèrasma ìti mono ìtan to dinusma sÔndesh enai kjeto sthdieÔjunsh didosh taant¸netai. To kÔma ena egkrsio ìqi mìno sth majhmatik perigraf tou, all kai sta fusik tou apotelèsmata.

1.4 Pìlwsh twn Barutik¸n Kumtwn 161.4 Pìlwsh twn Barutik¸n KumtwnGia thn exagwg twn paraktw sqèsewn akoloujoÔme ton d’ Inverno [11.'Estw èna eppedobarutikì kÔma, pou diaddetai kat ton xona z kai h sunist¸sa tou hxy = hyx = 0. Tìte oimìne mh-mhdenikè sunist¸se tou, sth bajmda TT, enaihTT

xx = −hTTyy = A+e−iω(t−z) (1.59)To kÔma paramorf¸nei to q¸ro apì th grammik morf tou kai to grammikì stoiqeo gnetai

ds2 = dt2 − [1 − hTTxx (t − z)]dx2 − [1 + hTT

xx (t − z)]dy2 − dz2 (1.60)'Estw ìti to hxx enai ma sunrthsh talantoÔmenh morf pou parnei timè jetikè hxx > 0kai arnhtikè hxx < 0. 'Estw t¸ra dÔo swmatdia sto eppedo x-y me suntetagmène (x0, y0)kai (x0 + dx, y0). Apì thn (1.60) parnoume thn idiapìstash tou ds2 = −(1 + hxx)dx2 (1.61)Apì thn (1.61) ìtan to hxx apì mhdèn kai jetikè hxx > 0 ta swmatdia plhsizoun metaxÔtou , en¸ ìtan parnei arnhtikè timè aut apomakrÔnontai. To antjeto sumbanei an jewr -soume eleÔjera swmatdia sti jèsei (x0, y0) kai (x0, y0) + dy sto eppedo x-y, afoÔ t¸ra hidiapìstash tou enaids2 = −(1 + hxx)dy2 (1.62)'Etsi an èqoume èna daqtuldi swmatidwn sto eppedo x-y me èna swmatdio sto kèntro tou,h èleush tou barutikoÔ kÔmato ja diatarxei to daqtuldi se ma talantoÔmenh èlleiyh, omeglo hmixona th opoa enai parllhlo diadoqik stou xone x kai y. Anaferìmastesthn katstash aut w pìlwsh +(cross) tou barutikoÔ kÔmato .'Estw t¸ra ìti oi mìne mh-mhdenikè sunist¸se tou barutikoÔ kÔmato enai oi:

hTTxy = hTT

yx = A+e−iω(t−z) (1.63)kai to grmmikì stoiqeods2 = dt2 − dx2 − dy2 − dz2 + 2hxy(t − z)dxdy (1.64)An strèyoume tou xone kat 45 sÔmfwna me ti sqèsei metasqhmatismoÔ gia strofè x → x =

1√2(x + y) kai y → y =

1√2(−x + y) (1.65)to grammikì stoiqeo sti nèe suntetagmène (t, x, y, z) gnetai

ds2 = dt2 − [1 − hTTxy (t − z)]dx2 − [1 + hTT

xy (t − z)]dy2 − dz2 (1.66)Sugkrnonta thn exswsh aut me thn (1.60) parathroÔme ìti to hxy-kÔma prokale to dioapotèlesma me to hxx-kÔma, mìno pou oi xone th èlleiyh enai stramènoi kat 45 .Thn ka-tstash aut thn onomzoume × pìlwsh tou kÔmato . Oi dÔo katastsei grammik pìlwsh

1.4 Pìlwsh twn Barutik¸n Kumtwn 17

Sq ma 1.2: Anaparstash th paramìrfwsh enì daqtulidioÔ swmatidwn kat thn èleushenì barutikoÔ kÔmato + kai × katstash pìlwsh antstoiqa.fanontai sto sq ma 1.2(H eikìna enai apì thn ergasa [16, selda 8). Ston hlektromagnhti-smì ta monadiaa dianÔsmata grammik pìlwsh enai ex kai ey.Sth Sqetikìthta, oi monadiaoitanustè grammik pìlwsh enaie+ ≡ ex ⊗ ex − ey ⊗ ey

e× ≡ ex ⊗ ey + ey ⊗ ex (1.67)Kje kÔma enai ma upèrjesh twn dÔo parapnw katastsewn pìlwsh . Axzei na to-niste ìti oi dÔo katastsei pìlwsei twn hlektromagnhtik¸n kumtwn enai kjete , en¸twn barutik¸n kumtwn ìpw edame sqhmatzoun gwna 45 . Autì prokÔptei apì to gego-nì ìti h barÔthta antiproswpeÔetai apì èna summetrikì tanust deutèra txew hµν , en¸ ohlektromagnhtismì apì to dianusmatikì dunamikì Aµ [18.'Opw kai ta hlektromagnhtik kÔmata ètsi kai ta barutik mpore na enai kai kuklikpolwmèna se dÔo katastsei . Sthn perptwsh aut oi monadiaoi tanustè kuklik pìlwsh enaieR ≡ 1√

2(e+ + ie×)

eL ≡ 1√2(e+ − ie×) (1.68)'Ena swmatido pou pnw tou pèftei se èna barutikì kÔma kuklik pìlwsh eR kinetai kuklikme dexiìstrofh for (ìpw oi xone tou rologioÔ, gia kÔma pou kinetai pro ton anagn¸sth).

1.5 Paragwg twn Barutik¸n Kumtwn 18Se perptwsh pou to sunant sei kÔma kuklik pìlwsh eL peristrèfetai aristerìstrofa. An-tstoiqa èna barutikì kÔma pìlwsh eR ja anagkzei ènan daktÔlio swmtwn na peristrèfetaidexiìstrofa , en¸ èna kÔma katstash pìlwsh eL aristerìstrofa. Sto sq ma 1.4(H eikìnaenai apì thn ergasa [12, selda 953) fanontai ìle oi katastsei pìlwsh .

Sq ma 1.3: Anaparstash th paramìrfwsh daktulou swmatidwn kat thn èleush mèsa apìautì enì barutikoÔ kÔmato kuklik katstash pìlwsh R kai L antstoiqa gia thn pnwkai ktw parstash.Apì to sq ma 1.3(H eikìna enai apì thn ergasa [16, selda 10) parathroÔme ìti, seopoiad pote qronik stigm , to barutikì kÔma mènei anallowto Ôstera apì ma peristrof 180 gÔrw apì th dieÔjunsh didosh tou. H antstoiqh gwna gia ta hlektromagnhtik kÔmataenai 360 kai gia ta kÔmata netrnwn 720 .H sumperifor aut sqetzetai me thn katstashspin swmatidwn mhdenik mza th kbantomhqanik perigraf twn kumtwn aut¸n: tagkrabitìnia(gravitons) èqoun spin 2, ta fwtìnia spin 1 kai ta netrna spin 1

2. To klasikì pedoaktinobola enì swmatidou me spin s mènei anallowto ktw apì peristrofè 3600

sgÔrw apìth dieÔjunsh didosh tou [12.1.5 Paragwg twn Barutik¸n KumtwnSta prohgoÔmena melet jhke h didosh twn barutik¸n kumtwn sto kenì, pou perigrfetaiapì thn kumatik exswsh (1.32), kaj¸ kai thn epdrash tou se eleÔjera swmatdia stoqwroqrìno. Shmantikì jèma ìmw sthn èreuna twn barutik¸n kumtwn apotele kai h melèthth paragwg tou apì difore phgè .

1.5 Paragwg twn Barutik¸n Kumtwn 19

Sq ma 1.4: Anaparstash ìlwn twn katastsewn grammik kai kuklik pìlwsh gia difo-re timè th fsh tou barutikoÔ kÔmato .H paragwg twn barutik¸n kumtwn dnetai apì th lÔsh th exswsh :¤hµν = −16πTµν (1.69)H lÔsh th parapnw exswsh enai kpw perplokh. Gia to lìgo autì ja akolouj soumetou Misner et al [12 se ènan aploÔstero trìpo, o opoo mpore na mh dnei akribe lÔsei ,all ma dnei ma polÔ kal ektmhsh th txh twn diafìrwn megej¸n pou perigrfoun thnparagwg ki ekpomp twn barutik¸n kumtwn. Oi ektim sei autè prokÔptoun apì ti parathroÔmene analoge th Sqetikìthta me thn hlektromagnhtik jewra kai baszontaise ma Neut¸neia prossèggish th barÔthta .Sthn klasik hlektrodunamik jewra, h enèrgeia pou ekpèmpetai sth monda tou qrìnou(fwteinìthta), L, apì èna talantoÔmeno hlektrikì dpolo d, dnetai apì ton tÔpo

LhlektrikoÔ dipìlou =(ekpempìmenh enèrgeia)(monda qrìnou)

=2

3q2a2 =

2

3d2 (1.70)ìpou

~d = q~x kai ~d = q~x = q~a (1.71)me q to hlektrikì forto kai thn telea na upodhl¸nei thn olik qronik pargwgo.To barutikì anlogo enì sust mato N swmatidwn mza mA(A = 1, 2, 3, . . . , N) tou

1.5 Paragwg twn Barutik¸n Kumtwn 20hlektrikoÔ dipìlou enai h dipolok rop mza :~d ≡

N∑

A=1

(mA~xA) (1.72)H pr¸th pargwgo th w pro to qrìno enai h olik orm tou sust mato :~d ≡

N∑

A=1

(mA˙~xA) = ~p (1.73)Ex' aita ìmw th arq diat rhsh th orm enì sust mato h deÔterh pargwgo w pro to qrìno mhdenzetai, ~d = ~p = 0. 'Etsi den mporei na uprxei barutik aktinobola dipìlou.To epìmeno edo th hlektromagnhtik aktinobola enai to magnhtikì dpolo kai tohlektrikì tetrpolo. H aktinobola magnhtikoÔ dipìlou pargetai apì th deÔetrh qronik pargwgo th magnhtik rop , ~µ. 'Omw ki ed¸ to barutikì anlogo enai ma stajer th knhsh h stroform

~µ =N∑

A=1

(jèsh tou A)× (reÔma ex' aita tou A)=

N∑

A=1

~rA × (m~uA) = ~J (1.74)th opoa h qronik pargwgo mhdenzetai. Epomènw oÔte èna tètoio sÔsthma mpore naaktinobole. Den uprqei barutik dipolik aktinobola kama morf .To pr¸to mh-mhdenikì apotèlesma to brskoume sthn aktinobola tetrapolik rop . Hhlektromagnhtik jewra dnei:LhlektrikoÔ tetrapìlou =

1

20

d ~Q2

dt≡ 1

20

dQjk

dt

dQjk

dt(1.75)ìpou

Qjk ≡N∑

A=1

eA(xAjxAk −1

3δjk~r

2) (1.76)To barutikì anlogo enaiLtetrapìlou mza =

1

5〈d

3I

dt3

2

〉 ≡ 1

5〈d

3Ijk

dt3d3Ijk

dt3〉 (1.77)kai

Ijk ≡N∑

A=1

mA(xAjxAk −1

3δjkr

2) =

∫(xjxk −

1

3δjkr

2)d3x (1.78)

1.5 Paragwg twn Barutik¸n Kumtwn 21O tÔpo (1.76) perièqei to swstì pargonta 15, o opoo proèrqetai apì thn tanustik anlush,ant tou ljou ìrou 1

20th hlektromagnhtik jewra . To dex mèlo th (1.77) perièqeito mèso ìro ( 〈 〉 ) Ôstera apì merikè periìdou th phg ,sÔmfwna kai me th mh-dunatìthta twn parathrht¸n na entopsoun thn enèrgeia th barutik aktinobola se ènam ko kÔmato .O ìro Ijk onomzetai mhdenikoÔ qnou tetrapolik rop ki èna aplì upologismì th trth qronik th parag¸gou dnei

d3

dt3Ijk ∼ (mza kinoÔmenou sust mato )(mègejo sust mato )2

(qrìno )3

∼ MR2

T 3∼ Mu2

T(1.79)ètsi ¸ste:

Ltetrapìlou mza ∼ G

c5

Mu2

T(1.80)An kai oi tÔpoi (1.79) kai (1.80) enai arket aplopoihmènoi perièqoun ti dÔo pio shmantikè plhrofore gia thn paragwg twn barutik¸n kumtwn. H pr¸th enai ìti h enèrgeia pouekpèmpetai me thn barutik kÔmata ellat¸netai shamntik lìgw tou ìrou G

c5∼ 10−59, na tojèsoume alli¸ , metatrop opoiasd pote morf enèrgeia se barutik kÔmata, enai genik¸ ,mh efikt . H deÔterh plhrofora enai h qronik metabol tou tetrapìlou mza , h opoa mporena gnei shmantik mìno gia polÔ megle mze kinoÔmene me sqetikistikè taqÔthte . Apìta parapnw gnetai emfanè ìti, oi sunj ke autè den mporoÔn na pragmatopoihjoÔn se g inaergast ria, all sunant¸ntai suqn se astrofusik sumpag antikemena, pou apoteloÔn ti pio uposqìmene phgè barutik¸n kumtwn.

Keflaio 2Diataraqè tou qwroqrìnouSchwarzschild kai barutik kÔmataSto prohgoÔmeno keflaio ègine ma eisagwg sth jewra twn barutik¸n kumtwn kaith didosh tou ston eppedo qwroqrìno. Sto keflaio autì ja melethjoÔn oi diataraqè pou prokaloÔntai se ma melan kai pio sugkekrimèna se ma mh peristrefìmenh melan op ,akolouj¸nta tou Regge kai Wheeler [1. 'Ena tètoio antikemeno onomzetai melan op Schwarzschild kai apotele ma statik lÔsh twn exis¸sewn pedou tou Einstein se èna sfairiksummetrikì kai kenì qwroqrìno. 'Ena basikì je¸rhma pou isqÔei gia th lÔsh Schwarzschildenai to je¸rhma tou Birkhoff, sÔmfwna me to opoo h lÔsh Schwarzschild enai h monadik sfairik summetrik statik , asumptwtik eppedh lÔsh twn exis¸sewn Einstein gia to kenì.H èreuna pnw sti twn diataraqè twn melan¸n op¸n enai arket shmantik ki endia-fèrousa kai apotele ènan taqèw anaptussìmeno kldo th jewrhtik kai upologistik astrofusik . O lìgo pou knei aut thn èreuna tìso shmantik enai to gegonì ìti h pa-rousa twn diataraq¸n mpore na spsei ti statikè idiìthte tou qwroqrìnou ma melan op , kajist¸nta thn phg barutik¸n kumtwn. Epsh ta ekpempìmena barutik kÔmata apìma melan op metafèroun plhrofore gia difore idiìthte th , ìpw h mza, h stroform kai to forto th . Tèlo melet¸nta ti antidrsei ma melan op se diataraqè sthnperioq th , mporoÔme na bgloume sumpersmata gia th eustjeia aut¸n twn antikeimènwn(toìti apoteloÔn lÔsh twn exis¸sewn tou Einstein enai ma ikanopoihtik sunj kh eustjeia ).2.1 Grammikè Diataraqè tou Qwroqrìnou Schwarz-

schildTo shmeo ekknhsh th anlush twn diataraq¸n melan¸n op¸n enai h adiatraqth lÔshtwn melan¸n op¸n Schwarzschild me grammikì stoiqeo, pou kajorzei th metrik tou Schwarz-schild, kai dnetai apì th sqèsh:

ds2 = gµνdxµdxν = −(

1 − 2M

r

)dt2 +

(1 − 2M

r

)−1

dr2 + r2dΩ2 (2.1)

2.1 Grammikè Diataraqè tou Qwroqrìnou Schwarzschild 23ìpou gµν o metrikì tanust (metrik Schwarzschild) tou statikoÔ adiatraktou qwroqrìnoukai M enai h mza th melan op ekfrasmènh se gewmetrikopoihmène monde m kou ,M = Gm

c2, ìpou m enai h mza se kil. H suntatagmènh r èqei ma gewmetrik ermhneapou prokÔptei apì th sfairik summetra.Den enai h apìstash apì kpoio kèntro. Aplsqetzetai me thn epifneia A ma sfara kajorismènou r kai t me th sqèsh

r =

(A

4π

) 1

2To dΩ enai h stoiqei¸dh stere gwna pou dnetai apì th sqèshdΩ2 = dθ2 + sin2 θdφ2Epeid o qwroqrìno enai kenì , oi exis¸sei tou Einstein gnontai pio sumpage kai parnounth morf

Rµν = 0 (2.2)ìpou Rµν o tanust tou Ricci kataskeuasmèno apì th metrik gµν .An eisaqjoÔn mikrè diataraqè hµν ,h telik metrik pou prokÔptei enaig′

µν = gµν + hµν (2.3)ìpou h sunj kh na enai oi diataraqè mikrè shmanei pw |hµν

gµν

| ≪ 1 (2.4)O tanust tou Ricci R′µν gia to diataragmèno qwrìqrono ja dnetai apì th sqèsh

R′µν = Rµν + δRµν = 0 (2.5)ìpw kai sth sqèsh (2.2), afoÔ o diataragmèno qwroqrìno den perièqei Ôlh enèrgeia.O ìro δRµν ≡ Rµν(hµν), pou apotele ton ìro th diataraq ston tanust tou Ricci,dnetaiapì th sqèsh

δRµν = −δΓβµν;β + δΓβ

µβ;ν (2.6)ìpou ta diataragmèna sÔmbola Christoffel orzontai w δΓβ

µν =1

2gαβ(hµα,ν + hνα,µ − hµν,α) (2.7)Sundizonta ti exis¸sei (2.2) kai (2.5) parnoume thn exswsh

δRµν = 0 (2.8)Sundizonta t¸ra thn (2.8) me ti (2.6) kai (2.7) parnoume ma diaforik exswsh deutèroubajmoÔ w pro hµν (ousiastik èqoume 10 exis¸sei ). Aut h exswsh apotele genkeush seèna kampÔlo qwroqrìno th exswsh tou Schrodinger gia swmatdio mhdenik mza kai spin2 se èna eppedo q¸ro. Qrhsimopoi¸nta thn exswsh (2.8) ja prospaj soume na broÔme ti diataraqè th metrik , hµν .

2.2 Anlush se Sfairikè Armonikè 242.2 Anlush se Sfairikè Armonikè 'Opw kai h lÔsh th exswsh Schrodinger mpore na qwriste se èna ginìmeno sunart sewnma metablht , gia kja metablht , to dio mpore na gnei kai me th lÔsh hµν th exswsh (2.8), ìpou ta hµν enai sunart sh twn metablht¸n t, r, θ, φ. Autì mpore na dikaiologhjeapì to gegonì ìti h adiatrakth pollaplìthta M enai qwrik sfairik summetrik , meapotèlesma na mpore na grafe w ginìmeno M = M2 × S

2, ìpou M2 enai h disdistathpollaplìthta Lorentz twn suntetagmènwn (t, r) kai S

2 enai h epifneia sfara monadiaa aktna kai suntetagmènwn (θ, φ). W apotèlesma autoÔ tou diaqwrismoÔ, oi diataraqè hµνmporoÔn na qwristoÔn se èna kommti pou periorzetai sto M2 ki èna kommti periorismènosth sfara S

2,th opoa h metrik enai γµν = gµν

r2 = diag(1, sin2 θ) kai ìpou ta µ, ν parnoungia timè ti metablhtè θ kai φ. Dhlad oi diataraqè hµν ja qwristoÔn se èna kommti pouexarttai mìno apì ta t kai r ki èna gwniakì kommti pou exarttai apì ta θ kai φ. Enaignwstì apì th jewra twn orjogwnwn sunart sewn pw kje sunrthsh twn θ kai φ mporena analuje se èna jroisma sfairik¸n armonik¸n. Ma tètoia anlush dieukolÔnei th lÔshtou probl mato , afoÔ plèon oi idiìthte tou gwniakoÔ mèrou tou hµν ja enai gnwstè .Efarmìzonta ma peristrof tou sust mato anafor ma gÔrw apì thn arq tou, oi dèkasunist¸se tou diataragmènou tanust hµν metasqhmatzontai w 3 bajmwt (h00; h01; h11),2dianÔsmata (h02, h03; h12, h13) ki èna tanust deutèra txh (h22, h23, h32, h33), ìtan jew-rhjoÔn sunallowte posìthte sth sfara. Ma bajmwt sunrthsh mpore na analuje sesfairikè armonikè w ex :φm

l = stajerY ml (θ, φ) (2.9)O parapnw ìro an kei se kÔma me parity (−1)l kai stroform l, me sunist¸sa ston xona z, m. Sth (2.9) ennoetai jroisma w pro l kai m, to sÔmbolo ìmw th jroish paralÔpetaigia lìgou suntomografa .Ta dianÔsmata, ìpw kai oi tanustè , mporoÔn na èqoun dÔo katastsei parity, anlogame to pw sumperifèrontai se metasqhmatismoÔ th parity ìpou (θ, φ) → (π − θ, π + φ). 'Osametasqhmatzontai w (−1)l+1 onomzontai peritt(odd) aktinik(radial), en¸ ìsa metasqh-matzontai w (−1)l onomzontai rtia(even) polik(polar). 'Etsi anloga me thn parity tou,èna dinusma analÔetai se sfairikè armonikè w ex :

ψml,µ = stajer ∂

∂xµY m

l (θ, φ) , parity (−1)l

φml,µ = stajerǫν

µ

∂

∂xνY m

l (θ, φ) , parity (−1)l+1 (2.10)Oi dekte µ, ν parnoun ti timè 2, 3, ìpou x2 = θ, x3 = φ. Oi posìthte ǫνµ dnontai apìti sqèsei ǫθ

θ = ǫφφ = 0; ǫφ

θ = − 1sin θ

, ǫθphi = sin θ. Kai ta dÔo edh twn dianusmtwn èqounstroform l. Anloga isqÔoun kai gia tou tanustè

ψmlµν = stajerY m

l;µν , parity (−1)l (2.11)φm

lµν = stajerγµνYml , parity (−1)l (2.12)

χmlµν =

stajer2

[ǫλµψ

mlµν + ǫλ

νψmlλµ] , parity (−1)l+1 (2.13)

2.2 Anlush se Sfairikè Armonikè 25Na shmei¸soume ìti èna tanust me parity (−1)l enai to jroisma twn (2.10) kai (2.11).Kajèna apì autoÔ tou ìrou mpore na pollaplasiaste me mia aujareth sunrthsh twn(t, r), qwr na allxoun oi idiìthte tou ktw apì peristrofè .'Etsi gia ma genik diataraq dosmènh stroform l,m kai parity (−1)l+1 èqoume

h00 = h01 = h11 = 0 (2.14)h02 = ǫν

2

∂

∂xνY m

l = ǫ22

∂

∂θY m

l + ǫ32

∂

∂φY m

l = − 1

sin θ

∂

∂φY m

l (2.15)h22 =

1

2[ǫλ

2ψmlλ2 + ǫλ

2ψmlλ2] = ǫλ

2Yml;λ2

= ǫ32Y

ml;32 = − 1

sin θ

(∂2Y m

l

∂θ∂φ− cos θ

sin θ

∂Y ml

∂φ

) (2.16)ìpou jews same th stajer sh me monda. Omow upologzontai kai oi upìloipe sunist¸se .'Etsi h genik diataraq h gnetaihµν =

0 0 −h0(t, r)∂∂φ

Y ml

1sin θ

h0(t, r) sin θ ∂∂θ

Y ml

0 0 −h1(t, r)∂∂φ

Y ml

1sin θ

h1(t, r) sin θ ∂∂θ

Y ml

∗ ∗ h2(t, r)(∂2

sin θ∂θ∂φ− cos θ

sin2 θ∂∂φ

)Y ml ∗

∗ ∗ 12h2(t, r)(

∂2

sin θ∂θ∂φ+ −h2(t, r)(sin θ ∂2

∂θ∂φ−

+ cos θ ∂∂θ

− sin θ ∂2

∂θ2 )Yml − cos θ ∂

∂φ)Y m

l

(2.17)Oi st le kai oi grammè arijmoÔntai me th seir 0, 1, 2, 3(t, r, θ, φ). To sÔmbolo ∗ deqnei ìtiautè oi sunist¸se tou tanust h brskontai apì to gegonì ìti o tanust enai summetrikì ,hµν = hνµ.Oi ìroi rtia parity ma dnoun ti rtie diataraqè :

hµν =

(1 − 2Mr

)H0Yml H1Y

ml h0

∂∂θ

Y ml h0

∂∂φ

Y ml

∗ (1 − 2Mr

)−1H2Yml h1

∂∂θ

Y ml h1

∂∂φ

Y ml

∗ ∗ r2[K ∗+G ∂2

∂θ∂φ]Y m

l

∗ ∗ r2G( ∂2

∂θ∂φr2[K sin2 θ

− cos θ ∂sin θ∂φ

)Y ml +G( ∂2

∂θ∂φ

+ sin θ cos θ ∂∂θ

)]Y ml

(2.18)Oi sunart sei H0(t, r), H1(t, r), H2(t, r), h0(t, r), h1(t, r), G(t, r), K(t, r) enai ìle sunart -sei twn (t, r) mìno. Epsh oi sunart sei h0(t, r), h1(t, r) enai diaforetikè apì ti antstoi-qe th (2.17) gia perittè diataraqè .

2.3 Anlush Suqnot twn:Eidkeush gia M = 0 262.3 Anlush Suqnot twn:Eidkeush gia M = 0Ex' aita th sfairik summetra th adiatrakth metrik , oi exis¸sei (2.6) kai (2.7)den mporoÔn na anamxoun ìrou diaforetikoÔ l kai parity. Efarmìzonta kbantik gl¸ssase èna klasikì prìblhma, mporoÔme na poÔme ìti ta l,m kai h parity enai stajerè th knhsh .H Ôparxh ma akìma stajer èpetai apì to gegonì ìti h metrik (2.1) enai anexrthth touidioqrìnou t. Me bsh ta parapnw mporoÔme na jewr soume ma diataraq ma dedomènh su-qnìthta , ω = k, ètsi ¸ste kje sunist¸sa tou tanust hµν na èqei ma qronik exrthsh th morf e−iωt = e−ikt. Sth sunèqeia ja prosdioriste apìluta h morf th atomik lÔsh sug-kekrimènh parity, tim¸n l,m kai suqnìthta . H genik lÔsh ja apotele ma upèrjesh aut¸ntwn sugkekrimènwn lÔsewn me stajerè prosarmosmène ¸ste na ikanopoioÔn ti katllhle oriakè sunj ke kai arqikè timè .H anlush sti sfairikè armonikè sunart sei Y ml katèsthse dunatì na mhn asqolhjoÔmello me th gwniak exrthsh afoÔ aut prosdiorsthke pl rw apì ti (2.17) kai (2.18). Maakìma aplopohsh apotele to gegonì ìti gia kje sugkekrimènh epilog twn l, kkai ìlwn twntim¸n tou m(m = −l,−l +1, . . . , 0, . . . , l− 1, l) ja odhg sei sti die aktinikè exis¸sei . Giato lìgo autì epilègoume thn perptwsh ìpou m = 0, me apotèlesma h gwna φ na exafanistetelew apì ti exis¸sei .2.4 Metasqhmatismo Bajmda Oi prohgoÔmene aplopoi sei pou èginan den enai oi mìne dunatè gia ti diataraqè hµν .H eleujera epilog tou sust mato suntetagmènwn apì th Genik Sqetikìthta eisgei kille aplousteÔsei pou sumazeÔoun akìma perissìtero ti ekfrsei twn hµν .'Estw o apeirostì metasqhmatismì suntetagmènwn(metasqhmatismì bajmda )

x′a = xa + ξa (ξa ≪ xa) (2.19)Oi apeirostè metatopsei ξa metasqhmatzontai w dianÔsmata. Sti nèe suntetagmène ,ìpw deqthke sto prohgoÔmeno keflaio, ja enaig′

µν + h′µν = gµν + hµν − ξµ;ν − ξν;µ (2.20)Ef' ìson ta hµν orzontai w h diafor th diataragmènh metrik apì th metrik Schwarz-

schild se sfairikè suntetagmène , ja enaihnèoµν = hpalaiìµν − ξµ;ν − ξν;µ (2.21)Dhlad to hµν èqei uposte èna metasqhmatismì bajmda . MporoÔme t¸ra na qrhsimopoi -soume autì to gegonì gia na aplopoi soume thn perigraf th diataraq kai na thn knoumemonadik , na dialèxoume dhlad èna kajorismèno sÔsthma anafor .H sunrthsh ξa den mpore na epilege telew tuqaa. Ki autì giat, par' ìlo pou ometasqhmatismì mpore na efarmoste se kje xeqwristì kÔma, mìno gia nèa kÔmata pouan koun sti arqikè idiotimè ja èqei nìhma h aplopohsh pou fèrnei o metasqhmatismì .

2.4 Metasqhmatismo Bajmda 27Epomènw to dinusma ξa prèpei na enai diou l kai parity, ìpw kai h diataraq (kÔma). 'Etsigia perittè diataraqè h genik morf enì aktinikoÔ(perittoÔ) dianÔsmato enaiξa = Λ(t, r)[0, 0,

1

sin θ

∂

∂φY m

l ,− sin θ∂

∂θY m

l ] (2.22)Qrhsimopoi¸nta ti exis¸sei (2.21) kai (2.22) oi Regge kai Wheeler dilexan ma bajmda sthnopoa h2(t, r) = 0. Aut h bajmda enai gnwst w bajmda Regge-Wheeler. Kat sunèpeia hmorf twn aktinik¸n diataraq¸n gia kÔmata olik stroform l kai probol m = 0 gnetaihµν =

0 0 0 h0(r)0 0 0 h1(r)0 0 0 0∗ ∗ 0 0

× e−ikt sin θ

∂

∂θPl(cos θ) (2.23)ìpou Pl(cos θ) enai ta polu¸numa Legendre.Kat antistoiqa gia rtie diataraqè , to polikì(rtio) dinusma me parity (−1)l, èqei thgenik morf

ξa = [M0(t, r)Yml ,M1(t, r)Y

ml ,M(t, r)

∂

∂θY m

l ,M(t, r)∂

∂φY m

l ] (2.24)Me bsh th morf tou polikoÔ dianÔsmato kai th sqèsh (2.21), oi diataraqè hµν metasqh-matzontai w h′

tt = htt − 2ξt,t + 2Γαttξα = htt − 2M0 +

2M

r2

(1 − 2M

r

)M1Y

ml

h′tφ = htφ − ξt,φ − ξφ, t + 2Γα

tφξα = htφ − M0∂Y m

l

∂φ− M2

∂Y ml

∂φ

h′tθ = htθ − ξt,θ − ξθ, t + 2Γα

tθξα = htθ − M0∂Y m

l

∂θ− M2

∂Y ml

∂θ

h′φr = hφr − ξr,φ − ξφ, r + 2Γα

rφξα = hφr − M ′2

∂Y ml

∂φ− M1

∂Y ml

∂φ+

+2

rM2

∂Y ml

∂φ (2.25)Omow upologzontai kai oi upìloipoi ìroi, ìpw fusik ètsi upologsthkan kai oi ìroi giati perittè sunist¸se . Akolouj¸nta kai pli tou Regge kai Wheeler dialègoume ètsi ti sunart sei M0,M1,M ¸ste na apalefontai oi sunart sei G, h0 kai h1 apì thn èkfrash(2.18), me apotèlesma na dnetai h morf twn artwn diataraq¸n apì thn èkfrashhµν =

H0(1 − 2Mr

) H1 0 0H1 H2(1 − 2M

r)−1 0 0

0 0 r2K 00 0 0 r2 sin2 θK

× e−iktPl(cos θ) (2.26)Aplopoi jhke ètsi h morf twn diataraq¸n, ¸ste oi perittè na exart¸ntai apì dÔo plèonsunart sei h0 kai h1 kai apì tèsseri , ant gia eft, oi rtie , ti H0, H1, H kai K.

2.5 Aktinikè Exis¸sei 282.5 Aktinikè Exis¸sei H parapnw doulei ègine gia na aplopoihje to prìblhma th lÔsh th exswsh (2.8).Epomènw antikajist¸nta ti telikè ekfrsei twn (2.23) kai (2.26) gia ta hµν , lÔnonta thn (2.8) brskoume ti ekfrsei gia ti aktinikè exis¸sei twn (2.23) kai (2.26). Me bshth (2.6) h (2.8) gnetaiδΓβ

µν;β − δΓβµβ;ν = 0 (2.27)H melèth twn dÔo diaforetik¸n diataraq¸n, gia ti dÔo parity, ja gnei qwrist.2.5.1 Perittè Diataraqè Sthn perptwsh aut h (2.27) aplopoietai akìma perissìtero, afoÔ δΓβ

µβ;ν = 0. H apìdeixhenai polÔ eÔkolh. Apì th (2.7) einaiδΓβ

µβ =1

2gβκ(hκµ,β + kκβ,µ − hµβ,κ)Epeid ìmw h metrik gβκ(metrik tou Schwarzschild) enai diag¸nia, dhlad β = κ ja enaihκµ;β = hµβ;κkai

hκβ;µ = 0afoÔ h (2.23) den èqei diag¸niou ìrou . Epomènw kai δΓβµβ = 0.'Ara apì th (2.27)

δRµν = δΓβµν;β = 0 (2.28)Na shmeiwje ìti par' ìlo pou ta sÔmbola Christoffel den enai tanustè , oi diataraqè tou enai. 'Ara apì th (2.28) ja enai

δRµν = δΓβµν;β = δΓβ

µν,β + ΓβαβδΓα

µν − ΓαµβδΓβ

αν − ΓανβδΓβ

µα = 0 (2.29)kai ìpou oi dekte µ, ν, α, β parnoun ti timè 0, 1, 2, 3 pou antistoiqoÔn sta t, r, θ, φ.Antikajist¸nta t¸ra th (2.7) sth (2.29), mìno trei exis¸sei parnoume, oi opoe pro-kÔptoun apì ti δR23 = δR13 = δR03 = 0, oi opoe na mhn enai ek tautìthto mhdèn.Autè oi exis¸sei enai(1 − 2M

r)−1kh0 +

d

dr[(1 − 2M

r)h1] = 0 , gia δR23 = 0

(1 − 2M

r)−1k(

dh0

dr− kh1 −

2h0

r) + (l − 1)(l + 2)

h1

r2= 0 , gia δR13 = 0

d

dr[kh1 −

dh0

dr] + 2k

h1

r= r−2(1 − 2M

r)·

[4Mh0

r− l(l + 1)h0] , gia δR03 = 0 (2.30)

2.5 Aktinikè Exis¸sei 29H teleutaa exswsh den enai anexrthth all mpore na prokÔyei apì ti lle dÔo.Sundizonta ti dÔo pr¸te exis¸sei enai dunatìn na apaleifje h ma gnwsth sunrthshkai na menei ma sunrthsh mìno se ma exswsh.Xekin¸nta apì thn pr¸th exswsh epilÔoume w pro h0

kh0 = −[ d

dr(1 − 2M

r)]h1(1 − 2M

r) − (1 − 2M

r)2 dh1

drkai thn antikajistoÔme sth deÔterh, parnonta ètsi ma diaforik exswsh deutèrou bajmoÔw pro h1:−

[ d2

dr2(1 − 2M

r)]h1 − (1 − 2M

r)d2h1

dr2− 3

[ d

dr(1 − 2M

r)]dh1

dr− h1(1−

2M

r)−1

( d

dr(1 − 2M

r))− (1 − 2M

r)−1k2h1 +

2h1

r

d

dr(1 − 2M

r)+

+(1 − 2M

r)2

r

dh1

dr+ l(l + 1)

h1

r2− 2h1

r2= 0 (2.31)Orzonta thn posìthta Q w

Q(r) = (1 − 2M

r)h1

r(2.32)kai me qr sh ma akìma sqèsew orismoÔ

dr∗ ≡ (1 − 2M

r)−1dr

⇒ r∗ = r + 2M ln(1 − 2M

r) (2.33)h (2.31) gnetai

d2Q

dr∗2+ k2

eff(r)Q = 0 (2.34)ìpouk2

eff = k2 − l(l + 1)

r2(1 − 2M

r) +

6M

r3(1 − 2M

r) (2.35)H exswsh (2.34) mpore na prei kai ma pio genik morf ¸ste na isqÔei kai gia pio genikè qronikè exart sei , ìqi mìno gia e−ikt, all genik¸ gia ma h1(r, t). Tìte h (2.32) gnetai

∂2Q

∂t2− ∂2Q

∂r∗2+ (1 − 2M

r)[ l(l + 1)

r2− 6M

r3

]Q = 0 (2.36)kai gnetai fanerì ìti gia th sugkekrimènh perptwsh th qronik exrthsh e−ikt, èqoume

∂2Q∂t2

= −k2.O ìro V (r) ≡ (1 − 2M

r)[ l(l + 1)

r2− 6M

r3

] (2.37)

2.5 Aktinikè Exis¸sei 30

Sq ma 2.1: Anaparstash tou dunamikoÔ Regge-Wheeler V kai th epdrash tou se ènaeiserqìmeno barutikì kÔma.onomzetai dunamikì Regge-Wheeler, afoÔ h (2.32) kai (2.36) enai kumatikè exis¸sei ,ìpou to V (r) ekfrzei to dunamikì skèdash . To dunamikì V (r) èqei mègisto lgo èxw apì tonorzonta gegonìtwn, se apìstash r ∼ 3.3M se suntetagmène Schwarzschild, ìpw fanetaisto sq ma 2.1 (H eikìna enai apì thn ergasa [16, selda 17).Na anaferje ìti h qr sh th suntetagmènh r∗ enai polÔ bolik , ènanti th r. Ki autì giatgia r → ∞, r∗ → r kai r → 2M+, r∗ → −∞, kajist autè ti suntetagmène katllhle giamelèth th didosh th diataraq kont ston orzonta gegonìtwn ma mean op , afoÔautì topojetetai sto −∞, exalefonta to prìblhma th idiomorfa twn suntetagmènwnSchwarzschild.H exswsh (2.30) (2.34) onomzetai exswsh twn Regge-Wheeler kai h sunrthshQ,sunrthshRegge-Wheeler.Maz me thn exswsh (2.30) apì to sÔsthma twn exis¸sewn (2.28) mpore na prokÔyei kaih exswsh

∂h0

∂t=

∂

∂r∗(r∗Q) (2.38)h opoa ìmw de qrhsimopoietai idiatera, gia lìgou pou ja gnoun emfane sto epìmenokeflaio.H exswsh Regge-Wheeler perièqei ìle ti idiìthte ma kumatik exswsh se skedzondunamikì kai ìla ta apotelèsmata pou èqoun breje gia tètoiou edou exis¸sei (p.q. exswshtou Schrodinger) mporoÔn na efarmostoÔn sth didosh twn diataraq¸n tou qwroqrìnou ma melan op Schwarzschild.W pardeigma, ma diataraq th metrik h opoa plhsizei th melan op apì peirhapìstash mpore na jewrhje san kumatopakèto pou ja skedaste sto frgma dunamikoÔ V .'Opw sthn kbantomhqanik , èna mìno mèro tou kumatopakètou ja ftsei sth melan op kai

2.5 Aktinikè Exis¸sei 31èna llo mèro , pou exarttai apì ti idiìthte tou diou tou kumatopakètou, ja anaklastepsw sto peiro. To apotèlesma autì deqnei th diafor sth sumperifor twn barutik¸nkumtwn ènanti ulik¸n swmtwn, ìpw gia pardeigma èna sfairikì ulikì kèlufo pou pèfteiaktinik pro th melan op . Aut h shmantik diafor tonzei th shmasa ma anlush twndiataraq¸n tou qwroqrìnou ma mela op .'Olh h anlush pou ègine mèqri t¸ra aforoÔse tanustikè diataraqè , afoÔ ta diataragmènamegèjh tan tanustik megèjh. Par' ìla aut h exswsh twn Regge-Wheeler parousizei mapolÔ endiafèrousa sumperifor, afoÔ parousizetai me thn dia morf tìso gia bajmwtè ìsokai gia dianusmatikè diataraqè . H mình diafor emfanzetai sto dunamikì V th (2.37), ìpouo ìro p = −6Mr3 enai p = 2M

r3 kai p = 0 gia bajmwtè kai dianusmatikè diataraqè antstoiqa.2.5.2 'Artie Diataraqè Gia ti polikè diataraqè h exswsh (2.8) gnetaiδRµν = δΓβ

µν;β − δΓβµβ;ν = 0

⇒ δΓβµν,β − δΓβ

µβ,ν + ΓβαβδΓα

µν − ΓαµβδΓβ

αν −−Γα

νβδΓβµα − Γβ

ανδΓαµβ + Γα

µνδΓβαβ − Γα

βνδΓβαµ = 0

⇒ δΓβµν,β − δΓβ

µβ,ν + ΓβαβδΓα

µν − ΓαµβδΓβ

αν +

+ΓαµνδΓ

βαβ − Γβ

ανδΓαµβ = 0 (2.39)Apì th (2.37) trei exis¸sei enai ek tautìthta se me mhdèn

δR03 = 0

δR13 = 0

δR23 = 0 (2.40)Apì ti upìloipe eft exis¸sei prokÔptoun ma algebrik sqèsh, trei diaforikè exis¸sei pr¸th txew kai trei diaforikè exis¸sei deutèra txew :H2 = H0 ≡ Hkai qrhsimopoi¸nta aut oi lle èxi gnontai

dK

dr+ r−1(K + H) − l(l + 1)

H1

2kr2− (1 − 2M

r)−1 MK

r2= 0

d

dr

[(1 − 2M

r)H1

]= k(K − H)

kH1 − (1 − 2M

r)

d

dr(K + H) − 2M

r2H = 0

d

dr

[(1 − 2M

r)(2rH +

d

dr(r2K)

)]− l(l + 1)K −

2.5 Aktinikè Exis¸sei 32−2krH1 + k2r2(1 − 2M

r)−1K = 0

2k(1 − 2M

r)dH1

dr+ 2kM

H1

r2+ k2H −

−(1 − 2M

r)2

[ 1

r2

d

dr(r2 dH

dr) +

2

r2

d

dr(r2 dK

dr)]−

−(1 − 2M

r)[l(l + 1)

H

r2+

4M

r2

dH

dr+

2M

r2

dK

dr

]= 0

(1 − 2M

r)2 d2H

dr2+

2

r(1 − 2M

r)dH

dr− k2H − l(l + 1)(1 − 2M

r)H

r2− 2kMH1

r2−

−2k(1 − 2M

r)

1

r2

d

dr(r2H1) + 2k2K + 2(1 − 2M

r)M

r2

dK

dr= 0 (2.41)Oi parapnw sqèsei apoteloÔn èna sÔsthma èxi exis¸sewn me trei agn¸stou . Para-thr¸nta ti exis¸sei ja mporoÔsame na upojèsoume ìti oi trei prwtobjmie diaforikè exis¸sei th (2.41) arkoÔn gia th lÔsh tou sust mato , dojèntwn twn katllhlwn oriak¸nsunjhk¸n. Oi trei deuterobjmie exis¸sei ìmw emperièqoun èna prìsjeto kommti plhro-fora gia th lÔsh, pou enai sÔmfwno me ti trei pr¸te . Sugkekrimèna, mia pio prosektik èreuna deqnei ìti oi trei deuterobjmie exis¸sei mporoÔn na exaqjoÔn ìle apì ti trei prwtobjmie , sun thn algebrik sqèsh:

[6M

r+ (l − 1)(l + 2)

]H +

[2kr − l(l + 1)

M

r2K

]H1+

+[2M

r+ (l − 1)(l + 2) − 2(1 − 2M

r)−1(

M2

r2+ k2r2)

]K ≡ F = 0 (2.42)kai antistrìfw , oi trei teleutae th (2.41) mporoÔn na d¸soun ti trei prwtobjmies(trei pr¸te th (2.41) ) sun thn sqèsh (2.42), me thn propìjesh h suqnìthta k 6= 0.Enai entupwsiakì to gegonì ìti h (2.42) enai ma algebrik sqèsh se sumfwna me ti diaforikè exis¸sei th (2.41). Ja mporoÔse na onomaste w pr¸to olokl rwma. 'Estwìti K ′, H ′, H ′

1 enai lÔsei twn tri¸n pr¸twn diaforik¸n exis¸sewn th (2.41) kai F ′ h sunr-thsh th (2.42) pou antistoiqe sti lÔsei K ′, H ′, H ′1.ApodeiknÔetai ìti h F ′ upakoÔei sthnparaktw exswsh:

dF ′

dr+

M

r2(1 − 2M

r)−1F ′ = 0 (2.43)An h F ′ exafanzetai se kpoio shmeo (me katllhlh eklog twn aujairètwn stajer¸n) tìteh F ′ enai mhdèn pantoÔ.MporoÔme kai gia ti rtie diataraqè na kataskeusoume ma kumatik exswsh anloghth (2.34) [3. Orzoume ma nèa sunrthsh Z, gnwst w sunrthsh tou Zerilli, h opoasundèetai me ti H,H1 kai K me ti sqèsei

K =q(q + 1)r2 + 3qMr + 6M2

r2(qr + 3M)Z +

r − 2M

r

dZ

dr

2.6 Enallaktik 'Ekfrash se Sfairikè Armonikè 33H1 = −ik

qr2 − 3qMr − 3M

(r − 2M)(qr + 3M)Z − ikr

dZ

dr

H =qr(r − 2M) − k2r4 + M(r − 3M)

(r − 2M)(qr + 3M)K +

M(q + 1) − k2r3

ikr(qr + 3M)H1 − Blm (2.44)ìpou

q =1

2(l − 1)(l + 2) (2.45)kai

Blm =8πr2(r − 2M)

qr + 3M

Alm +

[1

2l(l + 1)

]− 1

2

Blm

− 4π

√2

qr + 3M

Mr

kA

(1)lm (2.46)ìpou oi posìthte Alm, Blm kai A

(1)lm ja oristoÔn lgo paraktw. To mìno pou arke napoÔme t¸ra enai pw oi posìthte autè enai oi suntelestè tou anaptÔgmato enì tanust deutèra txh se sfairikè armonikè . H akrib morf th sunrthsh tou Zerilli ja dwjesto epìmeno keflaio, kaj¸ h exagwg th ja gnei pio eÔkola. Me bsh ìle ti parapnwsqèsei kai ti trei pr¸te exis¸sei th (2.41), parnoume thn kumatik exswsh gia polikè diataraqè d2Z

dr∗+ [k2 − V ]Z = 0 (2.47)gnwst ki w exswsh tou Zerilli. H (2.47) èqei thn dia morf me thn exswsh twn Regge-

Wheeler, enai dhlad ma kumatik exswsh se skedzon frgma dunamikoÔ V , to opoo orzetaiw V ≡ (1 − 2Mr)

[2q(q + 1)r3 + 6q2r2M + 18qM2r + 18M3

r3(qr + 3M)2

] (2.48)Enai endiafèron pw oi exis¸sei Regge-Wheeler kai Zerilli sundèontai sten kai enai duna-tìn na metasqhmatsoume thn pr¸th gia perittè diataraqè sth deÔterh gia rtie diataraqè me th bo jeia katllhlwn diaforik¸n telest¸n [16.2.6 Enallaktik 'Ekfrash se Sfairikè Armonikè H pr¸th ergasa pou dhmosieÔthke pnw sti diataraqè th metrik Schwarzschild tanaut twn Regge-Wheeler to 1957, th opoa to formalismì akolouj same mèqri stigm .'Ena nèo formalismì eis qjei to 1970 (arqik qrhsimopoi jhke to 1962 apì ton Mathews)apì tou Mathews kai Zerilli [3, o opoo qrhsimopoietai mèqri kai s mera, me merikè allagè .H diafor twn dÔo formalism¸n ofeletai ston trìpo anptuxh enì tanust se sfairikè armonikè kai sthn onomatologa pou qrhsimopoietai gia ti dÔo diaforetikè katastsei th parity. 'Etsi sto formalismì twn Mathews kai Zerilli oi perittè diataraqè (odd) onom-zontai magnhtikè kai oi rtie (even) diataraqè hlektrikè . Me bsh autìn to formalismì

2.6 Enallaktik 'Ekfrash se Sfairikè Armonikè 34èna genikì sunallowto summetrikì tanust T deutèra txh mpore na anaptuqje sesfairikè armonikè me bsh ton tÔpoT =

∑

l,m

[A

(0)lma

(0)lm + A

(1)lma

(1)lm + Almalm + B

(0)lm b

(0)lm + Blmblm+

+Q(0)lmc

(0)lm + Qlmclm + Glmglm + Dlmdlm + Flmflm

] (2.49)ìpoua

(0)lm =

Ylm 0 0 00 0 0 00 0 0 00 0 0 0

a(1)lm =

i√2

0 Ylm 0 0Ylm 0 0 00 0 0 00 0 0 0

alm =

0 0 0 00 Ylm 0 00 0 0 00 0 0 0

b(0)lm = ir

[2l(l + 1)

]− 1

2

0 0 ∂∂θ

Ylm∂∂φ

Ylm

0 0 0 0∗ 0 0 0∗ 0 0 0

blm = r[2l(l + 1)

]− 1

2

0 0 0 00 0 ∂

∂θYlm

∂∂φ

Ylm

∗ 0 0 0∗ 0 0 0

c(0)lm = r

[2l(l + 1)

]− 1

2

0 0 1sin θ

∂∂φ

Ylm − sin θ ∂∂θ

Ylm

0 0 0 0∗ 0 0 0∗ 0 0 0

clm = ir[2l(l + 1)

]− 1

2

0 0 0 00 0 1

sin θ∂∂φ

Ylm − sin θ ∂∂θ

Ylm

0 ∗ 0 00 ∗ 0 0

dlm = −ir2[2l(l + 1)(l − 1)(l + 2)

]− 1

2

0 0 0 00 0 0 00 0 − 1

sin θXlm sin θWlm

0 0 ∗ sin θXlm

2.6 Enallaktik 'Ekfrash se Sfairikè Armonikè 35glm =

r2

√2

0 0 0 00 0 0 00 0 1 00 0 0 sin2 θ

Ylm

flm = r2[2l(l + 1)(l − 1)(l + 2)

]− 1

2

0 0 0 00 0 0 00 0 Wlm Xlm

0 0 ∗ sin2 θWlm

(2.50)ìpou

Xlm = 2∂

∂φ

( ∂

∂θ− cot θ

)Ylm

Wlm =( ∂2

∂θ2− cot θ

∂

∂θ− 1

sin2 θ

∂2

∂φ2

)YlmTo parapnw set twn tanust¸n enai orjokanonikì sto eswterikì ginìmeno

(T, S) ≡∫∫

T ∗ : SdΩìpouT : S ≡ ηµληνκTµνSλκkai ηµν h metrik tou Minkowski. Oi suntelestè tou anaptÔgmato dnontai apì ti sqèsei :

A(0)lm = (a

(0)lm , T ), B

(0)lm = (b

(0)lm , T ) k.t.l. T¸ra gnetai safè apì thn prohgoÔmenh pargrafopoia h shmasa twn Alm, A(1)lm , Blm th sqèsh (2.46).Oi pr¸toi pènte kai oi dÔo teleutaoi ìroi th (2.49) odhgoÔn to hlektrikì kommti(even

parity) kai oi trei enapomenante ìroi to magnhtikì(odd parity). 'Etsi o tanust h ana-ptÔssetai w h =

∑

l,m

(h(e)lm + h

(m)lm ) (2.51)ìpou oi dekte (e), (m) dhl¸noun to hlektrikì kai to magnhtikì mèro th diataraq ant-stoiqa. Oi magnhtiko ìroi((−1)l+1parity) th (2.51) dnontai apì th sqèsh

h(m)lm =

i

r[2l(l + 1)]1/2

[iho(r, t)c

(0)lm(θφ) + h1(r, t)clm(θ, φ) −

− i

2r2[(l − 1)(l + 2)]1/2h2(r, t)dlm(θ, φ)

] (2.52)kai oi hlektriko ìroi (−1)lparity

h(e)lm = (1 − 2M

r)H0a

(1)lm −

√2iH1a

(1)lm + (1 − 2M

r)−1H2alm −

−1

r[(l − 1)(l + 2)]1/2(ih0b

(0)lm − h

(e)1 blm) +

+[1

2l(l + 1)(l − 1)(l + 2)]1/2Gflm +

√2[K − 1

2l(l + 1)G]glm (2.53)

2.6 Enallaktik 'Ekfrash se Sfairikè Armonikè 36H ìlh doulei apì ed¸ kai pèra se autìn ton formalismì enai dia me prin. Met kai thnepilog th bajmda Regge-Wheeler katal goume sto ìti oi magnhtiko ìroi tou h gnontaih

(m)lm =

i

r[2l(l + 1)]1/2(ih0c

(0)lm + h1clm) (2.54)kai o hlektrikì ìro

h(e)lm = (1 − 2M

r)H0a

(0)lm −

√2iH1a

(1)lm + (1 − 2M

r)−1H2alm +

√2Kglm (2.55)ìpou, ìpw eÔkola mpore na diapistwje, oi (2.52), (2.53) tautzontai me ti (2.17), (2.18) kaioi (2.54), (2.55) me ti (2.13) kai (2.15) antstoiqa, ìpw kai anamenìtan. Tèlo h jroish∑

l,m shmanei ìti (l = 0, 1, . . .) . . . kai (m = −l,−l + 1, . . . , 0, . . . , l − 1, l).'Ena akìma formalismì ,pou ja fane polÔ qr simo sto epìmeno keflaio, baszetaisth legìmenh 3+1 ditmhsh, sthn opoa o qwroqrìno temaqzetai se ma oikogèneiaqwrik¸n uperepifanei¸n (sunart sewn tou r, θ, φ) parametropoihmènh apì ma suntetagmènht = stajerì [16. Me aut n th ditmhsh to grammikì stoiqeo parnei th morf

ds2 = −a2dt2 + γij(dxi + βidt)(dxj + βidt)

= −(a2 + βjβj)dt2 + 2βidxidt + γijdxidxj (2.56)ìpou a enai h sunrthsh qronik dibash (lapse function), pou ekfrzei to rujmì stonopoo rolìgia se diaforetikè uperepifneie qtupoÔn, kai βi enai oi sunist¸se tou dia-nÔsmato qwrik metatìpish (shift vector), pou susqetzei allagè sti suntetagmène dÔodiaforetik¸n uperepifanei¸n.Me th bo jeia aut th ditmhsh , oi diataraqè th metrik ekfrzontai se ìrou enì kajar qronikoÔ tm mato (h00)(bajmwtì mègejo ), enì kajar qwrikoÔ (tanust ) hij(ìpouta i, j parnoun ti timè r, θ, φ) ki enì anmiktou qwroqronikoÔ tm mato h0i(dianusmatikìmègejo )hµν =

(h00 hi0

h0i hij

) (2.57)Me bsh ta parapnw, oi sunist¸se th (2.57) gia perittè (magnhtikè ) diataraqè d-nontai apì th sqèshh00 = 0

h0i = h0(t, r)[0,− 1

sin θ

∂Ylm

∂φ, sin θ

∂Ylm

∂θ

]

hij = h1(t, r)(e1)ij + h2(t, r)(e2)ij (2.58)ìpou(e1)ij =

0 − 1sin θ

∂Ylm

∂φsin θ ∂Ylm

∂θ

∗ 0 0∗ 0 0

(2.59)

2.6 Enallaktik 'Ekfrash se Sfairikè Armonikè 37kai(e2)ij =

0 0 0

0 1sin θ

(∂2

∂θ∂φ− cot θ ∂

∂φ

)Ylm

12

[1

sin2 θ∂2

∂φ2−− cos θ ∂

∂θ− sin θ ∂2

∂θ2

]Ylm

0 ∗ −[sin θ ∂2

∂θ∂φ− cos θ ∂

∂φ

]Ylm

(2.60)Sthn perptwsh rtiwn (hlektrik¸n) diataraq¸n enaih00 = −1

2(1 − 2M

r)1/2H0(t, r)Ylm

h0i = [H1Ylm, h(e)0

∂

∂θYlm, h

(e)0

∂

∂φYlm]

hij = h(e)1 (f1)ij +

H2

1 − 2Mr

(f2)ij + r2K(f3)ij + r2G(f4)ij (2.61)ìpou(f1)ij =

0 ∂Ylm

∂θ∂Ylm

∂φ

∗ 0 0∗ 0 0

(2.62)

(f2)ij =

Ylm 0 00 0 00 0 0

(2.63)

(f3)ij =

0 0 00 Ylm 00 0 sin2 θYlm

(2.64)

(f4)ij =

0 0 0

0 ∂2Ylm

∂θ2

(∂2

∂θ∂φ− cot θ ∂

∂φ

)Ylm

0 ∗(

∂2

∂φ2 − sin θ cos θ ∂∂θ

)Ylm

(2.65)Qr sh autoÔ tou formalismoÔ ja gnei sto epìmeno keflaio ìpou ja doÔme pw proqwreiakìma parapèra h melèth twn diataraq¸n tou qwroqrìnou ma melan op , dhlad touqwroqrìnou Schwarzschild.

Keflaio 3Anallowte Bajmda touQwroqrìnou SchwarzschildSto prohgoÔmeno keflaio melet jhke h sumperifor tou qwroqrìnou Schwarzschild ìtanautì diataraqte gia kpoio lìgo. H melèth ègine se èna sugkekrimèno sÔsthma anafor ,to opoo kajorzetai apì th bajmda twn Regge-Wheeler. Enai ìmw dunatì na melethje todio prìblhma qwr na qrhsimopoi soume kpoio sugkekrimèno sÔsthma anafor , all me thbo jeia anallowtwn posot twn oi opoe enai anexrthte apì kje sÔsthma anafor (apìed¸ kai sto ex ja anafèrontaai apl w anallowte ).H shmasa th qr sh anallowtwn posot twn fanetai apì to gegonì ìti akìma ki anto sÔsthma suntatagmènwn tou adiatraqtou qwroqrìnou èqei kajoriste, h eleujera sun-tetagmènwn th Genik Jewra th Sqetikìthta eisgei èna prìblhma ìtan prosjètoumesto qwroqrìno grammikè diataraqè . Pio sugkekrimèna, den enai dunatì na xeqwrsoumema apeirost fusik diataraq apì ma pou prokÔptei w apotèlesma enì apeirostoÔ me-tasqhmatismoÔ suntetagmènwn (metasqhmatismoÔ bajmda ) [9. Aut h duskola mpore naexalhfje ete me to na kajorsoume ma bajmda (ìpw ègine sth doulei twn Regge-Wheelerpou edame kai sto prohgoÔmeno keflaio) ete eisgonta grammik anallowte posìthte apì metasqhmatismoÔ bajmda (ìpw pr¸to prìteine o Moncrief [2 kai uiojet jhke apìpolloÔ llou ).Pio sugkekrimèna, dojènto enì tanustikoÔ pedou X se ma adiatrakth metrik g kaiδX h apeirost diataraq tou, èna apeirstì metasqhmatismì suntetagmènwn, th morf xµ → xµ′ ≡ xµ + ξµ, me ξµ ≪ 1,ja diegerei èna nèo tanustikì pedo

δX → δX′ = δX + LξXìpou Lξ h pargwgo Lie kat m ko tou ~ξ sth metrik g. To δX tìte ja jewretai anallo-wto(gauge-invariant) an kai mìno an LξX = 0. Autì shmanei, me bsh ti idiìthte th para-g¸gou Lie, ìti h ikanìthta na kataskeusoume anallowte diataraqè th metrik , exarttaiapì thn Ôparxh summetri¸n th adiatrakth metrik . Sthn perptwsh enì genik sfairiksummetrikoÔ adiatraktou qwroqrìnou kai o opoo èqei analuje se anptugma polupìllwn(ìpw ègine me ti diataraqè th metrik Schwarzschild), h kataskeu posot twn anallo-wtwn se metasqhmatismoÔ bajmda enai dunat gia polÔpolla txew l ≥ 2 mìno. Sthn

3.1 Arq Metabol¸n sth Jewra Diataraq¸n 39prxh, to pleonèkthma th qr sh anallowtwn posot twn enai ìti sundèontai fusiologikme bajmwt metr sima megèjh, ìpw h enèrgeia kai h orm twn barutik¸n kumtwn. Tautìqro-na, aut h epilog egkutai ìti pijanè exart¸mene apì th bajmda posìthte apokleontaiek kataskeu [9.3.1 Arq Metabol¸n sth Jewra Diataraq¸nTa probl mata th Genik Jewra th Sqetikìthta mporoÔn na lujoÔn ete ap' eujea apì ti exis¸sei pedou, Gµν = 8πTµν , ete xekin¸nta apì ma arq metabol¸n sthn opoadialègoume ma katllhlh Lagkrasian . Sth sunèqeia, akolouj¸nta ton Moncrief [2, jajemeli¸soume th jewra twn barutik¸n kumtwn mèsw th jewra twn metabol¸n.'Estw L (ϕA; ϕA,µ) ma bajmwt posìthta pou sqhmatzetai apì ta bajmwt tanustikpeda ϕA kai ti merikè parag¸gou tou ϕA

,µ(µ = 0, 1, 2, 3; A = 1, . . . , N). H apathsh toolokl rwmaI =

∫

Ω

d4xL (ϕA; ϕA,µ) (3.1)na enai stsimo se aujarete metabolè tou ϕA, oi opoe exafanzontai sto sÔnoro ∂Ω touq¸rou Ω, katal gei se èna sÔsthma exis¸sewn sto eswterikì tou Ω, gnwstè w exis¸sei

Euler-Lagrange∂L

∂ϕA− ∂

∂xµ

[ ∂L

∂ϕA,µ

]= 0 (3.2)Oi exis¸sei (3.2) apoteloÔn ti exis¸sei knhsh tou dunamikoÔ sust mato . Efarmìzonta t¸ra th mèjodo tou Jacobi th deÔterh metabol mporoÔme na proume ma akìma arq metabol¸n, th opoa oi exis¸sei Euler-Lagrange perigrfoun ti mikrè diataraqè twnlÔsewn sto arqikì prìblhma.JewroÔme ma monoparametrik oikogèneia sunart sewn

ϕA(e) ≡ ϕA(xµ; e)kai upolgzoume to olokl rwma (3.1) gia ti sunart sei autè :I(e) =

∫

Ω

d4xL (ϕA(e); ϕA,µ(e)) (3.3)Paragwgzonta to I(e) w pro e parnoume

dI(e)

de≡ I ′(e) =

∫

Ω

d4x[FA(e)ϕ′A(e)] +

∫

Ω

∂

∂xµ

[ ∂L

∂ϕA,µ

ϕ′A(e)] (3.4)ìpou

FA(e) ≡ ∂L

∂ϕA− ∂

∂xµ

[ ∂L

∂ϕA,µ

]= 0 (3.5)kai ϕ′A(e) ≡ ∂ϕA

∂e(xµ; e).

3.1 Arq Metabol¸n sth Jewra Diataraq¸n 40Ma deÔetrh parag¸gish tou I(e) dneiI ′′(e) =

∫

Ω

d4x[FA(e)ϕ′′A(e)] +

∫

Ω

∂

∂xµ

[ ∂L

∂ϕA,µ

ϕ′′A(e)]

+ 2J(e) (3.6)ìpouJ(e) =

1

2

∫

Ω

d4x[ ∂2L

∂ϕA∂ϕBϕ′Aϕ′B + 2

∂2L

∂ϕA∂ϕB,µ

ϕ′Aϕ′B,µ +

∂2L

∂ϕA,µ∂ϕB

,ν

ϕ′A,µ ϕ′B

,ν

] (3.7)H apathsh ìtiI ′(0) ≡ I ′(e)|e=0 = 0 (3.8)gia aujareto ϕ′A(0) ≡ ∂ϕA

∂e|e=0, to opoo exafanzetai sto ìrio ∂Ω, upodhl¸nei ìti ta ϕA(0)upakoÔoun sti exis¸sei Euler-Lagrange

FA(0) = 0Upologzonta to I ′′(0) ≡ d2I(e)de2 |e=0 sthn perptwsh pou ta ϕA(0) upakoÔoun sti exis¸sei

Euler-Lagrange, parnoume apì thn (3.6)I ′′(0) = 2J(0) +

∫

Ω

d4x∂

∂xµ

[ ∂L

∂ϕA,µ

ϕ′′A(0)] (3.9)To olokl rwma J(0) enai deutèrou bajmoÔ w pro ti sunart sei ϕ′A(xµ; 0) me sun-telestè anlogou twn ϕA(xµ; 0). Oi exis¸sei Euler-Lagrange twn mikr¸n diataraq¸n japrokÔyoun apì thn apathsh to I ′′(0) na enai stsimo gia aujarete metabolè tou ϕ′A(xµ; 0),oi opoe exafanzontai sto ∂Ω, kai enai

[ ∂2L

∂ϕA∂ϕBϕ′B +

∂2L

∂ϕA∂ϕB,µ

ϕ′B,µ − ∂

∂xµ

( ∂2L

∂ϕA,µ∂ϕB

ϕ′B +∂2L

∂ϕA,µ∂ϕB

,ν

ϕ′B,µ

)]e=0

= 0 (3.10)Oi exis¸sei (3.10) enai isodÔname me∂FA(e)

∂e|e=0 = 0 (3.11)Ef' ìson oi exis¸sei (3.10) enai autè pou dnoun ti mikrè diataraqè twn lÔsewn

ϕ′A(xµ; 0), to I ′′(0) apotele to olokl rwma metabol¸n aut¸n.Me bsh ta parapnw enai eÔkolo t¸ra na gnei h metbash apì to Lagkrasianì formali-smì, ston Qamiltonianì. To olokl rwma (3.1) ètsi gnetaiI =

∫

Ω

d4x[πAϕA,0 − H (πA, πA,i; ϕ

A, ϕA,i )] (3.12)ìpou πA = ∂L

∂ϕA,0

h genikeumènh orm , suzug th genikeumènh suntetagmènh ϕA,kai x0 = t hqronik suntetagmènh kai xi(i = 1, 2, 3) oi qwrikè suntetagmène . H apathsh kai pli to I

3.2 Diataraqè twn Ken¸n kai Statik¸n Metrik¸n 41na enai stsimo odhge sti exis¸sei Hamilton

∂ϕA

∂t=

∂H

∂πA

− ∂

∂xi

( ∂H

∂πA,i

)

∂πA

∂t= −∂H

∂ϕA+

∂

∂xi

(∂H

∂ϕA,i

) (3.13)pou enai isodÔname twn exis¸sewn Euler-Lagrange, me th diafor ìti èqoume N diaforikè exis¸sei 2ou bajmoÔ sto Lagkranzianì formalismì en¸ ed¸ èqoume 2N diaforikè exis¸sei pr¸tou bajmoÔ. Me ton dio akrib¸ trìpo ìpw kai sthn perptwsh tou LagkranzianoÔ for-malismoÔ, jewr¸nta thn monoparametrik oikogèneia sunart sewn πA(xµ; e), ϕA(xµ; e) upo-logzonta ta I ′(0) kai I ′′(0) parnoume apì th sunj kh I ′(0) = 0 th (3.13) kai apì thn I ′′(0)na enai stsimo ti exis¸sei Hamilton twn mikr¸n diataraq¸n∂ϕ′A(0)

∂t=

∂

∂e

[∂H

∂πA

− ∂

∂xi

( ∂H

∂πA,i

)]|e=0

∂π′A(0)

∂t= − ∂

∂e

[∂H

∂ϕA− ∂

∂xi

(∂H

∂ϕA,i

)]|e=0 (3.14)3.2 Diataraqè twn Ken¸n kai Statik¸n Metrik¸nOi Arnomitt,Deser kai Misner(ADM) br kan ma morf tou oloklhr¸mato metabol¸nsthn Genik Jewra th Sqetikìthta , to opoo enai

I =

∫d4x[πijγij,0 − NH − NiH

i] (3.15)ìpouH = γ−1/2[πijπij −

1

2(πi

i)2] − γ1/2R(3) (3.16)

Hi = −2πij

|j (3.17)kai sto opoo ja erfarmostoÔn ta apotelèsmata th prohgoÔmenh paragrfou gia th melèthtwn mikr¸n diataraq¸n. H H enia h uperqamiltonian , ma genkeush th qamiltonian kaiH i h uperorm . H exagwg th sqèsh (3.15) bassthke sthn anlush th 3+1 ditmhsh pou anafèrjhke sto prohgoÔmeno keflaio, sthn pargrafo (2.6) [12. H 3+1 ditmhsheis qjh apì tou Arnomitt, Deser kai Misner w h plèon leitourgik mèjodo gia th lÔsh twnexis¸sewn Einstein

Gµν = 8πTµνH sumbatik ermhnea th pnw exswsh enai ìti doj sa th katanom th mza enèrgeia sto qwroqrìno mporoÔme na lÔsoume thn exswsh Einstein kai na broÔme th gew-metra tou qwroqrìnou. H jewra ìmw th gewmetrodunamik proseggzei diaforetik thn

3.2 Diataraqè twn Ken¸n kai Statik¸n Metrik¸n 42parapnw exswsh. Dnoume ta peda pou pargoun thn katanom th mza -enèrgeia kai ti qronikè metabolè tou kai dnoume thn 3-gewmetra tou q¸rou kai th qronik metabol tou, ìla gia ma stigm , kai lÔnoume gia thn 4-gewmetra tou qwroqrìnou ekenh th qronik stigm . Kai mìno tìte problèpoume me ti exis¸sei th gewmetrodunamik gia kje qronik stigm tìso th gewmetra tou qwroqrìnou ìso kai th ro th enèrgeia -mza gia ekeno toqwroqrìno.Epistrèfonta sthn exswsh (3.15) oi metablhtè (γij, N,Ni) sqetzontai me thn tertadi-stath metrik g(4)µν me ti sqèsei

γij ≡ g(4)ij

N ≡ (−g(4)00 )−1/2

Ni ≡ g(4)0i i, j = (1, 2, 3) (3.18)'Etsi h metrik γij enai h metrik tou trisdistatou q¸rou pou pèrnoume apì thn 3+1ditmhsh kai R(3) to bajmwtì Ricci pou kataskeuzetai apì th metrik aut .To N enai hsunrthsh qronik dibash (lapse function), pou ekfrzei th metabol tou idioqrìnou me-taxÔ th kat¸terh kai th an¸terh uperepifneia (trisdistato q¸ro ) pou qwrsame tontetradistato qwroqrìno. To Ni enai to dinusma qwrik metatìpish (shift vector) kaisusqetzei ti allagè twn suntetagmènwn anmesa sti uperepifneie . Ta πij enai h orm tou gewmetrodunamikoÔ pedou,pou enai h suzug orm twn genikeumènwn suntetagmènwn

gij tou gewmetrodunamikoÔ pedou, ki ekfrzei thn allag th drsh an stoiqei¸dh metabol th 3-gewmetra twn uperepifanei¸n tou q¸rou. Sundèetai me thn kanonik morf Kij, thnkampulìthta dhlad th exwgenoÔ gewmetra (ìpw o tanust tou Riemann ekfrzei thnkampulìthta th endogenoÔ gewmetra ) me th sqèshπij = −γ1/2[Kij − γijK l

l ] (3.19)H anabbash kai katabbash twn latinik¸n deikt¸n gnetai me th metrik γij kai h kjeth gramm ekfrzei sunallowth parag¸gish w pro thn dia metrik .Anexrthte metablhtè tou oloklhr¸mato (3.15) enai oi πij, γij, N kai N i, w pro taopoa parnoume ti metabolè kai dnoun èna sÔsthma exis¸sewn isodÔnamo me ti exid¸dei pedou tou Einstein. Pio sugkekrimèna, metabol w pro N kai Ni dnei tou paraktwperiorismoÔ H = 0 ,Hi = 0 (3.20)oi opooi epibllontai pnw sti arqikè timè twn πij kai γij. Suneq ikanopohsh th (3.20)makri apì thn arqik uperepifneia thretai apì ti tautìthte Bianchi.H deÔterh metabol tou I dnei ti exis¸sei twn mikr¸n diataraq¸n. Gia na melet soumeti diataraqè ma statik , adiatrakth metrik (h opoa ikanopoie ti exis¸sei pedoutou Einstein sto kenì) mporoÔme na jèsoume, afoÔ proume th deÔterh metabol ,πij = 0 , Ni = 0 (3.21)

3.2 Diataraqè twn Ken¸n kai Statik¸n Metrik¸n 43Oi sunart sei N kai γij enai anexrthte th qronik suntetagmènh kai upakoÔoun sti exis¸sei N|ij = NR

(3)ij , R(3) = 0 (3.22)H deÔterh metabol tou I,upologismènh se ma statik lÔsh sto kenì, dnei

J =

∫d4x

[pij ∂hij

∂t− NH

∗ − N ′iH

′i − N ′H

′] (3.23)ìpou

hij ≡ ∂γij

∂e|e=0

pij ≡ ∂πij

∂e|e=0

N ′ ≡ ∂N

∂e|e=0

N ′i ≡ ∂Ni

∂e|e=0 (3.24)kai

H′i = −2pij

|j (3.25)H

′ = −γ1/2[h|ijij − h

|i|i − hijR

ij(3)] (3.26)Epsh isqÔei∫

d3xNH∗ ≡ H

=

∫d3x

(Nγ1/2[pijpij −

1

2p2] +

1

2Nγ1/2

[1

2hij|kh

ij|k −

−hij|khik|j − 1

2h|ih

|i + 2h|ihijj + hhij

|ij − hhijRij(3)

]) (3.27)Sti exs¸sei (3.25)-(3.27) èqoume orseip = γijp

ij , h = γijhij (3.28)Metabol tou J w pro th diataragmènh sunrthsh olsjhsh N ′ kai to diataragmèno-dinusma allag N ′i dnei ti perioristikè sunj ke

H′ = H

′i = 0 (3.29)Metabol tou J w pro ta pij kai hij dnei ti exis¸sei tou Hamilton

∂hij

∂t=

δHT

δpij

∂pij

∂t= −δHT

δhij(3.30)

3.3 Perittè Diataraqè 44ìpouHT = H +

∫d3x[N ′

H′ + N ′

iH′i] (3.31)kai δ()

δ()ekfrzei th sunarthsioeid parag¸gish. Oi periorismo (3.29) prèpei na epibllontaisti arqikè timè twn pij kai hij.Sti epìmene paragrfou ja jewr soume diataraqè me peritt kai rtia parity th metrik Schwarzschild

ds2 = −(1 − 2M

r)dt2 + (1 − 2M

r)−1dr2 + r2(dθ2 + sin2 θdφ2) (3.32)ìpou ja jèsoume

x0 = t, x1 = r, x2 = θ, x3 = φ.Qrhsimopoi¸nta ta apotelèsmata aut th paragrfou ja exgoume ti exis¸sei twn Regge-Wheeler kai Zerilli, dnonta tou ìmw ma diaforetik shmasa.3.3 Perittè Diataraqè Xekinme me th morf twn Regge-Wheeler gia ti perittè diataraqè . H diataraq txew (l,m) enai

hij = h1(r, t)(e1)ij + h2(r, t)(e2)ij (3.33)thn opoi d¸same sto prohgoÔmeno keflaio sth sqèsh (2.58). Ta (e1)ij kai h2(r, t)(e2)ijdnontai apì ti sqèsei (2.59) kai (2.60) antstoiqa. Oi perittè diataraqè th sunrthsh qronik dibash N ′ enai mhdèn kai to diataragmèno dinusma qwrik metatìpish ekfrzetaiapì th sqèsh[N ′

i ] = h0(r, t)[0,− 1

sin θ

∂Y ml

∂φ, sin θ

∂Y ml

∂θ

] (3.34)ìpou sugkrnonta me ti (2.58) parathroÔme ìti ìntw N ′ = h00 kai N ′i = h0i, sÔmfwname th jewra. Sto olokl rwma metabol¸n mìno ta pragmatik mèrh twn (3.33) kai (3.34)ja qrhsimopoihjoÔn. 'Omw , afoÔ oi exis¸sei gia ta h0, h1 kai h2 enai anexrthte tou m,mporoÔme na jèsoume m = 0 kai na knoume ti sunart sei h0, h1 kai h2 kajar pragmatikè .Tìte kai oi (3.33) kai (3.34) enai kajar pragmatikè .'Opw èqoume dh dei oi metasqhmatismo bajmda th grammikopoihmènh jewra dnontaiapì th sqèsh

δhµν = ξµ;ν + ξν;µ (3.35)ìpou δhµν = hpaliìµν − hnèoµν kai ξµ èna aujareto dinusma kai me to (;) sumbolzetai h sunal-lowth pargwgo w pro thn tetradistath, adiatrakth metrik g(4)µν . 'Otan h adiatrakthmetrik enai statik , gia ta δhij isqÔei

δhij = ξi;j + ξj;i

= ξi|j + ξj|i (3.36)

3.3 Perittè Diataraqè 45ìpou h teleutaa isìthta isqÔei mìno gia statik metrik . Oi mìnoi metasqhmatismo bajm-da pou ephrezoun ti perittè diataraqè , enai auto pou pargontai apì peritt paritydianusmatik peda ki èqoun (ìpw edame sto prohgoÔmeno keflaio) th morf [ξi] = ξ(r, t)

[0,− 1

sin θ

∂Y ml

∂φ, sin θ

∂Y ml

∂θ

] (3.37)Sundizonta thn (3.36) me thn (3.37) parnoume, gia th metrik tou Schwarzschild, gia ti metabolè bajmda twn h1, h2

δh1 = ξ,r −2

rξ

δh2 = −2ξ (3.38)Orzonta dÔo nèe sunart sei diataraq¸n k1, k2 me ti sqèsei k1 = h1 +

1

2[h2,r −

2

rh2]

k2 = h2 (3.39)parathroÔme ìti ktw apì tou metasqhmatismoÔ bajmda (3.38)δk1 = 0

δk2 = −2ξ (3.40)Dhlad h k1 enai ma anallowth . Epiplèon, ef' ìson ìle oi perittè diataraqè ikanopoioÔnthn perioristik sunj kh (3.26), h k1 enai ma mh-periorismènh, anallowth posìthta th diataragmènh 3-gewmetra .Gia na proume ìmw ti exis¸sei Hamilton twn mikr¸n diataraq¸n, prèpei na broÔme kaithn èkfrash twn diataragmènwn orm¸n pij. Ef' ìson ta pij metasqhmatzontai w oi sunist¸se ma tanustik puknìthta , mporoÔme na ekfrsoume to γ−1/2pij w anptugma sfairik¸narmonik¸n peritt parity, ìpw akrib¸ kai me ta hij.'Etsiγ−1/2pij = p1(r, t)(e1)ij + p2(r, t)(e2)ij (3.41)Orzonta ti posìthte

p1 = 2l(l + 1)(1 − 2M

r

)−1

p1

p2 =1

2

l(l + 1)(l − 1)(l + 2)

r2(1 − 2M

r

)1/2p2 (3.42)'Eqoume

Θ ≡∫

d3x[pij

∂hij

∂t

]

=

∫d3x

[p1

∂h1

∂t+ p2

∂h2

∂t

] (3.43)

3.3 Perittè Diataraqè 46H antikatstash sth sqèsh (3.43) twn suntetagmènwn h1, h2 me ti k1, k2, apaite kai thnantikatstash twn p1, p2 me ti suzuge ormè twn k1, k2, ¸ste na ikanopoietai h sqèshΘ =

∫d3x

[π1

∂k1

∂t+ π2

∂k2

∂t

] (3.44)Me bsh ti (3.43) kai (3.39), o metasqhmatismì p1 = π1

p2 = π2 −1

2π1,r −

1

rπ1 (3.45)epalhjeÔei thn (3.44). 'Ara ta π1, π2 enai oi suzuge ormè twn k1, k2. Sti nèe metablhtè

(k1, k2; π1, π2) oi perioristikè sunj ke (3.29) gnontaipj

i|j =r2 sin θπ2

l(l + 1)

[0,− 1

sin θ

∂Y ml

∂φ, sin θ

∂Y ml

∂θ

]= 0 (3.46) aploÔstera

π2 = 0 (3.47)Enai t¸ra dunatìn na upologiste to olokl rwma metabol¸n gia perittè diataraqè an-tikajist¸nta ta hij kai pij sti exis¸sei (3.23) kai (3.27). Oloklhr¸nonta w pro ti gwniakè metablhtè kai metasqhmatzonta ti arqikè metablhtè me ti k1, k2, π1kai π2 me thbo jeia twn sqèsewn (3.39).(3.42) kai (3.45) parnoumeJ =

∫dt

∫dr

[π1

∂k1

∂t+ π2

∂k2

∂t− H

] (3.48)ìpou∫

drH ≡ HT

=1

l(l + 1)

∫dr

1

2π2

1 +2r2(1 − 2M

r)

(l − 1)(l + 2)

[π2 −

1

2π1,r −

1

rπ1

]2+

+1

2l(l + 1)

∫dr

(l − 1)(l + 2)

r2(1 − 2M

r)k2

1

− 2

∫dr[h0π0] (3.49)Metabol tou J w pro h0 dnei ton periorismì twn arqik¸n sunjhk¸n (3.47), opoo diathretai sto qrìno afoÔ h k2 enai kuklik . DouleÔonta me ti exis¸sei tou Hamiltonparnoume ma kumatik exswsh gia to k1 h opoa mpore na grafe

∂2Q

∂t2− ∂2Q

∂r∗2+

(1 − 2M

r

)[ l(l + 1)

r2− 6M

r3

]Q = 0 (3.50)ìpou

Q ≡ k1

r

(1 − 2M

r

) (3.51)

3.4 'Artie Diataraqè 47kair∗ ≡ r + 2M ln

(2M

r− 1

) (3.52)H (3.50) enai h gnwst ma exswsh twn Regge-Wheeler. Antikajist¸nta to k1 apì thn(3.39)Q =

1

r

(1 − 2M

r

)[h1 +

1

2(h2,r −

2

rh2)] (3.53)H (3.53) tautzetai me thn exswsh twn Regge-Wheeler sth bajmda Regge-Wheeler,dhlad gia

h2 = 0, sthn opoaQ =

1

r

(1 − 2M

r

)h∗

1 (3.54)ìpou o astersko dhl¸nei ìti to h1(pou exarttai apì th bajmda) enai ed¸ upologismènosth bajmda Regge-Wheeler. 'Omw h Q th (3.54) enai exarthmènh apì th bajmda. Antijètw h Q th (3.51) enai anexrthth th bajmda ,prgma pou thn knei polÔ pio eÔqrhsth, afoÔikanopoie thn dia exswsh, (3.50), se kje bajmda.3.4 'Artie Diataraqè Gia ti 'Artie diataraqè ,to anptugma se sfairikè armonikè dneihij = h1(f1)ij +

H2

1 − 2Mr

(f2)ij + r2K(f3)ij + r2G(f4)ij (3.55)me diataragmènh sunrthsh qronik dibash kai dinusma qwrik metatìpish N ′ = −1

2

(1 − 2M

r

)1/2

H0(t, r)Yml (3.56)

[N ′i ] =

[H1(t, r)Y

ml , h0(t, r)

∂Y ml

∂θ, h0(t, r)

∂Y ml

∂φ

] (3.57)kai ìpou ta f1, f2, f3, f4 dnontai apì ti (2.62)-(2.65) tou kefalaou 2. Epsh na toniste giama akìma for ìti oi sunart sei h0, h1 enai diaforetikè apì autè twn peritt¸n diataraq¸n.Oi metasqhmatismo bajmda twn hij dnontai apì ti (3.36) ìpou mìno dianÔsmata me rtiaparity ta ephrezoun ki èqoun th morf

[ξi] =[ξ1Y

ml , ξ0

∂Y ml

∂θ, ξ0

∂Y ml

∂φ

] (3.58)DouleÔonta ìpw kai sthn prohgoÔmenh pargrafo,mporoÔme na proume gia ti auja-rete sunart sei ξ1, ξ2 antstoiqe sqèsei gia ti metablhtè twn H2, h1, K,G, ìpw sthn(3.38), mìno pou h morf tou enai algebrik pio perplokh. Ef' ìson t¸ra uprqoun dÔoaujarete sunart sei pou emplèkontai sth bajmda, ja enai dunatì na kataskeuastoÔn dÔo

3.4 'Artie Diataraqè 48mìno sundiasmo apì ti H2, h1, K kai G, pou na enai anallowtoi se metasqhmatismoÔ baj-mda . Eisgoume ant twn tessrwn sunart sewn H2, h1, K kai G tèsseri nèe k1, k2, k3, k4pou orzontaiG = k3

h1 = k4

K = k1 −1

re−2λ[r2k3,r − 2k4]

H2 = 2e−2λk2 + rk1,r + (1 + rl,r)k1 −

−e−λ ∂

∂r[r2e−λk3,r − 2e−λk4] (3.59)ìpou

e−2λ ≡(1 − 2M

r

)Oi metasqhmatismo (3.59) orsthkan ètsi ¸ste ta k1 kai k2 na enai anallowte δk1 = δk2 = 0 (3.60)Oi allagè sth bajmda twn k3 kai k4 enai

δk3 =2ξ0

r2

δk4 = ξ1 + r2 ∂

∂r

(ξ0

r2

) (3.61)O periorismì H ′ = 0, o opoo gia ma statik adiatrakth metrik aplopoietai ston[−γ1/2R(3)]

′ = 0parnei th morf −γ1/2[h

|ijij − h

|i|i − hijR

ij(3)] = 0 (3.62)O parapnw periorismì sunart sei twn k1, k2, k3, k4 gnetai ∂

∂r[4re−4λk2 + l(l + 1)rk1] + l(l + 1)[2e−2λk2 +

+(1 + rλ,r)k1](Y m

l

r2

)= 0 (3.63)H morf tou periorismoÔ odhge se ma teleutaa allag twn metablht¸n apì k1 kai k2 se

q1, q2, pou orzontai w q1 = 4re−4λk2 + l(l + 1)rk1

q2 =∂

∂r[4re−4λk2 + l(l + 1)rk1] + l(l + 1)[2e−2λk2 + (1 + rλ,r)k1] (3.64)

3.4 'Artie Diataraqè 49kai gia na uprqei èna omoiìmorfo sumbolismì orzoumeq3 = k3, q4 = k4 (3.65)Me thn allag aut petÔqame h morf tou periorismoÔ (3.63) na aplopoietai sthn

q2 = 0 (3.66)ìpou q1 enai h mình mh-periorismènh, anallowth sunrthsh pou mpore kane na kataskeuseiapì ta hij sthn perptwsh twn rtiwn diataraq¸n. Oi metasqhmatismo (3.64) enai antistrè-yimoi sthn perioq tou q¸rou ìpou r > 2M kai oi antstrofoi metasqhmatismo enaik1 =

1

Λ

[q1

r−

2(1 − 2M

r

)

l(l + 1)(q2 − q1,r)

]

k2 =1

2Λ(1 − 2M

r

)[q2 − q1,r −

q1

r

(1 − 3M

r

)

(1 − 2M

r

)] (3.67)ìpou

Λ ≡ l(l + 1) − 2 +6M

rGia ton upologismì twn orm¸n pij, anaptÔsoume ton tanust puknìthta γ1/2pij sti sfai-rikè armonikè , w ex γ1/2pij = Ph(r, t)(f1)ij + PH(r, t)

(1 − 2M

r

)−1

(f2)ij +

+r2PK(r, t)(f3)ij + r2PG(r, t)(f4)ij (3.68)Apì ti (3.55) kai (3.68) parnoumeΘ ≡

∫d3x[pij ∂hij

∂t]

=

∫dr

[r2

(1 − 2M

r2

)]−1/2

PHH2,0 + 2l(l + 1)(1 − 2M

r2

)1/2

Phh1,0 +

+r2(1 − 2M

r2

)−1/2

[2PK − l(l + 1)PG]K,0 +

+l(l + 1)[−PK + (l(l + 1) − 1)PG]G,0

] (3.69)Oi suzuge ormè π1, π2, π3, π4 twn q1, q2, q3, q4 ja brejoÔn apì thn apathsh na ikanopoioÔnth sqèshΘ =

∫dr[π1q1,0 + π2q2,0 + π3q3,0 + π4q4,0] (3.70)Antikajist¸nta sthn (3.69) tou metsqhmatismoÔ twn (H2, h1, K,G) se (q1, q2, q3, q4) (3.59)kai (3.64) kai exis¸nonta aut n me th (3.70) parnoume tou metasqhmatismoÔ pou sundèoun

3.4 'Artie Diataraqè 50ta dÔo set twn sunart sewn twn orm¸n pα kai πα. Sti nèe metablhtè orm¸n, oianexrthtoi periorismo twn orm¸n gnontaiπ

|jrj = −1

2

(1 − 2M

r

)−1

π4 sin θY ml = 0 (3.71)kai

π|j

θj =[ 1

2l(l + 1)(r2π4),r −

1

l(l + 1)π3

]sin θ

∂Y ml

∂θ= 0 (3.72) pio apl

π3 = π4 = 0 (3.73)Antikajist¸nta ti (3.55)-(3.57) kai thn (3.68) sth genik sqèsh (3.23) kai epanekfr-zonta to apotèlesma sunart sei twn nèwn metablht¸n (π1, . . . , π4) kai (q1, . . . , q4), parnoumeto olokl rwma metabol¸n gia ti diataraqè rtia parity

J =

∫dtdr

[ 4∑

a=1

πaqa,0 − H

] (3.74)ìpou∫

drH ≡ HT

=

∫dr

l(L + 1)(1 − 2M

r

)

2(l − 1)(l + 2)Λ2π2

1 +π2

4

2l(l + 1)−

−2

r

(1 − 2M

r

)π4[r(π1 − π2,r) + (1 + rλ,r] +

+

(1 − 2M

r

)

r2(l − 1)(l + 2)

[ 2π23

l(L + 1)+ 2π3[rΛπ1 + π2(rΛ),r]

]+

+

∫dr

(l − 1)(l + 2)(1 − 2M

r

)

2l(l + 1)Λ2(q2 − q1,r)

2 −

−(l − 1)(l + 2)

2rΛ2q1q2 +

(l − 1)2(l + 2)2

2r2Λ3q21 −

−Mq2(q2 − q1,r)

rΛl(l + 1)− Mq2

2r

(rq3,r −

2

rq4

)+

+

∫dr

H0q2

2+ H1π4 + h0

[2π3

r2− 1

r2(r2π4),r

] (3.75)Metabol tou J w pro ti sunart sei H0, H1 kai h0 dnei ti perioristikè exis¸sei q2 = 0

π4 = 0

0 =2π3

r2− 1

r2(r2π4),r (3.76)

3.4 'Artie Diataraqè 51En¸ h qamiltonian èqei ma perplokh morf , pollo ìroi th exafanzontai ìtan oi perio-rismo epibllontai. ApodeiknÔetai ìti ìtan oi periorismo ikanopoioÔntai, h qamiltonian enaima jetik sunrthsh twn q1 kai π1, h opoa mhdenzetai mìno ìtan autè oi dÔo sunart sei mhdenzontai.Oi exis¸sei tou Hamilton gia q1 kai π1 dnoun∂2Q

∂t2− ∂2Q

∂r∗2+

(1 − 2M

r

) 1

Λ2

[72M3

r5−

−12M

r3(l − 1)(l + 2)

(1 − 3M

r

)]+

(l − 1)(l + 2)l(l + 1)

r2Λ

Q = 0 (3.77)ìpou

Q ≡ q1

Λ(3.78)kai

r∗ = r + 2M ln( r

2M− 1

)H exswsh (3.77) enai h gnwst exswsh tou Zerilli. 'Etsi h Q, pou enai anallowth ,upakoÔei sthn dia exswsh ìpw h exarthmènh apì th bajmda sunrthsh Z tou Zerilli.LÔnonta ti exis¸sei tou Hamilton qA,0 = δHT

δπAgia th metablht π4 ki ekfrzonta to apo-tèlesma sti metablhtè twn Regge-Wheeler gia th bajmda Regge-Wheeler(pou sumbolzetaime ènan astersko), prokÔptei

π4

l(l + 1)= −H∗

1 +2

l(l + 1)

∂

∂t

[r2K∗, r − rH∗

2 + r(1 − 3M

r

)(1 − 2M

r

)−1

K∗]

= 0 (3.79)Ekfrzonta th sunrthsh Q sti metablhtè Regge-Wheeler prokÔpteiQ =

rl(l + 1)

ΛK∗ −

2r2(1 − 2M

r

)

Λ

[K∗

,r +1

rK∗ − MK∗

r2(1 − 2M

r

) − H∗2

r

] (3.80)Jewr¸nta ma lÔsh me qronik exrthsh e−iωt, sundizonta ti (3.79) kai (3.80), prokÔpteiQ =

1

Λ

[− i

ω

(1 − 2M

r

)l(l + 1)H∗

1 + rl(l + 1)K∗] (3.81)pou enai isodÔnamh me thn posìthta gia thn opoa o Zerilli ex gage thn exswsh tou.

3.5 Eustjeia Melan¸n Op¸n 523.5 Eustjeia Melan¸n Op¸n'Ena shmantikì apotèlesma tìso twn ergasi¸n twn Regge-Wheeler kai Zerilli ìso kai touMoncrief tan h katstash th eustjeia twn melan¸n op¸n. S' aut n thn pargrafo jamelethje h eustjeia twn melan¸n op¸n, akolouj¸nta ton Moncrief, kaj¸ h mèjodo tou,an kai ftnei sta dia sumpersmata me tou prohgoÔmenou enai pio komy , kaj¸ èqei kai topleonèkthma na baszetai se annallowte ktw apì metasqhmatismoÔ bajmida .Gia th melèth th eustjeia enai qr simo na ekfraste h qamiltonian sunart sei tou Qkai th suzugoÔ orm tou P = Λπ1. Uiojet¸nta kpoiou arijmhtikoÔ pargonte apotou epanorismoÔ

P → P[(l − 1)(l + 2)

l(l + 1)

]1/2

Q → Q[ l(l + 1)

(l − 1)(l + 2)

]1/2kai apalofonta ènan epifaneiakì ìro, pou prokÔptei apì thn olokl rwsh kat pargonte ,prokÔptei∫

drH ∗ ≡ H∗

=1

2

∫dr

(1 − 2M

r

)P 2 +

(1 − 2M

r

)(Q,r)

2 +

+(Q

Λ

)2[72M3

r5+ (l − 1)(l + 2)

[36M2

r4+ (l − 1)(l + 2)

6M

r3+

+1

r2l(l + 1)(l − 1)(l + 2)

]] (3.82)ìpou o astersko dhl¸nei ìti oi periorismo kai o epifaneiakì ìro èqoun paraleifje. Hqronik pargwgo th qamiltonian puknìthta H ∗ enaiH

∗,0 =

∂

∂r

[(1 − 2M

r

)2

PQ,r

]

=∂

∂r[Q,0Q,r] (3.83)Oloklhr¸nonta thn parapnw exswsh apì r → 2M èw r → ∞

dH∗

dt= Q,0Q,r∗|r

∗=+∞r∗=−∞ (3.84)pou ekfrzei to dH∗

dtse ìrou ma ro sto sÔnoro.Oi lÔsei th exswsh tou Zerilli (3.77) pou antistoiqoÔ se kajar exerqìmena kÔmata gia

r → ∞ kai kajar eiserqìmena gia r → 2M èqoun thn asumptwtik morf Q = f(t − r∗), r → +∞Q = g(t + r∗), r∗ → −∞ (3.85)

3.5 Eustjeia Melan¸n Op¸n 53Apì thn (3.85) kai (3.84) prokÔptei ìti gia ma lÔsei pou upakoÔei sti oriakè sunj ke exerqìmenwn kumtwn, isqÔei dH∗

dt≤ 0.Ma astaj lÔsh, kanonik¸n trìpwn talntwsh , th exswsh Zerilli ( th exswsh

Regge-Wheeler) enai ma pou èqei qronik exrthsh eikt kai gia thn opoa h suqnìthta k èqeiarnhtikì migadikì mèro . Tètoie lÔsei megal¸noun ekjetik me to qrìno. Ma tètoia lÔshpou upakoÔei ti sunoriakè sunj ke gia exerqìmena kÔmata (3.85) èqei thn asumptwtik morf Q = Ae(ikt−ikr∗), r → +∞Q = Be(ikt+ikr∗), r∗ → −∞ (3.86)ìpou A kai B stajerè . AfoÔ to k èqei èna arnhtikì migadikì mèro , h lÔsh pèftei ekjetikkaj¸ r∗ → ±∞. Upologzonta to H∗ gia ta pragmatik mèrh twn sunart sewn Q,PprokÔptei apì thn (3.84)dH∗

dt= 0 ⇒ H∗ = C = stajer (3.87)Par' ìla aut to H∗ ìpw dnetai apì thn (3.82) enai èna jroisma mh-arnhtik¸n ìrwn, pougia ma astaj lÔsh ja auxanìtan ekjetik me to qrìno. Gi' autì h H∗ den mpore na enaistajer , ektì ki an mhdenzetai ki autì gneti mìno ìtan kai oi diataraqè mhdenzontai. 'Etsiastaje lÔsei pou upakoÔoun sti oriakè sunj ke exerqìmenwn kumtwn den uprqoun.Anlogh apìdeixh mpore na doje kai gia ti diataraqè rtia parity.H ìlh diadikasa eÔresh anallowtwn posot twn se matasqhmatismoÔ bajmda enai du-nat an kai mìno an h adiatrakth metrik èqei ti katllhle summetre ktw apì tètoiou metasqhmatismoÔ suntetagmènwn (ìpw sumbanei me to qwrìqrono Schwarzchsild) kai èna anallowto apì bajmde formalismì twn exis¸sewn Einstein gia ti diataraqè enì geni-koÔ qwroqrìnou, den enai pijanì . Wstìso, mia kai kje asumptwtik eppedo qwroqrìno mpore genik na tautiste me ènan qwrìqrono Schwarzchsild se arket megle apostsei ,èna anallowto se bajmde formalismì mpore na enai èna qr simo ergaleo gia thn exagw-g fusik¸n plhrofori¸n gia ta barutik kÔmata pou pargontai se èna arijmhtik exelximo,asumptwtik eppedo qwroqrìno.

Keflaio 4Barutik Aktinobola apìPeristrefìmenou Astère Sta prohgoÔmena autì pou kname tan na parajèsoume kpoia kommtia th jewra twnbarutik¸n kumtwn, me skopì na ta efarmìsoume argìtera se pragmatik montèla th upologi-stik sqetikìthta . Sto keflaio autì ja exgoume ta barutik kÔmata pou ekpèmpontai apìarg peristrefìmenou sqetikistikoÔ astère ,ìpw enai oi leuko nnoi, oi astère netronwnkai oi astère polÔ meglh mza (supermassive stars).To prìblhma diafèrei apì aut pou antimetwpsthkan sta prohgoÔmena dÔo keflaia, ka-j¸ o qwrìqrono èxw apì tètoia astèria den enai kajar Schwarzschild, afoÔ oi peristrofè prosjètoun kai mh-sfarik kommtia sth metrik . Arqik ja analuje h mèjodo pou ja qrh-simopoi soume gia thn exagwg twn barutik¸n kumtwn apì tètoie metrikè . Sth sunèqeiaja efarmìsoume aut th mèjodo gia na broÔme th suneisfor twn mh sfairik¸n ìrwn sthnekpempìmenh barutik aktinobola twn arg peristrefìmenwn astèrwn.4.1 Exagwg Kumatomorf¸n se mh-sfairikoÔ Qwro-qrìnou Uprqei èna genikì prìblhma sthn upologistik sqetikìthta ìson afor thn exagwg twnekpempìmenwn barutik¸n kumtwn. Autì ofeletai sto ìti me tou k¸dike ma prospajoÔmena lÔsoume ti pl rei mh-grammikè exis¸sei tou Einstein gia ti metrikè sunart sei twnqwroqrìnwn pou douleÔoume, en¸ h barutik aktinobola melettai me ti grammikè exis¸sei [6. An ki èqoun protaje arketè mèjodoi gia thn exagwg kumatomorf¸n barutik¸n kumtwnsthn upologistik sqetikìthta, eme ja akolouj soume th mèjodo pou perigrfetai apì tou Abrahams,Bernstein,Hobill,Seidel kai Smarr [6. H mèjodo aut qrhsimopoie ma teqnik exagwg anallowtwn apì metsqhmatismoÔ bajmda kumatomorf¸n (ìpw perigrfthke stoprohgoÔmeno keflaio) kai mpore, jewrhtik, na efarmoste se kje qwroqrìno pou perièqeima apomonwmènh phg barutik¸n kumtwn. H teqnik aut baszetai sto gegono ìti o qw-roqrìno pou peribllei èna diataragmèno astèri melan op , enai basik sfairikì . An hmetaferìmenh enègeia apì ta barutik kÔmata enai mikr se sÔgkrish me th mza tou astèra

4.1 Exagwg Kumatomorf¸n se mh-sfairikoÔ Qwroqrìnou 55th melan op , ìpw sumbanei sto sugkekrimèno prìblhma pou ja melet soume, to ìlo sÔ-sthma mpore na jewrhje san èna ìpou oi diataraqè diaddontai se èna sfairikì adiatraktoqwroqrìno. 'Etsi mporoÔme na exetsoume to sÔsthma apì th skopi th jewra diataraq¸ntwn sfairik¸n qwroqrìnwn, pou enai h gnwst jewra apì ta dÔo prohgoÔmena keflaia.Sth sunèqeia ja qrhsimopoi soume to formalismì pou protenoun oi Allen,Camarda, Seidel[15. JewroÔme th metrik gαβ tou qwroqrìnou na ekfrzetai se èna genikì sÔsthma sunte-tagmènwn (t, η, θ, φ), ìpou η enai h aktinik sunist¸sa. Aut h metrik mpore na prokÔyeiete apì analutikè lÔsei twn exis¸sewn Einstein, ete apì arijmhtikè proswmei¸sei . Seautì to sÔsthma suntetagmènwn mporoÔme pnta na gryoume th metrik w gαβ = g

sfαβ + hαβ (4.1)ìpou g

sfαβ enai o sfairik summetrikì ìro l = 0 , se èna anptugma th gαβ se sfairikè armonikè , kai hαβ perièqei ta an¸tera polÔpolla pou perigrfoun ti apoklsei apì tonsfairik summetrikì ìro l = 0. Oi ìroi hαβ ikanopoioÔn, mèqri kai ìrou deutèra txh , ti grammikè exis¸sei pedou, grammikopoihmène sthn g

sfαβ . 'Etsi an oi ìroi hαβ einai mikro,dhlad an hαβ ≪ g

sfαβ , mporoÔn na proseggistoÔn ikanopoihtik apì ma lÔsh twn grammik¸nexis¸sewn. H metrik tou sfairikoÔ mèrou dnetai apì th sqèsh

gsfαβ =

−N2(t, η) 0 0 00 A2(t, η) 0 00 0 R2(t, η) 00 0 0 R2(t, η) sin2 θ

(4.2)Epìmeno b ma enai na perigryoume ti diataraqè hαβ, pou perigrfoun thn apìklish apìth sfairik summetra, kaj¸ autè antiproswpeÔoun to mh-sfairikì mèro th metrik kiepomènw perièqoun ìle ti plhrofore gia kje barutikì kÔma pou uprqei sto sÔsthma.Autì to knoume anaptÔssonta se sfairikè armonikè :

htt = −N2H(lm)0 Ylm

htη = H(lm)1 Ylm

htθ = h(lm)0 Ylm,θ − c

(lm)0 Ylm,φ sin θ

htφ = h(lm)0 Ylm,φ − c

(lm)0

Ylm,θ

sin θ

hηη = A2H(lm)2 Ylm

hηθ = h(lm)1 Ylm,θ − c

(lm)1

Ylm,φ

sin θ

hηφ = h(lm)1 Ylm,φ + c

(lm)1 sin θYlm,θ

hθθ = R2K(lm)Ylm + R2G(lm)Ylm,θθ +

+c(lm)2

1

sin θ(Ylm,θφ − cot θYlm,φ

hθφ = R2G(lm)(Ylm,θφ − cot θYlm,φ) −

4.1 Exagwg Kumatomorf¸n se mh-sfairikoÔ Qwroqrìnou 56−c

(lm)2

sin θ

2(Ylm,θθ − cot θYlm,θ −

1

sin2 θYlm)

hφφ = R2K(lm) sin2 θYlm +

+R2G(lm)(Ylm,φφ + sin θ cos θYlm,θ) −−c

(lm)2 sin θ(Ylm,θφ − cot θYlm,φ) (4.3)Sta parapnw uprqoun eft sunart sei rtia parity (H

(lm)0 , H

(lm)1 , h

(lm)0 , h

(lm)1 , h

(lm)2 , K(lm),

G(lm)) kai trei peritt parity (c(lm)0 , c

(lm)1 , c

(lm)2 ), oi opoe enai sunart sei mìno twn t kai η.Sti exis¸sei (4.3) ennoetai jroish w pro l,m (l ≥ 1,−l ≤ m ≥ l).'Eqonta to anptugma th olik metrik gαβ, qrhsimopoi¸nta thn orjogwniìthta twnsfairik¸n armonik¸n, mporoÔme na gryoume ti sunist¸se th adiatrakth sfairik me-trik g

sfαβ me katllhle oloklhr¸sei sth 2-sfara:

N2 = − 1

4π

∫gttdΩ

A2 =1

4π

∫gηηdΩ

R2 =1

8π

∫ (gθθ +

gφφ

sin2 θ

)dΩ (4.4)MporoÔme na proume apì th genik metrik gαβ ti sunart sei twn diataraq¸n (c0, c1, c2,

H0, h1, h2h0, h1, K,G). Oi posìthte autè brskontai probllonta thn olik metrik sthndiatargmènh metrik , me oloklhr¸sei se ma suntetagmènh 2-sfara pou peribllei thn phg .'Etsi oi posìthte autè dnontai apì ti sqèsei c(lm)0 (t, η) =

1

l(l + 1)

∫1

sin θ(gtφY

∗lm,θ − gtθY

∗lm,φ)dΩ

c(lm)1 (t, η) =

1

l(l + 1)

∫1

sin θ(gηφY

∗lm,θ − gηθY

∗lm,φ)dΩ

c(lm)2 (t, η) = − 2

l(l + 1)(l − 1)(l + 2)

∫ (− 1

sin2 θgθθ +

1

sin4 θgφφ

)(sin θY ∗

lm,θφ −

−Y ∗lm,φ) +

1

sin θgθφ(Y

∗lm,θθ − cot θY ∗

lm,θ −1

sin2 θYlm,φφ)

dΩ

h(lm)0 (t, η) =

1

l(l + 1)

∫ gtθY

∗lm,θ +

1

sin2 θgtφY

∗lm,φ

dΩ

h(lm)1 (t, η) =

1

l(l + 1)

∫ gηθY

∗lm,θ +

1

sin2 θgηφY

∗lm,φ

dΩ

H(lm)0 (t, η) =

1

N2

∫gttY

∗lmdΩ

H(lm)1 (t, η) =

∫gtηY

∗lmdΩ

H(lm)2 (t, η) =

1

A2

∫gηηY

∗lmdΩ

4.1 Exagwg Kumatomorf¸n se mh-sfairikoÔ Qwroqrìnou 57G(lm)(t, η) =

1

R2l(l + 1)(l − 1)(l + 2)

∫ (gθθ −

gφφ

sin2 θ

)(Y ∗

lm,θθ − cot θY ∗lm,θ −

− 1

sin2 θYlm,φφ

)+

4

sin2 θgθφ(Ylm,θφ − cot θYlm,φ)

dΩ

K(lm)(t, η) =l(l + 1)

2G(lm)(t, η) +

1

2R2

∫ (gθθ +

1

sin2 θgφφ

)Y ∗

lmdΩ (4.5)Oi sunart sei (4.5) twn diataraq¸n th metrik , upologismène kat' eujean apì th me-trik , exart¸ntai apì th bajmda. MporoÔme ìmw na qrhsimpoi soume th jewra diataraq¸nanallowtwn gia sfairikoÔ qwroqrìnou , ìpw anaptÔqjhke apì ton Moncrief [2 k.a. kaiexetsame sto prohgoÔmeno keflaio, me skopì na broÔme lle sunart sei diataraq¸n sthjèsh twn (4.5), oi opoe ìmw mènoun anephrèaste apì apeirostoÔ metasqhmatismoÔ sun-tetagmènwn. 'Eqei deiqje apì ton Abrahams ìti èna tètoio formalismì enai dunatìn naqrhsimopoihje sthn upologistik sqetikìthta gia na apomon¸sei ti rtia kai peritt parityanallowte sunart sei twn barutik¸n kumtwn. Sthn perptwsh pou to sfairikì mèro th metrik enai se suntetagmène Schwarzschild, h kataskeu twn anallowtwn sunart sewn jad¸sei dÔo sunart sei Q×lm(t, r), Q+

lm(t, r) oi opoe upakoÔoun ti exis¸sei Regge-Wheelerkai Zerilli antstoiqa, pou perigrfoun th didosh barutik¸n kumtwn se ènan adiatraktosfairikì qwrìqrono Schwarzschild. Sthn perptwsh pou to sfairikì mèro th metrik ma den enai statikì, all èqei ma qronik exrthsh, mporoÔn na kataskeuastoÔn anallowte sunart sei polÔpolla me l ≥ 2 kai sth sunèqeia na breje h exèlixh aut¸n twn sunart sewnqrhsimopoi¸nta ti anallowte grammikopoihmène exis¸sei ,pou katal goun se genikeÔsei twn sunart sewn kai exis¸sewn Regge-Wheeler kai Zerilli.Sthn efarmog pou ja melet soume mporoÔme na akolouj soume th suntag tou Moncrief,ìpw knoun oi Allen et al [15, kai na orsoume nèe anallowte sunart sei diataraq¸n, oiopoe gia perittè diataraqè ja d¸soun th sunrthsh peritt parity Q×lm

Q×lm =

√2(l + 2)

(l − 2)

[c(lm)1 +

1

2

(∂rc

(lm)2 − 1

2c(lm)2

)]S

r(4.6)kai gia ti rtie diataraqè thn rtia parity sunrthsh Q+

lm

Q+lm =

1

Λ

√2(l + 2)(l − 1)

(l − 2)

(l(l + 1)S(r2∂rG

(lm) − 2h(lm)1 ) +

+2rS(H(lm)2 − r∂rK

(lm)) + ΛrK(lm)) (4.7)Sti dÔo parapnw sqèsei qrhsimopoi same tou orismoÔ

S = 1 − 2M

rkaiΛ = (l − 1)(l + 2) +

6M

r

4.2 H Metrik twn Hartle-Thorne 58'Opw anafèrame kai parapnw oi dÔo exis¸sei Q×lm kai Q+

lm upakoÔn sti exis¸sei Regge-Wheeler kai Zerilli antstoiqa, dhlad :

(∂2t − ∂2

r∗)Q×lm + VRW (r)Q×

lm = 0 (4.8)(∂2

t − ∂2r∗)Q

+lm + VZ(r)Q+

lm = 0 (4.9)ìpou VRW (r) enai to dunamikì Regge-Wheeler, pou dnetai apì th sqèshVRW (r) = S

[ l(l + 1)

r2− 6M

r3

] (4.10)kai VZ(r) to dunamikì Zerilli

VZ = S[ 1

Λ2

(72M3

r5− 12M

r3(l − 1)(l + 2)

(1 − 3M

r

))+

+l(l − 1)(l + 1)(l + 2)

r2Λ

] (4.11)kair∗ = r + 2M ln

( r

2M− 1

)H parapnw doulei mpore na qrhsimopoihje me dÔo trìpou . Pr¸ton, me th mèjodo au-t mporoÔme na exgoume kumatomorfè apì ma arijmhtik proswmewsh se ma peperasmènhaktna. DeÔteron, qrhsimeÔei sto na d¸sei arqik dedomèna gia ti grammikopihmène kumatikè exis¸sei pou d¸jhkan arqik apì tou Regge-Wheeler [1 kai Zerilli [3 gia peritt kai rtiaparity antstoiqa.Autè oi exis¸sei mporoÔn na qrhsimopoihjoÔn gia th sÔgkrish me apotelè-smata pou prokÔptoun me thn pl rh mh-grammik exèlixh twn qwroqrìnwn twn melan¸n opwn.Sthn efarmog ma ja akolouj soume ton pr¸to drìmo.4.2 H Metrik twn Hartle-ThorneSth sunèqeia ja melet soume th suneisfor sth barutik aktinobola twn mh-sfairik¸nìrwn se ma metrik pou perigrfei arg peristrefìmenou sqetikistikoÔ astère . H metrik aut eis qjei to 1968 apì tou James Hartle kai Kip Thorne [8 gia astère pou peristrèfontaiarg kai san stere s¸mata (h ènnoia tou stereoÔ s¸mato afor mìno thn peristrof tou kiìqi thn katstash th Ôlh tou). H arg peristrof allzei th morf th metrik ,h opoaqwr thn peristrof enai h metrik tou Schwarzschild, prosjètonta ìrou peristrof sti diag¸nie sunist¸se th metrik kai prosjètonta mh diag¸niou ìrou sth metrik . Hanlush ègine me akrbeia mèqri kai deutèra txh w pro th gwniak taqÔthta. 'Etsi tobarutikì pedo sto exwterikì enì arg peristrefìmenou astèra enai

ds2 = −(1 − 2M

r+

2J2

r4

)1 + 2

[ J2

Mr3

(1 +

M

r

)+

+5

8

Q − J2/M

M3Q2

2

( r

M− 1

)]P2(cos θ)

dt2

4.2 H Metrik twn Hartle-Thorne 59+

(1 − 2M

r+

2J2

r4

)−11 − 2

[ J2

Mr3

(1 − 5M

r

)+

+5

8

Q − J2/M

M3Q2

2

( r

M− 1

)]P2(cos θ)

dr2

+r2(1 + 2

⟨− J2

Mr3

(1 +

2M

r

)+

5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2×

×Q12

( r

M− 1

)− Q2

2

( r

M− 1

)⟩P2(cos θ)

)×

×

dθ2 + sin θ[dφ −

(2J

r3

)dt

]2 (4.12)H parapnw metrik enai gnwst me to ìnoma metrik twn Hartle-Thorne. SÔmfwna me thnprohgoÔmenh pargrafo mporoÔme na gryoume th metrik (4.12) sth morf th sqèsh (4.1),ìpou to stsimo sfairikì mèro enai h metrik tou Schwarzschild

gsfαβ =

−(1 − 2M

r

)0 0 0

0(1 − 2M

r

)−1

0 0

0 0 r2 00 0 0 r2 sin2 θ

(4.13)sun tou ìrou pou an koun sthn peristrof kai gia autì ja anaferìmaste se autoÔ w toperistrofikì mèro , pou enaigpertt = −2

J2

r4− 2

(1 − 2M

r+ 2

J2

r4

)[ J2

Mr3

(1 +

M

r

)+

+5

8

Q − J2/M

M3Q2

2(r

M− 1)

]P2(cos θ) + r2 4J2

r6sin2 θ + 2r2

⟨− J2

Mr3

(1 +

2M

r

)+

+5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2Q1

2(r

M− 1) − Q2

2(r

M− 1)

⟩4J2

r6sin2 θP2(cos θ)(4.14)

gperrr = − 2(

1 − 2Mr

)(1 − 2M

r+ 2J2

r4

)(

1 − 2M

r

)[ J2

Mr3

(1 − 5M

r

)+

+5

8

Q − J2/M

M3Q2

2(r

M− 1)

]P2(cos θ) − J2

r4

(4.15)gperθθ = 2r2

⟨− J2

Mr3+

5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2Q1

2(r

M− 1) −

−Q22(

r

M− 1)

⟩P2(cos θ) (4.16)

gperφφ = 2r2

⟨− J2

Mr3+

5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2Q1

2(r

M− 1) −

−Q22(

r

M− 1)

⟩P2(cos θ) sin2(θ) (4.17)

4.3 EÔresh twn sunart sewn twn diataraq¸n 60gpertφ = −4J

rsin2 θ − 8J

r

⟨− J2

Mr3+

5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2Q1

2(r

M− 1) −

−Q22(

r

M− 1)

⟩P2(cos θ) sin2(θ) (4.18)ìpou M,Q, J enai h mza, h tetrapolik rop kai h stroform tou astèra antstoiqa,

P2(cos θ) = 3 cos2 θ−12

enai ta polu¸numa Legendre txh 2 kai Q22, Q

12 èinai ta polu¸numa

Legendre deutèrou edou Q1

2(x) = (x2 − 1)1/2[3x2 − 2

x2 − 1− 3

2x ln

(x + 1

x − 1

)] (4.19)Q2

2(x) =3

2(x2 − 1) ln

(x + 1

x − 1

)− 3x3 − 5x

x2 − 1(4.20)kai to x = ( r

M− 1) sthn perptwsh ma . Na shmeiwje ìti g

perφφ = sin2 θg

perθθ .4.3 EÔresh twn sunart sewn twn diataraq¸nEpìmeno b ma enai na upologsoume ti sunart sei twn diataraq¸n (4.5) gia th metrik twn Hartle-Thorne. Apì th morf twn (4.14)-(4.18) kai ti (4.5) èqoume amèsw ìti c

(lm)1 =

h(lm)1 = H

(lm)1 = 0. Oi upìloipe sunart sei , gia peritt parity, dnontai apì ti sqèsei :c(lm)0 = − 1

l(l + 1)

[4J

r

∫sin θY ∗

lm,θdΩ − 4J

r3T (r)

∫sin θP2(cos θ)Y ∗

lm,θdΩ (4.21)c(lm)2 = c

sf2 + c

per2 =

[− 2

l(l + 1)(l − 1)(l + 2)

∫ (− r2

sin2 θ+

+r2

sin2 θ

)(sin θY ∗

lm,θφ − Y ∗lm,φ)dΩ

]sf +

+[− 2

l(l + 1)(l − 1)(l + 2)

∫ (− T (r)

sin2 θ+

+T (r)

sin2 θ

)(sin θY ∗

lm,θφ − Y ∗lm,φ)dΩ

]per = 0 (4.22)ìpouT (r) = 2r2

⟨− J2

Mr3(1 +

2M

r) +

5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2Q1

2(r

M− 1) − Q2

2(r

M− 1)

⟩To gegonì ìti oi sunart sei c(lm)1 kai c

(lm)2 mhdenzontai odhge, sÔmfwna me th (4.6), ìtih sunrthsh peritt parity Q×

lm = 0. Oi upìloipe pènte sunart sei rtia parity pou

4.3 EÔresh twn sunart sewn twn diataraq¸n 61apomènoun enaih

(lm)0 = − 1

l(l + 1)

[4J

r

∫Y ∗

lm,φdΩ +4J

r3T (r)

∫P2(cos θ)Y ∗

lm,φdΩ]per (4.23)

H(lm)0 =

[− 1

N2

(1 − 2M

r

) ∫Y ∗

lmdΩ]sf +

[ 1

N2

2J2

r4

∫Y ∗

lmdΩ − K(r)

N2

∫P2(cos θ)Y ∗

lmdΩ

+1

N2

4J2

r4

∫sin2 θY ∗

lmdΩ +T (r)

N2

4J2

r6

∫sin2 θP2(cos θ)Y ∗

lmdΩ]per (4.24)

H(lm)2 =

[ 1

A2

(1 − 2Mr

) ∫Y ∗

lmdΩ]sf +

+[− L(r)

∫P2(cos θ)Y ∗

lmdΩ − 1

A2

2J2/r4

(1 − 2M

r

)(1 − 2M

r+ 2J2

r4

)∫

Y ∗lmdΩ

]per(4.25)Glm = 0 (4.26)

K(lm) =[ ∫

Y ∗lmdΩ

]sf +[T (r)

r2

∫P2(cos θ)Y ∗

lmdΩ]per (4.27)ìpou oi sunart sei T (r) kai L(r) orzontai w ex

T (r) = 2r2⟨− J2

Mr3

(1 +

2M

r

)+

5

8

Q − J2/M

M3

2M

[r(r − 2M)]1/2Q1

2(r

M− 1) − Q2

2(r

M− 1)

⟩

L(r) =1

A2

2(1 − 2M

r+ 2J2

r4

)( J2

Mr3(1 − 5M

r) +

5

8

Q − J2/M

M3Q2

2(r

M− 1)

)ki epsh N2 =

(1 − 2M

r

)

A2 =(1 − 2M

r

)−1Gia na upologsoume th sunrthsh Q+lm qreiazìmaste ti sunart sei H2 kai K. Ta olo-klhr¸mata pou emperièqontai se autè ti sunart sei exart¸ntai mìno apì ti gwne . Giaton upologismì tou knoume qr sh twn sqèsewn orjogwniìthta twn sfairik¸n armonik¸n[13[14 ∫

YlmY ∗l′m′dΩ = δl′lδm′m (4.28)Epsh apì ti sqèsei

P2(cos θ) =3 cos2 θ − 1

2

Y20 =

√5

4π

3 cos2 θ − 1

2(4.29)

4.3 EÔresh twn sunart sewn twn diataraq¸n 62prokÔptei ìtiP2(cos θ) =

√4π

5(4.30)Gnwrzonta akìma ìti h sfairik armonik Y00 isoÔtai me ma stajer, gia thn akrbeia Y00 =

1√4π, gnetai emfanè ìti oi ìroi pou an koun sto sfairikì kommti twn sunart sewn diataraq¸nja paraleifjoÔn, afoÔ ìloi ja perièqoun ton ìro

∫Y ∗

lmdΩ =√

4π

∫Y00Y

∗lmdΩdhlad antistoiqoÔn se l = m = 0. H talntwsh aut enai ma kajar aktinik talntwsh ,h opoa ìmw enai stsimh kai de metaddetai. Gia to lìgo autì oi ìroi auto paralepontaiafoÔ den paristnoun kÔmata.'Ara sthn sunrthsh Q+

lm ja emfanzontai mìno oi ìroi pou ofelontai sto peristrofi-kì mèro th metrik . Ed¸ na shmei¸soume ìti to prìblhma pou meletme xefeÔgei lgoapì ma apì ti apait sh th mejìdou tou Abrahams [6, pou propujètei ìti a) to stsimomèro th metrik , oi ìroi dhlad twn monopìlwn gia l = 0, l = 1 pou den apoteloÔn dia-dimìmene diataraqè , enai kajar sfairikì kai b) oi diadidìmenoi ìroi twn pollupìlwn pouperisseÔoun apoteloÔn ti diataraqè ma . 'Omw to stsimo mèro th metrik ma den enaikajar sfairikì afoÔ perièqei kai ti peristrofè . An prosjètame diataraqè sth metrik ma tìte oi exis¸sei pou ja parname ja èdinan w kÔmata ti diataraqè th metrik sunènan statikì ìro, exart¸meno mìno apì thn apìstash, o opoo proèrqetai apì mh sfairikìstatikì ma kommti th metrik . Autì to olikì kÔma enai oi sunart sei Qlm. An t¸ra denbloume diataraqè all èqoume skèth th metrik ma (4.12), tìte oi exis¸sei dnoun w Qmìno ton statikì ìro twn peristrof¸n. Sti proswmei¸sei pou knoume o statikì autì ìro emfanzetai san èna katèbasma anèbasma twn kumatomorf¸n pou parnoume, gi' autìsthn upologistik enai gnwstì w statikì sflma kai sumbolzetai w δQlm. Autì ton ìroyqnoume ki eme na broÔme, giat an enai gnwstì autì o ìro , tìte apì thn olik kuma-tomorf Q′lm pou parnoume apì ti proswmei¸sei ma gia ènan diataragmèno peristrefìmenoastèra, mporoÔme na anaferìmaste to statikì sflma kai na proume thn kajar morf toukÔmato pou prokÔptei. Sta paraktw ja anaferìmaste loipìn sth sunrthsh Q+

lm pou dnounoi peristrofè w stsimo sflma th sunrthsh rtia parity kai ja th sumbolzoume δQ+lm. Me bsh ti sqèsei (4.25), (4.27) kai (4.28) to mìno stsimo sflma δQlm pou suneisfèreisth barutik aktinobola enai autì pou antistoiqe sti timè l = 2,m = 0. Sundizonta ti sqèsei (4.7), (4.25) kai (4.27) parnoume to sflma δQ+

lm, gia ti rtia parity sunart sei diataraq¸n, pou apotele th suneisfor th peristrof sthn ekpomp barutik¸n kumtwn.'Etsi:δQ+

lm =1

Λ

√2(l − 1)(l + 2)

l(l + 1)2r

(1 − 2M

r

)×

×− 2

(1 − 2M

r

)

(1 − 2M

r+ 2J2

r4

)[ J2

Mr3

(1 − 5M

r

)+

5

8

Q − J2/M

M3Q2

2

]√4π

5δ2lδ0m −

4.4 Poluwnumikì anptugma tou statikoÔ sflmato 63−r

[ 2J2

Mr4+

8J2

r5+ 5

M

r

Q − J2/M

M3

1

[r(r − 2M)]1/2Q1

2 +

+5

2

Q − J2/M

M3

M

[r(r − 2M)]1/2

∂Q12

∂r− 5

2

Q − J2/M

M3

M

[r(r − 2M)]3/2(r − M)Q1

2 −

5

2

Q − J2/M

M3

Q22

r− 5

4

Q − J2/M

M3

∂Q22

∂r

]√4π

5δ2lδ0m +

+4[− J2

Mr3

(1 +

2M

r

)+

5

8

Q − J2/M

M3

( 2M

[r(r − 2M)]1/2Q1

2 − Q22

)]√4π

5δ2lδ0m

+

√2(l − 1)(l + 2)

l(l + 1)2r

[− J2

Mr3

(1 +

2M

r

)+

+5

8

Q − J2/M

M3

( 2M

[r(r − 2M)]1/2Q1

2 − Q22

)]√4π

5δ2lδ0m (4.31)ìpou to sflma δQ+ enai ma sunrthsh twn (r,M, J,Q), δQ+

20(r,M, J,Q) kai fusik l =2,m = 0.4.4 Poluwnumikì anptugma tou statikoÔ sflmato H morf th (4.32) enai profan¸ duskolìqrhsth. Gia to lìgo autì melet same th grafik parstash th sunrthsh (4.31) me th bo jeia th Mathematica [20. H grafik parstashègine gia ènan peristrefìmeno astèra netronwn, aktna lgo mikrìterh apì 10 km, giaapostsh apì 10 km, lgo pio èxw apì ton astèra, èw ta 100 km. Ta qarakthristikautoÔ tou astèra fanontai ston pnaka (4.1) ( o astèra pou qrhsimopoi same enai o pr¸to astèra autoÔ tou pnaka). H grafik parstash pou parnoume fanetai sto sq ma 4.1. H

Sq ma 4.1: Grafik parstash stsimou sflmato gia ènan astèra mza 2.075 km kaiaktna r = 9.8km.

4.4 Poluwnumikì anptugma tou statikoÔ sflmato 64morf th kampÔlh pou parnoume deqnei ma omal, fjnousa sunrthsh qwr akrìtata.Apì ti proswmei¸sei pou knoume, èqoume thn upoya ìti aut h kampÔlh prèpei na pèfteikaj¸ megal¸nei to r w ma sunrthsh 1r3 me 1

r4 . Epomènw dokimsame na knoume èna mh-grammikì fitting sth sunrthsh ma gia 1r4 gia 19 shmea th kampÔlh pou p rame me tonprohgoÔmeno astèra, apì 10-100 km me disthma 5km. Dokimsame kai gia 1

r3 , ìmw h 1r4 ma èdwse kalÔtera apotelèsmata.Sto sq ma 4.2 fanontai oi dÔo kampÔle , h analutik morf tousflmato kai aut pou prokÔptei apì to fitting. 'Opw diapist¸noume parnoume ma sqedìn

Sq ma 4.2: SÔgkrish metaxÔ th analutik sunrthsh tou stsimou sflmato kai aut pou prokÔptei apì to fitting. Me mple enai h analutik morf kai me kìkkino to fitting.tèleia taÔtish. To eutuqè autì apotèlesma ma odhge sth dunatìthta na anaptÔxoume tostatikì sflma san sunrthsh Taylor w pro 1rmèqri tetrth txh . Gia ton upologismìautì kname ènan metasqhmatismì metablht¸n 1

r= x, gia na mporèsei h Mathematica [20na brei autì to anptugma. Bebaw gia na anaptÔxoume se Taylor qreiazìmaste èna stajerìshmeo gÔrw apì to opoo ja proume autì to anptugma. Ef' ìson oi upologismo pou knoumeenai gia apostsei polÔ makrinè apì ton astèra, praktik moroÔme na ti jewr soume peire kai me th skèyh aut na anaptÔxoume sto r = ∞ x = 0. Upologzonta to anptugma autìbr kame

δQ+anapt = −[8

√3π

5Q

] 1

r2+

[4

√π

15[11J2 − 4MQ]

] 1

r3+

[2M

7

√π

15[114J2 − 261MQ]

] 1

r4(4.32)Apì thn exswsh (4.32) fanetai ìti o epikratèstero ìro tou anaptÔgmato enai o 1/r2, mesuntelest −8

√3π5

Q. Me bsh autì perimènoume ìti h kÔria morf tou statikoÔ sflmato

4.5 SÔgkrish th kampÔlh stsimou sflmato gia diaforetikoÔ astère 65ja exarttai apì ton ìro autì kai epeid autì o ìro enai eujèw anlogo th tetrapolikh rop Q, aut enai pou sthn ousa ja ephrezei ta kÔria qarakthristik th sunrthsh toustatikoÔ sflmato .Gia na doÔme thn akrbeia tou anaptÔgmato , sqedisame sto dio sq ma thn analutik morf tou statikoÔ sflmato kai aut n pou prokÔptei apì to anptugma. 'Opw fanetai kai stosq ma 4.3 h akrbeia enai exairetik gia 3 aktne kai pnw. Akìma kai gia apostsei arketkont ston astèra, h apìklish tou anaptÔgmato apì thn analutik morf tou stsimousflmato enai mèsa sta ìria sflmato pou mporoÔme na èqoume. Par' ìla aut stou upologismoÔ pou knoume potè den plhsizoume kont ston astèra. Oi metr sei ma xekinoÔnmet ti 4 aktne apì ton astèra, ìpou h morf tou anaptÔgmato den diafèrei kajìlou apìthn analutik sqèsh (4.31).

Sq ma 4.3: SÔgkrish analutik sunrthsh stsimou sflmato kai tou anaptÔgmato tou.Me mple enai h analutik morf kai kìkkino to anptugma.4.5 SÔgkrish th kampÔlh stsimou sflmato giadiaforetikoÔ astère Telikì stdio th ergasa ma enai na doÔme pw metablletai h morf toustsimousflmato apì astèra se astèra kai ti kajorzei thn exrthsh aut . Oi astère pou qrhsi-mopoi jhkan enai peristrefìmenoi astère netronwn, pou tou p rame apì thn ergasa twnBerti kai Stergioulas [5. Apì autoÔ dialèxame ènan sugkekrimèno arijmì, ta qarakthristiktwn opown dnontai ston pnaka (1).

4.5 SÔgkrish th kampÔlh stsimou sflmato gia diaforetikoÔ astère 66EOS M r Q J

(M⊙) (km) km3 km2

MB = 1.589M⊙1.405 9.741 -1.001 0.8121EOS A 1.424 11.12 -8.063 2.3071.435 12.64 -13.13 2.925

MB = 2.038M⊙1.742 8.892 -3.489 2.598EOS A 1.751 9.633 -5.653 3.0761.782 11.76 -14.09 4.462

MB = 1.510M⊙1.404 15.14 -3.671 0.9726EOS L 1.415 17.05 -21.42 2.4661.422 19.97 -34.06 3.179

MB = 3.470M⊙2.929 14.86 -24.16 8.839EOS L 2.952 16.22 -38.85 10.442.996 18.54 -69.78 13.32Pnaka 4.1: Pnaka tessrwn akolouji¸n astèrwn netronwn diaforetik¸n baruonik¸n ma-z¸n, ìpou oi dÔo pr¸te antistoiqoÔn sthn EOS A kai oi lle dÔo sthn EOS L.Ston pnaka autì blèpoume tèsseri akolouje astèrwn netronwn pou an dÔo upakoÔounsti die katastatikè exis¸sei (oi dÔo pr¸te akolouje upakoÔoun sthn katastatik ex-swsh A kai oi lle dÔo sthn katastatik exswsh (EOS-Equations Of State) L(Gia peris-sìtere plhrofore mporete na dete thn ergasa twn Berti kai Stergioulas [5) ).Apì ti akolouje twn astèrwn th ergasa twn Berti kai Stergioulas [5 p rame mìno trei astè-re w endeiktikoÔ gia na knoume th melèth ma . H kje akolouja qarakthrzetai apì thbaruonik mza MB, pou enai to jroisma twn maz¸n hrema twn baruonwn pou apoteloÔnton astèra qwr na upologzoume thn enèrgeia sÔndesh , kaj¸ kje astèra kaj¸ qneienèrgeia kai stroform gia diforou lìgou , ìpw barutik aktinobola hlektromagnhtik aktinobola, ja prèpei na exelssetai diathr¸nta ton olikì arijmì baruonwn, ki epomènw thn baruonik tou mza.Sqedisame tèsseri grafikè parastsei , ma gia kje akolouja, oi opoe perièqountrei kampÔle , ma gia kje astèra. Apì ti grafikè parastsei enai emfan h epdrashth tetrapolik rop tou astèra sto pìso apèqei apì ton xona twn r, kai apì poia apì-stash arqzei na gnetai orizìntia w pro autìn. Fusik oi timè pou parnoun oi kampÔle exart¸ntai kai apì ta megèjh th mza M kai th stroform Q. Tèlo to gegonì ìti oitimè tou stsimou sflmato enai th ida txh megèjou me autè pou parnoume apì ti proswmei¸sei , ma exasfalzei akìma perissìtero ìti to apotèlesma pou p rame tan swstì.

4.5 SÔgkrish th kampÔlh stsimou sflmato gia diaforetikoÔ astère 67

Sq ma 4.4: SÔgkrish th kampÔlh stsimou sflmato gia tou trei astère th pr¸th akolouja tou pnaka (4.1), gia baruonikì arijmì MB = 1.589M⊙. H kìkkinh kampÔlh an-tistoiq ston pr¸to astèra th akolouja , h prsinh sto deÔtero kai h mple ston trto(barÔtero) astèra th akolouja .

Sq ma 4.5: Omow gia thn deÔterh akolouja tou pnaka (4.1), gia baruonikì arijmì MB =2.038M⊙. H kìkkinh kampÔlh antistoiq ston pr¸to astèra th akolouja , h prsinh stodeÔtero kai h mple ston trto (barÔtero) astèra th akolouja .

4.5 SÔgkrish th kampÔlh stsimou sflmato gia diaforetikoÔ astère 68

Sq ma 4.6: Omow gia thn trth akolouja tou pnaka (4.1), gia baruonikì arijmì MB =1.510M⊙. H kìkkinh kampÔlh antistoiq ston pr¸to astèra th akolouja , h prsinh stodeÔtero kai h mple ston trto (barÔtero) astèra th akolouja .

Sq ma 4.7: Omow gia thn tètarth akolouja tou pnaka (4.1), gia baruonikì arijmì MB =3.470M⊙. H kìkkinh kampÔlh antistoiq ston pr¸to astèra th akolouja , h prsinh stodeÔtero kai h mple ston trto (barÔtero) astèra th akolouja .

4.6 Sumpersmata 694.6 SumpersmataTo basikìtero sumpèrasma th ergasa ma enai ìti katafèrame na broÔme ènan akrib tÔpo pou na upologzei to stsimo sflma twn peristrof¸n sthn ekmpempìmenh barutik akti-nobola enì peristrefìmenou astèra. Autì to sflma br kame ìti exarttai apì th mza Mtou astèra, thn tetrapolik tou rop Q kai th stroform tou J . Epsh epalhjeÔsame thnparat rhsh pou parname apì ti proswmei¸sei , ìti to stsimo sflma pèftei ìso apoma-krunìmaste apì thn epifneia tou astèra w 1r4 . AnaptÔssonta se èna polu¸numo tetrth txh w pro 1

r4 , diapist¸same ìti gia megle apostsei , sti apostsei perpou pou me-trme, o kurarqo ìro tou anaptÔgmato enai o 1r2 . Kaj¸ o ìro autì enai anlogo th tetrapolik rop , katal goume sto sumpèrasma ìti h kÔria morf tou stsimou sflmato enì peristrefìmenou astèra exarttai apì thn tim th tetrapolik tou rop .

Bibliografa[1 T. Regge, J. A. Wheeler, Stability of a Schwarzschild Singularity, Phys. Rev. 108, 1063(1957).[2 V. Moncrief, Gravitational Perturbations of Spherically Symmetric Systems.I.The Exter-

rior Problem, Annals of Physics, 88, 323 (1974)[3 F. J. Zerilli, Gravitational Field of a Particle Falling in a Schwarzschild Geometry Analy-

zed in Tensor Harmonics, Phys. Rev. D 2, 2141 (1970)[4 G. Allen, Extracting Gravitational Waves and Other Quantities from Numerical Space-

times[5 E. Berti, N. Stergioulas, Aproximate Matching of Analytic and Numerical Solutions for

Rapidly Rotating Neutron Stars, Mon. Not. R. Astron. Soc. 350, 1416 (2004)[6 A. Abrahams, D. Bernstein, D. Hobill, E. Seidel, L. Smarr, Numerically Generated Black-

Hole Spacetimes: Interactions with Gravitational Waves, Phys. Rev. D 45, 3544 (1992)[7 M. Shibata, Y. Sekiguchi, Three-dimensional Simulations of Stellar Core Collapse in Full

General Relativity: Nonaxisymmetric Dynamical Instabilities, Phys. Rev D 71, 024014(2005)[8 J. B. Hartle, K. S. Thorne, Slowly Rotating Relativistic Stars. II. Models for Neutron

Stars and Supermassive Stars, The Astrophysical Journal 153, 807 (1968)[9 A. Nagar, L. Rezzolla, Gauge-Invariant Non Spherical Metric Perturbations of Schwarz-

schild Spacetime (2005)[10 J. B. Hartle, Gravity:An Introduction to Einstein’s General Relativity, Addison-Wesley,San Francisco (2003)[11 R. D’ Inverno, Introducing Einstein’s Relativity, Oxford University Press, Oxford (1992)[12 C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Frieman and Co, NewYork (1970)[13 Hobson, Spherical and Ellipsoidal Harmonics, Cambridge University Press (1951)

BIBLIOGRAFIA 71[14 Mc Robert, Spherical Harmonics, Seconf Revised Edition, Dover Publications .Inc, NewYork (1948)[15 G. Allen, K. Camarda, E. Seidel, Evolution of Distorted Black Holes: Perturbative Ap-

proach (2004)[16 L. Rezzolla, Gravitational Waves from Perturbed Black Holes and Relativistic Stars(2002)[17 M. Kawamura, K. Oohara,Gauge-Invariant Gravitational Wave Extraction from Coale-

scing Binary Neutron Stars (2004)[18 B. F. Schutz, Genik Sqetikìthta, P. Traulì -E. Kwstarkh (1997)[19 N. K. SpÔrou, Eisagwg sth Genik jewra th Sqetikìthta , Ekdìsei Gartagnh ,Jessalonkh (1989)[20 S. Traqan , Mathematica kai Efarmogè , Panepisthmiakè Ekdìdei Kr th (2001)