Kef laio 1 TETRANUSMATA · 2012. 10. 31. · Kef laio 1 TETRANUSMATA 1.1 Eisagwg O q¸roc tou Minkowski (qwrìqronoc) M4 eÐnai ènac grammikìc q¸roc tes- s rwn diast sewn. Ta dianÔsmata

Kef�laio 1

TETRANUSMATA

1.1 Eisagwg

O q¸roc tou Minkowski (qwrìqronoc) M4 eÐnai ènac grammikìc q¸roc tes-s�rwn diast�sewn. Ta dianÔsmata tou q¸rou autoÔ onom�same tetranÔsmata(four-vectors). Profan¸c ta tetranÔsmata apoteloÔn basikì ergaleÐo sthmelèth thc gewmetrÐac tou q¸rou tou Minkowski kai sthn perigraf twnsqetikistik¸n fusik¸n fainomènwn, gi' autì kai ja asqolhjoÔme idiaÐtera meaut�.

Kat' arq n ja prèpei na tonÐsoume èna shmeÐo, to opoÐo den tonÐzetai i-diaÐtera sthn perÐptwsh twn EukleÐdeiwn dianusm�twn. Ta tetranÔsmata wcstoiqeÐa enìc grammikoÔ q¸rou tess�rwn diast�sewn de mporoÔme na ta qa-rakthrÐsoume pl rwc grafik� me ta gnwst� bel�kia twn EukleÐdeiwn trisdi�-statwn dianusm�twn. Epomènwc prèpei na ta kajorÐsoume �leitourgik�� kaiautì gÐnetai wc akoloÔjwc1.

Ac jewr soume arqik� èna tuqaÐo tetradi�stato grammikì q¸ro. Ta dia-nÔsmata tou q¸rou autoÔ èqoun p�li tèsseric sunist¸sec, �ra pia eÐnai hdiafor� touc apì ta tetranÔsmata tou q¸rou tou Minkowski? Me �lla lìgia�ti eÐnai tetr�nusma�? Prokeimènou na apant soume se autì to ousiastikì kaileptì er¸thma jètoume to nèo er¸thma: se ti diafèrei o q¸roc touMinkowskiapì to genikì grammikì q¸ro tess�rwn diast�sewn? H ap�nthsh eÐnai ìti oq¸roc tou Minkowski fèrei th metrik tou Lorentz h opoÐa orÐzei thn om�datwn metasqhmatism¸n Lorentz. Oi metasqhmatismoÐ Lorentz epilègoun touctanustèc Lorentz kai kat� sunèpeia ta tetranÔsmata. Profan¸c sto q¸rotou Minkowski up�rqoun kai �lla dianÔsmata ta opoÐa de mac endiafèroun.Sumpèrasma:

Ta tetranÔsmata qarakthrÐzontai apì to gegonìc ìti k�tw apì

1Βλέπε και σχετική συζήτηση στο εδάϕιο ;;.

1

2 KEF�ALAIO 1. TETRANUSMATA

metasqhmatismoÔc Lorentz oi sunist¸sec touc metasqhmatÐzontaime tic sugkekrimènec exis¸seic metasqhmatismoÔ (??). AntÐstro-fa, tetr�nusma eÐnai k�je tetraposìthta sto q¸ro touMinkowski,h opoÐa metasqhmatÐzetai me b�sh touc metasqhmatismoÔc (??).

'Opwc eÐdame sto ed�fio ?? ta tetranÔsmata diakrÐnontai se treic kleistèckl�seic an�loga me to prìshmo tou m kouc touc ta qronik�, ta qwrik� kaita mhdenik� tetranÔsmata. Ta stoiqeÐa thc k�je kl�shc èqoun diaforetikècqarakthristikèc idiìthtec, oi opoÐec ta k�noun kat�llhla gia diaforetikoÔcrìlouc sth sqetikistik Fusik . To antikeÐmeno tou parìntoc kefalaÐou eÐ-nai na anaptuqjoÔn oi qarakthristikèc idiìthtec k�je kl�shc tetranusm�twnkai na dojeÐ h shmasÐa touc sth JewrÐa thc Eidik c Sqetikìthtac. Oi kÔriecènnoiec pou ja eis�goume eÐnai h ènnoia tou idiosust matoc enìc qronikoÔ te-tranÔsmatoc kaj¸c kai tou tupikoÔ sust matoc enìc qwrikoÔ tetranÔsmatoc.

1.2 Qronik� tetranÔsmata

'Estw Σ èna tuqaÐo LKS sto opoÐo èna qronikì tetr�nusma Ai èqei anapar�-stash: [

A0

A

]Σ

. (1.1)

Epeid to Ai eÐnai qronikì isqÔei:

AiAi < 0 A2 − (A0)2 < 0.

'Otan h mhdenik sunist¸sa A0 > 0 (antÐstoiqa < 0) lème ìti to Ai èqeikateÔjunsh sto mellontikì (antÐstoiqa pareljontikì) k¸no fwtìc. Profa-n¸c up�rqei LKS, to Σ+ èstw, sto opoÐo to Ai èqei an�lush:[

(A0)+

0

]Σ+

A

. (1.2)

Thn an�lush aut , h opoÐa eÐnai monadik , onom�zoume kanonik anapar�-stash (normal form) tou tetranÔsmatoc Ai. To sÔsthma suntetagmènwnΣ+

A onom�zoume idiosÔsthma tou qronikoÔ tetranÔsmatoc Ai.Prokeimènou na kajorÐsoume to Σ+

A arkeÐ na upologÐsoume ton par�gon-ta β tou Σ+

A wc proc to Σ. Proc toÔto qrhsimopoioÔme to gegonìc ìti oidÔo analÔseic (1.1) kai (1.2) tou Ai sta Σ kai Σ+

A antÐstoiqa sundèontai meèna (orjìqrono) metasqhmatismì Lorentz, opìte sÔmfwna me th sqèsh (??)èqoume:

A0 = γ(A0)+, A = γβ(A0)+

1.2. QRONIK�A TETRAN�USMATA 3

ìpou γ = (1− β2)−1/2. Apì tic sqèseic autèc prokÔptei:

γ =A0

(A0)+(1.3)

β =A

A0(1.4)

Oi sqèseic (1.3) kai (1.4) qarakthrÐzoun pl rwc to idiosÔsthma Σ+A tou te-

tranÔsmatoc Ai wc proc to tuqaÐo LKS Σ.Apì th sqèsh AiAi = −(A0)+2 kai epeid to AiAi eÐnai èna analloÐwto

èqoume ìti kai to (A0)+ èna analloÐwto epomènwc den eÐnai apl� h sunist¸saenìc tetranÔsmatoc all� eÐnai ènac tanust c. To gegonìc autì qrhsimo-poieÐtai sth Fusik prokeimènou na orisjoÔn qronik� tetranÔsmata ìtan eÐnaikajorismènh h qronik sunist¸sa touc sto idiosÔsthm� touc. P.q., ìpwc jadoÔme sto Mèroc II tou biblÐou, to mètro thc taqÔthtac tou fwtìc c eÐnaianalloÐwto. Sunep¸c sto idiosÔsthma tou tetranÔsmatoc jèshc, Σ+ enìcsqetikistikoÔ ulikoÔ shmeÐou orÐzoume èna nèo tetr�nusma me thn apaÐthsh:[

c0

]Σ+

. (1.5)

To tetr�nusma autì onom�zoume tetrataqÔthta (four-velocity) tousqetikistikoÔ ulikoÔ shmeÐou. Aut th mejodologÐa orismoÔ �fusik¸n tetranu-sm�twn� qrhsimopoioÔme ektetamèna sth sqetikistik melèth twn fusik¸n fai-nomènwn.

Mia qr simh efarmog tou idiosust matoc enìc qronikoÔ tetranÔsmatoceÐnai h apìdeixh prot�sewn sto sÔsthma autì kai h genÐkeush tou sumper�-smatoc se k�je �llo, me b�sh thn arq tou sunalloi¸tou. Pr�gmati sÔmfwname thn arq tou sunalloi¸tou e�n mia tanustik exÐswsh/sqèsh isqÔei se ènaLKS tìte isqÔei se k�je LKS. Ac doÔme mÐa tètoia efarmog .

Par�deigma 1.2.1 DeÐxte ìti to prìshmo thc mhdenik c sunist¸sac enìcqronikoÔ tetranÔsmatoc paramènei analloÐwto k�tw apì th dr�sh tou orjì-qronou metasqhmatismoÔ Lorentz.

LÔsh'Estw Ai èna qronikì tetr�nusma to opoÐo sto tuqaÐo LSK Σ èqei a-

n�lush2 Ai = (l, r)t, kai èstw ìti sto idiosÔsthm� tou Σ+ èqei an�lushAi = (l+,0)t. Apì ton orjìqrono metasqhmatismì Lorentz (??) èqoume gia thmhdenik sunist¸sa l = γl+, to opoÐo apodeiknÔei ìti oi sunist¸sec l, l+ eÐnaiomìshmec . 2

2Δηλαδή συνιστώσες.


To apotèlesma tou ParadeÐgmatoc (1.2.1) eÐnai ìti o orjìqronoc metasqh-matismìc Lorentz (??) de metab�llei to prìshmo thc mhdenik c sunist¸sac enìcqronikoÔ tetranÔsmatoc, sunep¸c epitrèpetai na diakrÐnoume kat� Lo- rentzsunalloÐwto trìpo ta qronik� tetranÔsmata peraitèrw se ekeÐna to opoÐabrÐskontai sto mellontikì k¸no fwtìc (l > 0) kai se ekeÐna pou brÐskontaiston pareljontikì k¸no fwtìc (l < 0). Ta pr¸ta onom�zoume mellon-tik� tetranÔsmata (future directed) kai ta deÔtera pareljontik�tetranÔsmata (past directed)

1.3 Qwrik� tetranÔsmata

'Estw Bi èna qwrikì tetr�nusma kai èstw Σ èna LKS sto opoÐo to Bi èqeianapar�stash:

Bi =

[B0

B

]Σ

. (1.6)

AnazhtoÔme èna LKS Σ− wc proc to opoÐo h anapar�stash tou Bi na èqeithn anhgmènh ( kanonik ) morf ):

Bi =

[0B−

]Σ−

. (1.7)

Tètoio LKS up�rqei p�nta giatÐ to BiBi = (B−)2 > 0, to opoÐo alhjeÔei giak�je B−. Apomènei na upologÐsoume ton par�gonta β tou Σ− wc proc to Σ.QrhsimopoioÔme p�li to gegonìc ìti oi dÔo analÔseic tou Bi sta Σ− kai Σsundèontai me ton orjìqrono metasqhmatismì Lorentz (??) kai èqoume:

B0 = γβ ·B− (1.8)

B = B− + (γ − 1)β ·B−

β2β. (1.9)

Apì thn (1.9) èqoume se ènan profan sumbolismì3 :

B⊥ = B−⊥ (1.10)

B∥ = B−∥ +

γ − 1

β2(β ·B∥)β = γB∥. (1.11)

Sunep¸c to B− dÐnetai apì th sqèsh:

B− = B⊥ +1

γB∥ (1.12)

3Ο όρος B⊥ είναι η προβολή του B κάθετα στο β και ο όρος B∥ η προβολή στη διεύθυνσητου β. Προϕανώς ισχύει: B = B⊥ +B∥.

1.3. QWRIK�A TETRAN�USMATA 5

kai h sqèsh (1.9) dÐnei:

B0 = γβB∥ ⇒ β =B0

B∥. (1.13)

Sun�goume ìti den eÐnai dunatìn na kajorÐsoume ton par�gonta β touΣ− wc proc to Σ monos manta sunart sei twn sunistws¸n tou Bi sto Σ(kajorÐzoume mìnon to mètro β tou β). Autì sunep�getai ìti se èna qwrikìtetr�nusma de mporeÐ na susqetisteÐ èna monos manto (�endogenès�) LKS,ìpwc sumbaÐnei sthn perÐptwsh twn qronik¸n tetranusm�twn. H shmasÐaautoÔ tou gegonìtoc eÐnai meg�lh.

Pr�gmati, ìpwc ja doÔme sto Mèroc II tou biblÐou ta qronik� tetranÔ-smata èqoun jemeliak shmasÐa sto susqetismì thc JewrÐac thc Eidik c Sqe-tikìthtac me th Neut¸neia Fusik . H monadikìthta tou idioparathrht enìcqronikoÔ tetranÔsmatoc kai o sunalloÐwtoc qarakt rac tou mac epitrèpounna k�noume dÔo susqetismoÔc:

• Na jewr soume ìti ta qronik� tetranÔsmata paristoÔn megèjh pou qa-rakthrÐzoun (sqetikistik�) ulik� shmeÐa

• Na tautÐsoume tic sunist¸sec tou tetranÔsmatoc sto idiosÔsthm� toume tic sunist¸sec tou fusikoÔ megèjouc pou parist� to tetr�nusma kaimetr� o Neut¸neioc Adraneiakìc Parathrht c4

• Na ekmetalleutoÔme to sunalloÐwto qarakt ra tou tetranÔsmatoc kaina upologÐsoume tic sunist¸sec tou tetranÔsmatoc wc proc k�je �lloLKS.

'Oson afor� ta qwrik� tetranÔsmata, ìpwc ja deÐxoume sto MEROS IItou biblÐou, aut� antistoiqoÔn sta di�fora pedÐa (p.q (tetra)hlektrikì kai(tetra)magnhtikì pedÐo) all� kai se mh topik� (non-local) megèjh swmatidÐwn(p.q spin).

DÔo gegonìta orÐzoun (ìpwc akrib¸c ta dianÔsmata sth sun jh epÐpedhGewmetrÐa) èna tetr�nusma. E�n to tetr�nusma autì eÐnai qronikì lème ìtita gegonìta eÐnai tautìqwra . E�n to tetr�nusma eÐnai qwrikì èqoumedi�forec dunatìthtec, ìpwc faÐnetai apì ton epìmeno orismì.

Orismìc 1.3.1 DÔo gegonìta A,B ta opoÐa orÐzoun èna qwrikì tetr�nusmalème ìti eÐnai tautìqrona wc proc èna LKS Σ− e�n, kai mìnon e�n,

4Δηλαδή ταυτίζουμε (Αξίωμα!) τη Φυσική του ιδιοπαρατηρητή του χρονικού τετρανύσμα-τος με εκείνη του Νευτώνειου αδρανειακού παρατηρητή για τα σχετικιστικά ϕαινόμενα πουέχουν Νευτώνειο ανάλογο. ΄Οπως θα δούμε στο ΜΕΡΟΣ ΙΙ του βιβλίου υπάρχουν σχετικι-στικά ϕυσικά μεγέθη που δεν έχουν αντίστοιχο Νευτώνειο, π.χ. η τετραταχύτητα.


1. Ta gegonìta èqoun Ðsec qronikèc suntetagmènec wc proc to Σ− ,

2. Ta gegonìta keÐntai sto Ðdio qwrikì epÐpedo tou Σ−, , isodÔnama,

3. H anapar�stash tou tetranÔsmatoc ABi sto Σ− eÐnai thc morf c:

ABi =

[0AB

]Σ−

. (1.14)

To LKS Σ− onom�zoume qarakthristikì sÔsthma (standard frame)tou qwrikoÔ tetranÔsmatoc ABi.

To qarakthristikì sÔsthma enìc tetranÔsmatoc den eÐnai monos mantaorismèno. Pr�gmati eÐnai dunatìn na deÐxoume eÔkola ('Askhsh) ìti se k�je�llo LKS Σ−′, to opoÐo kineÐtai me taqÔthta u (EukleÐdeia) k�jeth procth qwrik dieÔjunsh AB sto Σ−, to qwrikì tetr�nusma AB èqei p�li thnanhgmènh anapar�stash (1.14).

To sÔnolo twn qarakthristik¸n susthm�twn enìc qwrikoÔ tetranÔsmatocmporoÔme na diakrÐnoume se �lla qarakthristik� uposust mata. Pr�gmati acjewr soume ta gegonìta A, B ta opoÐa orÐzoun èna qwrikì tetr�nusma kaièstw Σ− èna qarakthristikì sÔsthma tou tetranÔsmatoc ABi. 'Estw ìti stoΣ− ta dianÔsmata jèshc twn gegonìtwn A kai B èqoun an�lush Ai = (A0,A)t

kai Bi = (A0,B)t. E�n |A| = |B| tìte to Σ− to onom�zoume sugqroni-smèno qarakthristikì sÔsthma tou (qwrikoÔ) tetranÔsmatoc ABi.Eidikìtera e�n to qarakthristikì sÔsthma eÐnai tètoio ¸ste A = −B tìteto onom�zoume sÔsthma hremÐac tou tetranÔsmatoc ABi. Profan¸c tosÔsthma hremÐac enìc qwrikoÔ tetranÔsmatoc eÐnai monos manta orismèno.

Sto Sq ma 1.1 deÐqnoume sto qwrìqrono tic di�forec kl�seic twn qara-kthristik¸n susthm�twn enìc qwrikoÔ tetranÔsmatoc.

1.4 Trigwnometrik anapar�stash tetra-

nÔsmatoc

Mia t�sh pou up rxe metaxÔ twn fusik¸n pou asqoloÔntan me th JewrÐathc Eidik c Sqetikìthtac se epÐpedo didaskalÐac (gi autì anafèroume to jè-ma ed¸) tan h anapar�stash enìc tetranÔsmatoc sthn EukleÐdeia GewmetrÐa¸ste na eÐnai dunatìn na ermhneÔsoun tic sqetikistikèc ènnoiec kai ta di�-fora apl� (sqetikistik�) fainìmena me sq mata. Kat� th gn¸mh mac tètoiecermhneÐec den prosfèroun ousiastik� all� o plouralismìc potè de bl�ptei5.

5Εϕόσον υπόκειται στην αρχή του ‘παν μετρον αριστον’!

1.4. TRIGWNOMETRIK�H ANAPAR�ASTASH TETRAN�USMATOS 7

Σ Σ′ Σ′′

O′′: Mèson tou AB kaiarq tou Σ′′

Σ: QarakthristikìsÔsthma tou AB

Σ′: SugqronismènosÔsthma tou AB

Σ′′: SÔsthma hremÐactou AB

Sq ma 1.1: Qarakthristik� sust mata qwrikoÔ tetranÔsmatoc

Kat' arq n parathroÔme ìti gia na parast soume èna tetr�nusma (kaigenikìtera mia sqetikistik ènnoia) sthn EukleÐdeia GewmetrÐa ja prèpei namporeÐ na qarakthristeÐ monos manta sunart sei EukleÐdeiwn megej¸n (m -kh, gwnÐec). 'Opwc deÐxame èna qronikì tetr�nusma kajorÐzetai monos mantasto idiosÔsthm� tou sunart sei enìc bajmwtoÔ megèjouc mìnon, thc qroni-k c idosunist¸sac. Epomènwc e�n antistoiq soume th sunist¸sa aut se ènaEukleÐdeio m koc tìte pr�gmati ja mporèsoume na parast soume to qroni-kì tetr�nusma gewmetrik�. 'Oson afor� ta qwrik� tetranÔsmata, aut� denkajorÐzontai apì èna bajmwtì mègejoc all� apì mia dianusmatik dieÔjunsh(th dieÔjunsh tou β) opìte h gewmetrik tou anapar�stash (e�n up�rqei) japrèpei na eÐnai sunart sei poio sÔnjetwn gewmetrik¸n antikeimènwn apì tasun jh thc EukleÐdeiac GewmetrÐac.

Ac perioristoÔme sunep¸c sta qronik� tetranÔsmata kai èstw ìti èqoumeto qronikì tetr�nusma Ai to opoÐo sto tuqaÐo LKS Σ kai sto idiosÔsthm�tou Σ+ èqei an�lush:

ABi =

[(A0)+

0

]Σ+

=

[A0

A

]Σ

. (1.15)

Oi sunist¸sec tou tetranÔsmatoc sundèontai me th sqèsh:

−(A0)+2 = −(A0)2 +A2 =⇒ (A0)2 = (A0)+2 +A2. (1.16)

H sqèsh aut mac upodhl¸nei ìti e�n jewr soume èna orjog¸nio trÐgw-no me k�jetec pleurèc ta (A0)+, |A| kai upoteÐnousa to A0 tìte to trÐgwnopr�gmati parist� to qronikì tetr�nusma (blèpe Sq ma 1.2).


(A0)+

A0

|A|

θ

Sq ma 1.2: Trigwnometrik anapar�stash qronikoÔ tetranÔsmatoc

H gwnÐa θ tou trig¸nou dÐnei to mètro thc sqetik c taqÔthtac twn Σ, Σ+

dhlad ton par�gonta β. Pr�gmati isqÔei:

sin θ =|A|A0

= β, cos θ =(A0)+

A0= γ. (1.17)

(DeÐxte ìti tan θ = βγ.)Ac doÔme ti kerdÐzoume apì aut n thn anapar�stash. ParathroÔme ìti ìso

to β aux�nei tìso h gwnÐa θ aux�nei kai ìti gia β = 1 h θ = π2. Epomènwc h tim

β = 1 eÐnai mia oriak tim . Thn tim aut antistoiqoÔme sthn taqÔthta toufwtìc kai èqoume deÐxei ìti ta swmatÐdia de mporoÔn na èqoun mètro taqÔthtacc en¸ e�n èqoun c tìte ja èqoun c gia ìla ta LKS ( up�rqei trÐgwno denup�rqei!).

'Ena deÔtero stoiqeÐo pou kerdÐzoume eÐnai ìti mporoÔme na ermhneÔsoumeto metasqhmatismì Lorentz gewmetrik�. Pr�gmati o metasqhmatismìc Lorentzautì pou k�nei eÐnai na metafèrei thn koruf tou trig¸nou pou brÐsketai a-pènanti apì thn k�jeth pleur� (A0)+ kat� m koc thc �llhc k�jethc pleur�c,ètsi ¸ste h pleur� (A0)+ na paramènei h Ðdia (jumhjeÐte ìti eÐnai analloÐwto�ra to Ðdio gia ìla ta LKS!).

1.4.1 SÔmbash antistoiqÐac sunistws¸n tanust

me dÔo deÐktec me pÐnaka

Oi upologismoÐ twn tanustik¸n ekfr�sewn pou perièqoun tanustèc me dÔodeÐktec kai tetranÔsmata dieukolÔnetai polÔ sthn pr�xh e�n qrhsimopoihjoÔn

1.4. TRIGWNOMETRIK�H ANAPAR�ASTASH TETRAN�USMATOS 9

(A0)+

A0u

A0v

|Au|

|A|v

θuθv

v > u

Sq ma 1.3: Trigwnometrik anapar�stash tou metasqhmatismoÔ Lorentz

pÐnakec. Gia na gÐnei autì apaiteÐtai mÐa sÔmbash pou ja afor� thn antistoiqÐatwn sunistws¸n tou tanust me ta stoiqeÐa tou pÐnaka. Gia ta tetranÔsmataèqoume dh k�nei mÐa sÔmbash sÔmfwna me thn opoÐa èna antalloÐwto tetr�-nusma (p�nw deÐkthc) parÐstatai me èna pÐnaka st lh kai èna sunalloÐwtotetr�nusma (k�tw deÐkthc) mè èna pÐnaka gramm . Gia touc tanustèc me dÔosunalloÐwtouc deÐktec k�noume thn akìloujh sÔmbash:

O pr¸toc deÐkthc metr� grammèc kai o deÔteroc deÐkthc st -lec.

SÔmfwna me aut n th sÔmbash o tanust c Tij antistoiqeÐtai ston akìloujopÐnaka:

Tij =

T11 T12 T13

T21 T22 T23

T31 T32 T33

.

Mia eidik periptwsh tanust¸n me dÔo deÐktec eÐnai to tanustikì ginìmenodÔo tetranusm�twn AµBν . SÔmfwna me ton orismì tou tanustikoÔ ginomènouoi sunist¸sec tou tanust AµBν (sto sÔsthma suntetagmènwn pou dÐnon-tai oi sunist¸sec twn tetranusm�twn) upologÐzontai en� pollaplasi�soumediadoqik� tic sunist¸sec tou Aµ me ìlec tic sunist¸sec tou Bν . P.q e�n

Aµ =

120

, Bν =

111


eÐnai oi sunist¸sec twn Aµ, Bν sthn Ðdia(!) b�sh tou grammikoÔ q¸rou R3,tìte to tanustikì ginìmeno

Aµ ⊗Bν =

1 1 12 2 20 0 0

.

DeÐxte ìti me b�sh aut n th sÔmbash o pÐnakac pou antistoiqeÐ ston tanust Aν⊗Bµ eÐnai o an�strofoc tou pÐnaka pou antistoiqeÐ ston tanust Aµ⊗Bν .

MÐa �llh perÐptwsh pou sunant�me suqn� stouc upologismoÔc eÐnai oupologismìc twn sunistws¸n tou tanust TijA

j. SÔmfwna me th sÔmbashpou k�name autèc upologÐzontai apì to ginìmeno twn pin�kwn:

TijAj =

T11 T12 T13

T21 T22 T23

T31 T32 T33

A1

A2

A3

=

T11A1 + T12A

2 + T13A3

T21A1 + T22A

2 + T23A3

T31A1 + T32A

2 + T33A3

.

Shmei¸noume ìti gia ton upologismì twn sunistws¸n tou tanust TijAi

upologÐzoume to ginìmeno twn pin�kwn:

[TijAi] =

A1

A2

A3

t T11 T12 T13

T21 T22 T23

T31 T32 T33

=(A1, A2, A3

) T11 T12 T13

T21 T22 T23

T31 T32 T33

=

(A1T11 + A2T21 + A3T31, A1T12 + A2T22 + A3T32

,A1T13 + A2T23 + A3T33

).

Sta epìmena ja qrhsimopoi soume ekten¸c aut n th sÔmbash kai ja d¸-soume kai�lla paradeÐgmata efarmog c thc. P�ntwc shmei¸noume ìti kalìeÐnai na elègqontai ta apotelèsmata enìc upologismoÔ kai me th sun jh ana-lutik mèjodo twn sunistws¸n.

1.5 H 1+3 an�lush wc proc mh mhdenikì

tetr�- nusma

Se k�je qronikì kai qwrikì tetr�nusma mporoÔme na susqetÐsoume èna mona-dikì probolikì telest o opoÐoc prob�lei k�jeta sto tetr�nusma. O telest cautìc orÐzetai kai gia ta sun jh EukleÐdeia dianÔsmata, all� sun jwc denanafèretai parìlo pou h qrhsimìtht� tou eÐnai meg�lh.

Prokeimènou na k�noume thn katanìhsh autoÔ tou telest eukolìterhja asqolhjoÔme pr¸ta me thn EukleÐdeia perÐptwsh me th diafor� ìti deja perioristoÔme stic treic diast�seic, all� ja jewr soume ìti h di�stash

1.5. 1+3 AN�ALUSH 11

tou q¸rou eÐnai n(≥ 2). 'Estw Aµ èna di�nusma sto q¸ro autì me mètroA2 = gEµνA

µAν > 0. H gEµν eÐnai h (EukleÐdeia) metrik h opoÐa se ènaEKS6 èqei sunist¸sec δµν . JewroÔme ton EukleÐdeio tanust t�xhc (2,0):

hµν = δµν −1

A2AµAν . (1.18)

EÐnai eÔkolo na deiqjeÐ ìti o tanust c hµν ikanopoieÐ tic sqèseic:

hµν = hνµ summetrikìc stouc deÐktec µν (1.19)

hµνδµν = hµ

µ = n− 1 (1.20)

hµνAν = 0 prob�llei k�jeta sto Aµ. (1.21)

'Askhsh 1.5.1 DeÐxte ìti oi sunist¸sec tou probolikoÔ tanust hµν sek�je EKS kai gia tuqaÐo di�nusmaAµ eÐnai diag(1− 1

A2 (A1)2, . . . , 1− 1

A2 (An)2).

EpÐshc deÐxte ìti e�n to Aµ èqei m koc mon�da o tanust c hµν = δµν −AµAν .

Prokeimènou na doÔme th qrhsimìthta kai thn praktik efarmog tou ta-nust hµν jewroÔme èna tuqaÐo di�nusma Bµ kai gr�foume:

Bµ = δµνBν = (hµ

ν +1

A2AµAν)B

ν = hµνB

ν +1

A2(AνB

ν)Aµ. (1.22)

H sqèsh aut apoteleÐ tautìthta. To mèroc B⊥ ≡ hµνB

ν onom�zoume thnk�jeth sunist¸sa tou Bµ wc proc to Aµ kai to mèroc 1

A2 (AνBν)Aµ thn

par�llhlh sunist¸sa tou Bµ wc proc to Aµ.

Par�deigma 1.5.1 AnalÔste to di�nusma Bν =

111

par�llhla kai

k�jeta wc proc to di�nusma Aµ =

120

.

LÔshTo mètro tou |A|2 = 5. AntikajistoÔme ston orismì tou h(A)µν kai

upologÐzoume:

h(A)µν = δµν −1

A2AµAν =

1 0 00 1 00 0 1

− 1

5

1 2 02 4 00 0 0

=

1

5

4 −2 0−2 1 00 0 5

.

6ΕΚΣ= Ευκλείδειο Καρτεσιανό Σύστημα συντεταγμένων


(Elègxte ìti isqÔei h(A)µνAν = 0!). Gia thn an�lush tou Bµ èqoume:

Bµ⊥ = h(A)µνB

ν =1

5

2−15

, Bµ∥ = Bµ −Bµ

⊥ =1

5

360

.

(Elègxte ìti isqÔei Bµ = Bµ⊥ +Bµ

∥ !). 2

1.6 An�lush tanust deÔterhc t�xhc wc

proc di�nusma

Ja proqwr soume t¸ra se èna polÔ shmantikì jèma to opoÐo ja faneÐ polÔqr simo sth melèth thc Sqetikìthtac (all� kai thc GewmetrÐac). Ac k�noumeto akìloujo aplì er¸thma:

EÐnai dunatìn na analÔsoume èna tanust deÔterhc t�xhc wcproc èna di�nusma?

H ap�nthsh sto er¸thma autì eÐnai shmantik giatÐ polloÐ tanustèc sth Je-wrÐa thc Eidik c (kai thc Genik c) Sqetikìthtac all� kai thc Neut¸neiacFusik c eÐnai deÔterhc t�xhc. To pio shmantikì par�deigma eÐnai o tanust cenèrgeiac orm c all� kai h par�gwgoc ui,j thc tetrataqÔthtac ktl.7

1.6.1 An�lush se EukleÐdeio q¸ro

Prokeimènou na apant soume sto er¸thma ergazìmaste ìpwc kai sthn perÐ-ptwsh thc an�lushc twn dianusm�twn. 'Estw Aµ èna di�nusma (shmei¸steìti mil�me akìma gia EukleÐdeia dianÔsmata!) kai èstw h(A)µν o probolikìctanust c pou orÐzei to Aµ tou opoÐou to mètro jewroÔme ìti eÐnai A2.

'Estw Tµν ènac tuqaÐoc tanust c deÔterhc t�xhc. Tìte èqoume thn tautì-thta:

Tµν = δ αµ δ β

ν Tαβ =

(h αµ +

1

A2AµA

α

)(h βν +

1

A2AνA

β

)Tαβ

=1

A4

[AαAβAµAν +

1

A2h αµ AβAν +

1

A2h βν AαAµ + h α

µ h βν

]Tαβ

=1

A4

(TαβA

αAβ)AµAν+

7Και για όσους έχουν διαβάσει περισσότερο Διαϕορική Γεωμετρία μια άλλη σημαντικήποσότητα είναι η παράγωγος Lie της μετρικής LXgab, η οποία περιγράϕει τις συμμετρίες τηςμετρικής.

1.6. 1+3 AN�ALUSH TANUST�H (0,2) 13

+1

A2h αµ AβTαβAν +

1

A2h βν AαTαβAµ + h α

µ h βν Tαβ. (1.23)

To apotèlesma thc an�lushc tou tanust eÐnai:

• Mia bajmwt posìthta: 1A4TαβA

αAβ

• DÔo dianÔsmata k�jeta sto Aµ: 1A2h

αµ AβTαβ kai 1

A2hβ

ν AαTαβ

• 'Enac tanust c dÔterhc t�xhc me sunhst¸sec mìno ston upìqwro tonk�jeto sto di�nusma Aµ: h α

µ h βν Tαβ

Pollèc forèc thn anwtèrw an�lush gr�foume se morf pÐnaka pin�kwn(block matrix) wc akoloÔjwc:(

1A4TαβA

αAβ 1A2h

αµ AβTαβ

1A2h

βν AαTαβ h α

µ h βν Tαβ

)(1.24)

ìpou oi upopÐnakec (1,2) kai (2,1) eÐnai pÐnakec st lec di�stashc 1×n kai n×1(n= di�stash tou q¸rou) kai o upopÐnakac (2,2) eÐnai tetragwnikìc di�stashc(n− 1)× (n− 1). H sqèsh (1.24) ja qrhsimopoihjeÐ ektetamèna sth melèthtou tanust enèrgeiac orm c all� kai thc kinhmatik c thc JewrÐac thc Eidik cSqetikìthtac. Bèbaia de qrei�zetai na tonÐsoume ìti eÐnai jemeliak kai gia thNeut¸neia Fusik (Kinhmatik kai Dunamik ) all� den qrhsimopoieÐtai sqedìnpotè!

Par�deigma 1.6.1 AnalÔste ton EukleÐdeio tanust t�xhc (0,2) Tµν = 1 0 11 2 10 0 1

wc proc to EukleÐdeio di�nusma Aµ =

120

.

LÔshSto Par�deigma 1.5.1 upologÐsame ìti o probolikìc telest c tou dianÔ-

smatoc Aµ (sto EKS suntetagmènwn pou dÐnontai oi suntetagmènec tou Aµ!)eÐnai:

h(A)µν =1

5

4 −2 0−2 1 00 0 5

UpologÐzoume diadoqik� ta anag¸gima mèrh tou tanust Tµν ìpwc dÐnontaisthn (1.24). Epeid o skopìc tou ParadeÐgmatoc eÐnai na d¸sei praktikècupologismoÔ ja perigr�youme ton upologismì se mikr� b mata prospaj¸ntacna deÐxoume me pio trìpo efarmìzontai oi upologismoÐ twn sunistws¸n me thqr sh pin�kwn. SumbolÐzoume me [h], [T ], [A] touc pÐnakec pou antistoiqoÔnstouc tanustèc hαβ, Tαβ, A

α kai dÐnoume thn ap�nthsh wc ginìmeno pin�kwn.


Profan¸c oi sqèseic pou dÐnoume eÐnai anex�rthtec apì th di�stash tou q¸roukai mporoÔn na qrhsimopoihjoÔn se k�je an�logo par�deigma8. Shmei¸noumeìti gia thn EukleÐdeia metrik δµν oi sunist¸sec de metab�llontai e�n anebo-kateb�soume touc deÐktec dhlad èqoume [hαβ] = [hβ

α] = [hαβ ]. Autì den isqÔei

sth metrik tou Lorentz!To pr¸to an�gwgo mèroc eÐnai bajmwtì kai o upologismìc tou eÐnai �me-

soc:

1

A4(TαβA

αAβ) =1

A4[A]t[T ][A]

=1

25

(1 2 0

) 1 0 11 2 10 0 1

120

=11

25.

To deÔtero an�gwgo mèroc eÐnai ènac pÐnakac 3 × 1. JumÐzoume ìti oi k�twdeÐktec qarakthrÐzoun pÐnakec gramm c (sunalloÐwta) kai oi p�nw deÐktecpÐnakec st lh (antalloÐwta). Gr�foume:

1

A2hµαAβTαβ =

1

A2(hµ1AβT1β + hµ2AβT2β + hµ3AβT3β)

=1

A2

(hµ1 hµ2 hµ3

) AβT1β

AβT2β

AβT3β

=

1

A2[h][T ][A]

=1

25

4 −2 0−2 1 00 0 5

1 0 11 2 10 0 1

120

=1

25

-630

.

'Omoia gia to 1× 3 anag¸gimo mèroc èqoume:

1

A2h αµ AβTβα =

1

A2(h 1

µ AβTβ1 + h 2µ AβTβ2 + h 3

µ AβTβ3)

=1

A2

(AβTβ1 AβTβ2 AβTβ3

) h 1µ

h 2µ

h 3µ

=1

A2[A]t[T ][h]

=1

25

(1 2 0

) 1 0 11 2 10 0 1

4 −2 0−2 1 00 0 5

=

1

25

(4 -2 15

).

8Προσοχή όμως! Στην περίπτωση της μετρικής Lorentz τα πράγματα αλλάζουν ως προςτα πρόσημα. Το τι ακριβώς συμβαίνει θα το δούμε παρακάτω.


Tèloc gia to 3× 3 an�gwgo mèroc èqoume:

hαµh

βνTαβ = [h][T ][h] =

1

25

4 −2 0−2 1 00 0 5

1 0 11 2 10 0 1

4 −2 0−2 1 00 0 5

=

1

25

16 −8 10−8 4 −50 0 25

Prèpei na epalhjeÔsete ta anwtèrw apotelèsmata upologÐzontac tic suni-

st¸sec twn an�gwgwn mer¸n tou tanust kai analutik� ìpwc k�name sto ke-f�laio ??. EÐnai mia qr simh �skhsh na deÐxoume ìti to �jroisma twn an�gwgwnmer¸n pou upologÐsame pr�gmati dÐnei ton arqikì tanust . UpologÐzoume ar-qik� ta tanustik� ginìmena (dhlad to k�je stoiqeÐo tou enìc pÐnaka me ìlata stoiqeÐa tou �llou) twn pin�kwn. 'Eqoume diadoqik�:

AµAν =(1 2 0

)⊗

(1 2 0

)=

1 2 02 4 00 0 0

(

1

A2h αµ AβTαβ

)Aν =

1

25

(−6 3 0

)⊗(1 2 0

)=

1

25

−6 −12 03 6 00 0 0

(

1

A2h αµ AβTβα

)Aν =

1

25

(1 2 0

)⊗(4 -2 15

)=

1

25

4 −2 158 −4 300 0 0

.

To �jroisma aut¸n twn pin�kwn kai tou pÐnaka

h αµ h β

ν Tαβ =1

25

16 −8 10−8 4 −50 0 25

dÐnei pr�gmati ton pÐnaka pou antistoiqeÐ ston tanust Tµν . 2

1.6.2 An�lush sto q¸ro tou Minkowski

H an�lush enìc tanust deÔterhc t�xhc wc proc di�nusma pou anaptÔqjhkesto prohgoÔmeno ed�fio aforoÔse thn EukleÐdeia perÐptwsh, dhlad tìso


o tanust c ìso kai to di�nusma tan EukleÐdeioi tanustèc. Autì ìmwc pouendiafèrei sth JewrÐa thc Eidik c Sqetikìthtac eÐnai h an�lush twn sqeti-kistik¸n tanust¸n.

H kÔria diafor� metaxÔ thc EukleÐdeiac kai thc sqetikistik c perÐptwshceÐnai sto prìshmo thc mhdenik c sunist¸sac, h opoÐa ìtan metatrèpetai apìsunalloÐwth (k�tw deÐkthc) se antalloÐwth (p�nw deÐkthc) all�zei prìshmo.H diafor� aut den afor� anebokatèbasma deikt¸n se genikèc tanustikècsqèseic all� efarmìzetai mìnon ìtan oi sqèseic autèc ekfrastoÔn se sunist¸-sec.

Mia deÔterh diafor� eÐnai ìti sth sqetikistik perÐptwsh èqoume tetranÔ-smata me m koc AiAi < 0. To gegonìc autì den epidr� stic tanustikècsqèseic par� mìnon ìtan h posìthta AiAi antikatastajeÐ me to sign(A)A2

ìpou A2 > 0 afoÔ to A ∈ R kai sign(A) = 0, < 0, > 0 gia mhdenik�, qronik�kai qwrik� tetranÔsmata antÐstoiqa.

H kÔria an�lush pou mac endiafèrei9 eÐnai ekeÐnh wc proc th tetrataqÔthtaenìc reustoÔ opìte sta epìmena ja sumbolÐzoume to tetr�nusma Ai wc ui (poueÐnai o tupikìc sumbolismìc gia thn tetrataqÔthta), h opoÐa eÐnai èna monadiaÐoqronikì tetr�nusma dhlad uiui = −1 (jewroÔme c = 1). Gia to tetr�nusmaautì oi sqèseic pou par�game gia thn EukleÐdeia perÐptwsh (kai paramènounoi Ðdiec!) gÐnontai10 :

• Probolikìc tanust c:

h(u)ab = ηab + uaub (1.25)

• An�lush tetranÔsmatoc:

wa = −(wbub)ua + h(u)abu

b. (1.26)

An�lush tanust deÔterhc t�xhc:

Tab =(Tcdu

cud)uaub−h(u) c

a udTcdub−h(u) db ucTcdua+h(u) c

a h(u) db Tcd

(1.27)

ParathroÔme ìti oi tÔpoi autoÐ sumpÐptoun me touc antÐstoiqouc tÔpoucpou par�game sto prohgoÔmeno ed�fio, me mình antikat�stash tou A2 me −1.O upologismìc ìmwc twn sunistws¸n eÐnai mia �llh istorÐa. Sto epìmenopar�deigma deÐqnoume pwc douleÔei k�poioc sthn pr�xh. SunistoÔme stonanagn¸sth na epalhjeÔsei analutik� ìla ta apotelèsmata.

9Την ανάλυση ως προς γενικό διάνυσμα θα αναπτύξουμε στο κεϕάλαιο ;; όπου θα θεω-ρήσουμε την 1+3 ανάλυση ως προς γενικό μη χρονικό τετράνυσμα.

10Τις σχέσεις (1.26), (1.27) μπορείτε να τις αποδείξετε κατευθείαν θεωρώντας την ταυτό-τητα Tab = η c

a η db Tcd και να αντικαταστήσετε το ηab = h(u)ab + uaub.


Par�deigma 1.6.2 AnalÔste to tetr�nusma

wi =

√3

210

kaj¸c kai ton tanust Lorentz tanust t�xhc (0,2)

Tab =

1 0 0 10 1 0 10 1 2 10 0 0 1

wc proc to monadiaÐo qronikì tetr�nusma Lorentz ua =

√3110

.

LÔshH metrik Lorentz eÐnai η = diag(−1, 1, 1, 1). UpologÐzoume11:

ua ⊗ ub =

3 −

√3 −

√3 0

−√3 1 1 0

−√3 1 1 0

0 0 0 0

uaua = −1 (MonadiaÐo qronikì tetr�nusma).

h(u)ab =

2 −

√3 −

√3 0

−√3 2 1 0

−√3 1 2 0

0 0 0 1

'Oson afor� to anebokatèbasma twn deikt¸n èqoume:

h(u) ba = ηbch(u)ac =

−2 −

√3 −

√3 0√

3 2 1 0√3 1 2 00 0 0 1

11Θυμηθείτε: Πρώτος δείκτης γραμμή δεύτερος στήλη!


kai

h(u)ab = ηach(u)cb =

−2

√3

√3 0

−√3 2 1 0

−√3 1 2 0

0 0 0 1

.

Parathr ste ìti [h(u) ba ] = [h(u) b

a ]t.'Oson afor� to tetr�nusma wa èqoume diadoqik�:

wa|| = waua = 0

wa⊥ = h(u)abw

b =

√3210

'Ara wa = wa|| + wa

⊥ =

√3210

.

'Omoia gia ton tanust Tab èqoume diadoqik�:

Tabuaub = 7

h(u) ca Tcdu

d = [h(u) ca ][Tcd][u

d]

=

−2 −

√3 −

√3 0√

3 2 1 0√3 1 2 00 0 0 1

1 0 0 10 1 0 10 1 2 10 0 0 1

√3110

= −

(6√3,−8,−10, 0

)h(u)daTcdu

c = [uc]t[Tcd][h(u)da]

=(√

3, 1, 1, 0)

1 0 0 10 1 0 10 1 2 10 0 0 1

−2√3

√3 0

−√3 2 1 0

−√3 1 2 0

0 0 0 1

= −

(6√3, −9, −9, −

√3− 2

).


h(u) ca h(u) d

b Tcd = [h(u) ca ][Tcd][h(u)

db ]

=

−2 −

√3 −

√3 0√

3 2 1 0√3 1 2 00 0 0 1

1 0 0 10 1 0 10 1 2 10 0 0 1

−2√3

√3 0

−√3 2 1 0

−√3 1 2 0

0 0 0 1

=

16 −8

√3 −8

√3 −2

√3− 2

−7√3 11 10

√3 + 3

−9√3 13 14

√3 + 3

0 0 0 1

.

EpalhjeÔste ìti h anwtèrw an�lush eÐnai swst efarmìzontac th sqèsh(1.27). 2

Sthn perÐptwsh pou to tetr�nusma den eÐnai monadiaÐo oi sqèseic pou dÐnounthn 1+3 an�lush diaforopoioÔntai wc akoloÔjwc.

'Estw Aa èna tetr�nusma me m koc AaAa = ϵ(A)A2 ìpou ϵ(A) eÐnai odeÐkthc tou Aa o opoÐoc eÐnai ±1 an�loga me to e�n to tetr�nusma eÐnai qwrikì qronikì kai A > 0. O probolikìc tanust c eÐnai:

h(A)ab = ηab −ϵ(A)

A2AaAb (1.28)

kai èqei tic gnwstèc idiìthtec:

habAb = 0, habh

bc = hac, h

aa = 3. (1.29)

'Estw Ba èna tuqaÐo tetr�nusma (qwrikì, qronikì mhdenikì). Tìte h 1+3an�lush tou Ba wc proc to Aa eÐnai:

Ba = δabBb = ηabB

b = (hab (A) +

ε(A)

A2AaAb)B

b =ε(A)

A2(AbB

b)Aa + hab (A)B

b.

(1.30)Gia èna genikì tanust Tab t�xhc (0,2) ergazìmenoi an�loga èqoume12 thn

12Η απόδειξη έχει ως ακολούθως:

Tab = δcaδdbTcd =

(hca −

1

A2AcAd

)(hdb −

1

A2AdAb

)Tcd

=1

A4

(TcdA

cAd)AaAb −

1

A2

(hcaA

dAbTcd + hdbA

cAaTcd

)+ hc

ahdbTcd.


akìloujh genik sqèsh/tautìthta:

Tab =1

A4

(TcdA

cAd)AaAb −

1

A2

(h(A)caTcdA

dAb + h(A)dbTcdAcAa

)+h(A)cah(A)

dbTcd. (1.31)

Ac doÔme èna par�deigma kai gia aut n thn perÐptwsh.

Par�deigma 1.6.3 AnalÔste to tetr�nusma Ba =

3211

kai ton tanust

Tab =

1 0 0 10 1 0 10 1 2 10 0 0 1

wc proc to tetr�nusma Aa =

3210

.

LÔshUpologÐzoume AaAa = −4, epomènwc to tetr�nusma Aa eÐnai qronikì me

mètro A = 2. To tanustikì ginìmeno

Aa ⊗ Ab =

9 6 3 06 4 2 03 2 1 00 0 0 0

kai o probolikìc tanust c:

h(A)ab =

5

4

−3

2

−3

40

−3

22

1

20

−3

4

1

2

5

40

0 0 0 1

.

Gia to tetr�nusma Ba èqoume:

BaAa = −4

h(A)abBb =

0001

1.7. MHDENIK�A TETRAN�USMATA 21

'Ara Ba = Aa + h(A)abBb.

'Omoia gia ton tanust Tab èqoume diadoqik�:

1

A4TabA

aAb =17

16

1

A2h(A)caTcbA

b =

[39

16,−21

8,−33

16, 0

]1

A2h(u)baTcbA

c =

[39

16, −23

8, −25

16, −3

2

]

h(A)cah(A)dbTcd =

97

16

−57

8

−63

16

−7

2

−51

8

31

4

29

84

−87

16

47

8

73

16

5

2

0 0 0 1

EpalhjeÔste ìti h anwtèrw an�lush eÐnai swst efarmìzontac th sqèsh

(1.31). 2

1.7 Mhdenik� tetranÔsmata

Ta mhdenik� tetranÔsmata apoteloÔn mia idiaÐterh kathgorÐa tetranusm�twnme pollèc idiomorfÐec kai duskolÐec sto qeirismì touc. Apì thn �llh eÐnaishmantik� tetranÔsmata mia kai perigr�foun fwtìnia opìte prèpei na anafè-roume merik� genik� stoiqeÐa.

'Estw Ai =

(A0

A

)èna mhdenikì tetr�nusma. Tìte isqÔei:

−(A0)2 +A2 = 0 ⇒ A = A0e (1.32)

ìpou e eÐnai èna monadiaÐo 3-di�nusma. Epomènwc k�je mhdenikì tetr�nusmamporeÐ na grafeÐ sth morf :

Ai = A0

(1e

)(1.33)

dhlad kajorÐzetai pl rwc apì èna bajmwtì (ìqi ìmwc analloÐwto!) mègejoc(to A0) kai èna (EukleÐdeia) monadiaÐo di�nusma e.


Apì fusik �poyh e�n to tetr�nusma Ai perigr�fei èna fwtìnio tìtejewroÔme ìti to bajmwtì mègejoc A0 antistoiqeÐ sthn enèrgeia tou fwtonÐoukai to monadiaÐo e orÐzei th dieÔjunsh di�doshc tou fwtonÐou. To gegonìcìti oÔte h posìthta A0 oÔte h kateÔjunsh e eÐnai sunalloÐwta wc proc tometasqhmatismì Lorentz shmaÐnei ìti sth JewrÐa thc Eidik c Sqetikìthtacprèpei na perimènoume ìti h suqnìthta enìc fwtonÐou kaj¸c kai h dieÔjunshdi�dos c tou ja exart¸ntai apì ton parathrht . H metabol tou A0 apìparathrht se parathrht odhgeÐ sto fainìmeno Doppler kai h metabol toue sto fainìmeno thc apopl�nhshc tou fwtìc (aberration of light) toopoÐo èqei eureÐa efarmog stic optikèc parathr seic.

Merikèc qr simec, kai aplèc na apodeiqtoÔn, idiìthtec enìc mhdenikoÔ te-tranÔsmatoc dÐnontai sthn �skhsh 1.7.1.

'Askhsh 1.7.1 DeÐxte ìti k�je mhdenikì tetr�nusma:

1. EÐnai k�jeto ston eautì tou

2. MporeÐ na grafeÐ (ìqi monos manta!) wc to �jroisma enìc qronikoÔ kaienìc qwrikoÔ tetranÔsmatoc.

1.8 H an�lush wc proc zeÔgoc qronik¸n

tetranusm�twn

Sto ed�fio 1.6 jewr same thn 1 + 3 an�lush wc proc èna mh mhdenikì te-tr�nusma. H pr�xh ìmwc èqei deÐxei ìti eÐnai anagkaÐo na jewr soume stoqwrìqrono thn an�lush tanust¸n wc proc zeÔgh mh mhdenik¸n tetranusm�-twn. Profan¸c ta tetranÔsmata prèpei na eÐnai mh suggrammik� kai na orÐzounsto q¸roqrono èna epÐpedo13. Up�rqoun treic peript¸seic an�loga me to e�nto zeÔgoc twn tetranusm�twn eÐnai (qroniko, qronikì), (qronikì, qwrikì) kai(qwrikì, qwrikì). Sto ed�fio autì ja mac apasqol sei h pr¸th perÐptwshkai sto epìmeno h deÔterh. H trÐth perÐptwsh de qrhsimopoieÐtai mia kai p�ntah an�lush perièqei thn tetrataqÔthta, h opoÐa eÐnai èna qronikì tetr�nusma.

Sthn 1 + 3 an�lush orÐsame èna probolikì tanust o opoÐoc prìbalek�jeta kai par�llhla proc to basikì di�nusma. 'Otan èqoume zeÔgoc tetra-nusm�twn orÐzoume an�loga èna (summetrikì) probolikì tanust , o opoÐoc

13Θυμίζουμε ότι ο χωρόχρονος είναι επίπεδος και κατά συνέπεια ένα διάνυσμα μπορεί ναμεταϕερθεί παράλληλα όπως στην Ευκλείδεια Γεωμετρία (ελεύθερα διανύσματα). Επομέ- νωςακόμα και εάν δύο διανύσματα δεν έχουν κοινό σημείο εϕαρμογής, μπορούμε να τα μετα-ϕέρουμε ώστε να δημιουργήσουμε μια βάση που θα ορίσει ένα επίπεδο 2-διαστάσεων. Στοχωρόχρονο υπάρχουν και επιϕάνειες επίπεδα τριών διαστάσεων που τις ονομάζουμε υπερεπί-πεδα.

1.8. 1+1+2 AN�ALUSH QRONIKO�U ZE�UGOUS 23

prob�llei k�jeta sto epÐpedo pou orÐzoun ta dÔo tetranÔsmata, isodÔna-ma, prob�llei k�jeta kai sta dÔo tetranÔsmata. Me b�sh ton tanust autìnmporoÔme na analÔsoume peraitèrw touc tanustèc se trÐa mh an�gwga komm�-tia, dhlad èna komm�ti gia k�je basikì di�nusma kai èna komm�ti k�jeto stoepÐpedo twn dianusm�twn.

'Estw Aa, Ba dÔo qronik� tetranÔsmata me mètra A,B antÐstoiqa. Zht�meèna summetrikì tanust t�xhc (0, 2) (Pab = Pba) o opoÐoc ja prob�llei k�jetakai sta dÔo tetranÔsmata. O tanust c autìc prèpei na èqei th genik morf :

Pab(A,B) = ηab + a1AaAb + a2BaBb + a3(AaBb +BaAb) (1.34)

ìpou a1, a2, a3 eÐnai k�poioi suntelestèc pou prèpei na prosdioristoÔn. OrÐ-zoume arqik� thn posìthta:

γ = −ηabAaBb = −AaB

a. (1.35)

Oi sunj kec pou ja prosdiorÐsoun touc suntelestèc a1, a2, a3 eÐnai:

Pab(A,B)Ab = Pab(A,B)Bb = 0. (1.36)

H sunj kh Pab(A,B)Ab = 0 dÐnei tic exis¸seic:

1− a1A2 = a3γ

a2γ = −a3A2 (1.37)

kai h sunj kh Pab(A,B)Bb = 0 tic:

1− a2B2 = a3γ

a1γ = −a3B2. (1.38)

H lÔsh tou sust matoc aut¸n twn tess�rwn exis¸sewn eÐnai:

a1 = − B2

γ2 − A2B2, a2 = − A2

γ2 − A2B2, a3 =

γ

γ2 − A2B2.

Epomènwc o probolikìc tanust c pou zht�me dÐnetai apì th sqèsh:

Pab(A,B) = ηab − B2

γ2 − A2B2AaAb −

A2

γ2 − A2B2BaBb

+γ

γ2 − A2B2(AaBb +BaAb). (1.39)

IdiaÐtero endiafèron parousi�zei h perÐptwsh pou ta tetranÔsmata Aa, Ba

eÐnai monadiaÐa, dhlad èqoume Aa = ua, Ba = va ìpou uaua = vava = −1.Tìte h sqèsh (1.39) gÐnetai:

Pab(u, v) = ηab −1

γ2 − 1[uaub + vavb − γ(uavb + vaub)] . (1.40)


'Askhsh 1.8.1 ApodeÐxte ìti o tanust c Pab(A,B) pou upologÐsame ikano-poieÐ tic sqèseic (1.36). EpÐshc deÐxte ìti èqei Ðqnoc (trace) Ðso me 2, dhlad :

paa(A,B) = 2. (1.41)

Tèloc apodeÐxte ìti isqÔei14:

pca(A,B)hbc = pab(A,B). (1.42)

'Askhsh 1.8.2 JewreÐste èna genikì tetr�nusma Ca kai analÔste to wcproc ta dianÔsmata Aa, Ba wc akoloÔjwc:

Ca = a4Aa + a5Ba + Pab(A,B)Cb (1.43)

ìpou a4, a5 eÐnai suntelestèc pou prèpei na prosdioristoÔn. Jewr¸ntac su-stol me Aa kai Ba deÐxte ìti:

a4 =B2(CA)− γ(CB)

γ2 − A2B2, a5 =

A2(CB)− γ(CA)

γ2 − A2B2

ìpou (CA) = CaAa, (CB) = CaBa. Sumper�nete ìti h an�lush tou Ca wcproc ta dianÔsmata Aa, Ba dÐnetai apì th sqèsh:

Ca =B2(CA)− γ(CB)

γ2 − A2B2Aa +

A2(CB)− γ(CA)

γ2 − A2B2Ba + Pab(A,B)Cb. (1.44)

Tèloc sthn perÐptwsh pou ta tetranÔsmata Aa, Ba eÐnai ta monadiaÐa ua, va

deÐxte ìti h sqèsh aut gÐnetai:

Ca =(Cu)− γ(Cv)

γ2 − 1ua +

(Cv)− γ(Cu)

γ2 − 1va + Pab(u, v)C

b. (1.45)

1.9 Dipl� tetranÔsmata

'Otan lème diplì tetr�nusma ennooÔme èna zeÔgoc enìc qronikoÔ kai enìcqwrikoÔ tetranÔsmatoc. H qr sh twn dipl¸n tetranusm�twn den eÐnai eurèwcgnwst en¸ eÐnai idiaÐtera qr simh sth melèth tou hlektromagnhtikoÔ (all�kai �llwn) pedÐou. Sta epìmena anafèroume merik� genik� stoiqeÐa qwrÐc nampoÔme se idiaÐterec leptomèreiec.

Ac suzht soume arqik� thn anagkaiìthta twn dipl¸n tetranusm�twn apìmajhmatik kai apì fusik �poyh.

14Η απόδειξη είναι πολύ απλή. Πράγματι έχουμε pca(A,B)hbc = pca(A,B) (δca +AaA

c) =pab(A,B).

1.9. DIPL�A TETRAN�USMATA 25

'Opwc eÐdame sto ed�fio 1.3 to qarakthristikì LKS Σ− enìc qwrikoÔtetranÔsmatoc Bi den eÐnai monos manto kai mporeÐ na kajoristeÐ mìnon e�nkajoristeÐ o par�gontac β tou Σ− wc proc k�poio LKS Σ sto opoÐo gnw-rÐzoume thn an�lush (dhlad tic sunist¸sec) tou Bi. ToÔto shmaÐnei ìti hmelèth enìc qwrikoÔ tetranÔsmatoc proôpojètei thn Ôparxh enìc qronikoÔtetranÔsmatoc (to opoÐo kajorÐzei ton par�gonta β).

Mil¸ntac apì th skopi� thc Fusik c h melèth p.q. enìc magnhtikoÔ pedÐou(to opoÐo perigr�fetai sthn jewrÐa thc Sqetikìthtac me èna qwrikì tetr�nu-sma) proôpojètei thn Ôparxh enìc parathrht (to β), o opoÐoc ja parathreÐto magnhtikì pedÐo.

Ta anwtèrw mac odhgoÔn na jewr soume ta dipl� tetranÔsmata , taopoÐa eÐnai zeÔgh thc morf c (Ai, Bi) ìpou Ai eÐnai èna qronikì tetr�nusmakai Bi èna qwrikì tetr�nusma. Prin af soume thn ènnoia tou diploÔ tetra-nÔsmatoc orÐzoume èna nèo probolikì telest , o opoÐoc prob�llei k�jeta kaista dÔo tetranÔsmata enìc diploÔ tetranÔsmatoc.

'Askhsh 1.9.1 'Estw (Ai, Bi) èna diplì tetr�nusma tètoio ¸ste to Ai naeÐnai monadiaÐo kai qronikì (AiAi = −1) kai to Bi monadiaÐo kai qwrikì(BiBi = 1) kai èstw ϕ = AiBi h gwnÐa pou sqhmatÐzoun. OrÐste thn po-sìthta ∆ = 1 + ϕ2 kai deÐxte ìti o tanust c t�xhc (0,2):

pij(A,B) = ηij +1

∆(AiAj − ϕ(AiBj +BiAj)−BiBj) (1.46)

èqei tic akìloujec idiìthtec:

1. EÐnai summetrikìc

2. Prob�llei k�jeta kai sto Ai kai sto Bi, dhlad isqÔei:

pab(A,B)Ab = pab(A,B)Bb = 0. (1.47)

3. 'Eqei Ðqnoc (trace) Ðso me 2, dhlad :

paa(A,B) = 2 (1.48)

4.pab(A,B)hb

c(A) = pac(A,B). (1.49)

O tanust c pab(A,B) onom�zetai o telest c probol c ojìnhc (screenprojection operator) twn tetranusm�twn Aa kai Ba. 2


Kef laio 1 TETRANUSMATA · 2012. 10. 31. · Kef laio 1 TETRANUSMATA 1.1 Eisagwg O q¸roc tou Minkowski (qwrìqronoc) M4 eÐnai ènac grammikìc q¸roc tes- s rwn diast sewn. Ta dianÔsmata

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