❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦
▼♦❞❡❧♦s ❈♦s♠♦❧ó❣✐❝♦s ♥✉♠❛ ❚❡♦r✐❛
●❡♦♠étr✐❝❛ ❊s❝❛❧❛r✲❚❡♥s♦r✐❛❧ ❞❛
●r❛✈✐t❛çã♦✿ ❆s♣❡❝t♦s ❈❧áss✐❝♦s ❡
◗✉â♥t✐❝♦s
❋r❛♥❝✐s❝♦ ❆rt✉r P✐♥❤❡✐r♦ ❆❧✈❡s ❏ú♥✐♦r
❏♦ã♦ P❡ss♦❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦
▼♦❞❡❧♦s ❈♦s♠♦❧ó❣✐❝♦s ♥✉♠❛ ❚❡♦r✐❛
●❡♦♠étr✐❝❛ ❊s❝❛❧❛r✲❚❡♥s♦r✐❛❧ ❞❛
●r❛✈✐t❛çã♦✿ ❆s♣❡❝t♦s ❈❧áss✐❝♦s ❡
◗✉â♥t✐❝♦s
❋r❛♥❝✐s❝♦ ❆rt✉r P✐♥❤❡✐r♦ ❆❧✈❡s ❏ú♥✐♦r
❚❡s❡ s✉❜♠❡t✐❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛✱ s♦❜
♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❆✉❣✉st♦
❘♦♠❡r♦ ❋✐❧❤♦✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s
♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❋í✲
s✐❝❛✳
❏♦ã♦ P❡ss♦❛
A474m Alves Júnior, Francisco Artur Pinheiro.
Modelos cosmológicos numa teoria geométrica escalar - tensorial da gravitação: aspectos clássicos e quânticos / Francisco Artur Pinheiro Alves Júnior.- João Pessoa, 2016.
96 f. :
Orientador: Profº. Drº. Carlos Augusto Romero Filho. Tese (Doutorado) – UFPB/CCEN
1. Teoria Escalar Tensorial Geométrica. 2. Geometria de Weyl. 3. Transição de Fase Geométrica. 4. Dimensões Extras.
5. Compactação Dinâmica. I. Título.
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Doutorado do aluno Francisco Artur
Pinheiro Alves Júnior, candidato ao Título de
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Física das Partículas Elementares e Campos.
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1 Aos vinte e sete dias do mês de setembro do ano de dois mil e dezesseis, às 10hOO,no
2 Auditório da Pós-Graduação em Física do Centro de Ciências Exatas e da Natureza da
3 Universidade Federal da Paraíba, reuniram-se os membros da Banca Examinadpra
4 constituída para examinar o candidato ao grau de Doutor em Física na área de Física das
5 Partículas Elementares e Campos, Francisco Artur Pinheiro Alves Júnior. A
6 comissão examinadora foi composta pelos professores doutores: Ca rlos Augusto
7 Romero F ilho (UFPB), orientador e presidente da ban a examinadora, Va ldir Ba rbosa
8 Bezerra (UFPB), F á bio Lea l de Melo Da hia (UFPB), _'fa ria Emília Xa vier Guima rã es
9 (UFF) e Henrique P ereira de Oliveira (UERJ). Dand . 'cio aos trabalhos, o Prof.
lOCa ria s Augusto Romero F ilho comunicou aos presentes a finalidade da reunião. A
11 seguir, passou a palavra ao candidato para que o mesmo fiz~-e. oralmente, a exposição
12 do trabalho de tese intitulado ;'_'vJodeloscosmológicos nu 'oria geométrica esca la
r-13 tensoria l da gra vita çã o: a spectos clá ssicos e quâ ntico ". Concluída a exposição, o
14 candidato foi argüido pela Banca Examinadora, qUe o seguinte parecer:
15 "aprovado". Assim sendo, deye a Universidade Federa... ?anu a expedir o respectivo
16 diploma de Doutor em Física na forma da lei. E para onstar. Danilo Wilson Lemos
17 Menezes, servindo de Secretário, redigiu a presente a ue "ai assinada pelo mesmo e
18 pelos membros da Banca Examinadora. João Pessoa. Parruba, 27 de setembro de 2016.
✐✐✐
❆❣r❛❞❡❝✐♠❡♥t♦s
• ❆❣r❛❞❡ç♦ à ❉❡✉s✳
• ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉ ♣❛✐s ❡ ❢❛♠✐❧✐❛r❡s ♣♦r t♦❞♦ ❛♣♦✐♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♠ã❡ ❛ ♠❡✉
✐r♠ã♦✱ ❊♠❛♥✉❡❧ ❆❜❞❛❧❧❛✱ ❡ ❛ ♠❡✉ ♣❛✐✱ ❡ t❛♠❜é♠ ❛♦ ❚✐♦ ❍é❧❞❡r ❡ à ❚✐❛ ❊t✐❡♥❡✳ ❊st❡♥❞♦ ❡st❡s ❛❣r❛❞❡❝✐♠❡♥t♦s à ❚✐❛ ■❞ê✱ ❡ r❡❢♦rç♦ ♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛♦s ♣r✐♠♦s ❋✐❧✐♣❡✱ P❡❞r♦✱ ❉❛♥✐❡❧✱ ❘❛❢❛❡❧ ❡ ❛♦ ❚✐♦ ▼❡ss✐❛s ❡ à ❚✐❛ ❆❧❞❡♥✐❛ ❡ ❛s ♠✐♥❤❛s ❛✈ós✳
• ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❈❛r❧♦s ❘♦♠❡r♦ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ♣❡❧♦ ❛♣r❡♥✲
❞✐③❛❞♦✳ ❊ ❛♦s ♣r♦❢❡ss♦r❡s ❏♦❡❧ ❇❛t✐st❛✱ ❋á❜✐♦ ❉❛❤✐❛✱ ❊❞♠✉♥❞♦ ▼♦♥t❡✱ ❡ ❱❛❧❞✐r ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡♠ s❡♠♣r❡ ❛❥✉❞❛r✳
• ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❛♠✐❣❛ ▲❛✉r❛ P✉❝❤❡✉✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦✱ ❞❡t❡r♠✐♥❛çã♦ ❡ ♣♦r s✉❛
❛❥✉❞❛ ❡ ❝♦❧❛❜♦r❛çã♦ ♥♦ ♠❡✉ ❞♦✉t♦r❛❞♦✳
• ❆❣r❛❞❡ç♦ ❛ ♠❡✉ ❛♠✐❣♦ ❆❞r✐❛♥♦ ❇r❛❣❛ ♣♦r s❡r ✉♠ ❡①❝❡❧❡♥t❡ ♣r♦❢❡ss♦r ❞❡ ❈♦s♠♦❧♦❣✐❛
◗✉â♥t✐❝❛ ❡ ♣♦r s✉❛ ♣❛rt✐❝✐♣❛çã♦ ❡ss❡♥❝✐❛❧ ♥❛ s♦❧✉çã♦ ❞❡ q✉❡stõ❡s q✉❡ ❞❡r❛♠ ♦r✐❣❡♠ ❛♦s ♥♦ss♦s tr❛❜❛❧❤♦s✳
• ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ▲❛ér❝✐♦ ♣❡❧❛ ❣r❛♥❞❡ ❛❥✉❞❛ ♣❛r❛ ♠❡✉ ✐♥❣r❡ss♦ ♥♦ ❞♦✉t♦r❛❞♦✳
• ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❉✐♦♥ís✐♦ ❡ ❛♦ ♣r♦❢❡ss♦r ❙ér❣✐♦✳
• ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❯❋P❇✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❆❞✐❡❧ ▲❡♠♦s✱ ❚❤❛ís ❆❦❡♠✐✱
✐✈
❏ú❧✐♦ ❈és❛r✱ ●✐ ❱❛rã♦ ❡ P❛✉❧♦ ❍❡♥r✐q✉❡ ✭P❍✮✳ ●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r t❛♠❜é♠ ❛♦s ♠❡✉s ❛♠✐❣♦s ❋r❛♥❝✐♥❛❧❞♦✱ ❡ ❉❛♥♥✐❧♦✱ ❡st❡ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ♣♦r ❙❛❧❣❛❞✐♥❤♦✳
• ❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❯❋❈✳
• ❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❙♦♣❛r✐❛ ❡ ▲❛♥❝❤♦♥❡t❡ ❩♦♥❛ ❙✉❧✳
• ❆♦s ♠❡✉s ❛♠✐❣♦s q✉❡ ✜③ ❡♠ t♦❞♦s ♦s ❧✉❣❛r❡s✿ ♥♦ ▲ú♠❡♥✱ ♥♦ ❆✐❦✐❞♦✱ ♥♦ ❑❛r❛tê✱ ♥❛
♥❛t❛çã♦✳
• ❆❣r❛❞❡ç♦ à ❈❛♣❡s✳
✈
✧◆ã♦ ❤á ♣❡♥✐tê♥❝✐❛ ♠❡❧❤♦r ❞♦ q✉❡ ❛q✉❡❧❛ q✉❡ ❉❡✉s ❝♦❧♦❝❛ ❡♠ ♥♦ss♦ ❝❛♠✐♥❤♦ t♦❞♦s ♦s ❞✐❛s✳✧ ❉♦♠ ❍é❧❞❡r ❈â♠❛r❛
✧❙❡✐ q✉❡ ♦ ♠❡✉ tr❛❜❛❧❤♦ é ✉♠❛ ❣♦t❛ ♥♦ ♦❝❡❛♥♦✱ ♠❛s s❡♠ ❡❧❡ ♦ ♦❝❡❛♥♦ s❡r✐❛ ♠❡♥♦r✳✧ ▼❛❞r❡ ❚❡r❡s❛ ❞❡ ❈❛❧❝✉tá
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐✐
✶ ■♥tr♦❞✉çã♦ ✸
✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✻
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✷✳✶✳✶ ❱❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✷✳✶✳✷ ❈♦♥❡①ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷✳✶✳✸ ▼étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✶✳✹ ❈✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✶✳✺ ❉❡s✈✐♦ ❣❡♦❞és✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✷✳✶ ❈♦♥❡①ã♦ ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✷✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✷✳✸ ❲❡②❧ ✐♥t❡❣rá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✸✳✶ ❆ ♦❜❥❡çã♦ ❞❡ ❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
❙❯▼➪❘■❖ ✈✐✐
✸ ❚❡♦r✐❛s ❊s❝❛❧❛r❡s✲❚❡♥s♦r✐❛✐s ✷✸
✸✳✶ ❚❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✷ ❚❡♦r✐❛s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ❈♦♥❢♦r♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸✳✸✳✶ ❆s♣❡❝t♦s ❣❡♦♠étr✐❝♦s ❞❛ tr❛♥s❢♦r♠❛çã♦ ❝♦♥❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸✳✸✳✷ ❆ tr❛♥s❢♦r♠❛çã♦ ❝♦♥❢♦r♠❡ ♥❛ ❛çã♦ ❞❛ t❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✐❛❧ ✳ ✳ ✳ ✷✽
✸✳✹ ❊q✉❛çõ❡s ❞❡ ❈❛♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✺ ❊q✉❛çõ❡s ❝♦s♠♦❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✻ ▲✐♠✐t❡s s♦❜r❡ ω ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✹ ❚❡♦r✐❛ ❊s❝❛❧❛r✲❚❡♥s♦r✐❛❧ ●❡♦♠étr✐❝❛ ✸✹
✹✳✶ ❚❡♦r✐❛s ❲■❙❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✹✳✷ ❋♦r♠❛❧✐s♠♦ ❞❡ P❛❧❛t✐♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹✳✸ ❚❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❣❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✹ ❊q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✺ ❙♦❧✉çõ❡s ❡st✉❞❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✺ ❈♦s♠♦❧♦❣✐❛ ■✿ ❚r❛♥s✐çã♦ ❞❡ ❋❛s❡ ❣❡♦♠étr✐❝❛ ✹✶
✺✳✶ ▼♦❞❡❧♦s ❝♦s♠♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✺✳✶✳✶ ❙♦❧✉çõ❡s ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞♦ s✉♣❡r✲♣♦t❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✺✳✷ ❆ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✺✳✷✳✶ ❙♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✺✳✷✳✷ ❉✐❛❣r❛♠❛s ❞❡ ❢❛s❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✺✳✸ P♦t❡♥❝✐❛❧ mφ2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✺✳✹ P♦t❡♥❝✐❛❧ q✉árt✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✺✳✹✳✶ P♦t❡♥❝✐❛❧ ❡①♣♦♥❡♥❝✐❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✺✳✺ ❆ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❣❡♦♠étr✐❝❛ ❡ ❛ ❛❝❡❧❡r❛çã♦ t❛r❞✐❛ ❞♦ ✉♥✐✈❡rs♦ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❙❯▼➪❘■❖ ✈✐✐✐
✺✳✺✳✷ ❯♠❛ ✈❛r✐❛çã♦ ❞♦ ❝❛s♦ ♣❂✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✻ ❈♦s♠♦❧♦❣✐❛ ■■ ✿ ❯♥✐✈❡rs♦s ❝♦♠ ❞✐♠❡♥sõ❡s ❡①tr❛s ✻✶
✻✳✶ ❯♥✐✈❡rs♦ ♠♦❞❡❧❛❞♦ s❡❣✉♥❞♦ ❈❤♦❞♦s✲❉❡t✇❡✐❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✻✳✷ ❇r❛♥s✲❉✐❝❦❡ ❣❡♦♠étr✐❝♦ ❡♠ ♥ ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✻✳✸ ▼♦❞❡❧♦ ❝♦s♠♦❧ó❣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✻✳✹ ❙♦❧✉çõ❡s ❝❧áss✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✻✳✹✳✶ ❙♦❧✉çõ❡s s✐♥❣✉❧❛r❡s ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✻✳✹✳✷ ❙♦❧✉çõ❡s s✐♥❣✉❧❛r❡s ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✻✳✹✳✸ ❙♦❧✉çõ❡s t✐♣♦ ❞❡ ❙✐tt❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✻✳✹✳✹ ❆♥á❧✐s❡ ❞♦ ❢❛t♦r ❞❡ ❡①♣❛♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✼ ❈♦s♠♦❧♦❣✐❛ q✉â♥t✐❝❛ ✼✵
✼✳✶ ❆s♣❡❝t♦s ❣❡r❛✐s ❞❛ ❈♦s♠♦❧♦❣✐❛ ◗✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
✼✳✶✳✶ ■♥t❡r♣r❡t❛çõ❡s ❞❛ ❈♦s♠♦❧♦❣✐❛ ◗✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✼✳✶✳✷ ❖ ♣r♦❜❧❡♠❛ ❞♦ t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✼✳✷ ❊①❛♠✐♥❛♥❞♦ ♦ ✉♥✐✈❡rs♦ ♥ ❞✐♠❡♥s✐♦♥❛❧ ❛♥✐s♦tró♣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✼✳✷✳✶ ❖ ❤❛♠✐❧t♦♥✐❛♥♦ ❝❧áss✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✼✳✸ ◗✉❛♥t✐③❛♥❞♦ ♦ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
✼✳✹ ❙♦❧✉çõ❡s ❡ ✈❛❧♦r❡s ❡s♣❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✼✳✹✳✶ ❱❛❧♦r❡s ❡s♣❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽
✼✳✹✳✷ ❈❛s♦ n = 5✱ ♦ ✉♥✐✈❡rs♦ ❝♦♠ ✉♠❛ ❞✐♠❡♥sã♦ ❡①tr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾
✼✳✺ ❆ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ❞❡ ❇r♦❣❧✐❡✲❇♦❤♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
✼✳✺✳✶ ▼❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❇♦❤♠✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
✼✳✺✳✷ ❈♦s♠♦❧♦❣✐❛ q✉â♥t✐❝❛ s❡❣✉♥❞♦ ❞❡ ❇r♦❣❧✐❡✲❇♦❤♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
✼✳✺✳✸ ❯♥✐✈❡rs♦ ❡♠ ❝✐♥❝♦ ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸
❙❯▼➪❘■❖ ✐①
❘❊❙❯▼❖
◆❡st❛ t❡s❡✱ r❡❛♣r❡s❡♥t❛♠♦s ✉♠❛ t❡♦r✐❛ ❡s❝❛❧❛r t❡♥s♦r✐❛❧ ❣❡♦♠étr✐❝❛✱ q✉❡ é ✉♠❛ ✈❡rsã♦ ❞❛ ❣r❛✈✐t❛çã♦ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❢♦r♠✉❧❛❞❛ ❡♠ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ❈♦♠ ❡st❛ t❡♦r✐❛ ❢❛③❡♠♦s ❞✉❛s ❛♣❧✐❝❛çõ❡s ❡s♣❡❝í✜❝❛s✳ ❯♠❛ ❞❡❧❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✉♠ ❢❡♥ô♠❡♥♦✱ q✉❡ ❝❤❛♠❛♠♦s ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❣❡♦♠étr✐❝❛✱ ✉♠❛ ♠✉❞❛♥ç❛ ❝♦♥tí♥✉❛ ♥❛ ❡str✉t✉r❛ ❣❡✲ ♦♠étr✐❝❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❊st❡ ❢❡♥ô♠❡♥♦ ♣❛r❡❝❡ ♦❝♦rr❡r q✉❛♥❞♦ ♦ ✉♥✐✈❡rs♦ s❡ ❡①♣❛♥❞❡ ❛❝❡❧❡r❛❞❛♠❡♥t❡✳ ❆ s❡❣✉♥❞❛ ❛♣❧✐❝❛çã♦ r❡s✐❞❡ ♥♦ ❡st✉❞♦ ❝❧áss✐❝♦ ❡ q✉â♥t✐❝♦ ❞♦ ❝♦♠♣♦r✲ t❛♠❡♥t♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ✉♥✐✈❡rs♦ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❛♥✐s♦tró♣✐❝♦✳ ❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡st❛ ✐♥✈❡st✐❣❛çã♦ é ❛ ❜✉s❝❛ ❞❡ s♦❧✉çõ❡s q✉❡ ❡①✐❜❡♠ ♦ ❝♦♠♣❛❝t❛çã♦ ❞✐♥â♠✐❝❛ ❞❛s ❞✐♠❡♥sõ❡s ❡①tr❛s✱ q✉❡ ♥ã♦ sã♦ ♦❜s❡r✈❛❞❛s✳
❆❇❙❚❘❆❈❚
■♥ t❤✐s t❤❡s✐s✱ ✇❡ ❞❡❛❧ ✇✐t❤ ❛ ♣❛rt✐❝✉❧❛r ❣❡♦♠❡tr✐❝ s❝❛❧❛r t❡♥s♦r t❤❡♦r②✱ ✇❤✐❝❤ ✐s ❛ ✈❡rs✐♦♥ ♦❢ t❤❡ ❇r❛♥s✲❉✐❝❦❡ ❣r❛✈✐t❛t✐♦♥✱ ❢♦r♠✉❧❛t❡❞ ✐♥ ❛ ❲❡②❧ ✐♥t❡❣r❛❜❧❡ s♣❛❝❡✲t✐♠❡✳ ❚❤✐s ❢♦r♠✉❧❛t✐♦♥ ✐s ❞♦♥❡ ✉s✐♥❣ t❤❡ P❛❧❛t✐♥✐✬s ✈❛r✐❛t✐♦♥ ♣r♦❝❡❞✉r❡✳ ❚❤❡ ♠❛✐♥ ♣♦✐♥t ♦❢ ♦✉r ✇♦r❦ ✐s t♦ ♣❡r❢♦r♠ t✇♦ ♣❛rt✐❝✉❧❛r ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ❇r❛♥s✲❉✐❝❦❡ t❤❡♦r②✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ st✉❞② ♦❢ ❣❡♦♠❡tr✐❝ ❢❛s❡ tr❛♥s✐t✐♦♥ ♣❤❡♥♦♠❡♥❛✱ t❤❛t✬s r❡❧❛t❡❞ t♦ ❛ ❝♦♥t✐♥✉♦✉s ❝❤❛♥❣❡ ✐♥ t❤❡ s♣❛❝❡✲t✐♠❡ str✉❝t✉r❡ ♦❢ t❤❡ ✉♥✐✈❡rs❡ ❢r♦♠ ❛ ❘✐❡♠❛♥♥✬s ❣❡♦♠❡tr② t♦ ❛ ❲❡②❧✬s ❣❡♦♠❡tr②✱ ♦r ✐♥ t❤❡ ✐♥✈❡rs❡ s❡♥s❡✱ ❢r♦♠ ❲❡②❧✬s ❣❡♦♠❡tr② t♦ ❘✐❡♠❛♥♥✬s ❣❡♦♠❡tr②✳ ❚❤✐s ♣❤❡♥♦♠❡♥❛ s❡❡♠s t♦ t❛❦❡ ♣❧❛❝❡ ✇❤❡♥ t❤❡ ✉♥✐✈❡rs❡ st❛rts t♦ ❡①♣❛♥❞ ✐♥ ❛ ❛❝❝❡❧❡r❛t❡❞ r❛t❡✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❜❡❤❛✈✐♦✉r ♦❢ ❛ ❛♥✐s♦tr♦♣✐❝ ♥✲❞✐♠❡♥s✐♦♥❛❧ ✉♥✐✈❡rs❡ ✳ ❚♦ ✜♥❞ s♦❧✉t✐♦♥s t❤❛t ❞✐s♣❧❛② t❤❡ ❞②♥❛♠✐❝❛❧ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ ♥♦♥ ♦❜s❡r✈❡❞ ❡①tr❛ ❞✐♠❡♥s✐♦♥s ✐s t❤❡ ♠❛✐♥ ♠♦t✐✈❛t✐♦♥ t♦ st✉❞② s✉❝❤ ✉♥✐✈❡rs❡✳
✸
❈❆P❮❚❯▲❖
✶
■♥tr♦❞✉çã♦
❊♠ ❋ís✐❝❛✱ ❛ ♣❛❧❛✈r❛cosmologiar❡❢❡r❡✲s❡ ❛ ✉♠❛ ❝✐ê♥❝✐❛ ❝✉❥♦ ♦❜❥❡t✐✈♦ ✏é ♦ ❡st✉❞♦ ❞♦ ✉♥✐✈❡rs♦ ❝♦♠♦ ✉♠ t♦❞♦✿ s✉❛ ❡✈♦❧✉çã♦✱ s✉❛ ❤✐stór✐❛✱ s✉❛ ❝♦♠♣♦s✐çã♦✑✱ ❬✶❪✳ ➱ ❡st❡ r❛♠♦ ❝✐❡♥tí✜❝♦ q✉❡ ❡①♣❧✐❝❛ ❛ ❢♦r♠❛çã♦ ❞❛s ❡str✉t✉r❛s ❝♦♠♦ ❛s ❣❛❧á①✐❛s✱ ❛s ❡str❡❧❛s ❡ ♦s ♣❧❛♥❡t❛s✳
❖s ❝♦s♠ó❧♦❣♦s r❡❛❧✐③❛♠ s❡✉s ❡st✉❞♦s ♣❛rt✐♥❞♦ ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠♦❞❡❧♦s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ✉♥✐✈❡rs♦✳ ❯♠❛ ❤✐♣ót❡s❡✱ ❜❛s❡❛❞❛ ♥❛ ♦❜s❡r✈❛çã♦ ❡ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❝♦♥s✲ tr✉çã♦ ❞❡st❡s ♠♦❞❡❧♦s✱ ♦ ♣r✐♥❝í♣✐♦ ❝♦s♠♦❧ó❣✐❝♦✱ ❛✜r♠❛ q✉❡ ♦ ✉♥✐✈❡rs♦ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛ é ❤♦♠♦❣ê♥❡♦ ❡ ✐s♦tró♣✐❝♦✳
❖✉tr♦ ✐♥❣r❡❞✐❡♥t❡ ✐♠♣r❡s❝✐♥❞í✈❡❧ ♥❛ r❡♣r❡s❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❞♦ ✉♥✐✈❡rs♦ é ❛ ❡s❝♦❧❤❛ ❞❛ ❧❡✐ ❞❡ ❣r❛✈✐t❛çã♦✳ ❈❛♥♦♥✐❝❛♠❡♥t❡✱ ♦s ♠♦❞❡❧♦s ❡♠ ❝♦s♠♦❧♦❣✐❛ ❛❞♦t❛♠ ❝♦♠♦ t❡♦r✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✱ q✉❡ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❆❧❜❡rt ❊✐♥st❡✐♥ ❡♠ ✶✾✶✺✳ ◆❡st❛ t❡♦r✐❛✱ ❛ ✐♥t❡r❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ é ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠❛ ♠❛♥✐❢❡st❛çã♦ ❞❛ ❞✐♥â♠✐❝❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✱ r❡✈❡❧❛♥❞♦ ♣♦rt❛♥t♦✱ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❣❡♦♠❡tr✐❛ ♥❛ ❣r❛✈✐t❛çã♦✳ ❆ ❣r❛♥❞❡ q✉❡stã♦ é q✉❡ ❛ ❝♦s♠♦❧♦❣✐❛✱ ♥❡st❡s ♠♦❧❞❡s✱ ✈❡♠ ❡♥❝♦♥tr❛♥❞♦ ❧✐♠✐t❛çõ❡s ♦✉ ❞✐✜❝✉❧❞❛❞❡s ❡♠ ❡①♣❧✐❝❛r ❞❡t❡r♠✐♥❛❞❛s q✉❡stõ❡s✳ ❉❡♥tr❡ ❡st❛s✱ ♣♦❞❡♠♦s ❝✐t❛r ♦s ♣r♦✲ ❜❧❡♠❛s ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ✐♥✐❝✐❛✐s ♥♦ ✉♥✐✈❡rs♦ ❬✷✱✸❪ ❡ ♦s ♣r♦❜❧❡♠❛s r❡❧❛t✐✈♦s à ❢❛s❡ ❞❡ ❛❝❡❧❡r❛çã♦ ❝ós♠✐❝❛ ❛t✉❛❧ ❬✹✱✺❪✳
✹
❝❛❞❛s ❡♠ ❝♦s♠♦❧♦❣✐❛✳ ❉❡♥tr❡ ❡ss❛s t❡♦r✐❛s ❡①✐st❡♠ ❛q✉❡❧❛s q✉❡ ✉t✐❧✐③❛♠ ❛ ❢♦r♠❛ ❞❛ ❛çã♦ ❞❛ ❊✐♥st❡✐♥✲❍✐❧❜❡rt✱ ♠❛s sã♦ ❛♠❜✐❡♥t❛❞❛s ❡♠ ❣❡♦♠❡tr✐❛s ♥ã♦✲r✐❡♠❛♥♥✐❛♥❛s✱ ❝♦♠♦ ♣♦r
❡①❡♠♣❧♦ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❢♦r♠✉❧❛❞❛ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❬✼❪✳ ❈♦♠♦ t❛♠❜é♠ ❤á
❢♦r♠✉❧❛çõ❡s q✉❡ ♠❛♥tê♠ ❛ ❡str✉t✉r❛ ❣❡♦♠étr✐❝❛ r✐❡♠❛♥♥✐❛♥❛✱ ♠❛s q✉❡ ♣♦ss✉❡♠ ❛çõ❡s ❞✐st✐♥t❛s ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✳ ❆s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❡①❡♠♣❧✐✜❝❛♠ ❜❡♠ ❡st❛ ❝❧❛ss❡✳
❆s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s t❡♠ ✉♠❛ ✈❛♥t❛❣❡♠ ❡♠ r❡❧❛çã♦ à ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✭❘●✮✱ ♣♦✐s ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛ tr❛t❛r♠♦s ❞❡ ❝❡♥ár✐♦s ✐♥✢❛❝✐♦♥ár✐♦s ✉s❛♥❞♦ ❛ ❘● ❞❡✲ ✈❡♠♦s ❛❝r❡s❝❡♥t❛r ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r ♣❛r❛ ❣❡r❛r ✐♥✢❛çã♦ ❬✽❪ ✱ ❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s t❡♥s♦r✐❛✐s ❥á ♣♦ss✉❡♠ ❡♠ s✉❛ ❝♦♥str✉çã♦ ❡st❡ ❝❛♠♣♦ ❡s❝❛❧❛r ❬✾❪✳
❖✉tr❛s t❡♦r✐❛s ❛❧t❡r♥❛t✐✈❛s ♣♦❞❡♠ s❡r ❢♦r♠❛❞❛s t❛♥t♦ ✉t✐❧✐③❛♥❞♦ ❛çõ❡s ❞✐❢❡r❡♥✲ t❡s ❞❛ t❡♦r✐❛ ❞❡ ❊✐♥st❡✐♥ q✉❛♥t♦ ✉t✐❧✐③❛♥❞♦ ❣❡♦♠❡tr✐❛s ♥ã♦✲r✐❡♠❛♥♥✐❛♥❛s✳ P❛r❛ ❝♦♥❢❡❝❝✐♦✲ ♥❛r t❛✐s ♠♦❞❡❧♦s✱ ♦s ❢ís✐❝♦s ❞✐s♣õ❡♠ ❞❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠✉✐t♦ út✐❧✱ ♦ ♠ét♦❞♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ P❛❧❛t✐♥✐✱ q✉❡ ♣❡r♠✐t❡ q✉❡ ❝❛❞❛ ❛çã♦ ❡st❡❥❛ r❡❧❛❝✐♦♥❛❞❛ ❛ ✉♠ t✐♣♦ ❡s♣❡❝í✜❝♦ ❞❡ ❣❡♦♠❡✲ tr✐❛ ❬✶✵✱✶✶❪✳
❆♦ ❛♣❧✐❝❛r♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞❡ P❛❧❛t✐♥✐ ❡♠ t❡♦r✐❛s ❡s❝❛❧❛r❡s t❡♥s♦r✐❛✐s✱ ❞❛♠♦s ❧✉③ às ❢♦r♠✉❧❛çõ❡s ❞✐st✐♥t❛s ❞❛ ❣r❛✈✐t❛çã♦✱ ❛♠❜✐❡♥t❛❞❛s ❡♠ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ❊st❛s sã♦ ❛s ❝❤❛♠❛❞❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❣❡♦♠étr✐❝❛s ❬✶✷❪✳
❆s ❣❡♦♠❡tr✐❛s ❞❡ ❲❡②❧ ❢♦r❛♠ ♣r✐♠❡✐r❛♠❡♥t❡ ♣r♦♣♦st❛s ❡♠ ✶✾✶✽ ❬✶✸❪ ❝♦♠♦ ♣❛rt❡ ❡ss❡♥❝✐❛❧ ❞❡ ✉♠❛ ♥♦✈❛ t❡♦r✐❛ ❞❛ ❣r❛✈✐t❛çã♦✳ ❊st❛ t❡♦r✐❛ t✐♥❤❛ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❣❡♦♠❡tr✐③❛çã♦ t❛♥t♦ ♣❛r❛ ❛ ❣r❛✈✐t❛çã♦ q✉❛♥t♦ ♣❛r❛ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦✳ ❈♦♥t✉❞♦✱ s❡✉ ♠♦❞❡❧♦ ❞❡ ❣r❛✈✐t❛çã♦ ♥ã♦ tr✐✉♥❢♦✉ ❞❡✈✐❞♦ às sér✐❛s ❝rít✐❝❛s ❧❡✈❛♥t❛❞❛s ♣♦r ❊✐♥st❡✐♥✳ ❊♠❜♦r❛ ❛ t❡♦r✐❛ ❞❡ ❲❡②❧ t❡♥❤❛ s✉❝✉♠❜✐❞♦✱ s✉❛ ❣❡♦♠❡tr✐❛ r❡❛♣❛r❡❝❡ ❝♦♠♦ ♣❛❧❝♦ ♣❛r❛ ♠✉✐t❛s ♣❡sq✉✐s❛s ❢ís✐❝❛s q✉❡ ✈ã♦ ❞❡s❞❡ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ às t❡♦r✐❛s ❞❡ ❝❛♠♣♦s ❬✶✹✱✶✺❪✳
❊♠ ❝♦s♠♦❧♦❣✐❛✱ ❛s ❣❡♦♠❡tr✐❛s ❞❡ ❲❡②❧ sã♦ ❛♣❧✐❝❛❞❛s ❡♠ ❝❡♥ár✐♦s ❞✐st✐♥t♦s✳ P♦r s❡r❡♠ ♣♦ssí✈❡✐s ❝❛♥❞✐❞❛t❛s ❛ ❡①♣❧✐❝❛r ❛ ❛❝❡❧❡r❛çã♦ ❞♦ ✉♥✐✈❡rs♦ ❬✶✻❪✱ ♠✉✐t♦ s❡ t❡♠ ♣❡sq✉✐s❛❞♦ s♦❜r❡ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ♥❛s t❡♦r✐❛s ❞❡ ✉♥✐✈❡rs♦s ✐♥✢❛❝✐♦♥ár✐♦s ❬✶✼✱✶✽❪✱ ❜❡♠ ❝♦♠♦ ♥❛ ❜✉s❝❛ ❞❡ s♦❧✉çõ❡s ❝♦♠♣❛tí✈❡✐s ❝♦♠ ❛s ♦❜s❡r✈❛çõ❡s ❞❛ ❡①♣❛♥sã♦ ❛❝❡❧❡r❛❞❛ ❬✶✾✱✷✵❪✳
✺
✻
❈❆P❮❚❯▲❖
✷
●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧
◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ❛ ❣❡♦♠❡tr✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❲❡②❧ ❝♦♠❡ç❛♥❞♦ ❝♦♠ ✉♠❛ r❡✈✐sã♦ ❞❛s ♣r✐♥❝✐♣❛✐s ❡str✉t✉r❛s ❞❛ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛✱ ♥♦s ❣✉✐❛♥❞♦ ♣♦r ❬✷✸❪ ❡ ♣❡❧❛s r❡❢❡rê♥❝✐❛s ❝♦♥t✐❞❛s ♥♦ ❝❛♣ít✉❧♦✳ ❊♠ s❡❣✉✐❞❛✱ ❛❜♦r❞❛♠♦s ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❡♠ s✉❛s ❞✉❛s ❢♦r♠❛s✿ ♥ã♦ ✐♥t❡❣rá✈❡❧ ❡ ✐♥t❡❣rá✈❡❧✳ P♦r ✜♠✱ ♠♦str❛♠♦s ❛ t❡♦r✐❛ ❞❛ ❣r❛✈✐t❛çã♦ ❡❧❛❜♦r❛❞❛ ♣♦r ❲❡②❧ ❡ ❛ ♦❜❥❡çã♦ ❞❡ ❊✐♥st❡✐♥ à ❡st❛ ❢♦r♠✉❧❛çã♦✳
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛
✷✳✶✳✶
❱❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧
❊♠ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❡ ♦ ♠♦❞❡❧♦ ♠❛t❡♠á✲ t✐❝♦ q✉❡ ✉t✐❧✐③❛♠♦s ✈❡♠ ❞❛ ❣❡♦♠❡tr✐❛✳ ▼♦❞❡❧❛♠♦s ♦ ❡s♣❛ç♦✲t❡♠♣♦ ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥✲❞✐♠❡♥s✐♦♥❛❧ é ✉♠ ❡s♣❛ç♦ ❝♦♥tí♥✉♦ q✉❡✱ ❧♦✲ ❝❛❧♠❡♥t❡✱ s❡ ❛ss❡♠❡❧❤❛ ❛♦ Rn.
❙♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡M ❞❡✜♥✐♠♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s ❝♦♠♦ ❢✉♥çõ❡s✱ ✈❡t♦r❡s ❡
t❡♥s♦r❡s✳ ❯♠❛ ❢✉♥çã♦ r❡❛❧f ❡♠M é ✉♠❛ ❛tr✐❜✉✐çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦P ❞❡ ▼✱ ✉♠
♥ú♠❡r♦ r❡❛❧ f(P)✳ ❯♠ ✈❡t♦rVP é ✉♠ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ s♦❜r❡ ✉♠ ♣♦♥t♦ P ❞❛ ✈❛r✐❡❞❛❞❡✱
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✼
t❛♥❣❡♥t❡ ❡♠ P✱ TPM✳ ❖ ❡s♣❛ç♦ ❞✉❛❧ ❛♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ TPM é ♦ ❡s♣❛ç♦ ❝♦t❛♥❣❡♥t❡✱
T∗
PM✳ ◆❡❧❡ t❡♠♦s ♦s ❝♦✲✈❡t♦r❡s ♦✉ ✶✲❢♦r♠❛s✳ ❚❡♠♦s ❛✐♥❞❛ ❡s♣❛ç♦s ❢♦r♠❛❞♦s ♣❡❧♦ ♣r♦❞✉t♦
t❡♥s♦r✐❛❧ s✉❝❡ss✐✈♦ ❞❡ m ❡s♣❛ç♦s TpM ❝♦♠ n ❡s♣❛ç♦s Tp∗M✱ ♦s ❡❧❡♠❡♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛
❡ss❡ ❝♦♥❥✉♥t♦s sã♦ ❝❤❛♠❛❞♦s t❡♥s♦r❡s ❞♦ t✐♣♦(m, n)✳ ◆❛ ✈❛r✐❡❞❛❞❡ ❛✐♥❞❛ ♣♦❞❡♠♦s ❞❡✜♥✐r ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s✱ ❝❛♠♣♦s ❞❡ ✶✲❢♦r♠❛s ❡ ❝❛♠♣♦s ❞❡ t❡♥s♦r❡s✳ ❖s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s é
❞❡♥♦t❛❞♦ ♣♦r T M✱ ♦ ❝❛♠♣♦ ❞❡ ✶✲❢♦r♠❛s é ❞❡♥♦t❛❞♦ ♣♦r T∗M ❡ ♦ ❝❛♠♣♦ ❞❡ t❡♥s♦r❡s ❞♦
t✐♣♦ (m, n) é ❞❡♥♦t❛❞♦ ♣♦rτm
n (M)✳
✷✳✶✳✷ ❈♦♥❡①ã♦
❊♠ ✉♠❛ ✈❛r✐❡❞❛❞❡M ♥ã♦ ♣♦❞❡♠♦s ❝♦♠♣❛r❛r ✈❡t♦r❡s ❡♠ ♣♦♥t♦s ❞✐st✐♥t♦s✱ ♣♦✐s
sã♦ ♣❡rt❡♥❝❡♥t❡s à ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞✐❢❡r❡♥t❡s✳ P❛r❛ ❢❛③❡r♠♦s ✐ss♦ ♣r❡❝✐s❛♠♦s ❞❡ ✉♠❛ r❡✲ ❣r❛ ♣❛r❛ ♠♦✈❡r ✈❡t♦r❡s ♥❛ ✈❛r✐❡❞❛❞❡ ❡ ❝♦❧♦❝á✲❧♦s ♥♦ ♠❡s♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ◆♦r♠❛❧♠❡♥t❡✱ ❞♦t❛♠♦s ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ✉♠❛ ❝♦♥❡①ã♦ ❡ ❛ss✐♠ ✉♠❛ ❝♦♠♣❛r❛çã♦ ♣♦❞❡ s❡r ❡st❛❜❡❧❡❝✐❞❛✳
❉❡✜♥✐çã♦ ✿ ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠✱ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ é ✉♠❛ ❛♣❧✐❝❛çã♦ ∇ :
T(M)× T(M) −→T(M)✱ ✐♥❞✐❝❛❞♦ ♣♦r (X, Y)−→ ∇XY ❡ q✉❡ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s✿
a)∇f X+gYU =f∇XU +g∇YU,
b)∇X(U +V) = ∇XU +∇XV,
c)∇V(f U) = V[f]U +f∇VU.
✭✷✳✶✮
❆ q✉❛♥t✐❞❛❞❡ ∇XY é ❝❤❛♠❛❞❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❞♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧Y ❝♦♠
r❡s♣❡✐t♦ ❛ ❞✐r❡çã♦ ❞❡ X✳
❉❛❞❛ ✉♠❛ ❜❛s❡ {E1, E2.., En} ♣❛r❛ T(M)✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛✲
r✐❛♥t❡ ❞❡ ✉♠ ✈❡t♦r ❞❛ ❜❛s❡ ❝♦♠ r❡s♣❡✐t♦ ❛ ♦✉tr♦✱ ♣❡❧❛ ❡①♣r❡ssã♦
∇EiEj = Γ
k
ijEk. ✭✷✳✷✮
❖ t❡r♠♦Γk
ij ❝♦rr❡s♣♦♥❞❡ às ❝♦♠♣♦♥❡♥t❡s ❞❛ ❝♦♥❡①ã♦ ❡♠ ✉♠❛ ❞❛❞❛ ❜❛s❡✳ ❊s❝♦✲
❧❤❡♥❞♦ ✉♠❛ ❜❛s❡ ❞❡ ❝♦♦r❞❡♥❛❞❛s {∂i}✱ t❡♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡
∇YX =∇Yi∂ i(X
j∂
j) =Yi ∂iXj + ΓjikXk
∂j. ✭✷✳✸✮
❆s ❝♦♠♣♦♥❡♥t❡s✱∂iXj+ ΓjikXk✱ ❞❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ sã♦ ✉t✐❧✐③❛❞❛s ♥♦s ❧✐✈r♦s ❞❡ ❝á❧❝✉❧♦
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✽
❆tr❛✈és ❞❛ ❝♦♥❡①ã♦ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡❥❛ ✉♠❛ ❝✉r✈❛α(λ)♥❛ ✈❛r✐❡❞❛❞❡M ❡V(λ)♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ss♦❝✐❛❞♦ ❛ ❡st❛
❝✉r✈❛✳ ❊♥tã♦✱ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ U é tr❛♥s♣♦rt❛❞♦ ♣❛r❛❧❡❧❛♠❡♥t❡ s❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦
❞❡ q✉❡ ∇VU = 0 ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛✳
➱ ✉s✉❛❧ ❞❡✜♥✐r ♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ❛tr❛✈és ❞❡ ✉♠ ♦♣❡r❛❞♦r D
dλ ❝♦♠♦
D
dλU =∇VU, ✭✷✳✹✮
♦✉ ❛✐♥❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
D dλU =
dxi
dλ∇∂iU, ✭✷✳✺✮
♦♥❞❡ dxi
dλ sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r t❛♥❣❡♥t❡ V(λ) ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳
❘❡❡s❝r❡✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ❡♠ t❡r♠♦s ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s✱ ♦❜t❡♠♦s✿ d
dλU
k+ Γk
ij
dxi
dλU
j = 0. ✭✷✳✻✮
❆ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ ♣❛r❛ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ k✱ ❞❡t❡r♠✐♥❛ ♦ ❝❛♠♣♦
✈❡t♦r✐❛❧ U(λ) tr❛♥s♣♦rt❛❞♦ ♣❛r❛❧❡❧❛♠❡♥t❡✳ ❉❛❞♦ ✉♠ ✈❡t♦r ❞❡✜♥✐❞♦ ❡♠ ✉♠ ♣♦♥t♦ P ❞❛
❝✉r✈❛✱ ♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ❞❡st❡ ♦❝♦rr❡ ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛✱ ❞❡✈✐❞♦ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♥❡❝❡ssár✐❛s ♣❛r❛ r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s r❡❧❛t✐✈❛s ❛ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡✳
❈♦♠ ♦ ❛✉①í❧✐♦ ❞♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ❣❡♦❞és✐❝❛s ❛✜♥s✱ q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ❡♠ ❣r❛✈✐t❛çã♦✱ ♣♦✐s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛s tr❛❥❡tór✐❛s ❞♦s r❛✐♦s ❞❡ ❧✉③ ❡♠ ✉♠ ❡s♣❛ç♦✲❝✉r✈♦✳ ❙❡❣✉✐♥❞♦ ❬✷✺❪✱ ❞❡✜♥✐♠♦s ❛ ❛❝❡❧❡r❛çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡ ❡♠ s❡❣✉✐❞❛ ❛ ❣❡♦❞és✐❝❛ ❛✜♠✳
❉❡✜♥✐çã♦✿ ❯♠❛ ❛❝❡❧❡r❛çã♦ ❞❡ ✉♠❛ ❝✉r✈❛α(λ)❡♠ M✱ é ❞❛❞❛ ♣♦r∇VV✱ ♦♥❞❡
V(λ) é ♦ ✈❡t♦r t❛♥❣❡♥t❡ à ❝✉r✈❛✳
❉❡✜♥✐çã♦✿ ❯♠❛ ❣❡♦❞és✐❝❛ ❛✜♠✱ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ é ✉♠❛ ❝✉r✈❛ ❝♦♠ ❛❝❡❧❡r❛çã♦ ♥✉❧❛✱ ♦✉ s❡❥❛✱ ✉♠❛ ❝✉r✈❛ q✉❡ s❛t✐s❢❛③ ∇VV = 0✳
❯s❛♥❞♦ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ s❡❥❛♠ dxµ
dλ ❛s ❝♦♠✲
♣♦♥❡♥t❡s ❞♦ ✈❡t♦r t❛♥❣❡♥t❡ V✳ ❊♥tã♦✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❣❡♦❞és✐❝❛ ❛✜♠ é ❡①♣r❡ss❛ ♣♦r d2xµ
dλ2 + Γ
µ αβ
dxα
dλ dxβ
dλ = 0. ✭✷✳✼✮
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✾
✷✳✶✳✸ ▼étr✐❝❛
❯♠ ❞♦s ♦❜❥❡t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❣❡♦♠❡tr✐❛ é ❛ ♠étr✐❝❛✳ ❊❧❛ ♥♦s ♣❡r♠✐t❡ r❡❛❧✐③❛r ❛s ♠❡❞✐❞❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦s✱ ❞❡ ár❡❛s ❡ t❛♠❜é♠ ❞❡ â♥❣✉❧♦s ♥❛ ✈❛r✐❡❞❛❞❡✳
❋♦r♠❛❧♠❡♥t❡✱ ❞❡✜♥✐♠♦s ✉♠❛ ♠étr✐❝❛ ❝♦♠♦ ✉♠ ❝❛♠♣♦ t❡♥s♦r✐❛❧ ❞♦ t✐♣♦τ0
2(M)✳
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ ♠étr✐❝❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r g q✉❡ ❛t✉❛ ❡♠ TpM ×TpM
❝✉❥❛ ✐♠❛❣❡♠ ♣❡rt❡♥❝❡ ❛♦s r❡❛✐s✳
❉❛❞♦ ✉♠❛ ❜❛s❡ {E1, E2, .., En}✱ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠étr✐❝❛ sã♦
g(Eα, Eβ) =gαβ. ✭✷✳✽✮
❈♦♠ ♦ ❛✉①í❧✐♦ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡g✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❛ss♦❝✐❛❞❛ ❛ ❞♦✐s
✈❡t♦r❡s U ❡ V✱ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦ P ❞❛ ✈❛r✐❡❞❛❞❡✱ ♣♦r
g(U, V) =g(Eα, Eβ)UαVβ =gαβUαVβ. ✭✷✳✾✮
❙❡❥❛♠ ❞♦✐s ♣♦♥t♦s P ❡Q✱ ❡♠ M✱ ✐♥✜♥✐t❡s✐♠❛❧♠❡♥t❡ ♣ró①✐♠♦s✱ ❡ s❡❥❛♠xα(P)
❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P ❡ xα(Q) ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ Q✳ ❖ ✈❡t♦r ❞✐stâ♥❝✐❛✱ U✱
❡♥tr❡ ♦s ♣♦♥t♦s P ❡ Q✱ ❞❛❞♦ ♣♦r U = ∆xα∂
α✱ ♣♦❞❡ ♥♦s ❢♦r♥❡❝❡r ♣♦r ♠❡✐♦ ❞❛ ♠étr✐❝❛ ♦
❡❧❡♠❡♥t♦ ❞❡ ❞✐stâ♥❝✐❛ ✐♥✜♥✐t❡s✐♠❛❧✱
ds2 =g(∆xα∂α,∆xβ∂β) = gαβ∆xα∆xβ. ✭✷✳✶✵✮
❖❜s❡r✈❛♠♦s q✉❡ ❡st❛ ❡①♣r❡ssã♦ r❡t♦♠❛ ❛ ❢♦r♠❛ ❝♦♥❤❡❝✐❞❛ ❞♦s ❧✐✈r♦s ❞❡ r❡❧❛✲ t✐✈✐❞❛❞❡ ❣❡r❛❧✳ ❆ ♠étr✐❝❛ é ✉♠❛ ❡str✉t✉r❛ q✉❡ ♥♦s ♣❡r♠✐t❡ ❢❛③❡r ✉♠❛ ❞✐st✐♥çã♦ ❡♥tr❡ ❛s
❣❡♦♠❡tr✐❛s✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❞❡ ❞✐♠❡♥sã♦ ✹✱ ❝✉❥❛ ♠étr✐❝❛ é gαβ = δαβ✱
❝❛r❛❝t❡r✐③❛ ✉♠❛ ❣❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉♠❛ ✈❛r✐❡❞❛❞❡M ❡♠ q✉❛❞r✐❞✐♠❡♥✲
s✐♦♥❛❧ ❝✉❥❛ ♠étr✐❝❛ é gαβ =diag(1,−1,−1,−1)✱ ❝❛r❛❝t❡r✐③❛ ✉♠❛ ❣❡♦♠❡tr✐❛ ▲♦r❡♥t③✐❛♥❛✳
❉❡st❛❝❛♠♦s ❞♦✐s t✐♣♦s ❜ás✐❝♦s ❞❡ ❣❡♦♠❡tr✐❛s✿ ❛s ❣❡♦♠❡tr✐❛s r✐❡♠❛♥♥✐❛♥❛s✱ q✉❡ sã♦ ❡①t❡♥sõ❡s ❞❛s ❣❡♦♠❡tr✐❛s ❡✉❝❧✐❞✐❛♥❛s✱ ❡ ❛s ❣❡♦♠❡tr✐❛s s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛s✱ q✉❡ sã♦ ❡①t❡♥sõ❡s ❞❛s ❣❡♦♠❡tr✐❛s ❞❡ ▲♦r❡♥t③✳
❉❡✜♥✐çã♦✿ ❯♠❛ ✈❛r✐❡❞❛❞❡ é r✐❡♠❛♥♥✐❛♥❛ s❡ ♣♦ss✉✐r ✉♠❛ ♠étr✐❝❛ s✐♠étr✐❝❛ ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ ❞❛❞♦ ❞♦✐s ✈❡t♦r❡s X, Y ∈ T(M)❛s ❝♦♥❞✐çõ❡s✿
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✶✵
✐✐✮ g(X, X)≥0✱ ❞❡✈❡♠ s❡r s❛t✐s❢❡✐t❛s✳
❊ss❡ t✐♣♦ ❞❡ ♠étr✐❝❛ é ♠✉✐t♦ ❝♦♠✉♠ ❡♠ ❢ís✐❝❛✱ ♣♦r ❡①❡♠♣❧♦ q✉❛♥❞♦ r❡s♦❧✈❡♠♦s ♣r♦❜❧❡♠❛s ❞❡ ♠❡❝â♥✐❝❛ ❛♥❛❧ít✐❝❛ ❬✷✽❪✳ ▼❛s ❡♠ ❣r❛✈✐t❛çã♦✱ ❛s ♠étr✐❝❛s ✉t✐❧✐③❛❞❛s sã♦ s❡♠✐✲ r✐❡♠❛♥♥✐❛♥❛s✳
❉❡✜♥✐çã♦✿ ❯♠❛ ✈❛r✐❡❞❛❞❡ é s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛ s❡ ♣♦ss✉✐r ✉♠❛ ♠étr✐❝❛ s✐♠é✲ tr✐❝❛ ❡ ♥ã♦ ❞❡❣❡♥❡r❛❞❛✳
❆ ❝♦♥❞✐çã♦ ❞❡ s❡r ♥ã♦ ❞❡❣❡♥❡r❛❞❛ s✐❣♥✐✜❝❛ q✉❡ ❛ ♠étr✐❝❛ é ✐♥✈❡rsí✈❡❧✳ ◆❡st❛s ❣❡♦♠❡tr✐❛s✱ t❡♠♦s três t✐♣♦s ❞❡ ✈❡t♦r❡s✿ t✐♣♦ ♥✉❧♦✱g(X, X) = 0✱ t✐♣♦ t❡♠♣♦✱g(X, X)>0✱ t✐♣♦ ❡s♣❛ç♦✱ g(X, X)<0✳
❯♠ ❝♦♥❝❡✐t♦ q✉❡ ❝♦♥str✉í♠♦s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ♠étr✐❝❛ é ♦ ❞❡ t❡♠♣♦ ♣ró♣r✐♦✳
❊st❡ é ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ♠❡❞✐❞♦ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❞♦ t✐♣♦✲t❡♠♣♦ ✶✱ ❝✉❥♦ ✈❡t♦r
t❛♥❣❡♥t❡ é U(λ)✳ ❊ss❡ ✐♥t❡r✈❛❧♦ t❡♠♣♦r❛❧✱ ∆τ✱ é ♠❡❞✐❞♦ ♣❡❧❛ ❡①♣r❡ssã♦
∆τ =
Z λ1
λ0
p
g(U(λ), U(λ))dλ. ✭✷✳✶✶✮
■♠♣❧✐❝✐t❛♠❡♥t❡ ❡st❛ ❡q✉❛çã♦ ♥♦s ✐♥❢♦r♠❛ q✉❡ ♦ t❡♠♣♦ ♠❡❞✐❞♦ ♥ã♦ é ✐♥✢✉❡♥❝✐✲ ❛❞♦ ♣❡❧❛ ❛❝❡❧❡r❛çã♦ ❞❛ ♣❛rtí❝✉❧❛✳ ❊st❛ ✐❞❡✐❛ é ❝❤❛♠❛❞❛ ❤✐♣ót❡s❡ ❞♦ r❡❧ó❣✐♦ ❬✷✻❪✳
❚❛♠❜é♠ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ♠étr✐❝❛ ❡ ❞❡ s✉❛s ❞❡r✐✈❛❞❛s ♣♦❞❡♠♦s✱ ❡♠ ❣❡♦♠❡tr✐❛ s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛✱ ❝♦♥str✉✐r ✉♠❛ ❝♦♥❡①ã♦ ♠✉✐t♦ ♣❛rt✐❝✉❧❛r✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦s ❝á❧❝✉❧♦s ❡♠ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✱ ❊st❛ ❝♦♥❡①ã♦ é ❞❛❞❛ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❬✷✼❪✱ ❬✷✾❪✱ q✉❡ tê♠ ❝♦♠♦ ♣ré✲r❡q✉✐s✐t♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡ ❞❡ s✐♠❡tr✐❛ ❞❡ ✉♠❛ ❝♦♥❡①ã♦✳
❉❡✜♥✐çã♦✿ ❯♠❛ ❝♦♥❡①ã♦∇❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛ ▼ é ❝♦♠♣❛✲
tí✈❡❧ ❝♦♠ ❛ ♠étr✐❝❛ s❡ ❡ s♦♠❡♥t❡ s❡
V [g(U, W)] = g(∇VU, W) +g(U,∇VW), ✭✷✳✶✷✮
♦♥❞❡ U, V, W sã♦ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❞❡✜♥✐❞♦s ♥❛ ✈❛r✐❡❞❛❞❡✳
❊st❛ ❞❡✜♥✐çã♦ ♥♦s ✐♥❢♦r♠❛ q✉❡ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s U ❡ W
é ♣r❡s❡r✈❛❞♦ ♣♦r ✉♠ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦✱ ♥❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦ V✳ ❙✐♠♣❧✐✜❝❛❞❛♠❡♥t❡✱
♣♦❞❡✲s❡ ♠♦str❛r q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ♠étr✐❝❛ é ❡q✉✐✈❛❧❡♥t❡ à ❝♦♥❞✐çã♦
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✶✶
❉❡✜♥✐çã♦✿ ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛ ▼ é
❞✐t❛ s✐♠étr✐❝❛ s❡ s❛t✐s❢❛③ ❛ r❡❧❛çã♦
[V, U] =∇VU − ∇UV, ✭✷✳✶✸✮
♣❛r❛ U, V ∈T M✳
❯s❛♥❞♦ ✉♠ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❛ ❝♦♥❞✐çã♦ ❞❡ s✐♠❡tr✐❛ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ♣♦r Γαµν = Γανµ✳ ❯♠❛ ✈❡③ ❡♥t❡♥❞✐❞♦ ❡ss❡s ❝♦♥❝❡✐t♦s✱ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ♦ t❡♦r❡♠❛ q✉❡
❞❡t❡r♠✐♥❛ ❛ ❝♦♥❡①ã♦ ❡♠ q✉❡stã♦✳
❚❡♦r❡♠❛ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✿ ❊♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛ ▼✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❝♦♥❡①ã♦ q✉❡ é ❛♦ ♠❡s♠♦ t❡♠♣♦ ❝♦♠♣❛tí✈❡❧ ❡ s✐♠étr✐❝❛✳ ❊st❛ ❝♦♥❡①ã♦ é ❝❤❛✲ ♠❛❞❛ ❞❡ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✳
P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ❢♦r♠❛ ❞❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❡q✉❛✲
çã♦ ∇αgµν = 0✱ ❥✉♥t♦ ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡ s✐♠❡tr✐❛✳ ❉❡t❡r♠✐♥❛♠♦s✱ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❡
♠❛♥✐♣✉❧❛çã♦✱ q✉❡ ❛ ❝♦♥❡①ã♦ t❡♠ ❛ ❢♦r♠❛
Γαµν = 1 2g
αλ(∂
µgνλ+∂νgλµ−∂λgµν). ✭✷✳✶✹✮
◆♦t❛♠♦s q✉❡ ❡st❛ é ✉♠❛ ❡①t❡♥sã♦ ❞❛ ❝♦♥❡①ã♦ ✉s❛❞❛ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡ ❝✉r✈❛s ❞❡ s✉♣❡r❢í✲ ❝✐❡s✷✳
❆✐♥❞❛ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ♠étr✐❝❛ ❡ ❝♦♠ ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s ❣❡♦❞és✐❝❛s ♠étr✐❝❛s✱ q✉❡ sã♦ ❛s ❝✉r✈❛s s❡❣✉✐❞❛s ♣❡❧❛s ♣❛rtí❝✉❧❛s ♠❛ss✐✈❛s q✉❛♥❞♦ ❡stã♦ s♦♠❡♥t❡ s♦❜ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳
❉❡✜♥✐çã♦✿ ❯♠❛ ❝✉r✈❛ α(λ)✱ q✉❡ ❧✐❣❛ ❞♦✐s ♣♦♥t♦s P ❡ Q ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡
M✱ é ✉♠❛ ❣❡♦❞és✐❝❛ ♠étr✐❝❛ s❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ❞❛❞♦ ♣♦r
s=
Z Q
P p
g( ˙α(λ),α(λ))dλ˙ ✭✷✳✶✺✮
❢♦r ❡st❛❝✐♦♥ár✐♦✱ ✐st♦ é✱ δs= 0✳
❆s ❡q✉❛çõ❡s q✉❡ ❛s ❣❡♦❞és✐❝❛s ♠étr✐❝❛s s❛t✐s❢❛③❡♠✱ ♣r♦✈❡♥✐❡♥t❡s ❞❛ ♥♦ss❛ ❞❡✲ ✜♥✐çã♦✱ t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛
d2xµ
dλ2 + Γ
µ
αβx˙
αx˙β = s¨
˙ sx˙
µ. ✭✷✳✶✻✮
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✶✷
❊s❝♦❧❤❡♥❞♦ ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❛❞❡q✉❛❞❛✱ λ=s✱ ❛s ❡q✉❛çõ❡s t♦♠❛♠ ❛ ❢♦r♠❛
d2xµ ds2 + Γ
µ
αβx˙αx˙β = 0, ✭✷✳✶✼✮
◆♦t❛♠♦s q✉❡ ❛s ❡q✉❛çõ❡s ✭✷✳✼✮ ❡ ✭✷✳✶✼✮ sã♦ ❛ ♠❡s♠❛ ❡q✉❛çã♦✳ ■st♦ r❡✢❡t❡ ♦ ❢❛t♦ ❞❡ q✉❡ ❡♠ ❣❡♦♠❡tr✐❛ s❡♠✐✲r✐❡♠❛♥♥✐❛♥❛ ❣❡♦❞és✐❝❛s ♠étr✐❝❛s ❡ ❣❡♦❞és✐❝❛s ❛✜♥s ❝♦✐♥❝✐❞❡♠✳
✷✳✶✳✹ ❈✉r✈❛t✉r❛
❆❧é♠ ❞❛ ♠étr✐❝❛ ❡ ❞❛ ❝♦♥❡①ã♦✱ ♦✉tr♦ ♦❜❥❡t♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❡♠ ❣❡♦♠❡tr✐❛ ❡ ❡♠ ❣r❛✈✐t❛çã♦ é ❛ ❝✉r✈❛t✉r❛✳
❉❡✜♥✐çã♦✿ ❆ ❝✉r✈❛t✉r❛ R❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛ ❛♣❧✐❝❛çã♦
q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣❛r ❞❡ ✈❡t♦r❡s(X, Y)∈T(M)×T(M)✉♠ ♦♣❡r❛❞♦rR(X, Y) :T(M)→
T(M)❞❛❞♦ ♣♦r✿
R(X, Y)Z =∇X∇YZ− ∇Y∇XZ− ∇[X,Y]Z. ✭✷✳✶✽✮
❉❡st❛ ❞❡✜♥✐çã♦ t❡♠♦s três ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❞❡st❛❝❛♠♦s ❛ s❡❣✉✐r✿
i) ❆ ❝✉r✈❛t✉r❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r ❡♠ T(M)×T(M)✱ ♦✉ s❡❥❛✱ ❞❛❞♦s
X, Y, Z ∈T(M)❡ f✱ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥❛ ✈❛r✐❡❞❛❞❡✱ t❡♠♦s q✉❡
R(f X+Y, Z) = f R(X, Z) +R(Y, Z) ✭✷✳✶✾✮
ii) P❛r❛ t♦❞♦ ♣❛r ❞❡ ✈❡t♦r❡s (X, Y) ∈ T(M)×T(M)✱ ♦ ♦♣❡r❛❞♦r R(X, Y) é ❧✐♥❡❛r✱ ♦✉ s❡❥❛✱ ❞❛❞♦ Z, W ∈T(M)✱ t❡♠♦s✿
R(X, Y) [Z+W] =R(X, Y)Z+R(X, Y)W. ✭✷✳✷✵✮
iii) ❖ ♦♣❡r❛❞♦r é ❛♥t✐✲s✐♠étr✐❝♦
R(X, Y) = −R(Y, X). ✭✷✳✷✶✮
❊①✐❜✐♥❞♦ ♦ t❡♥s♦r R ❡♠ ✉♠❛ ❜❛s❡{Ei}✱ t❡♠♦s ❛ ❡①♣r❡ssã♦
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✶✸
P♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ t❡♥s♦r R ❡♠ t❡r♠♦s ❞❛ ❝♦♥❡①ã♦Γl
jk✱ ♣❛r❛ ✐st♦ r❡♣r❡s❡♥✲
t❛♠♦s ♦ t❡♥s♦r ❞❡ ❝✉r✈❛t✉r❛ ❡♠ t❡r♠♦s ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s ♥✉♠❛ ❞❛❞❛ ❜❛s❡ Ei✱
R(Ei, Ej)Ek=Rijkl El, ✭✷✳✷✸✮
✉t✐❧✐③❛♠♦s ❛ ✐❞❡♥t✐❞❛❞❡
∇Ei∇EjEk =∇EiΓ
l
jkEl= Γljk,iEl+ ΓljkΓ
p
ilEp, ✭✷✳✷✹✮
❡ ♦ ❝♦♠✉t❛❞♦r [Ei, Ej] =CijlEl✱ ❝❤❡❣❛♠♦s ❢❛❝✐❧♠❡♥t❡ ♥❛ ❡①♣r❡ssã♦
Rijkl El = Γljk,iEl−Γlik,jEl+ ΓljkΓ p
ilEp−ΓlikΓ
p
ilEp−Cijl Γ
p
lmEp. ✭✷✳✷✺✮
❉❡st❛ ♠❛♥❡✐r❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ ❡♠ ✉♠❛ ❜❛s❡ ❞❡ ❝♦♦r✲ ❞❡♥❛❞❛s✱ {∂α}✱ ❛ss✉♠❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❝♦♥❤❡❝✐❞❛✿
Rαµβν = Γαβµ,ν −Γαµν,β+ Γρµ,να Γρβµ−ΓαρβΓρνµ. ✭✷✳✷✻✮
◗✉❛♥❞♦ ❡st❛♠♦s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ ✈❡r✐✜❝❛♠♦s q✉❡ ♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ é ③❡r♦✱ ❡ ❛ ❝✉r✈❛t✉r❛ é ♥✉❧❛✳ ❊st❡ ❢❛t♦ ♥♦s ❞á ❛ ✐♥❢♦r♠❛çã♦ ❞❡ q✉❡ ♦ ❡s♣❛ç♦ é ♣❧❛♥♦✳ ❯♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ ❞❛ ❝✉r✈❛t✉r❛ ✈❡♠ q✉❛♥❞♦ ❛♥❛❧✐s❛♠♦s ♦ ❞❡s✈✐♦ ❣❡♦❞és✐❝♦✱ ❝♦♠♦ ❡♠ ❬✸✵❪✳
✷✳✶✳✺ ❉❡s✈✐♦ ❣❡♦❞és✐❝♦
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ ❣❡♦❞és✐❝❛s✱γ(s, t)✱ t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛sr❡❛❧✱ γs(t) r❡♣r❡s❡♥t❛ ✉♠❛ ❣❡♦❞és✐❝❛ ♣❛r❛♠❡tr✐③❛❞❛ ❡♠ t✱ ❡ ♣❛r❛ ❝❛❞❛ t r❡❛❧✱ ❛ ❝✉r✈❛ γt(s)
r❡♣r❡s❡♥t❛ ✉♠❛ ❣❡♦❞és✐❝❛ ♣❛r❛♠❡tr✐③❛❞❛ ❡♠ s✳
❱❛♠♦s ❡①❛♠✐♥❛r ❛s ❝✉r✈❛s γs(t)✳ ❆❞♠✐t✐♠♦s q✉❡ ❡❧❛s ❡st❡❥❛♠ ✐♥✐❝✐❛❧♠❡♥t❡ ❡♠
♣❛r❛❧❡❧♦ ❡ q✉❡ ❝♦♠♣õ❡♠ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❜✐❞✐♠❡♥s✐♦♥❛❧✳ ❙❡❥❛♠ xµ(γ
s(t))❛s ❝♦♠♣♦♥❡♥✲
t❡s ❞❛ ❣❡♦❞és✐❝❛ ❡♠ q✉❡stã♦✳ P♦❞❡♠♦s ❞✐st✐♥❣✉✐r ❞♦✐s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s✿ T✱ ♦ ❝❛♠♣♦ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ ❝✉r✈❛ γs(t)✱ ❡ S✱ ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ ❝✉r✈❛ γt(s)✳
❊♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ T ❡ ❞❡ S sã♦ r❡s♣❡❝t✐✈❛✲
♠❡♥t❡
Tµ= ∂x
µ
∂t , S
µ= ∂xµ
✷✳✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✶✹
❖ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s S ♥♦s ❞á ✉♠❛ ❞✐stâ♥❝✐❛ r❡❧❛t✐✈❛ ❡♥tr❡ ❛s ❣❡♦❞és✐❝❛s γs(t)
❡ γs+δs(t)✳ ❆ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛ ❞❡ ❛❢❛st❛♠❡♥t♦ é ♦❜t✐❞❛ ♣♦r
V = DS
dλ =∇TS. ✭✷✳✷✽✮
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛❝❡❧❡r❛çã♦ r❡❧❛t✐✈❛ A ❞❡ ❛❢❛st❛♠❡♥t♦
♣♦r
A= DV
dλ =∇TV =∇T∇TS. ✭✷✳✷✾✮
❯t✐❧✐③❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✽✮✱ ❡ ❧❡♠❜r❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ [∂µ, ∂ν] = 0✱ ❡ ❛✐♥❞❛ q✉❡ ∇ST = ∇TS✱ ♣♦❞❡♠♦s r❡s❝r❡✈❡r ❛ ❛❝❡❧❡r❛çã♦ r❡❧❛t✐✈❛ ❝♦♠♦
A=∇TV =∇T∇TS =∇T∇ST =R(T, S)T, ✭✷✳✸✵✮
✉♠❛ ✈❡③ q✉❡ ∇TT = 0✱ ♣♦r ❝♦♥str✉çã♦✳
❱❡r✐✜❝❛♠♦s ❛ss✐♠ q✉❡ ♦ ♣❛r❛❧❡❧✐s♠♦ ❡♥tr❡ ❡ss❛s ❝✉r✈❛s só é ♠❛♥t✐❞♦ q✉❛♥❞♦ ❛ ❛❝❡❧❡r❛çã♦ r❡❧❛t✐✈❛ ❡♥tr❡ ❡❧❛s é ♥✉❧❛✱ ♦✉ s❡❥❛✱ s❡ R(T, S) = 0✳ ■st♦ é ❝❛r❛❝t❡ríst✐❝♦ ❞❡ ❡s♣❛ç♦s ✢❛t✳ ❊♠ ❡s♣❛ç♦s ❝✉r✈♦s ♥ã♦ ❡s♣❡r❛♠♦s q✉❡ ✐ss♦ ♦❝♦rr❛✱ ♣♦✐s ❣❡r❛❧♠❡♥t❡ ❤á ✉♠❛ ❛❝❡❧❡r❛çã♦ r❡❧❛t✐✈❛ ❡♥tr❡ ❛s ❣❡♦❞és✐❝❛s✳
✷✳✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧
❯♠❛ ✈❡③ q✉❡ ❧❡♠❜r❛♠♦s ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❝♦♥❡①ã♦✱ ❞❡ ♠étr✐❝❛ ❡ ❞❡ ❝✉r✈❛t✉r❛✱ ✈❛♠♦s tr❛t❛r ❞❛ ❣❡♦♠❡tr✐❛ ❡st❛❜❡❧❡❝✐❞❛ ♣♦r ❍✳ ❲❡②❧✳
❊♠ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛✱ ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ♣r❡s❡r✈❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r tr❛♥s♣♦rt❛❞♦ ♣❛r❛❧❡❧❛♠❡♥t❡✱ ♣♦ré♠ ♥ã♦ ❤á ❣❛r❛♥t✐❛s ❞❡ q✉❡ s✉❛ ❞✐r❡çã♦ ♣❡r♠❛♥❡ç❛ ❛ ♠❡s♠❛ ♥❡st❛ ♦♣❡r❛çã♦✳
P❛r❛ ❞❡s❡♥✈♦❧✈❡r♠♦s ✉♠❛ ✐♥t✉✐çã♦ s♦❜r❡ ❣❡♦♠❡tr✐❛s ❞❡ ❲❡②❧✱ ❞✐s❝✉t✐♠♦s ❛ s❡❣✉✐♥t❡ ✐❧✉str❛çã♦✳ ■♠❛❣✐♥❡♠♦s ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛S✳ ◆❡❧❛ tr❛ç❛♠♦s ✉♠❛ ❣❡♦❞és✐❝❛✱ γ✱ q✉❡ ❝♦rt❛ ♦s ❞♦✐s ♣♦❧♦s ❞❛ s✉♣❡r❢í❝✐❡✱ ♦✉ s❡❥❛✱ ✉♠ ❣r❛♥❞❡ ❝ír❝✉❧♦ ❞❛ ❡s❢❡r❛✳ ❯♠ ✈❡t♦r
V q✉❡ ❡st❡❥❛ ❡♠ ✉♠ ❞♦s ♣♦❧♦s✱ ❛♦ s❡r tr❛♥s♣♦rt❛❞♦ ♣❛r❛❧❡❧❛♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❝✉r✈❛✱
❛♣ós ✉♠❛ ✈♦❧t❛✱ ❛♦ ♣❛ss❛r ♣❡❧♦ s❡✉ ♣♦♥t♦ ❞❡ ♦r✐❣❡♠ ❡st❡ ✈❡t♦r t❡rá s✉❛ ❞✐r❡çã♦ ❞✐❢❡r❡♥t❡ ❞❛ ❞✐r❡çã♦ ✐♥✐❝✐❛❧✳ ❊ss❡ é ✉♠ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ s✉♣❡r❢í❝✐❡s✳
✷✳✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✶✺
❞❡✈❡♠ ♠✉❞❛r ❞❡✈✐❞♦ ❛♦ tr❛♥s♣♦rt❡✳ ❙❡❣✉♥❞♦ ❲❡②❧ q✉❡ ❛ ✈❛r✐❛çã♦ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❡
❞♦ ❝♦♠♣r✐♠❡♥t♦✱ L✱ ❞❡ ✉♠ ✈❡t♦r V tr❛♥s♣♦rt❛❞♦s ♣❛r❛❧❡❧❛♠❡♥t❡ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ M
❢♦ss❡♠ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱
dVα = ΓαλβdxλVβ, ✭✷✳✸✶✮
dL=Lσα 2 dx
α, ✭✷✳✸✷✮
♦♥❞❡ r❡♣r❡s❡♥t❛♠♦s ♣♦r Vα ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r V✳ ◆♦t❛♠♦s ❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠
♥♦✈♦ t❡r♠♦ σα✱ q✉❡ ❛❧t❡r❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r✱ ❝♦♠♦ s❡ ❢♦ss❡ ✉♠❛ ❝♦♥❡①ã♦ ♣❛r❛ ♦
❝♦♠♣r✐♠❡♥t♦ ❬✸✶❪✳ ❉❡ ♠❛♥❡✐r❛ ❛ ✐♥❝♦r♣♦r❛r ❡st❛ ✐❞❡✐❛✱ ❞❡✜♥✐♠♦s ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✸✷❪✳
❉❡✜♥✐çã♦✿ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❲❡②❧✱ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱M✱ ❞♦t❛❞❛
❞❡ três ♦❜❥❡t♦s✿ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠✱∇✱ ✉♠❛ ♠étr✐❝❛g ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✶✲❢♦r♠❛σ✱ ❝❤❛♠❛❞♦
❞❡ ❝❛♠♣♦ ❞❡ ❲❡②❧✳ ❖❜❡❞❡❝❡♥❞♦ ❛ ❞✉❛s ❝♦♥❞✐çõ❡s✿
✐✮ ❆ ♠étr✐❝❛ é s✐♠étr✐❝❛✳
✐✐✮ ❉❛❞♦s três ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s V, U, W✱ ♣❡rt❡♥❝❡♥t❡s ❛ T(M)✱ ❡♥tã♦ ❛ ❝♦♥✲ ❞✐çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❡ ❲❡②❧ é s❛t✐s❢❡✐t❛✱ ✐st♦ é✱
V[g(U, W)] =g(∇VU, W) +g(U,∇VW) +σ(V)g(U, V), ✭✷✳✸✸✮
♦✉ s✐♠♣❧✐✜❝❛❞❛♠❡♥t❡✸✱ ♣♦r
V(g(U, W)) =σ(V)g(U, W). ✭✷✳✸✹✮
❊s❝♦❧❤❡♥❞♦ ✉♠❛ ❜❛s❡ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❝♦♥❞✐çã♦ ❞❡ ❲❡②❧ ❞❛ ❢♦r♠❛ ∇αgµν =σαgµν✱ ❝♦♠♦ ♠✉✐t♦s ✈❡③❡s ❡❧❛ é ✉s❛❞❛ ♥❛ ❧✐t❡r❛t✉r❛✳
P❛r❛ ❡♥t❡♥❞❡r♠♦s ♦ ❡❢❡✐t♦ ❞♦ ❝❛♠♣♦ ❞❡ ❲❡②❧ ♥❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ ✈❛♠♦s ✐❧✉str❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦✳
Pr♦♣♦s✐çã♦ ✿ ❙❡❥❛ ▼ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❲❡②❧✱ ❝♦♠ ✉♠❛ ♠étr✐❝❛
g ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✉♠❛ ❢♦r♠❛ σ✳ ❊♥tã♦✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r U tr❛♥s♣♦rt❛❞♦
♣❛r❛❧❡❧❛♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❝✉r✈❛ α(λ)✱ s❡ ♠♦❞✐✜❝❛ s❡❣✉♥❞♦ ❛ ❝♦♥❞✐çã♦
∇V(λ)g(U, U) =σ(V(λ))g(U, U), ✭✷✳✸✺✮
✷✳✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✶✻
♦♥❞❡ V(λ) é ♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ t❛♥❣❡♥t❡ à ❝✉r✈❛✳
❊st❛ ❝♦♥❞✐çã♦ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ❛♣❧✐❝❛çã♦ ❞❡ ✭✷✳✸✹✮✱ q✉❡ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛
dL=Lσα 2 dx
α, ✭✷✳✸✻✮
r❡s❣❛t❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✸✷✮✱ q✉❡ ♥♦s ✐♥❢♦r♠❛ ❝♦♠♦ s❡ ❞á ❛ ✈❛r✐❛çã♦ ❝♦♠ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r U ♥❛ ❝✉r✈❛✳
✷✳✷✳✶ ❈♦♥❡①ã♦ ❞❡ ❲❡②❧
❊♠ ❣❡♦♠❡tr✐❝❛ r✐❡♠❛♥♥✐❛♥❛ ♦ t❡♦r❡♠❛ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❝♦♥❡①ã♦ ❡s♣❡❝✐❛❧ ❞❛❞❛ ❡♠ t❡r♠♦s ❞❛ ♠étr✐❝❛ ❡ ❞❡ s✉❛s ❞❡r✐✈❛❞❛s✳ ◆♦ ❝❛s♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧✱ t❛♠❜é♠ ❡①✐st❡ ✉♠❛ ❝♦♥❡①ã♦ ❡s♣❡❝✐❛❧ ❣❛r❛♥t✐❞❛ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ▲❡✈✐✲ ❈✐✈✐t❛ ❡st❡♥❞✐❞♦✳
❚❡♦r❡♠❛ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❡st❡♥❞✐❞♦ ✿ ❙❡❥❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M✱
❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ ∇✱ ✉♠❛ ♠étr✐❝❛ g ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✶✲❢♦r♠❛ σ ❞❡✜♥✐❞♦s ❡♠ M✳
❊①✐st❡ ✉♠❛ ú♥✐❝❛ ❝♦♥❡①ã♦ ❛✜♠ ∇✱ t❛❧ q✉❡ s❡❥❛ s✐♠étr✐❝❛✱ ❡ q✉❡ ♦❜❡❞❡ç❛ ❛ ❝♦♥❞✐çã♦ ❞❡
❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❡ ❲❡②❧✳
P❛r❛ ❞❡♠♦♥str❛r ❡ss❡ t❡♦r❡♠❛ ❞❡ ♠❛♥❡✐r❛ s✐♠♣❧✐✜❝❛❞❛✱ ❛❞♦t❛♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ r❡❡s❝r❡✈❡♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡✳ ❙♦♠❛♠♦s ❛ ❡q✉❛çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
∇αgµν+∇νgαµ− ∇µgνα = ✭✷✳✸✼✮
∂αgµν+∂νgαµ−∂µgνα−2Γλναgλµ= ✭✷✳✸✽✮
σαgµν+σνgαµ−σµgνα. ✭✷✳✸✾✮
❈♦♠♦ ❛ ❝♦♥❡①ã♦ é s✐♠étr✐❝❛ ❛❧❣✉♥s t❡r♠♦s sã♦ ❡❧✐♠✐♥❛❞♦s✱ ❡ ❛♣ós ✉♠❛ ♠❛♥✐♣✉❧❛çã♦✱ ❡①✐❜✐♠♦s ❛ ❝♦♥❡①ã♦ ❞❡ ❲❡②❧ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❝♦♠♦ s❡♥❞♦ ❞❛❞❛ ♣♦r
Γβαν ={β
αν} −
1 2g
βµ(σ
αgµν +σνgαµ−σµgαν). ✭✷✳✹✵✮
❊st❛ é ❛ ❝♦♥❡①ã♦ ❞❡ ❲❡②❧✱ q✉❡ é ❞❡t❡r♠✐♥❛❞❛ ♥ã♦ só ♣❡❧❛ ♠étr✐❝❛✱ ♠❛s t❛♠❜é♠ ♣❡❧♦
✷✳✷ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✶✼
✷✳✷✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡
P♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ✉♠❛ s✐♠❡tr✐❛ ♥❛ ❡q✉❛çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❡ ❲❡②❧✱ ❛♦ ❢❛③❡r♠♦s ❛s s❡❣✉✐♥t❡s tr❛♥s❢♦r♠❛çõ❡s ♥♦s ❝❛♠♣♦s
¯
g =efg, ✭✷✳✹✶✮
¯
σ=σ+df. ✭✷✳✹✷✮
❊ss❛s tr❛♥s❢♦r♠❛çõ❡s sã♦ ❞❡♥♦♠✐♥❛❞❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❲❡②❧✱ ❡ r❡❝❛❧✐❜r❛♠ ♦s ❝❛♠♣♦s ♠étr✐❝♦ ❡ ❞❡ ❲❡②❧✳ ❆ tr❛♥s❢♦r♠❛çã♦ ♥❛ ♠étr✐❝❛ é ❞♦ t✐♣♦ ❝♦♥❢♦r♠❡✱ ❡ ❛ tr❛♥s✲ ❢♦r♠❛çã♦ ❞♦ ❝❛♠♣♦ ❡s❝❛❧❛r é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♠♣❡♥s❛çã♦✱ ✐st♦ é✱ ❡❧❛ ❝♦♠♣❡♥s❛ ❛ ♠✉❞❛♥ç❛ ♥❛ ♠étr✐❝❛ ❬✸✹❪✳
❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❲❡②❧✱ M✱ é ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧❛ ♠étr✐❝❛✱ g✱ ❡ ♣❡❧♦ ❝❛♠♣♦
❞❡ ❲❡②❧✱ σ✳ ❆ ✉♠❛ trí♣❧✐❝❡ (M, g, σ)✱ s❡ ❞á ♦ ♥♦♠❡ ❞❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❲❡②❧✳ ❆♦
❢❛③❡r♠♦s ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❲❡②❧ ♠✉❞❛♠♦s ❞❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ (M, g, σ)♣❛r❛ ✉♠ r❡❢❡✲
r❡♥❝✐❛❧ (M,g,¯ σ)¯ ✱ ♣♦r ❡st❡ ♠♦t✐✈♦ ❡st❛s tr❛♥s❢♦r♠❛çõ❡s sã♦ ✐♥t❡r♣r❡t❛❞❛s ❝♦♠♦ ♠✉❞❛♥ç❛s ❞❡ r❡❢❡r❡♥❝✐❛✐s ♦✉ ❞❡ f rames✳
❊♠ ❝❛❞❛ r❡❢❡r❡♥❝✐❛❧ t❡♠♦s ✉♠❛ ♠étr✐❝❛ ❞✐❢❡r❡♥t❡✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r ♠✉❞❛ ❝♦♥❢♦r♠❡ ♠✉❞❛♠♦s ❞❡ r❡❢❡r❡♥❝✐❛❧✳ ❊♠ ❣❡r❛❧✱ ❛s ♠❡❞✐❞❛s ❞❡ ár❡❛ ❡ ❞❡ ✈♦❧✉♠❡ t❛♠❜é♠ ❞❡♣❡♥❞❡♠ ❞♦ r❡❢❡r❡♥❝✐❛❧ ❡♠ q✉❡ ❢♦r❛♠ ❢❡✐t❛s✳
P♦ré♠✱ t❡♠♦s ❡♠ ❲❡②❧ ❛❧❣✉♥s ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s q✉❡ r❡✈❡❧❛♠ s❡r ✐♥✈❛r✐❛♥t❡s ♣♦r ❡ss❛s tr❛♥s❢♦r♠❛çõ❡s✳ ❆ ❝♦♥❡①ã♦ é ✉♠ ❡①❡♠♣❧♦✳ ❙❡ ❡st✐✈❡r♠♦s ❡♠ ✉♠ r❡❢❡r❡♥❝✐❛❧ (M,¯g,σ)¯ ❛ ❡q✉❛çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ é ∇¯αgµν = ¯σαgµν✱ ❡♥tã♦ ❡♥❝♦♥tr❛rí❛♠♦s ✉♠❛
❝♦♥❡①ã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s Γ¯β
αν✳ P♦ré♠✱ ❛♦ ❛♣❧✐❝❛r♠♦s ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❲❡②❧ ♥♦t❛♠♦s
q✉❡ Γ¯β
αν = Γβαν é ❛ ❝♦♥❡①ã♦ ❡♠ ✉♠ r❡❢❡r❡♥❝✐❛❧ (M, g, σ)✳
❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠♣♦rt❛♥t❡ ❞❛ ❝♦♥❡①ã♦ s❡r ✐♥✈❛r✐❛♥t❡ é ♣❡r❝❡❜✐❞❛ ❛♦ ❡st✉✲ ❞❛r♠♦s ❛s ❣❡♦❞és✐❝❛s ❛✜♥s✳ ❖❜s❡r✈❡♠♦s ✉♠❛ ❣❡♦❞és✐❝❛ ❛✜♠✱
∇VV = 0, ✭✷✳✹✸✮
P♦r ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ r❡❢❡r❡♥❝✐❛❧✱ ♥ã♦ ❛❧t❡r❛♠♦s ❛ ❡str✉t✉r❛ ❞❛ ❝✉r✈❛✱ ♣♦✐s
∇= ¯∇✳ ❆ss✐♠✱ ❛s ❣❡♦❞és✐❝❛s ❛✜♥s sã♦ ✐♥✈❛r✐❛♥t❡s ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❲❡②❧✳
