❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛✐❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦
❚❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✐❛❧✿
❯♠❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛
❚♦♥② ❙✐❧✈❛ ❆❧♠❡✐❞❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦
❚❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✐❛❧✿
❯♠❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛
❚♦♥② ❙✐❧✈❛ ❆❧♠❡✐❞❛
❚❡s❡ s✉❜♠❡t✐❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡
❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛
P❛r❛í❜❛✱ s♦❜ ♦r✐❡♥t❛çã♦ ❞♦ ♣r♦❢❡ss♦r
❉r✳ ❈❛r❧♦s ❆✉❣✉st♦ ❘♦♠❡r♦ ❋✐❧❤♦✱
❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜✲
t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❋í✲
s✐❝❛✳
A447g Almeida, Tony Silva.
Teoria escalar-tensorial: Uma abordagem geométrica / Tony Silva Almeida.-- João Pessoa, 2016.
59f.
Orientador: Carlos Augusto Romero Filho Tese (Doutorado) – UFPB/CCEN
1. Física. 2. Teoria de Brans-Dicke. 3. Geometria de Weyl. 4. Buraco de minhoca. 5. Gravitaçãoem (2+1).
❙✉♠ár✐♦
❘❡s✉♠♦ ✐✐✐
❆❜str❛❝t ✐✈
■♥tr♦❞✉çã♦ ✶
✶ ❊s♣❛ç♦✲t❡♠♣♦ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧ ✭❲■❙❚✮ ✸ ✶✳✶ ●❡♥❡r❛❧✐③❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❈✉r✈❛t✉r❛ ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ❈✉r✈❛t✉r❛ ❞❡ ❲❡②❧ ♥ã♦✲✐♥t❡❣rá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐ ♣❛r❛ ❲❡②❧ ✐♥t❡❣rá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ■♥✈❛r✐â♥❝✐❛ ❞❛ ❛çã♦ ❡ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷ ❚❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ✶✵ ✷✳✶ ❚❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ ▲✐♠✐t❡ ❞❡ ❇❉ ♣❛r❛ ❘● ❖✭1
ω✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷✳✶ ▲✐♠✐t❡ ❞❡ ❇❉ ♣❛r❛ ❘● ❖✭√1
ω✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ❝♦♥❢♦r♠❡s ❡ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸✳✶ ❘❡❢❡r❡♥❝✐❛❧ ❞❡ ❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✹ ❙♦❧✉çõ❡s ❡stát✐❝❛s ❝♦♠ s✐♠❡tr✐❛ ❡s❢ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹✳✶ ❙♦❧✉çã♦ ❞❡ ❈❛♠♣❛♥❡❧❧✐✲▲♦✉st♦ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺ ❙✐♠✐❧❛r✐❞❛❞❡s ❡♥tr❡ ❛ t❡♦r✐❛ ❞❡ ❇❉ ❡ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺✳✶ ❆ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❲❡②❧ ❝♦♠♦ tr❛♥s❢♦r♠❛çã♦ ❞❡ ✉♥✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✺✳✷ ❈♦♥s❡r✈❛çã♦ ❞❡ ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ♠♦❞✐✜❝❛❞❛ ❡ ❣❡♦❞és✐❝❛s ❞❡ ❲❡②❧ ✳ ✷✺
✸ ❚❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✐❛❧ ❣❡♦♠étr✐❝❛ ✷✼ ✸✳✶ ◆ã♦✲♠❡tr✐❝✐❞❛❞❡ ❡♠ t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ✭❙❚✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✳✶ ❈♦♥str✉✐♥❞♦ ✐♥✈❛r✐❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✳✷ ❊q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✳✸ ▲✐♠✐t❡ ♥❡✇t♦♥✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❲❡②❧✿ r❡❧❛çã♦ ❝♦♠ ❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✶ ▲❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❛ ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❞❛ ♠❛tér✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ❉✐str✐❜✉✐çã♦ ❡s❢❡r✐❝❛♠❡♥t❡ s✐♠étr✐❝❛ ❡ ❡stát✐❝❛ ♥♦ ✈❛③✐♦✿ ❛ s♦❧✉çã♦ ❞❡ ❲②♠❛♥ ✸✺ ✸✳✹ ❆ s♦❧✉çã♦ ❞❡ ❲②♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✺ ❙✐♥❣✉❧❛r✐❞❛❞❡s ♥✉❛s ❡ ❜✉r❛❝♦s ❞❡ ♠✐♥❤♦❝❛ ❝♦♠♦ ❢❡♥ô♠❡♥♦s ❣❡♦♠étr✐❝♦s ✳ ✳ ✸✻ ✸✳✺✳✶ ❙✐♥❣✉❧❛r✐❞❛❞❡s ♥✉❛s ♣❛r❛ ω >˜ 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸✳✺✳✷ ❇✉r❛❝♦s ❞❡ ♠✐♥❤♦❝❛ ♣❛r❛−2M2 <ω <˜ 0 ✱ ♦✉ s❡❥❛✱S > 1 ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✻ ❆ s♦❧✉çã♦ ❞❡ ❲②♠❛♥ ♥❛ t❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r ❣❡♦♠❡tr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✹ ●r❛✈✐t❛çã♦ ❡♠ ✭✷✰✶✮❉✿ ❲■❙❚ ✹✷ ✹✳✶ ❆ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ❡♠ ✭✷✰✶✮ ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✶✳✶ ❖ ❜✉r❛❝♦ ♥❡❣r♦ ❇❚❩ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✶✳✷ ❈♦s♠♦❧♦❣✐❛ ❡♠ ✷✰✶ ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷ ❆❧t❡r♥❛t✐✈❛s à r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✸ ❆ t❡♦r✐❛ ❲■❙❚ ❡♠ n ❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹✳✹ ❲■❙❚ ❡♠ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ✭✷✰✶✮✲❞✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✺ ◆♦ ❧✐♠✐t❡ ♥❡✇t♦♥✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✻ ❯♠❛ s♦❧✉çã♦ ❡st❛t✐❝❛ ❝♦♠ s✐♠❡tr✐❛ ❝✐r❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✼ ❉❡s✈✐♦ ❣❡♦❞és✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✽ ▼♦❞❡❧♦s ❝♦s♠♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✽✳✶ ❙♦❧✉çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ✈❛③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✽✳✷ ❙♦❧✉çã♦ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ♠❛tér✐❛✱ ρ6= 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❈♦♥❝❧✉sã♦ ✺✷
❇✐❜❧✐♦❣r❛✜❛ ✺✹
❘❡s✉♠♦
❊st❛ t❡s❡ tr❛t❛ ❞❡ tó♣✐❝♦s r❡❧❛❝✐♦♥❛❞♦s às t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ◆♦ss❛ ❛❜♦r❞❛❣❡♠ s❡rá ♥♦ s❡♥t✐❞♦ ❞❡ ✐♥❞✐❝❛r ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡✲ ❣rá✈❡❧ ❝♦♠♦ s❡♥❞♦ ✉♠ ❛♠❜✐❡♥t❡ ♥❛t✉r❛❧ ♣❛r❛ ❛ ✐♥tr♦❞✉çã♦ ❞❡ t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛s✳ ◆♦ss❛ ❞✐s❝✉ssã♦ s❡rá ❡♠ t♦r♥♦ ❞❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✱ ❝♦♥s✐❞❡r❛❞❛ ♦ ♣r♦tót✐♣♦ ❞❛s t❡♦✲ r✐❛s ❡s❝❛❧❛r❡s t❡♥s♦r✐❛✐s✱ ♥♦ ❡♥t❛♥t♦ ❛ ❞✐s❝✉ssã♦ é ❢❛❝✐❧♠❡♥t❡ ❡st❡♥❞✐❞❛ ♣❛r❛ ❡ss❛s ✈❡rsõ❡s ♠❛✐s ❣❡r❛✐s✳ ❋❛③❡♠♦s ✐ss♦ ❡♠ ❞♦✐s ♠♦♠❡♥t♦s✳ Pr✐♠❡✐r♦✱ ✐♥❞✐❝❛♥❞♦✱ ♥♦ â♠❜✐t♦ ❞❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✱ q✉❡ ♥❛ ❡str✉t✉r❛ ❣❡♦♠étr✐❝❛ ❡ ❞❡ ❝❛♠♣♦s ❛❞♦t❛❞❛s ♣❡❧❛ t❡♦r✐❛ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❡str❡✐t❛ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧✱ ✐♥❝❧✉s✐✈❡ ❛ss♦❝✐❛♥❞♦ ♦ ❡❢❡✐t♦ ❞❡s❝r✐t♦ ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ ✧q✉✐♥t❛ ❢♦rç❛✧✭q✉❡ ✈✐♦❧❛r✐❛ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✮ ❝♦♠ ♦ ♠♦✈✐♠❡♥t♦ ❣❡♦❞és✐❝♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✱ r❡❢♦r♠✉❧❛♥❞♦ ♦ ♣♦st✉❧❛❞♦ ❣❡♦❞és✐❝♦✳ ❊✱ ♥✉♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ P❛❧❛t✐♥✐✱ ❛❝❛❜❛♠♦s ♣♦r ❢♦r♠✉❧❛r ✉♠❛ ♥♦✈❛ t❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✐❛❧✱ ❛❣♦r❛ ❝♦♠ ✐♥❣r❡❞✐❡♥t❡s ❝♦♠♣❧❡t❛♠❡♥t❡ ❣❡♦♠étr✐❝♦s✱ ❛♠❜✐❡♥✲ t❛❞❛ ♥✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ❊st✉❞❛♠♦s ❛✐♥❞❛ s♦❧✉çõ❡s ♥♦ ✈❛③✐♦ ❞♦ ♣r♦❜❧❡♠❛ ❡stát✐❝♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛ss❛ ❡s❢❡r✐❝❛♠❡♥t❡ s✐♠étr✐❝❛✱ ♦♥❞❡ s✉r❣❡♠ ♦❜❥❡t♦s ❞❡ ✐♥t❡r❡ss❡ ❛str♦❢ís✐❝♦ ❝♦♠♦ s✐♥❣✉❧❛r✐❞❛❞❡s ♥✉❛s ❡ ❜✉r❛❝♦s ❞❡ ♠✐♥❤♦❝❛✳ ❚❛♠❜é♠ ❢♦r♠✉❧❛♠♦s ❛ t❡♦r✐❛ ❝♦♥❤❡❝✐❞❛ ♣♦r W IST ✭❲❡②❧ ■♥t❡❣r❛❜❧❡ ❙♣❛❝❡t✐♠❡s✮ ❡♠ (2 + 1)D✱ ♦ q✉❡ r❡s✉❧t❛
♥✉♠❛ t❡♦r✐❛ ❝♦♥s✐st❡♥t❡✱ ♥ã♦ s♦❢r❡♥❞♦ ❞❛s ❢❛❧❤❛s ❛ss♦❝✐❛❞❛s à t❡♦r✐❛ ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ♥❡ss❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❚❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✱ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✱ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧✱ s✐♥❣✉❧❛r✐❞❛❞❡ ♥✉❛✱ ❜✉r❛❝♦ ❞❡ ♠✐♥❤♦❝❛✱ ❣r❛✈✐t❛çã♦ ❡♠ ✭✷✰✶✮D✳
❆❜str❛❝t
■♥ t❤✐s ❝♦♦❧ t❤❡s✐s✱ ✇❡ ❝♦♥s✐❞❡r ❛♥ ❛♣♣r♦❛❝❤ t♦ ❇r❛♥s✲❉✐❝❦❡ t❤❡♦r② ♦❢ ❣r❛✈✐t② ✐♥ ✇❤✐❝❤ t❤❡ s❝❛❧❛r ✜❡❧❞ ❤❛s ❛ ❣❡♦♠❡tr✐❝❛❧ ♥❛t✉r❡✳ ❇② ♣♦st✉❧❛t✐♥❣ t❤❡ P❛❧❛t✐♥✐ ✈❛r✐❛t✐♦♥✱ ✇❡ ✜♥❞ ♦✉t t❤❛t t❤❡ r♦❧❡ ♣❧❛②❡❞ ❜② t❤❡ s❝❛❧❛r ✜❡❧❞ ❝♦♥s✐sts ✐♥ t✉r♥✐♥❣ t❤❡ s♣❛❝❡✲t✐♠❡ ❣❡♦♠❡tr② ✐♥t♦ ❛ ❲❡②❧ ✐♥t❡❣r❛❜❧❡ ♠❛♥✐❢♦❧❞✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ❧❡❛❞s t♦ ❛ s❝❛❧❛r✲t❡♥s♦r t❤❡♦r② t❤❛t ❞✐✛❡rs ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ❇r❛♥s✲❉✐❝❦❡ t❤❡♦r② ✐♥ ♠❛♥② ❛s♣❡❝ts ❛♥❞ ♣r❡s❡♥ts s♦♠❡ ♥❡✇ ❢❡❛t✉r❡s✳ ❲❡ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ ❲❡②❧ ✐♥t❡❣r❛❜❧❡ ❣❡♦♠❡tr② t♦ ✐♥✈❡st✐❣❛t❡ ❣r❛✈✐t② ✐♥ ✭✷✰✶✮✲❞✐♠❡♥s✐♦♥s✳ ❲❡ s❤♦✇ t❤❛t✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ ❧❡❛❞✐♥❣ t♦ ❛ ◆❡✇t♦♥✐❛♥ ❧✐♠✐t✱ ❲■❙❚ ✐♥ ✭✷✰✶✮ ❞✐♠❡♥s✐♦♥s ♣r❡s❡♥ts s♦♠❡ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s t❤❛t ❛r❡ ♥♦t s❤❛r❡❞ ❜② ❊✐♥st❡✐♥ t❤❡♦r②✱ s✉❝❤ ❛s ❣❡♦✲ ❞❡s✐❝ ❞❡✈✐❛t✐♦♥ ❜❡t✇❡❡♥ ♣❛rt✐❝❧❡s ✐♥ ❛ ❞✉st ❞✐str✐❜✉t✐♦♥✳ ❋✐♥❛❧❧②✱ t❛❦✐♥❣ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❞✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ ❣❡♦♠❡tr✐❝❛❧ s❝❛❧❛r✲t❡♥s♦r t❤❡♦r② ❛♥❞ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t② ❝♦✉♣❧❡❞ ✇✐t❤ ❛ ♠❛ss❧❡ss s❝❛❧❛r ✜❡❧❞ ✇❡ st✉❞② ♥❛❦❡❞ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ✇♦r♠❤♦❧❡s✳
❑❡②✇♦r❞s✿ ❙❝❛❧❛r✲t❡♥s♦r t❤❡♦r✐❡s✱ ❇r❛♥s✲❉✐❝❦❡ t❤❡♦r②✱ ❲❡②❧ ✐♥t❡❣r❛❜❧❡ ❣❡♦♠❡tr②✱ ♥❛❦❡❞ s✐♥❣✉❧❛r✐t②✱ ✇♦r♠❤♦❧❡✱ ✭✷✰✶✮ ❣r❛✈✐t②✳
■♥tr♦❞✉çã♦
❊st❛ t❡s❡❧✐❛ tr❛t❛ ❞❡ tó♣✐❝♦s r❡❧❛❝✐♦♥❛❞♦s às t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ◆♦ss❛ ❛❜♦r❞❛❣❡♠ s❡rá ♥♦ s❡♥t✐❞♦ ❞❡ ✐♥❞✐❝❛r ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡✲ ❣rá✈❡❧ ❝♦♠♦ s❡♥❞♦ ✉♠ ❛♠❜✐❡♥t❡ ♥❛t✉r❛❧ ♣❛r❛ ❛ ✐♥tr♦❞✉çã♦ ❞❡ t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛s✳ ❆ ❛♠❜✐❡♥t❛çã♦ ❞❡ t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❡♠ ❣❡♦♠❡tr✐❛s ♥ã♦✲r✐❡♠❛♥♥✐❛♥❛s t❡♠ s✐❞♦ ♦❜❥❡t♦ ❞❡ ✈ár✐♦s tr❛❜❛❧❤♦s ♥❛ ❧✐t❡r❛t✉r❛✱ t♦❞♦s ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❞♦t❛r ♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❞❡ ✉♠❛ ♥❛t✉r❡③❛ ❣❡♦♠étr✐❝❛✳ P❛r❛ ❝✐t❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s t❡♠♦s ❛ ❣❡♦♠❡tr✐③❛çã♦ ❞♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❛tr❛✈és ❞❛ t♦rçã♦ ❬✶❪✱ ❧❡✈❛♥❞♦ ❛ ✉♠❛ t❡♦r✐❛ q✉❡ ❝♦♥té♠ ✐♥✈❛r✐â♥❝✐❛ ❝♦♥❢♦r♠❡ ♥♦ s❡t♦r ❣r❛✈✐t❛❝✐♦♥❛❧ ♠❛s q✉❡ é q✉❡❜r❛❞❛ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ♠❛tér✐❛✳ ❚❡♠♦s ❛ ♥ã♦✲♠❡tr✐❝✐❞❛❞❡ ♥❛ ❛✉s❡♥❝✐❛ ❞❡ ♠❛tér✐❛ ❬✷❪ ❀ ❡✱ ❛✐♥❞❛✱ ❛tr❛✈és ❞❛ ❛❞♦çã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❝♦♠ ❞✐♠❡♥sã♦ ♠❛✐♦r q✉❡ q✉❛tr♦✱ ❝♦♠♦ ❛s t❡♦r✐❛s ❞❡ ❑❛❧✉③❛✲❑❧❡✐♥✱ ♦♥❞❡ ♦ ❝❛♠♣♦ ❡s❝❛❧❛r s✉r❣❡ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♦ ❝♦♠♣♦♥❡♥t❡ ❞❛ ♠étr✐❝❛ ❞♦ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ♠❛✐♦r✳
❚❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s sã♦ ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ❛ ♠❛✐s s✐♠♣❧❡s ❡①t❡♥sã♦ ❞❛ r❡❧❛t✐✲ ✈✐❞❛❞❡ ❣❡r❛❧ q✉❡ ♣r❡s❡r✈❛ ♦ ❛❝♦♣❧❛♠❡♥t♦ ♠í♥✐♠♦ ❞❛ ❣r❛✈✐t❛çã♦ ❝♦♠ ❛ ♠❛tér✐❛✳ ◆❡ss❛s t❡♦r✐❛s✱ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ é ❞❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞♦ t❡♥s♦r ♠étr✐❝♦ gµν ❡ ❞❡ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r ϕ✳ ❈♦♠ ♦ ❛❝rés❝✐♠♦ ❞❡ ✉♠ ❣r❛✉ ❞❡ ❧✐❜❡r❞❛❞❡ ❛tr❛✈és ❞♦ ❝❛♠♣♦ ❡s❝❛❧❛r✱ ❡ss❛s
t❡♦r✐❛s tê♠ s✐❞♦ ❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ❛ ❝♦s♠♦❧♦❣✐❛ ♥♦s ❛♥♦s r❡❝❡♥t❡s ❡ ❡st✉❞❛❞❛s ❝♦♠♦ ❛❧t❡r✲ ♥❛t✐✈❛ à t❡♦r✐❛ ❞❡ ❊✐♥st❡✐♥✳ ❖ ✐♥t❡r❡ss❡ ❣❡r❛❞♦ ♣♦r t❛✐s t❡♦r✐❛s é ♠♦t✐✈❛❞♦ ♣♦r ✈❛r✐❛❞❛s r❛③õ❡s✳ ❯♠❛ ♣r❡✈✐sã♦ ❣❡♥ér✐❝❛ ❞❡ t♦❞♦s ♠♦❞❡❧♦s ❞❡ t❡♦r✐❛s ❞❡ ❝♦r❞❛s é ❞❡ q✉❡ ♦ ❣rá✈✐t♦♥✱ ❝♦♠ s♣✐♥ 2✱ t❡♥❤❛ ✉♠ ♣❛r❝❡✐r♦ ❝❤❛♠❛❞♦ ❞✐❧❛t♦♥✱ ❞❡ s♣✐♥✲0✳ ◆❡ss❡ s❡♥t✐❞♦ ❛ t❡♦r✐❛ ❞❡
❝♦r❞❛s ♣r❡✈ê q✉❡ ❛ ❝♦rr❡t❛ t❡♦r✐❛ ❞❛ ❣r❛✈✐❞❛❞❡ ❡♠ ❜❛✐①❛s ❡♥❡r❣✐❛s é ✉♠❛ t❡♦r✐❛ ❡s❝❛❧❛r✲ t❡♥s♦r✐❛❧ ❛♦ ✐♥✈és ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ❬✸❪✳ ❉❡ ❢❛t♦✱ ♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❞❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛ ❜♦sô♥✐❝❛ ❝♦rr❡s♣♦♥❞❡ à t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❝♦♠ ω=−1 ❬✹❪
❖ ❡st✉❞♦ ❞❡ ❝❛♠♣♦s q✉â♥t✐❝♦s ❡♠ ❡s♣❛ç♦s ❝✉r✈♦s ✐♥❞✐❝❛ q✉❡ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞♦ t❡♥s♦r ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❞❡ss❡s ❝❛♠♣♦s ❞✐✈❡r❣❡✱ ♠❡s♠♦ ♥♦ ❡st❛❞♦ ❞❡ ✈á❝✉♦✱ ♦ q✉❡ ♣♦❞❡ s❡r ❝♦♥t♦r♥❛❞♦ ♣❡❧♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ r❡♥♦r♠❛❧✐③❛çã♦✱ q✉❡✱ ♥❡ss❡ ❝❛s♦✱ ❝♦♥s✐st❡ ❞❛ ❛❞✐çã♦✱ ♥❛ ❛çã♦ ❣❛✈✐t❛❝✐♦♥❛❧✱ ❞❡ t❡r♠♦s ❞❡ ❝✉r✈❛t✉r❛ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❬✺❪✳ ❊ss❡s t❡r♠♦s ❛❞✐❝✐♦♥❛✐s s✉❣❡r❡♠ q✉❡ ❞❡s✈✐♦s ❞❛s ♣r❡❞✐çõ❡s ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ❞❡✈❡♠ ♦❝♦rr❡r ❡♥tr❡ ❛ ♠❡♥♦r ❡s❝❛❧❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❥á t❡st❛❞❛ ❡ ❛ ❡s❝❛❧❛ ❞❡ P❧❛♥❝❦✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ❡st✉❞♦ ❝❧áss✐❝♦ ❞❡ss❡s t❡r♠♦s ❛❞✐❝✐♦♥❛✐s t❡♠ s✐❞♦ ♠✉✐t♦ ❛♣❧✐❝❛❞♦ à ❝♦s♠♦❧♦❣✐❛✱ ❝♦♠ ❛s ❝❤❛♠❛❞❛s t❡♦r✐❛s
f(R)✱ ●❛✉ss✲❇♦♥♥❡t✱ f(T)✱ f(RµνR
µν, RαβµνRαβµν) ❡t❝✳ ❬✻❪✱ s❡❥❛ ♥♦ ✉♥✐✈❡rs♦ ♣r✐♠♦r❞✐❛❧ ♦✉ r❡❝❡♥t❡✱ s❡♥❞♦ ❝❛♣❛③❡s ❞❡ ♠♦❞❡❧❛r ♠❛tér✐❛ ❡ ❡♥❡r❣✐❛ ❡s❝✉r❛✱ ❛✜♥❛❧ ❡ss❛s ❝♦♠♣♦♥❡t❡s sã♦ ❞❡t❡❝t❛❞❛s s♦♠❡♥t❡ ❛tr❛✈és ❞❡ ❡❢❡✐t♦s ❣r❛✈✐t❛❝✐♦♥❛✐s✳✶✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ✐♥t❡r❡ss❡ ❡♠ ❣❡♦♠❡tr✐❛s ♥ã♦✲r✐❡♠❛♥♥✐❛♥❛s ♥❛ ❢ís✐❝❛ ♣♦❞❡ s❡r ❥✉st✐✲ ✜❝❛❞♦ ♣❡❧❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ ❡str✉t✉r❛s ♥ã♦ ❝♦♥t❡♠♣❧❛❞❛s ♣❡❧❛ ❣❡♦♠❡tr✐❛ ❞❡ ❘✐❡♠❛♥♥✱ ❛s q✉❛✐s ♣♦❞❡♠ s❡r ❛ss♦❝✐❛❞❛s ❛ ♣r♦♣r✐❡❞❛❞❡s ✜s✐❝❛♠❡♥t❡ ♠❡♥s✉rá✈❡✐s ❝♦♠♦ t♦rçã♦✱ ♦ s♣✐♥ ❞❡ ♣❛rtí❝✉❧❛s ♦✉ ❛ ♥ã♦✲♠❡tr✐❝✐❞❛❞❡ ✭❝♦♠♦ é ♦ ❝❛s♦ ❛q✉✐ ❡①♣❧♦r❛❞♦✮✳
✶❖ ♣r♦❜❧❡♠❛ ❝♦♠ ❡ss❛ ❡s♣❡❝t❛t✐✈❛ é q✉❡ r❡st❛ ❡①♣❧✐❝❛r ❝♦♠♦ ❡ss❡s ❞❡s✈✐♦s ❞❡s❛♣❛r❡❝❡♠ ❡♠ ❡s❝❛❧❛s ✐♥t❡r♠❡❞✐ár✐❛s✱ ♦♥❞❡ ❛ ❘● é ❜❡♠ t❡st❛❞❛ ❡ s✉❝❡❞✐❞❛ ❬✻❪
❚❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s sã♦ ❝❛r❛❝t❡r✐③❛❞❛s ♣♦r ❢✉♥çõ❡s ❞♦ ❝❛♠♣♦ ❡s❝❛❧❛r✱ s❡♥❞♦ q✉❡ ♣❛r❛ ❝❛❞❛ ❢✉♥çã♦ ❡s❝♦❧❤✐❞❛ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ t❡♦r✐❛✳ ❆q✉✐ ❞✐s❝✉t✐r❡♠♦s ♣r✐♥❝✐♣❛❧✲ ♠❡♥t❡ ✉♠❛ t❡♦r✐❛ q✉❡ é ❝♦♥s✐❞❡r❛❞❛ ♦ ♣r♦tót✐♣♦ ❞❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s✱ ❛ s❛❜❡r✱ ❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✳ ◆❛ ✐♥✈❡st✐❣❛çã♦✱ ❞❡s❝r✐t❛ ❛q✉✐✱ tr❛t❛r❡♠♦s ❞❡ r❡❧❛❝✐♦♥❛r t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❡♠ ❞♦✐s ♠♦♠❡♥t♦s✿ ♣r✐♠❡✐r♦ ✭♥♦ ❝❛♣ít✉❧♦ ✷✮ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ♥❛ s✉❛ ❢♦r♠✉❧❛çã♦ ♦r✐❣✐♥❛❧ ❞❡ ✶✾✻✶ ❬✼❪ ✭❢♦r♠✉❧❛❞❛ ♥♦ ❝❤❛♠❛❞♦ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❏♦r❞❛♥✮✱ ❡ ❞✐s❝✉t✐♠♦s✱ ❛ss✐♠ ❝♦♠♦ ❥á ❢♦✐ ❞✐s❝✉t✐❞♦ ♣♦r ❙❝❤♦❧③ ❡ ◗✉✐r♦s ❬✽✱✾❪✱ ❝♦♠♦ ❡ss❛ t❡♦r✐❛ ❝♦♥té♠ ♦s ✐♥❣r❡❞✐❡♥t❡s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ✉♠❛ ❡str✉t✉r❛ ❣❡♦♠étr✐❝❛ ❞❡ ❲❡②❧✳ ❚❛❧ ❡str✉t✉r❛ ✜❝❛ ❡①♣❧í❝✐t❛ q✉❛♥❞♦ ❛♥❛❧✐s❛♠♦s ❡ss❛ t❡♦r✐❛ ❡♠ s✉❛ ✈❡rsã♦ ❞❡ ✶✾✻✷ ✭❉✐❝❦❡✮ ✭♥♦ ❝❤❛♠❛❞♦ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❊✐♥st❡✐♥✮✱ ❡♠ q✉❡ é ❞✐s❝✉t✐❞❛ ❛ ❧✐❜❡r❞❛❞❡ ❞❡ s❡ r❡❛❧✐③❛r❡♠ tr❛♥s❢♦r♠❛çõ❡s ❝♦♥❢♦r♠❡s✱ ♦✉ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ✉♥✐❞❛❞❡s ❝♦♠♦ ❣❡r❛❧♠❡♥t❡ sã♦ ❝♦♥❤❡❝✐❞❛s ❬✶✵❪✳ ❆✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ❞✐s❝✉t✐♠♦s ❛ s❡♠❡❧❤❛♥ç❛ ❡♥✲ tr❡ ❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❡ ✉♠❛ t❡♦r✐❛ ❢♦r♠✉❧❛❞❛ ♥✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❲■❙❚✳ ▼♦str❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ❡✈✐❞❡♥❝✐❛♠ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ❡♥✲ tr❡ ❛s ❞✉❛s t❡♦r✐❛s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ q✉❡ é ❡♥t❡♥❞✐❞♦ ❡♠ ❇r❛♥s✲❉✐❝❦❡ ✭❢♦r♠✉❧❛❞❛ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❊✐♥st❡✐♥✮ ❝♦♠♦ ✉♠❛ ✧q✉✐♥t❛ ❢♦rç❛✧s❡rá ✐♥t❡r♣r❡t❛❞♦ ❡♠ ❲■❙❚ ❝♦♠♦ s❡♥❞♦ ♦ ♠♦✈✐♠❡♥t♦ ❡♠ ❣❡♦❞és✐❝❛s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❲❡②❧✳
◆✉♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❝♦♥s✐❞❡r❛♥❞♦ s♦♠❡♥t❡ ❛ ❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❡ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ P❛❧❛t✐♥✐✱ ❞❡❞✉③✐♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ♣❛r❛ ❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❝♦♥❝❧✉í♠♦s q✉❡✱ ❞❡ ❢❛t♦✱ t❡♠♦s ✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ❊ss❛ ✐♥❢♦r♠❛çã♦ ♥♦s ❧❡✈❛ ❛ s❡❧❡❝✐♦♥❛r ❛s ❣❡♦❞és✐❝❛s ❞❡ ❲❡②❧ ❝♦♠♦ s❡♥❞♦ ❛s ❝✉r✈❛s tr❛ç❛❞❛s ♣♦r ♣❛rtí❝✉❧❛s ❡♠ q✉❡❞❛ ❧✐✈r❡✳ ❊ss❛ ❡s❝♦❧❤❛ ❛❝❛rr❡t❛ ♥❛ ♠♦❞✐✜❝❛çã♦ ❞❛ ✐♥t❡r❛çã♦ ❞❛ ❣r❛✈✐t❛çã♦ ❝♦♠ ❛ ♠❛tér✐❛✱ ♦ ❝❛♠♣♦ ❡s❝❛❧❛r t❡♥❞♦ ♣❛♣❡❧ ❞✐r❡t♦ ♥❡ss❛ ♠♦❞✐✜❝❛✲ çã♦✳ ❈♦♠ ✐ss♦ ♦❜t❡♠♦s ✉♠❛ ✈❡rsã♦ ♠♦❞✐✜❝❛❞❛ ❞❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✱ ❝♦♠♣❧❡t❛♠❡♥t❡ ❣❡♦♠étr✐❝❛✳ ❊①♣❧♦r❛♠♦s ❡ss❛ t❡♦r✐❛ ♥♦ ♣r♦❜❧❡♠❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ❝♦♠ s✐♠❡tr✐❛ ❡s❢ér✐❝❛ ❡ ❛♥❛❧✐s❛♠♦s ❛s s✐♥❣✉❧❛r✐❞❛❞❡s ♥✉❛s ❡ ❜✉r❛❝♦s ❞❡ ♠✐♥❤♦❝❛ ❬✶✶❪ q✉❡ s✉r✲ ❣❡♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✈❛❧♦r❡s ❞♦ ♣❛râ♠❡tr♦ ω ❬✶✷❪✳ ❆♥❛❧✐s❛♠♦s ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧ ♦ s✐st❡♠❛
❞✐♥â♠✐❝♦ ❞❛s s♦❧✉çõ❡s ❝♦s♠♦❧ó❣✐❝❛s ❝♦♠ ♠❛tér✐❛✳
❆ t❡s❡ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥♦ ❝❛♣✐t✉❧♦ ✶ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❜r❡✈❡ r❡✲ ✈✐sã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❡ ❡♥tã♦ ♥♦s r❡str✐♥❣✐♥❞♦ ❛♦ ❝❛♠♣♦ ✐♥t❡❣rá✈❡❧ ❡ ❝♦♥str✉✐♠♦s ❛❧❣✉♠❛s q✉❛♥t✐❞❛❞❡s ✐♥✈❛r✐❛♥t❡s s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ❞❡ ❲❡②❧✳ ◆♦ ❝❛✲ ♣ít✉❧♦ ✷ tr❛t❛♠♦s ❞❡ ✐❞❡♥t✐✜❝❛r ❛s ✈❛r✐á✈❡✐s ❞❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❝♦♠ ❛s ✈❛r✐á✈❡✐s ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❢♦r♠✉❧❛♠♦s ❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ♥✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❲❡②❧ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦✱ ♦ q✉❡ ❧❡✈❛ à ❞✐❢❡r❡♥ç❛s ❢✉♥❞❛♠❡♥t❛✐s ❡♠ r❡❧❛çã♦ à ✈❡rsã♦ ♦r✐✲ ❣✐♥❛❧ ❞❡ss❛ t❡♦r✐❛✳❚❡♠♦s ❡♥tã♦ ✉♠❛ t❡♦r✐❛ ❡s❝❛❧❛r✲t❡♥s♦r✐❛❧ ❝♦♠♣❧❡t❛♠❡♥t❡ ❣❡♦♠étr✐❝❛✱ ♦♥❞❡ ✐♥✈❡st✐❣❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ s♦❧✉çõ❡s ❡stát✐❝❛s ❡ ❡s❢❡r✐❝❛♠❡♥t❡ s✐♠étr✐❝❛s✱ ❡✈✐✲ ❞❡♥❝✐❛♥❞♦ s✐♥❣✉❧❛r✐❞❛❞❡s ♥✉❛s ❡ ❜✉r❛❝♦✲❞❡✲♠✐♥❤♦❝❛✳ ◆♦ ❝❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ t❡♦r✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ W IST ❡♠ (2 + 1)D ❡ ♠♦str❛♠♦s q✉❡ ❡ss❡
✈❡rsã♦ ❞❛ t❡♦r✐❛ s❛t✐s❢❛③ ✉♠❛ sér✐❡ ❞❡ r❡q✉✐s✐t♦s ♣❛r❛ s❡r ❝♦♥s✐❞❡r❛❞❛ ✧❝♦♥s✐st❡♥t❡✧✱ ❝♦♠♦ ♣r♦♣❛❣❛çã♦ ❞♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ ❧✐♠✐t❡ ♥❡✇t♦♥✐❛♥♦✱ ❡t❝✱ q✉❡ ❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ♥ã♦ s❛t✐s❢❛③ ♥❡ss❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡✳ ❊♥❝♦♥tr❛♠♦s ❛❧❣✉♠❛s s♦❧✉çõ❡s ❝♦s♠♦❧ó❣✐❝❛s✱ ❝♦♠ ❞❡s✲ t❛q✉❡ ♣❛r❛ s♦❧✉çõ❡s s❡♠ s✐♥❣✉❧❛r✐❞❛❞❡ ✭❜♦✉♥❝✐♥❣✮ ❝♦s♠♦❧ó❣✐❝❛❀ ❡①✐❜✐♠♦s t❛♠❜é♠ ✉♠❛ s♦❧✉çã♦ ❡stát✐❝❛ ♣❛r❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝✐r❝✉❧❛r♠❡♥t❡ s✐♠étr✐❝❛✳ ◆♦ ❝❛♣ít✉❧♦ ✺✱ ❢❛③❡♠♦s ✉♠ s✉♠ár✐♦ ❞♦s r❡s✉❧t❛❞♦s ❞❛ t❡s❡✱ s❡❣✉✐❞♦ ❞❛s ❝♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s ♣❛r❛ ❡st❡♥❞❡r ♦ ♣r❡s❡♥t❡ ❡st✉❞♦✳
❈❛♣ít✉❧♦ ✶
❊s♣❛ç♦✲t❡♠♣♦ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧
✭❲■❙❚✮
❊♠ ✶✾✶✽✱ ❍❡r♠❛♥ ❲❡②❧ ❝♦♥❝❡❜❡✉ ✉♠❛ t❡♦r✐❛ q✉❡ t❡♥t❛✈❛ ✉♥✐✜❝❛r ❣r❛✈✐t❛çã♦ ❡ ❡❧❡tr♦✲ ♠❛❣♥❡t✐s♠♦✱ ❛s ❞✉❛s ❢♦rç❛s ❢✉♥❞❛♠❡♥t❛✐s ❝♦♥❤❡❝✐❞❛s ♥❛ é♣♦❝❛ ❬✶✸❪✳ P❛r❛ t❛♥t♦✱ ❢♦✐ ❞❡✲ s❡♥✈♦❧✈✐❞❛ ✉♠❛ ❣❡♦♠❡tr✐❛ q✉❡ ❣❡♥❡r❛❧✐③❛✈❛ ♦ ❡s♣❛ç♦✲t❡♠♣♦ r✐❡♠❛♥♥✐❛♥♦ ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✳ ❖ ♦❜❥❡t✐✈♦ ❡r❛ ❞❡ q✉❡ ❡ss❡s ♥♦✈♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❣❡♦♠étr✐❝♦s ❝♦rr❡s♣♦♥❞❡ss❡♠ ❛♦ ♣♦t❡♥❝✐❛❧ ❡❧❡tr♦♠❛❣♥ét✐❝♦ Aµ✳ P❛r❛ ✐ss♦✱ ❲❡②❧ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ ❣❡♦♠❡tr✐❛ ✐♥✈❛r✐❛♥t❡ ❞❡ ❝❛❧✐❜r❡✶ q✉❡✱ ❛♣❡s❛r ❞❛ ❡♥❣❡♥❤♦s✐❞❛❞❡ ❡ ❛♣❡❧♦ ❡stét✐❝♦✱ ♥ã♦ r❡s✉❧t♦✉ ♥✉♠❛ t❡♦r✐❛ ❝♦♠✲ ♣❛tí✈❡❧ ❝♦♠ ❛s ♦❜s❡r✈❛çõ❡s✳ ❆ ❞✐✜❝✉❧❞❛❞❡ ❡r❛ ♣♦r ❝♦♥t❛ ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❛❞✐❝✐♦♥❛✐s σµ ❝♦♠ ♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦✱ r❡♣r❡s❡♥t❛❞♦ ♣♦rAµ✳ P♦r ❡①❡♠♣❧♦✱ ❡ss❡ ❝❛♠♣♦ σµ ✐♥t❡r❡❣❡ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ ❝♦♠ ♣❛rtí❝✉❧❛s ❡ ❛♥t✐♣❛rtí❝✉❧❛s✱ ❝♦♥trár✐♦ ❛ t♦❞❛s ❡✈✐❞ê♥❝✐❛s ❡①♣❡r✐♠❡♥t❛✐s ❞♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❬✶✹❪✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❡ss❡s ♥♦✈♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❝♦rr❡s♣♦♥❞❡ss❡♠ à ♣ró♣r✐❛ ❣r❛✈✐t❛çã♦✱ ❝♦♠ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ❡s❝❛❧❛s ❝♦s♠♦❧ó✲ ❣✐❝❛s✱ ❛ ❣❡♦♠❡tr✐❛ ❢✉♥❞❛❞❛ ♣♦r ❲❡②❧ ♣✉❞❡ss❡ ❡①♣❧♦r❛r ♥♦✈♦s ❝❡♥ár✐♦s✳ ❊✱ ❞❡ ❢❛t♦✱ ❢♦✐ ✉s❛❞❛ ♣♦r ❉✐r❛❝ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ t❡♦r✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ❞❡ ❢♦rt❡ ❛♣❡❧♦ ❝♦s♠♦❧ó❣✐❝♦✱ ❜❛s❡❛❞❛ ♥❛ s✉❛ ❍✐♣ót❡s❡ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ❬✶✺❪✱ s❡♥❞♦✱ ❞❡♣♦✐s✱ r❡❢♦r♠✉❧❛❞❛ ♣❛r❛ ❡①♣❧♦r❛r ❡s✲ ❝❛❧❛s ❞❡ ❛❧t❛s ❡♥❡r❣✐❛s ♥♦ r❡✐♥♦ ❞❛s ❢♦rç❛s ♥✉❝❧❡❛r❡s ❡ ❛tr❛✈és ❞❛ q✉❡❜r❛ ❞❡ s✐♠❡tr✐❛ ❞❡ ❝❛❧✐❜r❡ ❝♦♥❢♦r♠❡ ♥❛ ❣r❛✈✐t❛çã♦ ❬✽✱✶✻❪✳
❙❡❣✉♥❞♦ ❛❧❣✉♥s ❛✉t♦r❡s✱ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ r✐✲ ❡♠❛♥♥✐❛♥❛ q✉❡ t❡♠ ❛ ✐♥t❡♥çã♦ ❞❡ ❢❛③❡r ❥✉st✐ç❛ ♠❛t❡♠át✐❝❛ ❛♦s ❝♦♠♣r✐♠❡♥t♦s ❬✶✼❪✳ ❆ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛ ❝♦♥s✐❞❡r❛ q✉❡ ♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ♥✉♠ ❡s♣❛ç♦ ❝✉r✈♦✷ ❛❧t❡r❛ ❛ ❞✐r❡çã♦ ❞❡ ✈❡t♦r❡s✱ ❡♥q✉❛♥t♦ s✉❛ ♥♦r♠❛✭ ♦✉ ❝♦♠♣r✐♠❡♥t♦✮ ♣❡r♠❛♥❡❝❡ ✐♥❛❧t❡r❛❞♦✱ ♣❡r✲ ♠✐t✐♥❞♦ ❛ss✐♠ q✉❡ ❢❛ç❛♠♦s ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ❝♦♠♣r✐♠❡♥t♦s ❞❡ ✈❡t♦r❡s ❡ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦✸ ❡♠ ♣♦♥t♦s ❞✐st❛♥t❡s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❆ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡✈✐❞❛ ❛ ❍✳ ❲❡②❧ r❡❧❛①❛✈❛ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡✱ ♣❡r♠✐t✐♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❛ ♥♦r♠❛ ✭❞♦ ❝♦♠♣r✐♠❡♥t♦✮ ❞❡✈✐❞♦ ❛♦ tr❛♥s✲ ♣♦rt❡ ♣❛r❛❧❡❧♦✳ ❈♦♠♦ ✈❡r❡♠♦s✱ ❲❡②❧ ♣ô❞❡ r❡❛❧✐③❛r t❛❧ ❣❡♥❡r❛❧✐③❛çã♦ ❛❧t❡r❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛ ❝♦♥❡①ã♦ ❡ ❛ ♠étr✐❝❛✱ ❝♦♥str✉✐♥❞♦ ❛ss✐♠ ✉♠❛ ❣❡♦♠❡tr✐❛ ✐♥✲ t❡✐r❛♠❡♥t❡ ✐♥✜♥✐t❡s✐♠❛❧ ♦✉ ❧♦❝❛❧ ✭◆ã♦ s❡r✐❛♠ ♣♦ssí✈❡✐s ❝♦♠♣❛r❛çõ❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✈❡t♦r❡s ❡♥tr❡ ♣♦♥t♦s ❞✐st❛♥t❡s ❞❛ ✈❛r✐❡❞❛❞❡ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✱ t❛❧ ❝♦♠♣❛r❛çã♦ só s❡r✐❛ ❢❛❝tí✈❡❧ ♠❡❞✐❛♥t❡ ❞❡t❡r♠✐♥❛❞❛ ❧❡✐ ❞❡ tr❛♥♣♦rt❡ ♣❛r❛❧❡❧♦ ❞♦s ❝♦♠♣r✐♠❡♥t♦s✮ ❬✶✸❪✳
✶❚❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ♣♦r ❣❡♦♠❡tr✐❛ ✐♥✈❛r✐❛♥t❡ ❞❡ ❡s❝❛❧❛✱ ♦♥❞❡ tr❛♥s❢♦r♠❛çõ❡s ❝♦♥❢♦r♠❡s ❞❛ ♠étr✐❝❛ ❢❛③❡♠ ♣❛rt❡ ❞❛ tr❛♥s❢♦r♠❛çã♦
✷❖ ❡q✉✐✈❛❧❡♥t❡ à tr❛♥s♣♦s✐çã♦ ❞❡ ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳
✸❊♠ ♣r✐♥❝í♣✐♦ ❛s ❧✐♥❤❛s ❡s♣❡❝tr❛✐s ♣❛❞rã♦✱ ❡♠ q✉❡ s❡ ❜❛s❡✐❛♠ r❡❧ó❣✐♦s ❛tô♠✐❝♦s✱ ♣♦❞❡r✐❛♠ ❡st❛r s✉❥❡✐t❛s ❛ ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s✱ ❞❡ ♠♦❞♦ q✉❡ ♥ã♦ ❤❛✈❡r✐❛ ♠❡❞✐❞❛s r✐❣♦r♦s❛♠❡♥t❡ ♣❛❞r♦♥✐③❛❞❛s ❞❡ t❡♠♣♦ ♣ró♣r✐♦✳
✶✳✶ ●❡♥❡r❛❧✐③❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛
❆ ❞✐❢❡r❡♥ç❛ ❡ss❡♥❝✐❛❧ ❡♥tr❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❘✐❡♠❛♥♥ ❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ é q✉❡ ♥❛ ♣r✐♠❡✐r❛ ❛❞♠✐t✐♠♦s q✉❡ ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❞❛ ♠étr✐❝❛ ∇αgµν é ③❡r♦✱ ❡♥q✉❛♥t♦ q✉❡ ♥❛ ú❧t✐♠❛ t❡♠♦s
∇αgµν =σαgµν, ✭✶✳✶✮ ♦♥❞❡ σα ❞❡♥♦t❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ 1✲❢♦r♠❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❜❛s❡ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛❧✳ ■ss♦ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛ ❝♦♥❡①ã♦∇❡ ❛ ♠étr✐❝❛ g✱ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❝♦♥❞✐çã♦ ❞❡ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r
s❡ ♠❛♥t❡♥❤❛ ✐♥❛❧t❡r❛❞♦ s♦❜ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ❬✶✽❪✳
❆ trí❛❞❡ (M, g, σ) ♦♥❞❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ✉♠❛ ♠étr✐❝❛ g ❡ ✉♠
❝❛♠♣♦ ❞❡ ❲❡②❧ σ✱ q✉❡ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ g ❛tr❛✈és ❞❡ ✭✶✳✶✮✱ s❡rá ❞❡♥♦t❛❞❛ ❝♦♠♦ r❡❢❡r❡♥❝✐❛❧
❞❡ ❲❡②❧✳ ➱ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❲❡②❧ ✭✶✳✶✮ ♣❡r♠❛♥❡❝❡ ✐♥❛❧t❡r❛❞❛ q✉❛♥❞♦ ✈❛♠♦s ♣❛r❛ ✉♠ ♦✉tr♦ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❲❡②❧ (M,g,¯ σ¯)s♦❜ ❛s tr❛♥s❢♦r♠❛çõ❡s s✐♠✉❧tâ♥❡❛s ❡♠
g ❡ σ✿
¯
g = efg
¯
σ = σ+df ✭✶✳✷✮
♦♥❞❡ f é ✉♠❛ ❢✉♥çã♦ ❡s❝❛❧❛r ❞❡✜♥✐❞❛ ❡♠M✳ ❆ss✐♠✱ ♦s r❡❢❡r❡♥❝✐❛✐s ❞❡ ❲❡②❧ ❝♦♥st✐t✉❡♠
✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ r❡❧❛❝✐♦♥❛❞♦s ♣♦r ✭✶✳✷✮✱ ♦✉ s❡❥❛✱ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ♠étr✐❝❛ ❡st❛rá ❞❡✜♥✐❞❛ à ♠❡♥♦s ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❲❡②❧✳
❆♥❛❧♦❣❛♠❡♥t❡ à ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛✱ ❛ ❝♦♥❞✐çã♦ ✭✶✳✶✮ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❝♦♥❡①ã♦ s❡♠ t♦rçã♦✱ ∇✱ ❡♠ t❡r♠♦s ❞❛ ♠étr✐❝❛g ❡ ❞♦ ❝❛♠♣♦ ❞❡ 1✲❢♦r♠❛ ❞❡ ❲❡②❧σ✳ ❆s
❝♦♠♣♦♥❡♥t❡s ❞❛ ❝♦♥❡①ã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ✉♠❛ ❜❛s❡ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛r❜✐trár✐❛s é✿
Γα µν =
α
µν −
1 2g
αβ(g
βµσν +gβνσµ−gνµσβ). ✭✶✳✸✮ P♦r ❞❡✜♥✐çã♦✱ ❛ ❝♦♥❡①ã♦ ❞❡ ❲❡②❧ é s✐♠étr✐❝❛✹✳ ❙❡ ❡①✐st❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ ❡♠ q✉❡ ♦ ❝❛♠♣♦ ❞❡ ❲❡②❧ s❡ ❛♥✉❧❛✱ σ= 0✱ ♦❜✈✐❛♠❡♥t❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ s❡ r❡❞✉③ à ❣❡♦♠❡tr✐❛ ❞❡ ❘✐❡♠❛♥♥
♥❡st❡ r❡❢❡r❡♥❝✐❛❧✳
❱❛♠♦s ✈❡r ❛❣♦r❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ tr❛♥s♣♦rt❡ ♣❛r❛❧❡❧♦ ❞❡ ❲❡②❧✳ ❙❡❥❛ M ✉♠❛
✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✱ ✉♠❛ ♠étr✐❝❛ g ❡ ✉♠ ❝❛♠♣♦ ❞❡ ❲❡②❧ σ✳ ❙❡
∇ ❢♦r ❝♦♠♣❛tí✈❡❧ ❝♦♠ g ♥♦ s❡♥t✐❞♦ ❞❡ ❲❡②❧ ✭✶✳✶✮✱ ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r ❝✉r✈❛ γ✱ ❞❡✜♥✐❞❛
♣❡❧❛s ❡q✉❛çõ❡s ♣❛r❛♠❡tr✐③❛❞❛s xα = xα(u)✱ s❡♥❞♦ u ✉♠ ♣❛râ♠❡tr♦✱ ❡ q✉❛❧q✉❡r ♣❛r ❞❡ ✈❡t♦r❡s tr❛♥s♣♦rt❛❞♦s ♣❛r❛❧❡❧❛♠❡♥t❡ V ❡ W ❛♦ ❧♦♥❣♦ ❞❡ γ✱ t❡r❡♠♦s
d
dug(V, W) = dxα
du ∇α(gµνV
µWν) = VµWνdxα
du ∇αgµν. ✭✶✳✹✮
P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛ ❝♦♥❞✐çã♦ ✭✶✳✶✮ s❡❣✉❡
dxα
du ∇αgµν = Dgµν
du =σα dxα
du gµν. ✭✶✳✺✮
P♦rt❛♥t♦
d
dug(V, W) =gµνV
µWνσ α
dxα
du =σ( d
dug(V, W)). ✭✶✳✻✮
✹❈♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛♥✉❧❛r ❧♦❝❛❧♠❡♥t❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❝♦♥❡①ã♦
❙❡❥❛W =V✱ ❡L(u)♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦rV(u)❡♠ ✉♠ ♣♦♥t♦ ❛r❜✐tár✐♦ ❞❛ ❝✉r✈❛✱ ❡♥tã♦
❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❝♦♥❞✉③ ❛
dL du =
σα
2
dxα
du L. ✭✶✳✼✮
■♥t❡❣r❛♥❞♦ ✭✶✳✼✮ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣♦♥t♦ P0 =xα(u0)♦❜t❡♠♦s
L(u) =L0e Ru
u0 σα2
dxα
du′du′. ✭✶✳✽✮
❱❡♠♦s ❝❧❛r❛♠❡♥t❡ q✉❡ ❛ ✈❛r✐❛çã♦ ♥♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r é r❡❣✉❧❛❞❛ ♣❡❧❛ ✐♥t❡❣r❛❧✱ q✉❡ ❞❡♣❡♥❞❡ ❞♦s ♣♦♥t♦s ✐♥✐❝✐❛❧ ❡ ✜♥❛❧ ❡ ❞❛ ❝✉r✈❛ ❡s❝♦❧❤✐❞❛ γ✳ ▼❛✐s ❛✐♥❞❛✱ ❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡
q✉❛✐sq✉❡r q✉❛♥t✐❞❛❞❡s ♠étr✐❝❛s ❡♠ ❞✐❢❡r❡♥t❡s ♣♦♥t♦s só s❡ ❞❛rá ♠❡❞✐❛♥t❡ ♦ tr❛♥s♣♦rt❡ ❞❡ s❡✉ ✧❝♦♠♣r✐♠❡♥t♦ ♣❛❞rã♦✧✭❛q✉✐✱ ❡st❛❜❡❧❡❝❡r ✉♠ ✧❝♦♠♣r✐♠❡♥t♦ ♣❛❞rã♦✧é ✜①❛r ✉♠❛ ♠é✲ tr✐❝❛ ❞❡♥tr♦ ❞❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❡s❝♦❧❤❡r ✉♠ ❝❛❧✐❜r❡✮✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ♠❡❞✐❛♥t❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣❡❧♦ ❢❛t♦r ❡①♣♦♥❡♥❝✐❛❧ ❛❝✐♠❛✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❢✉♥çã♦ ✧tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❡s❝❛❧❛✧✳ ❆ ✧❝✉r✈❛t✉r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✧é s✐♠♣❧❡s♠❡♥t❡ ❛ ❞❡r✐✈❛❞❛ ❡①t❡r✐♦r✱ F =dσ✱ ❝♦♠
❝♦♠♣♦♥❡♥t❡sFµν =∂µσν−∂νσµ✳ ❙❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ é ♥✉❧❛✱ ✐st♦ é✱ s❡ F = 0✱ ❛ ❢✉♥çã♦ tr❛♥s❢❡rê♥❝✐❛ ♣♦❞❡ s❡r ✐♥t❡❣r❛❞❛✳ ❱♦❧t❛♥❞♦ ❛♦ r❡s✉❧t❛❞♦ ✭✶✳✽✮✱ t❡♠♦s ✉♠❛ ♣❡❝✉✲ ❧✐❛r✐❞❛❞❡ ❛♣♦♥t❛❞❛ ♣♦r ❊✐♥st❡✐♥✿ ❞♦✐s ♦❜s❡r✈❛❞♦r❡s ♣❛rt✐♥❞♦ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ❝♦♠ r❡❧ó❣✐♦s s✐♥❝r♦♥✐③❛❞♦s ❛♦ t♦♠❛r❡♠ ❞✐❢❡r❡♥t❡s tr❛❥❡tór✐❛s ❡ ❛❧❝❛♥ç❛♥❞♦ ✉♠ ♦✉tr♦ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ♣❡r❝❡❜❡rã♦ q✉❡ s❡✉s r❡❧ó❣✐♦s ❛❣♦r❛ ❡stã♦ ❢✉♥❝✐♦♥❛♥❞♦ ❝♦♠ ✐♥t❡r✈❛❧♦s ❞✐❢❡r❡♥t❡s✳ ❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ xα(u) : [a, b]∈R→M✱ ♦✉ s❡❥❛ x(a) =x(b)✱ t❡♠♦s
L(u) = L0eH σα2
dxα
du′du′. ✭✶✳✾✮
❉♦ t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s s❡❣✉❡ q✉❡ s❡ σ ❢♦r ✉♠❛ ❢♦r♠❛ ❡①❛t❛✱ ✐st♦ é✱ s❡ ❡①✐st✐r ✉♠❛ ❢✉♥çã♦
❡s❝❛❧❛r φ t❛❧ q✉❡ σ =dφ✱ ❡♥tã♦
I
∂αφ
2
dxα
du′du ′ =
I
dφ du′du
′ =
I
dφ= 0. ✭✶✳✶✵✮
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♥❡ss❡ ❝❛s♦ ❛ ✐♥t❡❣r❛❧ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ tr❛❥❡tór✐❛ t♦♠❛❞❛✳ ❙❡♥❞♦ ❡ss❛ ✐♥t❡❣r❛❧ q✉❡ r❡❣✉❧❛ ❛ ♠❛r❝❤❛ ❞♦s r❡❧ó❣✐♦s ❛tô♠✐❝♦s ✱ ❡ss❛ ❝❧❛ss❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ♥ã♦ s♦❢r❡ ❞❛ ❢❛❧❤❛ ❛♣♦♥t❛❞❛ ♣♦r ❊✐♥st❡✐♥✱ ❡ t❡♠♦s ♦ q✉❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✈❛r✐❡❞❛❞❡ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢❛r❡♠♦s ✉s♦ ❡①❝❧✉s✐✈♦ ❞❡ss❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧✱ ❝❤❛♠❛❞❛ ●❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ■♥t❡❣rá✈❡❧✳ ❆q✉✐ ♣♦❞❡♠♦s ❢❛③❡r ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ r❡❞✉③✐r ❛ ✉♠❛ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛✱ ❜❛st❛ ✐❞❡♥t✐✜❝❛r ♦ ❝❛♠♣♦ ❡s❝❛❧❛r ❛❝✐♠❛ ❝♦♠ ♦ ❢❛t♦r ❝♦♥❢♦r♠❡ ❞❛ ♠étr✐❝❛✳ ❆ ❡ss❛ ❡s❝♦❧❤❛ ♥♦s r❡❢❡r✐r❡♠♦s ❝♦♠♦ r❡❢❡r❡♥❝✐❛❧ ❞❡ ❘✐❡♠❛♥♥✱
(M,g,˜ 0)✳ ❙❡ σ =dφ❡♥tã♦
∇αgµν =gµνφ,α, ✭✶✳✶✶✮ ♦♥❞❡ φ,α=∂αφ✱ ♦✉ ❛✐♥❞❛
∇(e−φg
µν) = ∇˜gµν = 0, ✭✶✳✶✷✮ q✉❡ é ❛ ❝♦♥❞✐çã♦ r✐❡♠❛♥♥✐❛♥❛✳ P♦rt❛♥t♦✱ ❞❡s❞❡ q✉❡ tr❛❜❛❧❤❡♠♦s ❝♦♠ ❛ ✈❡rsã♦ ✐♥t❡❣rá✲ ✈❡❧ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✭M, g, φ✮✱ ❡s❝♦❧❤❡♥❞♦ ❡♠ ✭✶✳✷✮ f = −φ✱ t❡r❡♠♦s ✉♠❛ tr❛♥s✲
❢♦r♠❛çã♦ ♣❛r❛ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦♥❞❡ ♦ ❝❛♠♣♦ ❞❡ ❲❡②❧ φ˜ é ♥✉❧♦✱ ♦✉ s❡❥❛✱ ♦ r❡❢❡r❡♥❝✐❛❧
r✐❡♠❛♥♥✐❛♥♦✭M,g,˜ φ˜= 0✮✳
❯♠ ❢❛t♦ ♠❛t❡♠át✐❝♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ é ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❛s ❣❡♦❞és✐❝❛s s♦❜ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❲❡②❧ ✭✶✳✷✮✳ P❛r❛ ✈❡r✐✜❝❛r ✐ss♦✱ ❝♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞❡ ❣❡♦❞és✐❝❛ ❛✜♠
∇VV = 0 ✭✶✳✶✸✮
❡♠ ✉♠ ❝❡rt♦ r❡❢❡r❡♥❝✐❛❧ (M, g, σ)✱ ♦♥❞❡V ❞❡♥♦t❛ ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❛ ❝✉r✈❛ ❣❡♦❞❡s✐❝❛✳ ❊♠
❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s
d2xµ
du2 + Γ
µ αβ
dxα
du dxβ
du = 0, ✭✶✳✶✹✮
♦♥❞❡ Γµαβ ❞❡♥♦t❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ❝♦♥❡①ã♦ ∇✱ u é ✉♠ ♣❛râ♠❡tr♦ ❛✜♠ ❡ xµ = xµ(u) r❡♣r❡s❡♥t❛ ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❧♦❝❛✐s ❞❛s ❣❡♦❞és✐❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ♠✉❞❛♠♦s ♣❛r❛ ✉♠ r❡❢❡r❡♥❝✐❛❧ (M,g,¯ σ¯) ❛tr❛✈és ❞❡ ✭✶✳✷✮✳ ❖❜✈✐❛♠❡♥t❡ ❡♠ ❝❛❞❛ r❡❢❡r❡♥❝✐❛❧✱ ❛s ❝♦♠♣♦♥❡♥t❡s
❞❡ ∇✱✐st♦ é✱ Γµαβ ❡ Γ¯µαβ✱ ♣♦❞❡♠ s❡r ❡①♣r❡ss❛s ❡♠ t❡r♠♦s ❞♦s s✐♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧✱µ αβ ❡ µ¯
αβ ✱ ❡ ❞♦s ❝❛♠♣♦s ❞❡ ❲❡②❧ σ ❡ σ¯✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠♦ ❡♠ ✭✶✳✸✮✳ ❖r❛✱ ❝♦♠♦∇ ✜❝❛ ✐♥❛❧t❡r❛❞❛ ♣❡❧❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❲❡②❧✱ s❡ xα =xα(u)é s♦❧✉çã♦ ❞❡ ✭✶✳✶✸✮ ♥♦ r❡❢❡r❡♥❝✐❛❧
(M, g, σ)✱ ❡♥tã♦ é ✉♠❛ s♦❧✉çã♦ ❞❡ss❛ ♠❡s♠❛ ❡q✉❛çã♦ ♥♦ ♦✉tr♦ r❡❢❡r❡♥❝✐❛❧ (M,g,¯ σ¯)✳ ❆s
❡q✉❛çõ❡s ❣❡♦❞❡s✐❝❛s sã♦ ✐♥✈❛r✐❛♥t❡s ♣♦rq✉❡ Γµαβ = ¯Γµαβ✱ ❡ ❛ ✈❡r❞❛❞❡ ❞❡ss❡ ❢❛t♦ ♣♦❞❡ s❡r
✈❡r✐✜❝❛❞❛ ❢❛❝✐❧♠❡♥t❡ ✉s❛♥❞♦ ✭✶✳✸✮✳
✶✳✷ ❈✉r✈❛t✉r❛ ❞❡ ❲❡②❧
❆ ♠♦❞✐✜❝❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ r✐❡♠❛♥♥✐❛♥❛ ❛ ♣❛rt✐r ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❲❡②❧ ✭✶✳✶✮ ✐♠♣❧✐❝❛ ♥❛ ♠♦❞✐✜❝❛çã♦ ❞♦ t❡♥s♦r ❞❡ ❝✉r✈❛t✉r❛ R✱ ♦✉ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥✱
R=❘+f(g, σ,∇σ), ✭✶✳✶✺✮
♦♥❞❡ R(g, σ) é ❛ ❝✉r✈❛t✉r❛ ❞❡ ❲❡②❧✱ ❘(g) é ❛ ❝✉r✈❛t✉r❛ r✐❡♠❛♥♥✐❛♥❛ ❡ f(g, σ,∇σ) ❡
✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞♦s t❡r♠♦s ❛❞✐❝✐♦♥❛✐s ❞❡✈✐❞♦ ❛ ✭✶✳✶✮✱ ❝✉❥❛ ❡①♣r❡ssã♦ ❡♥❝♦♥tr❛r❡♠♦s ❧♦❣♦ ❛ s❡❣✉✐r✳ ❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ é q✉❡ s❡❧❡❝✐♦♥❛♥❞♦ ♦ ❡s♣❛ç♦ ❞❡ ♠étr✐❝❛ ♣❧❛♥❛ ❞❡ ▼✐♥❦♦✇s❦✐✱ ηµν✱ ♦♥❞❡ ❛ ❝✉r✈❛t✉r❛ r✐❛♠❛♥♥✐❛♥❛ é ♥✉❧❛✱ ❘ = 0✱ t❡♠♦s ✉♠❛ ❝✉r✈❛t✉r❛ ❞❡ ❲❡②❧ ♥ã♦✲♥✉❧❛✱ R 6= 0✱ ❞❛❞❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ♣❡❧♦ ❝❛♠♣♦ ❞❡ ❲❡②❧
R=f(η, σ, ∂σ,σ), ✭✶✳✶✻✮
❚❛✐s ❡s♣❛ç♦s ❞❡ ❲❡②❧✱ ♥♦ ❝❛s♦ ✐♥t❡❣rá✈❡❧✱ ❢♦r❛♠ ❡st✉❞❛❞♦s ❡♠ ❬✶✾❪ ❡ ❬✷✵❪ ♦♥❞❡ ❢♦r❛♠ ❢♦r♠✉✲ ❧❛❞❛s t❡♦r✐❛s ❞❛ ❣r❛✈✐t❛çã♦ ❡s❝❛❧❛r ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✱ ❝♦♠♦ ❛ t❡♦r✐❛ ❞❡ ◆♦r❞strö♠ ❡ ❡s♣❡❝✉❧❛çõ❡s s♦❜r❡ ❛ ✈✐❛❜✐❧✐❞❛❞❡ ❞❡ q✉❛♥t✐③❛çã♦ ❞❡ss❛ ❣r❛✈✐❞❛❞❡ ❡s❝❛❧❛r✳ ◆❡ss❡ ❝♦♥t❡①t♦✱ t❛♠❜❡♠ ❡①✐st❡ ✉♠ ♣r♦❣r❛♠❛ q✉❡ r❡❧❛❝✐♦♥❛ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❝♦♠ ❛ ♠❡❝â♥✐❝❛ q✉❛♥t✐❝❛ r❡❧❛t✐✈íst✐❝❛ ❞❡ ❇♦❤♠✱ ♦ ❝❛♠♣♦ ❞❡ ❲❡②❧ ❢❛③❡♥❞♦ ♦ ♣❛♣❡❧ ❞❡ ♣♦t❡♥❝✐❛❧ q✉â♥t✐❝♦ ❬✷✶✱✷✷❪✳
✶✳✷✳✶ ❈✉r✈❛t✉r❛ ❞❡ ❲❡②❧ ♥ã♦✲✐♥t❡❣rá✈❡❧
P❛r❛ ♦❜t❡r ❛ ❝✉r✈❛t✉r❛ ❞❡ ❲❡②❧ ♦ ♣r♦❝❡❞✐♠❡♥t♦ é ✐❞ê♥t✐❝♦ ❛♦ ❝❛s♦ r✐❡♠❛♥♥✐❛♥♦✱ ♦✉ s❡❥❛✱ ❞❡✜♥✐♠♦s Rλ
µαβ ❬✷✸❪
∇β∇αVλ− ∇α∇βVλ =RλµαβVµ, ✭✶✳✶✼✮ ❞❡ ♦♥❞❡ r❡s✉❧t❛ q✉❡
Rλ
ραβ =∂βΓλρα−∂αΓρβλ + ΓλµβΓµρα−ΓλµαΓ µ
ρβ. ✭✶✳✶✽✮
P♦❞❡♠♦s ❛❣♦r❛ ❡①♣r❡ss❛r Rλ
µαβ ❡♠ t❡r♠♦s ❞❛ ❝✉r✈❛t✉r❛ r✐❡♠❛♥✐❛♥❛✱ ✉s❛♥❞♦ ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❝♦♥❡①ã♦ ♦❜t✐❞❛ ❡♠ ✭✶✳✸✮ ♥❛ ❢♦r♠❛ ♠❛✐s s✉❝✐♥t❛
Γαµν =α
µν −Wµνα ✭✶✳✶✾✮
♦♥❞❡
Wµνα = 1 2g
αβ(g
βµσν +gβνσµ−gνµσβ). ✭✶✳✷✵✮ P♦rt❛♥t♦✱ t❡♠♦s
Rλραβ = ❘λραβ +
1 2δ
λ ρFαβ−
1 2(δ
λ αδ
µ
β −δλβδαµ)
˜
∇µσρ+
1
2σµσρ− 1 2gρµσ
γσ γ +1 2 δ ν
βδµα−δανδ µ β
˜
∇ν(gρµσλ) +
1
2gρµσνσ
λ
✭✶✳✷✶✮
♦♥❞❡ Fαβ =∂ασβ −∂βσα é ❛ ✧❝✉r✈❛t✉r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✧ ❬✷✹❪ ❡ ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ∇˜ r❡♣r❡s❡♥t❛ ❛ ❝♦♥❡①á♦ r✐❡♠❛♥♥✐❛♥❛✳ ❖ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ é ❞❛❞♦ ♣♦r
Rρβ =Rαραβ = ❘ρβ+
1
2Fρβ−∇˜βσρ− 1
2gβρ∇˜ασ
α
− 12σβσρ+
1 2gρβσ
γσ
γ ✭✶✳✷✷✮
◆♦t❡ q✉❡ ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ t❡♠ ✉♠❛ ♣❛rt❡ ❛♥t✐ss✐♠étr✐❝❛✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ✧❝✉r✈❛t✉r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✧✳ ❈♦♥tr❛✐♥❞♦ ❝♦♠ ❛ ♠étr✐❝❛ t❡♠♦s ♦ ❡s❝❛❧❛r ❞❡ ❘✐❝❝✐
R=❘−3 ˜∇ασα+
3 2σ
γσ
γ. ✭✶✳✷✸✮
P❛r❛ ♦ ❝❛s♦ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✱σ =dφ✱ ❛s ❡①♣r❡ssõ❡s ❛❝✐♠❛ sã♦ ❢❛❝✐❧♠❡♥t❡ ❛❞❛♣t❛❞❛s
❡ ❛ ♣❛rt❡ ❛♥t✐ss✐♠étr✐❝❛ ❞♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ Fρβ✱ ❛ ✧❝✉r✈❛t✉r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✧✱ s❡ ❛♥✉❧❛✳
✶✳✷✳✷ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐ ♣❛r❛ ❲❡②❧ ✐♥t❡❣rá✈❡❧
❖ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ ❞❡✜♥✐❞♦ ♣♦rRαµβν = Γαβµ,ν−Γανµ,β + ΓαρνΓ ρ
βµ−ΓαρβΓρνµ ✭✶✳✷✹✮ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡
∇σRαµβν+∇βRµνσα +∇νRαµσβ = 0. ✭✶✳✷✺✮ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r δβ
α t❡♠♦s
∇σRβµβν+∇βRβµνσ+∇νRβµσβ = 0 ✭✶✳✷✻✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ Rµν =Rβµβν✱ ❡✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ t❡♥s♦rFµν = 0✱ ♣♦rt❛♥t♦ ♦ t❡♥♦sr ❞❡ ❘✐❝❝✐ ❛❣♦r❛ é s✐♠étr✐❝♦✱ ✉s❛♥❞♦ ❛ s✐♠❡tr✐❛ ❞♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ Rβµσβ = −Rβµβσ =−Rµσ✱ t❡♠♦s
∇σRµν+∇βRβµνσ − ∇νRµσ = 0. ✭✶✳✷✼✮ ❆❣♦r❛✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r gµσ✱ s❡❣✉❡
gµσ∇σRµν+gµσ∇βRβµνσ−gµσ∇νRµσ = 0. ✭✶✳✷✽✮ ❆ss✐♠✱
∇σRσν− Rµν∇σgµσ+∇βRβσνσ− Rβµνσ∇βgµσ− ∇νR+Rµσ∇νgµσ = 0. ✭✶✳✷✾✮
❖ t❡r❝❡✐r♦ t❡r♠♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❝♦♠♦ Rβσ
νσ = Rσβσν = Rβν ❡✱ s♦♠❛❞♦ ❝♦♠ ♦ ♣r✐♠❡✐r♦ t❡r♠♦✱ ♥♦s ❞á
2∇σRσν − ∇νR − Rµν∇σgµσ− Rβµνσ∇βgµσ+Rµσ∇νgµσ = 0, ✭✶✳✸✵✮ ♦✉ ❛✐♥❞❛
2[∇σ(Rσν −
1 2δ
σ
νR)] =∇βgµσ(Rβµνσ +δσβRµν −δνβRµσ). ✭✶✳✸✶✮ ❯s❛♥❞♦ ✭✶✳✶✶✮✱ t❡♠♦s
2[∇σ(Rσν−
1 2δ
σ
νR)] =−φ,β(gµσRβµνσ+gµβRµν −gµσδβνRµσ), ✭✶✳✸✷✮ ♦ q✉❡ ❝♦♥❞✉③ ❛
2[∇σ(Rσν−
1 2δ
σ
νR)] =−2φ,β(Rβν −
1 2δ
β
νR). ✭✶✳✸✸✮ P♦r ✜♠✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦
∇νGνµ=−φ,νGνµ ✭✶✳✸✹✮ q✉❡ ♠♦str❛ q✉❡ ❛ ❝♦♥tr✐❜✉✐çã♦ ♥ã♦✲♥✉❧❛ ♣❛r❛ ❛ ❞✐✈❡r❣ê♥❝✐❛ ❞♦ t❡♥s♦r ❞❡ ❊✐♥st❡✐♥ é ❞❡✈✐❞♦ ❛ ♥ã♦✲♠❡tr✐❝✐❞❛❞❡✱♦✉ ❝❛♠♣♦ ❞❡ ❲❡②❧
✶✳✸ ■♥✈❛r✐â♥❝✐❛ ❞❛ ❛çã♦ ❡ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧
❯♠ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡✱ ❝♦♥s❡qü❡♥❝✐❛ ❞❛ ✐♥✈❛r✐â♥❝✐❛ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❝♦♥❡①ã♦ s♦❜r❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❲❡②❧✱ é q✉❡ t❛♥t♦ ♦ t❡♥s♦r ❞❡ ❝✉r✈❛t✉r❛✱ q✉❛♥t♦ ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ sã♦ ✐♥✈❛r✐❛♥t❡s s♦❜ ❡st❛s tr❛♥s❢♦r♠❛çõ❡s✳ ❊ss❡ r❡s✉❧t❛❞♦ ❢❛❝✐❧✐t❛ ❡♠ ♠✉✐t♦ ♦s ❝á❧❝✉❧♦s ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ♦ ❝❛s♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♣✉r❛♠❡♥t❡ ❝♦♥❢♦r♠❡s ♥❛ ♠étr✐❝❛✱ ♦♥❞❡ t❡♠♦s q✉❡ ✉s❛r ❡①♣r❡ssõ❡s s❡♠❡❧❤❛♥t❡s✳ ❆♦ s❡ tr❛t❛r ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ♥ã♦ t❡♠♦s ❡ss❡ ✐♥❝♦♥✈❡♥✐❡♥t❡✳ ❚❛✐s ✐♥✈❛r✐❛♥t❡s sã♦ ❞❡ ✐♥t❡r❡ss❡ q✉❛♥❞♦ q✉❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ❢✉♥❝✐♦♥❛❧ ♣❛r❛ ❝♦♥str✉✐r ❛ ❛çã♦ ❞❡ ❛❧❣✉♠❛ t❡♦r✐❛✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ❛çã♦✱ ❞❡ ❛❧❣✉♠❛ t❡♦r✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧✱ ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❲❡②❧
S =
Z √
−gW. ✭✶✳✸✺✮
❖♥❞❡ W ❛q✉✐ ❞❡♥♦t❛ ✉♠ ❡s❝❛❧❛r ❝♦♥str✉✐❞♦ ❝♦♠ ❛s ❝✉r✈❛t✉r❛s ❞❡ ❲❡②❧✳ ❙♦❜ ✉♠❛ tr❛♥s✲
❢♦r♠❛çã♦ ❞❡ ❲❡②❧✱ ♦ ❞❡t❡♠✐♥❛♥t❡ s❡ tr❛♥s❢♦r♠❛ ❝♦♠♦
p
−g˜=e2f√−g. ✭✶✳✸✻✮
P♦rt❛♥t♦✱ ♦ ❢✉♥❝✐♦♥❛❧ ❡s❝♦❧❤✐❞♦ W ❞❡✈❡ s❡ tr❛♥s❢♦r♠❛r ❝♦♠ W˜ =e−2fW✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❛çã♦ s❡❥❛ ✐♥✈❛r✐❛♥t❡ ❞❡ ❲❡②❧✳ ❖ ❡s❝❛❧❛r ♠❛✐s s✐♠♣❧❡s✱ ❞❡♣♦✐s ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠étr✐❝❛✱ é ♦ ❡s❝❛❧❛r ❞❡ ❘✐❝❝✐✱ q✉❡ s❡ tr❛♥s❢♦r♠❛ ❝♦♠
˜
R= ˜gµνR˜µν =e−fR ✭✶✳✸✼✮ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r W = R2✱ q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛✳ ❉❡ ❢❛t♦✱ ♥❛
❢♦r♠✉❧❛çã♦ ♦r✐❣✐♥❛❧ ❞❛ t❡♦r✐❛ ❞❡ ❲❡②❧ q✉❡ ✉♥✐✜❝❛✈❛ ❛tr❛✈és ❞❛ ❣❡♦♠❡tr✐❛ ❛ ❣r❛✈✐t❛çã♦ ✭gµν✮ ❡ ♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ✭σµ✮✱ ❛ ❛çã♦ ❡s❝♦❧❤✐❞❛ ❢♦✐
S=
Z √
−g R2+FµνFµν
✭✶✳✸✽✮
♦♥❞❡Fµν =∂µσν−∂νσµé ❛ ✧❝✉r✈❛t✉r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦✧q✉❡ ♣❛ss❛ ❛ s❡r ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ♦ t❡♥s♦r ❡❧❡tr♦♠❛❣♥ét✐❝♦ ❞❡ ▼❛①✇❡❧❧✱ ✐♥✈❛r✐❛♥t❡ ❞❡ ❝❛❧✐❜r❡✳ ❲❡②❧ ❛✐♥❞❛ ❡①✐❣✐✉ ❛ ❡s❝♦❧❤❛ ❞❡ ✉♠ ❝❛❧✐❜r❡ ♣❛r❛ ♦❜t❡r ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦✱ R= Λ✱ ♦♥❞❡Λ é ✉♠❛ ❝♦♥st❛♥t❡✳ P❛r❛ ♠❛✐s
❞❡t❛❧❤❡s s♦❜r❡ ❛ t❡♦r✐❛ ❞❡ ❲❡②❧ ❡ s✉❛s ♠♦❞✐✜❝❛çõ❡s ❝♦♠ ❛ ✐♥t❡♥çã♦ ❞❡ ✉♥✐✜❝❛çã♦ ❞❡✈✐❞♦ ❛ ❊❞❞✐♥❣t♦♥✱ ❙❝❤r♦❡❞✐♥❣❡r✱ P❛✉❧✐✱ ❇❡r❣♠❛♥✱ ❊✐♥st❡✐♥ ❡ ♦✉tr♦s ✐♥❞✐❝❛♠♦s ♦s ❡①❝❡❧❡♥t❡s ❧✐✈r♦s ❞❡ ❝❛❞❛ ✉♠ ❞♦s ❛✉t♦r❡s ❝✐t❛❞♦s✱ ❬✶✸❪✳ ❆ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ❢♦✐ r❡✈✐s✐t❛❞❛ ♣♦r ❉✐r❛❝ ♥❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ♥✉♠❛ t❡♥t❛t✐✈❛ ❞❡ ✐♠♣❧❡♠❡♥t❛r q✉❡stõ❡s ❝♦s♠♦❧ó❣✐❝❛s✱ ❝♦♠ ❛ ❛çã♦
S =
Z √
−g
βR+ κ
βDαβD
αβ+ Λβ4+FµνF µν
✭✶✳✸✾✮
♦♥❞❡ β é ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r ❛✉①✐❧✐❛r✱ Dα = ∂α+wσα é ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❞❛ ❢✉♥çã♦ ❡s❝❛❧❛r β✱ ❞❡ ♣❡s♦ w s♦❜ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❲❡②❧✳ ◆❡ss❛ t❡♦r✐❛ ♥ã♦ t❡♠♦s ♠❛✐s ❛
✐♥t❡r♣r❡t❛çã♦ ❞❡ σµ❝♦♠♦ ♣♦t❡♥❝✐❛❧ ❡❧❡tr♦♠❛❣♥ét✐❝♦✱ ❝♦rr❡s♣♦♥❞❡♥❞♦ ❛❣♦r❛ ❛ ♥♦✈♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ ❣r❛✈✐t❛çã♦✳ ❱❛r✐❛çõ❡s ❞❡ss❛ t❡♦r✐❛✱ ❢♦r❛♠ ❡st✉❞❛❞❛s ♣♦r ▲❡❡ ❙♠♦❧❧✐♥ ❬✶✻❪ ❡ ❙✳ ❆❞❧❡r ❬✷✺❪ ✳ ❱❡rsõ❡s ♠❛✐s r❡❝❡♥t❡s sã♦ ✉s❛❞❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ q✉❛♥t✐③❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❞❡♣♦✐s q✉❡ ❢♦✐ ♠♦str❛❞❛ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ❝❛♠♣♦ ❞❡ ❲❡②❧ ✭❡♠ ❛❧❣✉♥s tr❛❜❛❧❤♦s ✐♥t❡❣rá✈❡❧ ❡ ❡♠ ♦✉tr♦s ♥ã♦ ✐♥t❡❣rá✈❡❧✮ ❡ ♦ ♣♦t❡♥❝✐❛❧ q✉â♥t✐❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❈❛rr♦❧ ❬✷✻❪✱ ◗✉✐r♦s ❬✾❪ ❡ ❙❤♦❥❛✐ ❬✷✶❪✳ ❖✉tr❛s ❛♣❧✐❝❛çõ❡s sã♦ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ♠❛tér✐❛ ❡s❝✉r❛ ❡ ❡♥❡r❣✐❛ ❡s❝✉r❛ ■sr❛❡❧✐t ❬✷✼❪✳
❱✐st♦ q✉❡ ♦s r❡s✉❧t❛❞♦s ❞❡ss❛ t❡s❡ s❡ r❡str✐♥❣❡♠ ❛♦ r❡✐♥♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✲ ✈❡❧✱ ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛ ♣❛ss❛r❡♠♦s ❛ ❞✐s❝✉t✐r ❡ss❛ ❡str✉t✉r❛✳ ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥✲ t❛♠♦s ✉♠❛ r❡✈✐sã♦ ❞❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✱ ♦ ♣r♦tót✐♣♦ ❞❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s✱ ❡ ♥❛ s❡çã♦ ✷✳✺ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❞✐s❝✉ssã♦ s♦❜r❡ ♦ ❝♦♥t❡ú❞♦ ❣❡♦♠étr✐❝♦ ❝♦♥t✐❞♦ ♥❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ ✈❡r❡♠♦s ❝♦♠♦ ❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❝♦♥té♠ ♦s ✐♥❣r❡❞✐❡♥t❡s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧ ✐♥t❡❣rá✈❡❧✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❛ t❡s❡✱ ✉♠❛ t❡♦r✐❛ ✐♥s♣✐r❛❞❛ ❝♦♠♣❧❡t❡♠❡♥t❡ ♥♦s ❛tr✐❜✉t♦s ❣❡♦♠étr✐❝♦s ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❲❡②❧✳
❈❛♣ít✉❧♦ ✷
❚❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s
❆ ❝❛r❛❝t❡ríst✐❝❛ ❜ás✐❝❛ ❞❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s é ❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r q✉❡✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ♠étr✐❝❛✱ ❞❡s❝r❡✈❡ ♦s ❢❡♥ô♠❡♥♦s ❣r❛✈✐t❛❝✐♦♥❛✐s✳ ◆❡ss❡ s❡♥t✐❞♦✱ t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s sã♦✱ ❡♠ ❝♦♥❥✉♥t♦✱ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ ✭❘●✮✱ ♥❛ q✉❛❧ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ é ❞❡s❝r✐t♦ s♦♠❡♥t❡ ♣❡❧♦ t❡♥s♦r ♠étr✐❝♦✳
❆ ✐❞é✐❛ ♦r✐❣✐♥❛❧ r❡♠♦♥t❛ ❛♦s tr❛❜❛❧❤♦s ❞❡ ❏♦r❞❛♥ ❬✷✽❪✱ ❇r❛♥s ❡ ❉✐❝❦❡ ❬✼❪✱ ❝♦♠ ❞❡st❛q✉❡ ❡s♣❡❝✐❛❧ ♣❛r❛ ❡ss❡ ❞♦✐s ú❧t✐♠♦s✱ ❝✉❥♦ ♦❜❥❡t✐✈♦ ❡r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ▼❛❝❤ ♥✉♠❛ t❡♦r✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧✱ q✉❡✱ ❡♠ ♣♦✉❝❛s ♣❛❧❛✈r❛s✱ s✉❣❡r❡ q✉❡ ♦s ❡❢❡✐t♦s ❧♦❝❛✐s ❞❡ ✐♥ér❝✐❛ sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♠❛tér✐❛ ♥♦ ✉♥✐✈❡rs♦✳ ❆tr❛✈és ❞❛ ❤✐♣ót❡s❡ ❞♦s ❣r❛♥❞❡s ♥ú♠❡r♦s✱ ❉✐r❛❝ s✉❣❡r❡ q✉❡ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❣r❛✈✐t❛❝✐♦♥❛❧ G ❞❡✈❡ ✈❛r✐❛r ❝♦♠ ♦
t❡♠♣♦ ❝ós♠✐❝♦ t✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡
G∝ 1
t. ✭✷✳✶✮
P❛r❛ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❝♦✈❛r✐❛♥t❡ ❞❡ss❛ ❤✐♣ót❡s❡✱ ❈❛r❧ ❇r❛♥s ❡ ❘♦❜❡rt ❉✐❝❦❡ ✐♠♣õ❡♠ q✉❡ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ♣❛ss❡ ❛ s❡r ❞❡t❡r♠✐♥❛❞♦ ♣♦r ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r✳
G= 1
ϕ. ✭✷✳✷✮
❆ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ é ❞❡ ❢❛t♦ ♦ ♣r♦tót✐♣♦ ❞❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s ❡ ❧✐✲ ♠✐t❛r❡♠♦s ❛ ❞✐s❝✉ssã♦ ❡♠ t♦r♥♦ ❞❡ss❛ t❡♦r✐❛✳ ❆ ❡①t❡♥sã♦ ❞♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s é ❞✐r❡t❛ ♣❛r❛ ♦ ❢♦r♠❛❧✐s♠♦ ❣❡r❛❧✱ ❡♥tr❡t❛♥t♦ ❛ ❛♥á❧✐s❡ ❞❡ s♦❧✉çõ❡s ❡①✐❣❡ ✉♠ ♥♦✈♦ ❡st✉❞♦✳
❆ r❡♥♦✈❛çã♦ ♥♦ ✐♥t❡r❡ss❡ ♥❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡ ❡ ♥❛ s✉❛ ❣❡♥❡r❛❧✐③❛çã♦✱ ❛s t❡♦r✐❛s ❡s❝❛❧❛r❡s✲t❡♥s♦r✐❛✐s s✉r❣✐✉ ♥♦ ❝♦♥t❡①t♦ ❝♦s♠♦❧ó❣✐❝♦ ✐♥✢❛❝✐♦♥ár✐♦✳ ❋♦✐ ♠♦str❛❞♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❝❤❛♠❛❞♦ ✧❣r❛❝❡❢✉❧❧ ❡①✐t✧✭✉♠❛ ✈❡③ ✐♥✐❝✐❛❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ✐♥✢❛çã♦ é ♥❡❝❡ssár✐♦ ❛❧❣✉♠ ♠❡❝❛♥✐s♠♦ ♣❛r❛ ✐♥t❡rr♦♠♣ê✲❧♦✱ ❞❛í ❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ✉♠❛ ✧s❛í❞❛ ❣r❛❝✐♦s❛✧❞❡ss❡ ♣r♦❝❡ss♦✮ ❡r❛ ❜❡♠ r❡s♦❧✈✐❞♦ ♥❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲❉✐❝❦❡✱ ❞❡s❞❡ q✉❡ ♦ ♣❛râ♠❡tr♦ ❞❛ t❡♦r✐❛ t✐✈❡ss❡ ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r✱ ω <25❬✷✾❪✳
❯♠❛ ❧✐❜❡r❞❛❞❡ ❛r❜✐trár✐❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❡ss❛ t❡♦r✐❛ é ❛ ♣r❡s❡♥ç❛ ❞♦ ♣❛râ♠❡tr♦ ❧✐✈r❡ ❛❝✐♠❛ ♠❡♥❝✐♦♥❛❞♦✱ω✱ q✉❡ r❡❣✉❧❛ ❛ ✐♥t❡r❛çã♦ ❞❛ ♠❛tér✐❛ ❝♦♠ ♦ ❝❛♠♣♦ ❡s❝❛❧❛r✱ ❡ ❡s♣❡r❛✲s❡
q✉❡ s❡❥❛ ❞❛ ♦r❞❡♠ ❞❛ ✉♥✐❞❛❞❡ω ∼1✶✳ ❆❝r❡❞✐t❛✲s❡ q✉❡ ♥♦ ❧✐♠✐t❡ω→ ∞❛ t❡♦r✐❛ ❞❡ ❇r❛♥s✲
❉✐❝❦❡ s❡❥❛ ✐❞ê♥t✐❝❛ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❬✸✵❪✳ ❖s ❡①♣❡r✐♠❡♥t♦s ♥♦ â♠❜✐t♦ ❞♦ s✐st❡♠❛ s♦❧❛r
✶❆ r❡♥♦✈❛çã♦ ❞❡ ✐♥t❡r❡ss❡ ♥❛ t❡♦r✐❛ ❞❡ ❇❉ ❢♦✐ t❛♠❜é♠ ❞❡✈✐❞❛ à ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❝❛♠♣♦s ❡s❝❛❧❛r❡s ♥❛s t❡♦r✐❛s ❞❡ ✉♥✐✜❝❛çã♦ ♠♦❞❡r♥❛s✱ ❡♠ ♣❛rt✐❝✉❧❛r ♦ ❞✐❧❛t♦♥ ❞❛s t❡♦r✐❛s ❞❡ ❝♦r❞❛s✳ ◆♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛ ❜♦sô♥✐❝❛ ♣r♦❞✉③ ✉♠❛ t❡♦r✐❛ ❞❡ ❇❉ ❝♦♠ ω=−1❬✶✵❪✳
