❯◆■❱❊❘❙■❉❆❉❊ ❉❊ ❇❘❆❙❮▲■❆
■◆❙❚■❚❯❚❖ ❉❊ ❋❮❙■❈❆
❚❊❙❊ ❉❊ ❉❖❯❚❖❘❆❉❖
❖ ▼❖▼❊◆❚❖ ❆◆●❯▲❆❘ ❉❖ ❈❆▼P❖
●❘❆❱■❚❆❈■❖◆❆▲ ❊ ❖ ●❘❯P❖ ❉❊ P❖■◆❈❆❘➱
❙➱❘●■❖ ❈❖❙❚❆ ❯▲❍❖❆
❖❘■❊◆❚❆❉❖❘✿
❏❖❙➱ ❲❆❉■❍ ▼❆▲❯❋
❆❣r❛❞❡❝✐♠❡♥t♦s
❙♦✉ ❣r❛t♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♦ ♣r♦❢❡ss♦r ❉r✳ ❏♦sé ❲❛❞✐❤ ▼❛❧✉❢✱ ♣♦r t❡r ♠❡ ❣✉✐❛❞♦ ❛♦ ❧♦♥❣♦ ❞❡st❡s ❛♥♦s ❡♠ q✉❡ s❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ❡ s❡ ❡st❛❜❡❧❡❝❡r❛♠ r❡❧❛çõ❡s ❞❡ ❛♠✐③❛❞❡✱ r❡s♣❡✐t♦ ❡ ♣r♦❢✉♥❞❛ ❛❞♠✐r❛çã♦✳ ❋♦✐ ❡❧❡ ♦ ❣r❛♥❞❡ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ♠✐♥❤❛ ❡s❝♦❧❤❛ ❡♠ tr❛❜❛❧❤❛r ♥❛ ár❡❛ ❞❡ ❘❡❧❛t✐✈✐❞❛❞❡✳ ◆✉♥❝❛ ✈♦✉ ❡sq✉❡❝❡r ♦ ♠❡✉ ♣r✐♠❡✐r♦ ❝♦♥t❛t♦ ❝♦♠ ❛ ♣❡sq✉✐s❛ ❛❝❛❞ê♠✐❝❛ q✉❡ s❡ ❞❡✉✱ ❛✐♥❞❛ ❝❛❧♦✉r♦✱ ❡♠ ✉♠ s❡♠✐♥ár✐♦ ♠✐♥✲ ✐str❛❞♦ ♣❡❧♦ ♣r♦❢✳ ▼❛❧✉❢✳ ●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♦✉ ♦ ✜③❡r❛♠ ❞❡ ❢♦r♠❛ ✐♥❞✐r❡t❛✳ ❙♦✉ ❣r❛t♦ ❛♦s ❛♠✐❣♦s q✉❡ ✜③ ❛♦ ❧♦♥❣♦ ❞❡st❡s ❛♥♦s✱ ♠❛s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ▼❛r✐❛♥♥❡ ▼❛❝✐❡❧ ❞❡ ❆❧♠❡✐❞❛✱ ♣♦r s❡♠♣r❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦✱ ❛♣♦✐❛♥❞♦✲♠❡ ❡ ❛✉①✐❛♥❞♦✲♠❡ ❛ s❡❣✉✐r ♣♦r ❡st❛ ✈✐❞❛✳
❘❡s✉♠♦
❖ t❡❧❡♣❛r❛❧❡❧✐s♠♦ ❡q✉✐✈❛❧❡♥t❡ à ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✭❚❊●❘✱ ♥❛ s✐❣❧❛ ❡♠ ✐♥✲ ❣❧ês✮ é ✉♠❛ ❞❡s❝r✐çã♦ ❛❧t❡r♥❛t✐✈❛ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡♠ t❡r♠♦s ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛s✱ q✉❡ ❝♦rr❡s♣♦♥❞❡♠ às ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s ❞♦ s✐st❡♠❛✳ ❖ ❚❊●❘ ♣❡r♠✐t❡✲ ♥♦s tr❛t❛r ❞❡ ♠❛♥❡✐r❛ ❛❞❡q✉❛❞❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡✜♥✐çã♦ ❞❛ ❡♥❡r❣✐❛✱ ♠♦♠❡♥t♦ ❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳ ◆❡st❛ t❡s❡ ♠♦str❛r❡♠♦s ❝♦♠♦ ❞❡s❝r❡✈❡r ♦ ❚❊●❘ ✉s❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ▲❛❣r❛♥❣❡❛♥♦ ❡ ❍❛♠✐❧t♦♥✐❛♥♦✳ ❯t✐❧✐③❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❝♦♥str✉✐r❡♠♦s ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❝❛♠♣♦ ❣r❛✈✲ ✐t❛❝✐♦♥❛❧ q✉❡ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❉✐s❝✉t✐r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♠❛♥❡✐r❛s ❞❡ ❞❡✜♥✐r ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❡①✐st❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛✱ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ♥♦ss❛ ❡①♣r❡ssã♦ ♣❛r❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ q✉❡ ❡①✐❜❡ s✐♠❡tr✐❛ ❛①✐❛❧✳ ❊st❛❜❡❧❡❝❡r❡♠♦s q✉❛❧ ❞❡✈❡ s❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞♦ t❡♥s♦r ♠étr✐❝♦ ♣❛r❛ q✉❡ ❛ ❡①♣r❡ssã♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r s❡❥❛ ❜❡♠ ❞❡✜♥✐❞❛✳ ❱❡r✐✜❝❛r❡♠♦s q✉❡ ❛s ♥♦ss❛s ❡①♣r❡ssõ❡s ♣❛r❛ ♦ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❢♦r♠❛♠ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❞❡✜♥✐r ♦s ✐♥✈❛r✐❛♥t❡s ❞❡ ❈❛s✐♠✐r✳ ❯t✐❧✐③❛♥❞♦ ❡ss❛s q✉❛♥t✐❞❛❞❡s t❡♥t❛r❡♠♦s ❝♦♥str✉✐r ❛ ❤❡❧✐❝✐❞❛❞❡ ❞❡ ♦♥❞❛s ❣r❛✈✐t❛❝✐♦♥❛✐s✱ ❛♥❛❧✐s❛♥❞♦ ❞♦✐s s✐st❡♠❛s✿ ❛ ♠étr✐❝❛ ❞❡ ❇♦♥❞✐ ❡ ♦♥❞❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ♣❧❛♥❛s ❝♦♠♦ s♦❧✉çõ❡s ❡①✲ ❛t❛s ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥✳ ❉✐s❝✉t✐r❡♠♦s q✉❛❧ é ❛ ✐♥t❡r♣r❡t❛çã♦ ❢ís✐❝❛ ❞♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ❡ ❡①❡♠♣❧✐✜❝❛r❡♠♦s ♥♦ss❛ ✐♥t❡r♣r❡t❛çã♦ ❛tr❛✈és ❞♦ ❝á❧❝✉❧♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❝♦♠♣❛r❛♥❞♦ ❞♦✐s ❝❛♠♣♦s ❞❡ tétr❛❞❛s ♣❛r❛ ❛ ❝❛s❝❛ ❡s❢ér✐❝❛ ❡♠ r♦t❛çã♦✳
❆❜str❛❝t
❚❤❡ t❡❧❡♣❛r❛❧❧❡❧ ❡q✉✐✈❛❧❡♥t ♦❢ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t② ✭❚❊●❘✮ ✐s ❛ ✈✐❛❜❧❡ ❛❧t❡r♥❛✲ t✐✈❡ ❣❡♦♠❡tr✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ✐♥ t❡r♠s ♦❢ t❤❡ t❡tr❛❞ ✜❡❧❞✳ ■♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤❡ ❚❊●❘ ✐t ❤❛s ❜❡❡♥ ♣♦ss✐❜❧❡ t♦ ❛❞❞r❡ss t❤❡ ❧♦♥❣st❛♥❞✐♥❣ ♣r♦❜❧❡♠ ♦❢ ❞❡✜♥✐♥❣ t❤❡ ❡♥❡r❣②✱ ♠♦♠❡♥t✉♠ ❛♥❞ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ♦❢ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞✳ ■♥ t❤✐s t❤❡s✐s ✇❡ s❤❛❧❧ s❤♦✇ ❤♦✇ t♦ ❞❡s❝r✐❜❡ t❤❡ ❚❊●❘ ❜② ♠❡❛♥s t❤❡ ▲❛❣r❛♥❣✐❛♥ ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r♠❛❧✐s♠s✳ ❯s✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r♠❛❧✐s♠ ✇❡ s❤❛❧❧ ❣✐✈❡ ❛ ❡①✲ ♣r❡ss✐♦♥ ❢♦r ❣r❛✈✐t❛t✐♦♥❛❧ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ t❤❛t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s✱ ✇❡ s❤❛❧❧ ❞❡s❝r✐❜❡ t❤❡ s❡✈❡r❛❧ ✇❛②s t♦ ❞❡✜♥✐♥❣ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛♥❞ ❝♦♠♣❛r❡ t❤❡♠ ✇✐t❤ ♦✉r ❞❡✜♥✐t✐♦♥ ❜② ❛♣♣❧②✐♥❣ ✐t t♦ ❛ ❝♦♥✜❣✲ ✉r❛t✐♦♥ t❤❛t ❡①❤✐❜✐ts ❛①✐❛❧ s②♠♠❡tr②✳ ❲❡ s❤❛❧❧ ✜① t❤❡ ❡①❛❝t ❛s②♠♣t♦t✐❝ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ♠❡tr✐❝ t❡♥s♦r ✐♥ ♦r❞❡r t♦ ❣❡t ❛ ✇❡❧❧ ❞❡✜♥❡❞ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠✳ ❲❡ ✜♥❞ t❤❛t t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❡♥❡r❣②✲♠♦♠❡♥t✉♠ ❛♥❞ ❛♥❣✉✲ ❧❛r ♠♦♠❡♥t✉♠ ❝♦rr❡s♣♦♥❞ t♦ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ P♦✐♥❝❛ré ❣r♦✉♣✳ ❚❤✐s r❡s✉❧t ❛❧❧♦✇s ✉s t♦ ❞❡✜♥❡ ❈❛s✐♠✐r t②♣❡ ✐♥✈❛r✐❛♥ts ❢♦r t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞✳ ❯s✐♥❣ t❤❡s❡ ✐♥✈❛r✐❛♥ts ✇❡ s❤❛❧❧ tr② t♦ ❜✉✐❧❞ t❤❡ ❤❡❧✐❝✐t② ♦❢ ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛✈❡s ❜② ❛♥❛❧②③✐♥❣ t✇♦ ❝♦♥✜❣✉r❛t✐♦♥s✿ ❇♦♥❞✐✬s ♠❡tr✐❝ ❛♥❞ ❣r❛✈✐t❛t✐♦♥❛❧ ♣❧❛♥❡✲✇❛✈❡s ❛s ❡①❛❝t s♦❧✉t✐♦♥s ♦❢ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥s✳ ❲❡ s❤❛❧❧ ❞✐s❝✉ss t❤❡ ♣❤②s✐❝❛❧ ♠❡❛♥✐♥❣ ♦❢ t❤❡ t❡tr❛❞ ✜❡❧❞ ❜② ✐♥✈❡st✐❣❛t✐♥❣ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ♦❢ t✇♦ ❞✐✛❡r❡♥t t❡tr❛❞ ✜❡❧❞s ❢♦r ❛ r♦t❛t✐♥❣ ♠❛ss s❤❡❧❧✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❖ ❚❡❧❡♣❛r❛❧❡❧✐s♠♦ ❊q✉✐✈❛❧❡♥t❡ à ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✼ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❆❧❣✉♠❛s ❈♦♥s✐❞❡r❛çõ❡s ❙♦❜r❡ ♦ ❈❛♠♣♦ ❞❡ ❚étr❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❆ ❋♦r♠✉❧❛çã♦ ▲❛❣r❛♥❣❡❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✹ ❆ ❋♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✸ ❖ ●r✉♣♦ ❞❡ P♦✐♥❝❛ré ✷✶
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷✳✶ ●r✉♣♦s ❈♦♥tí♥✉♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸ ❖ ●r✉♣♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✹ ❖♣❡r❛❞♦r❡s ❞❡ ❈❛s✐♠✐r ♣❛r❛ ♦ ❈❛♠♣♦ ●r❛✈✐t❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✹ ❙✐st❡♠❛s ❞❡ ❘❡❢❡rê♥❝✐❛ ❡ ♦ ▼♦♠❡♥t♦ ❆♥❣✉❧❛r ●r❛✈✐t❛❝✐♦♥❛❧ ✸✼ ✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷ ❈❛♠♣♦s ❞❡ ❚étr❛❞❛s ❝♦♠♦ ❙✐st❡♠❛s ❞❡ ❘❡❢❡rê♥❝✐❛ ❡ ❊①♣r❡ssõ❡s ❘❡✲
❣✉❧❛r✐③❛❞❛s ♣❛r❛ ♦ ▼♦♠❡♥t♦ ❆♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✸ ❖ ▼♦♠❡♥t♦ ❆♥❣✉❧❛r ❞❛ ❈❛s❝❛ ❊s❢ér✐❝❛ ❡♠ ❘♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✸✳✶ ❖❜s❡r✈❛❞♦r ❡♠ ❘♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✸✳✷ ❖❜s❡r✈❛❞♦r ❊stát✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✺ ❖ ▼♦♠❡♥t♦ ❆♥❣✉❧❛r ●r❛✈✐t❛❝✐♦♥❛❧ ✺✷ ✺✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ s♦❜r❡ ▼♦♠❡♥t♦ ❆♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✸ ❖ ▼♦♠❡♥t♦ ❆♥❣✉❧❛r ❞❡ ✉♠❛ ❙✐♠❡tr✐❛ ❆①✐❛❧ ♥♦ ❚❡❧❡♣❛r❛❧❡❧✐s♠♦ ❊q✉✐✲
✈❛❧❡♥t❡ à ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✸✳✶ ❖❜s❡r✈❛❞♦r ❊stát✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✺✳✸✳✷ ❖❜s❡r✈❛❞♦r ❡♠ ❘♦t❛çã♦ ♣❛r❛ ♦ ❇✉r❛❝♦ ◆❡❣r♦ ❞❡ ❑❡rr ✳ ✳ ✳ ✳ ✳ ✼✶ ✺✳✹ ❖ ❙✐❣♥✐✜❝❛❞♦ ❞❡ L(0)(i) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
✺✳✺ ❍❡❧✐❝✐❞❛❞❡ ❞❛s ❖♥❞❛s ●r❛✈✐t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✺✳✺✳✶ ❆ ▼étr✐❝❛ ❞❡ ❇♦♥❞✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✺✳✺✳✷ ❆ ❖♥❞❛ P❧❛♥❛ ◆ã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
✻ ❈♦♥❝❧✉sã♦ ❡ P❡rs♣❡❝t✐✈❛s ✽✼
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✾✶
◆♦t❛çã♦✿
❖ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦ s❡rá ❞❡s✐❣♥❛❞♦ ♣♦r ❧❡tr❛s ❣r❡❣❛s✱ ❞❡ ❢♦r♠❛ s✐♠✐❧❛r ♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡ s❡rá ❞❡s✐❣♥❛❞♦ ♣♦r ❧❡tr❛s ❧❛t✐♥❛s✳ ❮♥❞✐❝❡s ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦ µ, ν, ... ❡ í♥❞✐❝❡s ❙❖✭✸✱✶✮ ♦✉ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡ a, b, ... ✈❛r✐❛♠ ❞❡ ✵ ❛
✸✳ ❮♥❞✐❝❡s ❞❡ ❡s♣❛ç♦ ❡ t❡♠♣♦ sã♦ ✐♥❞✐❝❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ µ = 0, i, a = (0),(i)✳
❖ ❝❛♠♣♦ ❞❡ tétr❛❞❛s é ❞❡♥♦t❛❞♦ ♣♦r eaµ✱ ♦ t❡♥s♦r ♠étr✐❝♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❧❡✈❛♥t❛ ❡ ❛❜❛✐①❛ í♥❞✐❝❡s ❡ é ✜①❛❞♦ ♣♦r ηab = eaµebνgµν = (−+ ++)✳ ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s é ✐♥❞✐❝❛❞♦ ♣♦r e = det(eaµ)✳ ❆s ✉♥✐❞❛❞❡s sã♦ ✜①❛❞❛s ❝♦♠ ❛ ❡s❝♦❧❤❛G=c= 1✱ ❛ ♠❡♥♦s q✉❡ s❡ ❞✐❣❛ ♦ ❝♦♥trár✐♦✳
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❯♠ ❡♥t❡♥❞✐♠❡♥t♦ ♠❛✐s ❝♦♠♣❧❡t♦ ❡ ♣r♦❢✉♥❞♦ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❞❡ ❊✐♥✲ st❡✐♥ r❡q✉❡r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ❡str✉t✉r❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦✱ s♦❧✉çõ❡s ❡ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s✱ ❜❡♠ ❝♦♠♦ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s t❛✐s ❝♦♠♦✿ ❛ ❡♥❡r❣✐❛✱ ♠♦✲ ♠❡♥t♦ ❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❬✶❪✳ ❉❡✈✐❞♦ ❛♦ s✉r❣✐♠❡♥t♦ ❞❡ ♣r♦❜❧❡♠❛s ♥❛ ✐♥t❡r♣r❡t❛çã♦✱ ❡ ♠❡s♠♦ ♥❛ ❞❡✜♥✐çã♦ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s✱ q✉❡ sã♦ ✐♥✲ ❞✐s♣❡♥sá✈❡✐s ♣❛r❛ ❛ ❝♦♠♣❧❡t❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ t❡♦r✐❛✱ t♦r♥❛✲s❡ ♥❡❝❡ssár✐❛ ✉♠❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠✱ ♣♦ré♠ ❡q✉✐✈❛❧❡♥t❡✱ ♣❛r❛ ❛ ❞❡s❝r✐çã♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳
◆❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ❞❛ ❣r❛✈✐t❛çã♦ s✉r❣❡♠ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❝♦♥✲ ❝❡✐t✉❛✐s t❛✐s ❝♦♠♦ ❛ ✐♥❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ❡♥❡r❣✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧✱ ❡ ❡①✐st❡♠ sér✐❛s ❞✐✜❝✉❧❞❛❞❡s q✉❛♥❞♦ t❡♥t❛♠♦s ❝♦♥str✉✐r ✉♠❛ t❡♦r✐❛ ❞❡ ❝❛❧✐❜r❡ ♥❛ t❡♥✲ t❛t✐✈❛ ❞❡ s❡ ✉♥✐✜❝❛r ❛s q✉❛tr♦ ✐♥t❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♥❛t✉r❡③❛✳ P❛rt❡ ❞❡ss❛ ❞✐✜❝✉❧❞❛❞❡ ❛❞✈é♠ ❞❡ ❡st❡♥sõ❡s ❡q✉✐✈♦❝❛❞❛s ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❊q✉✐✈❛❧ê♥❝✐❛✳
▼♦❧❧❡r ❬✷❪ ❥á ❤❛✈✐❛ ♥♦t❛❞♦ q✉❡ é ✐♠♣♦ssí✈❡❧ ❛♥✉❧❛r ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ♣♦r ✉♠❛ s✐♠♣❧❡s tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♦✉ s❡❥❛✱ ❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s tê♠ q✉❡ s❡r ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ t❛✐s tr❛♥s❢♦r♠❛çõ❡s✳ P♦r ✐ss♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ♣s❡✉❞♦✲t❡♥s♦r❡s t♦r♥❛✲s❡ ✐♥✈✐á✈❡❧✱ ✉♠❛ ✈❡③ q✉❡ ❡♠ s✉❛ ❢♦r♠✉❧❛çã♦✱ ❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛ é ❡①♣❧í❝✐t❛✳ ❯♠ ♦✉tr♦ ♣r♦❜❧❡♠❛ é q✉❡ ❛ ✐♥t❡r❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ é ♠✉✐t♦ ❢r❛❝❛ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛s ♦✉tr❛s ✐♥t❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s✱ ❝❛r❛❝t❡r✐③❛♥❞♦ ❛ ❝❤❛♠❛❞❛ ✏❤✐❡r❛rq✉✐❛ ❞❛s ✐♥t❡r❛çõ❡s✑✳
❆ t❡♦r✐❛ ❞❡ ❨❛♥❣✲▼✐❧❧s ❬✸❪ ❞❡s❝r❡✈❡ ❝♦♠ s✉❝❡ss♦ três ❞❛s q✉❛tr♦ ✐♥t❡r❛çõ❡s
✷
❢✉♥❞❛♠❡♥t❛✐s✳ ❆ ❣r❛✈✐t❛çã♦ é ✉♠❛ ✐♥t❡r❛çã♦ q✉❡ ♣❡r♠❛♥❡❝❡ ❛❧❤❡✐❛ ❛ ❡ss❛ ✉♥✐✜❝❛çã♦✳ ❊①✐st❡♠ ❞✉❛s r❛③õ❡s q✉❡ ❡①♣❧✐❝❛♠ ❡ss❡ ❢❛t♦✳ ❆ ♣r✐♠❡✐r❛ é q✉❡ ❛ ▲❛❣r❛♥❣❡❛♥❛✱ ♥❛ ✈✐sã♦ ❣❡♦♠étr✐❝❛✱ é ❧✐♥❡❛r ♥❛ ❝✉r✈❛t✉r❛✱ s❡♥❞♦ q✉❡ ♥♦ t❡♦r❡♠❛ ❞❡ ◆♦❡t❤❡r ❬✹❪ ✭❢✉♥✲ ❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ❨❛♥❣✲▼✐❧❧s✮ ❛ ▲❛❣r❛♥❣❡❛♥❛ é q✉❛❞rát✐❝❛✳ ❆ s❡❣✉♥❞❛ é q✉❡ ♥ã♦ s❡ s❛❜❡ q✉❛❧ é r❡❛❧♠❡♥t❡ ❛ s✐♠❡tr✐❛ ❞❡ ❝❛❧✐❜r❡ ❞❛ ❣r❛✈✐t❛çã♦✱ ♠✉✐t♦ ❡♠❜♦r❛ ❤❛❥❛ ❛r❣✉♠❡♥t♦s ♠✉✐t♦ ❢♦rt❡s ❛ ❢❛✈♦r ❞♦ ❣r✉♣♦ ❞❛s tr❛♥s❧❛çõ❡s ❬✺❪✳ ▲♦❣♦ ♥ã♦ ♣♦❞❡♠♦s ❝♦♥str✉✐r ❛✐♥❞❛ ♥❡♥❤✉♠ ♦❜s❡r✈á✈❡❧ ❞❛ t❡♦r✐❛ ♥♦s ♠♦❧❞❡s ❞❛ t❡♦r✐❛ ◗✉â♥t✐❝❛✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❡①♣r❡ssã♦ ❞❡ ◆♦❡t❤❡r ♣❛r❛ ♦s ♦❜s❡r✈á✈❡✐s ♣r❡ss✉♣õ❡ ✉♠❛ ❝♦♥❡①ã♦ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❣r✉♣♦ ❞❡ ❝❛❧✐❜r❡ q✉❡ ♥❡ss❡ ❝❛s♦ ❛✐♥❞❛ é ❝♦♥tr♦✈❡rs♦✳ ❊ss❛ ❝♦♥❡①ã♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ r❡✢❡t❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❢ís✐❝❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❡❧❛ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❣r✉♣♦ ❞❡ ❝❛❧✐❜r❡ ❡♠ q✉❡stã♦✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ t❡♦r✐❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛✱ ♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦r Aµ ❢✉♥❝✐♦♥❛ ❝♦♠❛ ❛ r❡❢❡r✐❞❛ ❝♦♥❡①ã♦✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ❝✉r✈❛t✉r❛ ❝♦♥tr✉í❞❛ ❛ ♣❛rt✐r ❞❡❧❛ é ♦ t❡♥s♦r Fµν = ∂µAν −∂νAµ✳ ❯♠❛ ❛❜♦r❞❛❣❡♠ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ é ❛q✉❡❧❛ q✉❡ ❧✐❞❛ ❝♦♠ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❝♦♠♦ t❡♦r✐❛ ❞❡ ❝❛❧✐❜r❡ ♣❛r❛ ♦ ❣r✉♣♦ ❞❛s tr❛♥s❧❛çõ❡s ❞❡✜♥✐❞♦ ♥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ ❝❛❞❛ ♣♦♥t♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❬✻✱ ✼✱ ✽✱ ✺❪✳ ❊st❛ ❛❜♦r❞❛❣❡♠ t❡♠ ❡str❡✐t❛ r❡❧❛çã♦ ❝♦♠ ♦ q✉❡ ♣❛ss❛r❡♠♦s ❛ ❞✐s❝✉t✐r ♥❡st❛ t❡s❡✱ ❡♥tr❡t❛♥t♦ ❛ ♥♦ss❛ ✈✐sã♦ é ✉♠ ♣♦✉❝♦ ❞✐❢❡r❡♥t❡ ❝♦♠♦ ✈❡r❡♠♦s✳
❆ss✐♠✱ t❡♠♦s q✉❡ ❛❜♦r❞❛r ❛ ❣r❛✈✐t❛çã♦ s♦❜ ✉♠ ♦✉tr♦ ♣♦♥t♦ ❞❡ ✈✐st❛ q✉❡ ♥♦s ♣❡r♠✐t❛ r❡s♦❧✈❡r ❛❧❣✉♥s ❞♦s ♣r♦❜❧❡♠❛s ❝✐t❛❞♦s ❡ q✉❡ r❡❝✉♣❡r❡ ♦s ❣❛♥❤♦s ❡ ❝♦♥q✉✐st❛s ❞❛ ✈✐sã♦ ❣❡♦♠étr✐❝❛✳ ■ss♦ é ❢❡✐t♦ ❛tr❛✈és ❞♦ ❝❤❛♠❛❞♦ ❚❡❧❡♣❛r❛❧❡❧✐s♠♦ ❊q✉✐✈❛❧❡♥t❡ à ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✭❚❊●❘✮✳ ❆♥t❡s✱ ♣♦ré♠✱ t❡♠♦s q✉❡ ✐♥tr♦❞✉③✐r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s✳
✸
❞❡s❝r✐çã♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡♠ t❡r♠♦s ❞❡st❡s ❝❛♠♣♦s✱ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❊q✉✐✈❛❧ê♥✲ ❝✐❛ s✉r❣❡ ❞❡ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✳ ■ss♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♦s ❝❛♠♣♦s ❞❡ tétr❛❞❛s✱ q✉❡ ❞❡s❝r❡✈❡♠ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦ ❡ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡✱ ♣♦❞❡♠ s❡r ✐♥t❡r♣r❡t❛❞♦s ❝♦♠♦ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ ❡♥tr❡ ♦s ❞✐❢❡r❡♥❝✐❛✐s dxµ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦ ❡ dqa ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡✱ ❛tr❛✈és ❞❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ΛcaΛd
bηcd = ηab✱ ♦♥❞❡ Λab é ❛ ♠❛tr✐③ ❞❡ ▲♦r❡♥t③✱ ❡ eaµebνηab = gµν✳ ❈♦♠ ✐ss♦ ♣♦❞❡♠♦s s❡♠♣r❡ ❡s❝r❡✈❡r ❛ q✉❛♥t✐❞❛❞❡ ♣r♦❥❡t❛❞❛ea =ea
µdxµ✱ ♣♦ré♠ ♣♦❞❡♠♦s ❡s✲ ❝r❡✈❡r t❛♠❜é♠ dqa = ea
µdxµ✱ ♠✉✐t♦ ❡♠❜♦r❛ ♥ã♦ ♣♦ss❛♠♦s ✐♥t❡❣r❛r ❡st❛ r❡❧❛çã♦ ❡ ❡s❝r❡✈❡r qa = qa(xµ)✳ ❖ s❡♥t✐❞♦ ❢ís✐❝♦ ❞♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s s❡rá ❡①♣❧♦r❛❞♦ ♠❛✐s ♣r♦❢✉♥❞❛♠❡♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✺✳
◆♦ ❡s♣❛ç♦✲t❡♠♣♦ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ♦s ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ sã♦ ❞✐t♦s ♣❛r❛❧❡❧♦s s❡ s✉❛s ♣r♦❥❡çõ❡s ❡♠ ♣♦♥t♦s ❞✐st✐♥t♦s ❞❛ ✈❛r✐❡❞❛❞❡✱ ❝♦♠ r❡s♣❡✐t♦ ❛ ✉♠ ❝❛♠♣♦ ❧♦❝❛❧ ❞❡ tétr❛❞❛s✱ ❢♦r❡♠ ✐❞ê♥t✐❝♦s✳ ❈❧❛r♦ q✉❡ s❡ ♣✉❞❡r♠♦s ❝♦♥str✉✐r ✉♠❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ q✉❡✱ ❛♣❧✐❝❛❞❛ s♦❜r❡ ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛s✱ s❡ ❛♥✉❧❛ ✐❞❡♥t✐❝❛♠❡♥t❡✱ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❛♥t❡r✐♦r é s❛t✐s❢❡✐t❛ ❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❛r❛❧❡❧✐s♠♦ ❛❜s♦❧✉t♦ ♦✉ t❡❧❡♣❛r❛❧❡❧✐s♠♦✱ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❬✶✵❪✳
❙❡ ✉s❛r♠♦s ❛ ❝♦♥❡①ã♦ ❞❡ ❈❛rt❛♥✱ Γλ
µν = eaλ∂µeaν✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ❛ ❝❤❛♠❛❞❛ ❣❡♦♠❡tr✐❛ ❚❡❧❡♣❛r❛❧❡❧❛ q✉❡ é ♠❡♥♦s r❡str✐t✐✈❛ ❞♦ q✉❡ ❛ ❣❡♦♠❡tr✐❛ ❘✐❡✲ ♠❛♥♥✐❛♥❛✳ ❯♠❛ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ❣❡♦♠❡tr✐❛s ❚❡❧❡♣❛r❛❧❡❧❛s✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❞❛❞❛ ✉♠❛ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✭❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ✉♠ t❡♥s♦r ♠étr✐❝♦✮ ❡①✐st❡♠ ❞✐✈❡rs❛s ♠❛♥❡✐r❛s ❞❡ s❡ ❝♦♥str✉✐r ❣❡♦♠❡tr✐❛s ❚❡❧❡✲ ♣❛r❛❧❡❧❛s ✭❝❛r❛❝t❡r✐③❛❞❛s ♣♦r ❝❛♠♣♦s ❞❡ tétr❛❞❛s✮✳ ■ss♦ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ❛tr❛✈és ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦ t❡♥s♦r ♠étr✐❝♦ ❡ ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛sgµν =eaµeaν✳ ❯♠ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥str✉í❞♦ ❛ ♣❛rt✐r ❞❡ss❛ ❝♦♥❡①ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ ♦ q✉❡ ♣❡r✲ ♠✐t❡ ❡s❝r❡✈❡r ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ q✉❛❞rát✐❝❛ ❞♦ t❡♥s♦r ❞❡ t♦rçã♦ ✭q✉❡ é ❛ ♣❛rt❡ ❛♥t✐✲s✐♠étr✐❝❛ ❞❛ ❝♦♥❡①ã♦ ❞❡ ❈❛rt❛♥✮✳
✹
✐á✈❡✐s ❞✐♥â♠✐❝❛s ❞♦ s✐st❡♠❛✳ ❊ss❡s ♦❜❥❡t♦s sã♦ ❛❞❡q✉❛❞♦s ❛ ❡ss❛ ❞❡s❝r✐çã♦ ♣♦✐s ♣r♦❞✉③❡♠ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❡st❛❜❡❧❡❝❡♠ ✉♠ ❝❛♠♣♦ ❞❡ ♦❜✲ s❡r✈❛❞♦r❡s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❆♣❡s❛r ❞❡ ❛✐♥❞❛ s❡r ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❡ ✐♥✈❡st✐❣❛çã♦✱ ♦ ❚❊●❘ ♣❛r❡❝❡✲♥♦s s❡r ❛ ❛❧t❡r♥❛t✐✈❛ ♠❛✐s ✈✐á✈❡❧ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳ ❊♥t❡♥❞❡♠♦s s❡r ❛ss✐♠✱ ♣♦✐s ♥♦ ❝♦♥t❡①t♦ ❞♦ ❚❊●❘ t❡♠ s✐❞♦ ♣♦ssí✈❡❧ tr❛t❛r ❞❡ ♠❛♥❡✐r❛ ❛❞❡q✉❛❞❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡✜♥✐çã♦ ❞♦ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳
P♦❞❡✲s❡ ❞❡s❝r❡✈❡r ❛ ❣r❛✈✐t❛çã♦ ❝♦♠ ♦ ❚❊●❘ ❞❡ ❞✉❛s ♠❛♥❡✐r❛s✳ ❆ ❢♦r♠✉✲ ❧❛çã♦ ▲❛❣r❛♥❣❡❛♥❛ ❡ ❛ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ❞♦ ❚❊●❘✳ ❆ ♣r✐♠❡✐r❛ é ❝♦♥str✉✐❞❛ ♣❡♥s❛♥❞♦✲s❡ ♥❛s s✐♠❡tr✐❛s q✉❡ ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ❡①✐❜❡♠ ❡ ❛ s❡❣✉♥❞❛ é ♦❜t✐❞❛ ❞❛ ♣r✐♠❡✐r❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲❡❣❡♥❞r❡✳
❆ss✐♠✱ ♥❡ss❡ ❝♦♥t❡①t♦✱ ♥♦ ❡s❢♦rç♦ ❞❡ t❡♥t❛r ❝❛r❛❝t❡r✐③❛r ❛s s✐♠❡tr✐❛s ❞♦ s✐s✲ t❡♠❛✱ ❝♦♥str✉í♠♦s ❛ ▲❛❣r❛♥❣❡❛♥❛ ❬✶✷✱ ✶✸✱ ✶✹❪ ❡ ❛tr❛✈és ❞❛ ❡①❡❝✉çã♦ ❞❡ ✉♠❛ tr❛♥s✲ ❢♦r♠❛çã♦ ❞❡ ▲❡❣❡♥❞r❡ ❞❡✜♥✐♠♦s ♦ q✉❡ é ❝❤❛♠❛❞♦ ❞❡ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ❞❛ ❣r❛✈✐t❛çã♦ ❬✶✺❪✳ ❊♥tr❡t❛♥t♦ s❛❜❡♠♦s q✉❡ ❡ss❛ ❢♦r♠✉❧❛çã♦ ♥❡♠ s❡♠♣r❡ é ❜❡♠ ❞❡✜♥✐❞❛ ♣❛r❛ ✉♠❛ t❡♦r✐❛ ❣❡♦♠étr✐❝❛ ❛r❜✐trár✐❛ ❞❛ ❣r❛✈✐t❛çã♦✳ P❛r❛ q✉❡ s❡❥❛ ❜❡♠ ❞❡✜♥✐❞❛ é ♥❡❝❡ssár✐♦ q✉❡ ♦s ✈í♥❝✉❧♦s s❛t✐s❢❛ç❛♠ ✉♠❛ á❧❣❡❜r❛ ❡ q✉❡✱ ❛❧é♠ ❞✐ss♦✱ ❡ss❛ á❧❣❡❜r❛ s❡❥❛ ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❝❛❞❛ ✈í♥❝✉❧♦ ❞❡✈❡ ❝♦♠✉t❛r ❝♦♠ t♦❞♦s ♦s ♦✉tr♦s✱ ♦✉ s❡❥❛✱ ♦ ✏♣r♦❞✉t♦✑ ❡♥tr❡ ♦s ✈í♥❝✉❧♦s ❞❡✈❡ s❡r ❡s❝r✐t♦ ❛♣❡♥❛s ❡♠ t❡r♠♦s ❞♦s ♣ró♣r✐♦s ✈í♥❝✉❧♦s✳
❆♦ ❛♥❛❧✐s❛r♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥✱ ♥❛ ❢♦r♠✉❧❛çã♦ ♠étr✐❝❛ ❡ ❧❛❣r❛♥❣❡❛♥❛ ✉s✉❛✐s✱ ♥♦t❛♠♦s q✉❡ ❡❧❛s ♣♦❞❡♠ s❡r s❡♣❛r❛❞❛s ❡♠ ❞✉❛s ❝❛t❡❣♦r✐❛s✱ q✉❡ ♥ã♦ s✉r❣❡♠ ❞❡ ♠❛♥❡✐r❛ ❡①♣❧í❝✐t❛ ♥❛ t❡♦r✐❛✱ ❛ s❛❜❡r✿ s❡✐s ❡q✉❛çõ❡s ❞✐♥â♠✐❝❛s ✭❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐✲ ❛✐s ❤✐♣❡r❜ó❧✐❝❛s✮ ❡ q✉❛tr♦ ❡q✉❛çõ❡s ❞❡ ✈í♥❝✉❧♦ ✭❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡❧í♣t✐❝❛s✮✳ ◆♦ ❢♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦✱ q✉❛♥❞♦ ❡s❝r❡✈❡♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ✭❡❧❛s sã♦ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥✮✱ ❡ss❛ s❡♣❛r❛çã♦ ♦❝♦rr❡ ❞❡ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✱ ❛tr❛✈és ❞❡ ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ✸✰✶ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✳
✺
❡♥t❡♥❞✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❢ís✐❝❛ ♣♦r ✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❞✐❢❡r❡♥t❡✳ ◆❡ss❛ t❡s❡ ❧✐❞❛♠♦s ❝♦♠ ♦ ❢♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❝♦♠ ✈í♥❝✉❧♦s✱ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❉✐r❛❝ ❬✶✻❪ ❡ q✉❡ t❡♠ s❡ ♠♦str❛❞♦ ♠✉✐t♦ út✐❧ ♣♦r ♠♦str❛r ❡①♣❧í❝✐t❛♠❡♥t❡ ❛ ❢♦r♠❛ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❡ ❞♦ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❣r❛✈✐t❛❝✐♦♥❛✐s✳
◆♦s ❝❛♣ít✉❧♦s ✷ ❡ ✸✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ❜❡♠ ❞❡✜♥✐❞❛✱ ✐♥t❡r♣r❡t❛♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ✈í♥❝✉❧♦s ❝♦♠♦ ❞❡✜♥✐çõ❡s ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❡ ♠♦♠❡♥t♦✲ ❡♥❡r❣✐❛ ❣r❛✈✐t❛❝✐♦♥❛✐s ❞❡ ❢♦r♠❛ q✉❡ ❛s ❞❡✜♥✐çõ❡s s❡❥❛♠ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ❝♦♦r❞❡✲ ♥❛❞❛s✳ ■ss♦ s❡ ❥✉st✐✜❝❛✱ ❡♠ ♣❛rt❡✱ ♣❡❧♦ ❢❛t♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ✈í♥❝✉❧♦s ♣r✐♠ár✐♦s s❛t✲ ✐s❢❛③❡r❡♠ ❛ á❧❣❡❜r❛ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✳ ❈♦♠ ✐ss♦✱ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ ❞❡✜♥✐❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❛ t❡♦r✐❛✱ ❝♦♠♦ s❡♥❞♦ ♦ ♣r♦❞✉t♦ ❞❛ á❧❣❡❜r❛✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦rr❡s♣♦♥✲ ❞❛♠ ❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré✳ ❊ss❡ r❡s✉❧t❛❞♦ ♣❡r♠✐t❡ ❡s❝r❡✈❡r ♦s ♦♣❡r❛❞♦r❡s ❞❡ ❈❛s✐♠✐r ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✱ q✉❛♥t✐❞❛❞❡s q✉❡ sã♦ ✐♥✈❛r✐❛♥t❡s ❣❡r❛❞♦s ♣❡❧❛ t❡♦r✐❛ ❡ q✉❡ ❝❛r❛❝t❡r✐③❛♠ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s✳
◆♦ ❝❛♣ít✉❧♦ ✹✱ ❛❧é♠ ❞❡ ✐♥t❡r♣r❡t❛r♠♦s ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ té✲ tr❛❞❛s✱ ✈❛♠♦s ✐♥✈❡st✐❣❛r ❛ ❞❡✜♥✐çã♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♠❛✐s ♣r♦❢✉♥❞❛♠❡♥t❡✱ ❜❡♠ ❝♦♠♦ ❡①♣❧♦r❛r ❛ ❢♦r♠❛ ❞❛ ❡①♣r❡ssã♦ r❡❣✉❧❛r✐③❛❞❛ ♣❛r❛ ❡ss❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛✱ ♥♦ s❡♥✲ t✐❞♦ ❞❡ s❡ ❡❧✐♠✐♥❛r❡♠ ♣♦ssí✈❡✐s ✐♥✜♥✐t♦s q✉❛♥❞♦ ❛s ✐♥t❡❣r❛✐s sã♦ ❝❛❧❝✉❧❛❞❛s✳ ❯♠❛ ❡①✲ ♣r❡ssã♦ r❡❣✉❧❛r✐③❛❞❛ ♣❛r❛ ❛❧❣✉♠ ♦❜❥❡t♦ s✐❣♥✐✜❝❛✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ s✉❜tr❛✐r ✉♠❛ q✉❛♥✲ t✐❞❛❞❡ ✐♥✜♥✐t❛ ❞❡ss❡ ♦❜❥❡t♦ ♦❜t✐❞❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ♣❧❛♥♦✳ ■ss♦ s❡ t♦r♥❛ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛❢❛st❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✈❛❧♦r❡s ♥ã♦ ♥✉❧♦s ❞❡ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ♦✉ ♠♦♠❡♥t♦ ❛♥✲ ❣✉❧❛r ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳ ❱❛♠♦s✱ t❛♠❜é♠✱ ❝❛❧❝✉❧❛r ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞❡ ✉♠❛ ❝❛s❝❛ ❡s❢ér✐❝❛ ♣❛r❛ ❞✐❢❡r❡♥t❡s ♦❜s❡r✈❛❞♦r❡s✳
✻
❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ♥❡❝❡ssár✐❛s ♣❛r❛ s❡ ❞❡✜♥✐r ❡♥❡r❣✐❛✱ ♠♦♠❡♥t♦ ❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳ ❙③❛❜❛❞♦s ❬✷✵❪✱ ❛❧é♠ ❞✐ss♦✱ ❡♥❝♦♥tr♦✉ ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s q✉❡ ♣r♦❞✉③❡♠ ✈❛❧♦r❡s ✜♥✐t♦s ♣❛r❛ ❛s q✉❛♥t✐❞❛❞❡s ♠❡♥❝✐♦♥❛❞❛s ❛❝✐♠❛✳ ❊♠ t♦❞❛s ❡ss❛s ❛♥á❧✐s❡s ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ P♦✐♥❝❛ré sã♦ ❞❡✜♥✐❞❛s ❡ r❡❛❧✐③❛❞❛s ❡♠ r❡❣✐õ❡s ❛ss✐♥tót✐❝❛s ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❆ á❧❣❡❜r❛ ❞❡ P♦✐♥❝❛ré t❛♠❜é♠ é ✈❡r✐✜❝❛❞❛ ♥♦ ❧✐♠✐t❡ ❛ss✐♥tót✐❝♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ◆❡st❛ t❡s❡ ♠♦str❛r❡♠♦s q✉❡✱ ♥♦ â♠❜✐t♦ ❞❡ ♥♦ss❛ ❞❡✜♥✐çã♦ ♣❛r❛ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✱ ❛ á❧❣❡❜r❛ ❞❡ P♦✐♥❝❛ré é ✈❡r✐✜❝❛❞❛ ❡♠ t♦❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❛ t❡♦r✐❛✱ ♥ã♦ s❡♥❞♦ r❡str✐t❛ à r❡❣✐ã♦ ❞❡ ❢r♦♥t❡✐r❛✳
❈❛♣ít✉❧♦ ✷
❖ ❚❡❧❡♣❛r❛❧❡❧✐s♠♦ ❊q✉✐✈❛❧❡♥t❡ à
❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧
✷✳✶ ■♥tr♦❞✉çã♦
Pr♦❝✉r❛r❡♠♦s tr❛ç❛r ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s ❛ r❡s♣❡✐t♦ ❞♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ❡ ♠♦str❛r ❝♦♠♦ ❡st❛❜❡❧❡❝❡r ❛ ❢♦r♠✉❧❛çã♦ ▲❛❣r❛♥❣❡❛♥❛ ❡ ❍❛♠✐❧t♦♥✐❛♥❛ ❬✶✺❪ ❞♦ ❚❊●❘✳ ❊①✐❣✐r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ ❛ t❡♦r✐❛ ❡①✐❜❛ ✐♥✈❛r✐â♥❝✐❛ ❧♦❝❛❧ ❞❡ ▲♦r❡♥t③✱ ❛tr❛✈és ❞❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠❛ ❝♦♥❡①ã♦ ❞❡ s♣✐♥ωµab❞♦ ❣r✉♣♦ ❙❖✭✸✱✶✮ ❧♦❝❛❧✱ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ✈❛✲ ♠♦s ✐♠♣♦r q✉❡ ❡ss❛ ❝♦♥❡①ã♦ s❡❥❛ ♥✉❧❛ ♣❛r❛ ♦❜t❡r♠♦s ✉♠❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❣❧♦❜❛✐s ❞❡ ▲♦r❡♥t③✳ ❆ ♣❛rt✐r ❞❡ss❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛✲ ❣r❛♥❣❡❛♥❛ ♦❜t❡r❡♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦✳
◗✉❛♥❞♦ ✐♥tr♦❞✉③✐♠♦s ❛ ❝♦♥❡①ã♦ ❞❡ s♣✐♥ ❛ ❝♦♥❞✐çã♦ ❞❡ ❚❡❧❡♣❛r❛❧❡❧✐s♠♦ ❡①✐❣❡ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❞❛ tétr❛❞❛ s❡❥❛ ♥✉❧❛✱ ♦ q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿
∇µeaν = 0
∂µeaν−Γλµνeaλ +ωµabebν = 0, ✭✷✳✶✮ ■s♦❧❛♥❞♦ ❛ ❝♦♥❡①ã♦Γλ
✽
Γλµν =eaλebνωµab+eaλ∂µeaν. ✭✷✳✷✮ ❙✉❜st✐t✉✐♥❞♦ ❡ss❛ q✉❛♥t✐❞❛❞❡ ♥❛ ❞❡✜♥✐çã♦ ✉s✉❛❧ ❞♦ t❡♥s♦r ❞❡ ❝✉r✈❛t✉r❛ ♦❜t❡♠♦s✿
Rλγµν(e, ω) = eaλebγ(∂µωνab−∂νωµab +ωµacωνcb−ωνacωµcb). ✭✷✳✸✮ ❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✷✮✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ t❡♥s♦r ❞❡ t♦rçã♦Tλ
µν = Γλµν−Γλνµ✱ q✉❡ ❣❡r❛ ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿
Taµν(e, ω) =∂µeaν−∂νeaµ+ωµabebν −ωνabebµ. ✭✷✳✹✮ ❆ ❝♦♥❡①ã♦ ❞❡ s♣✐♥✱ ✉s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✹✮✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ✐❞❡♥t✐❝❛♠❡♥t❡ ❝♦♠♦✿
ωµab =♦ωµab+Kµab, ✭✷✳✺✮ ♦♥❞❡ Kµab é ♦ t❡♥s♦r ❞❡ ❝♦♥t♦rçã♦ ❡ ♦ωµab é ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ s❡♥❞♦ ❡ss❛s q✉❛♥t✐❞❛❞❡s ❞❡✜♥✐❞❛s ♣❡❧❛s ❡①♣r❡ssõ❡s✿
Kµab = 1 2ea
λe
bν(Tλµν +Tνλµ+Tµλν),
♦ω
µab = − 1 2e
cµ(Ωabc
−Ωbac−Ωcab), ✭✷✳✻✮
❝♦♠ Ωabc ❞❛❞♦ ♣♦r✿
Ωabc =eaν(ebµ∂µecν −ecµ∂µebν). ✭✷✳✼✮ ❉❡✈❡♠♦s ♥♦t❛r q✉❡♦ω
µab ♣♦ss✉✐ t♦rçã♦ ♥✉❧❛✳
✾
eR(e, ω) = eR(e) +e(1 4T
abcT abc+
1 2T
abcT
bac−TaTa)−2∂µ(eTµ). ✭✷✳✽✮ ❊ss❛ é ❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❡ s❡rá ✉s❛❞❛ ♣❛r❛ ❞❡✜♥✐r♠♦s ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛✲ ❣r❛♥❣❡❛♥❛ ♥♦ ❚❊●❘✳
❚r❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛ é ♦❜t✐❞❛ q✉❛♥❞♦ ❞❡❝♦♠♣♦✲ ♠♦s ♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡♠ ❤✐♣❡rs✉♣❡r❢í❝✐❡s tr✐❞✐♠❡♥s✐♦♥❛✐s ❞♦ t✐♣♦ ❡s♣❛ç♦ ❡ q✉❡ sã♦ ❞❡❢♦r♠❛❞❛s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛s ❢✉♥çõ❡s ❧❛♣s♦N ❡ s❤✐❢t Ni✱ ❛s q✉❛✐s ❛❣❡♠ ♥❛ ❞✐r❡çã♦ ♥♦r♠❛❧ ❡ t❛♥❣❡♥❝✐❛❧ ❞❡ss❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡s♣❛❝✐❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❣❡r❛♥❞♦ ♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦✳ ❊♥tr❡t❛♥t♦ ♥❡st❡ ❝❛♣ít✉❧♦ ❝♦♥str✉✐r❡♠♦s ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧✲ t♦♥✐❛♥❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲❡❣❡♥❞r❡ ❛♣❧✐❝❛❞❛ à ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛✲ ❣r❛♥❣❡❛♥❛ s❡♠ ❢❛③❡r♠♦s ❛ ❞❡❝♦♠♣♦s✐çã♦ 3 + 1 ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❆ ♣❛rt✐r ❞✐ss♦ ❞❡✜♥✐r❡♠♦s ❛s ❡①♣r❡ssõ❡s ♣❛r❛ ♦ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❣r❛✈✐t❛❝✐♦♥❛✐s✳ ❈♦♥str✉✐r ✉♠❛ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ❞❛ ❣r❛✈✐t❛çã♦ é ✐♠♣♦rt❛♥t❡ ♣♦✐s ❛s ❡q✉❛çõ❡s s❡ t♦r♥❛♠ ♠❡♥♦s ❝♦♠♣❧❡①❛s✱ ✉♠❛ ✈❡③ q✉❡ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡♥✲ ✈♦❧✈❡♠ ❞❡r✐✈❛❞❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ s✐st❡♠❛ sã♦ ♠❡❧❤♦r ♦❜s❡r✈❛❞❛s✱ ♣❡r♠✐t✐♥❞♦✲♥♦s r❡t✐r❛r ✐♥❢♦r♠❛çõ❡s q✉❡✱ ♠✉✐t❛s ✈❡③❡s✱ sã♦ ♦❜s❝✉r❛s ♥♦ ❝♦♥t❡①t♦ ❞♦ ❢♦r♠❛❧✐s♠♦ ▲❛❣r❛♥❣❡❛♥♦✳ ❊✱ ❢✉♥❞❛♠❡♥t❛❧♠❡♥t❡✱ ❡ss❡ ❡♥❢♦q✉❡ ♣♦❞❡ ♥♦s ♣❡r♠✐t✐r ❛ q✉❛♥t✐③❛çã♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳
✷✳✷ ❆❧❣✉♠❛s ❈♦♥s✐❞❡r❛çõ❡s ❙♦❜r❡ ♦ ❈❛♠♣♦ ❞❡ ❚étr❛❞❛s
❊♠ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦ ❛r❜✐trár✐♦ ❤á s❡♠♣r❡ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❡♠ ❝❛❞❛ ♣♦♥t♦✳ P♦❞❡♠♦s ♣r♦❥❡t❛r ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡✜♥✐❞❛ ♥❡ss❡ ❡s♣❛ç♦✲ t❡♠♣♦ ❛r❜✐trár✐♦✱ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡✳ P❛r❛ ✐ss♦✱ ✉s❛♠♦s ♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s✳ ❈♦♥s✐❞❡r❡ ✉♠ ✈❡t♦r ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ Vµ✱ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣r♦❥❡çã♦ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡ é ❞❛❞❛ ♣♦r✿
✶✵
s❡♥❞♦ q✉❡ ♣❛r❛ ✐ss♦ ✉t✐❧✐③❛♠♦s ♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛sea
µ✳ ❈❧❛r♦ q✉❡ ♣♦❞❡♠♦s ❢❛③❡r ♦ ❝❛♠✐♥❤♦ ♦♣♦st♦ ❡ ♣r♦❥❡t❛r ✉♠ ✈❡t♦r ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡Va ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦✱ ❛ss✐♠ t❡♠♦s✿
Vµ =eaµVa, ✭✷✳✶✵✮
♥❡ss❡ ❝❛s♦ ✉s❛♠♦s ♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ✐♥✈❡rs♦eaµ✳
❈♦♠♦ ❢♦✐ ❞✐t♦✱ ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛s é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s q✉❡ ♦❜❡❞❡❝❡♠ ✉♠❛ r❡❧❛çã♦ ❞❡ ♦rt♦❣♦♥❛❧✐❞❛❞❡✳ ❊ss❛ r❡❧❛çã♦ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ♣♦r✿
gµν = eaµeaν;
ηab = eaµebµ. ✭✷✳✶✶✮
❱❛♠♦s ♠♦str❛r ❝♦♠♦ ❝♦♥str✉✐r ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ❡♠ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ♣❧❛♥♦✱ ❝❧❛r♦ q✉❡ ♥❡st❡ ❝❛s♦ ♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡ s❡rá ♦ ♣ró♣r✐♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❞♦✐s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s✱qa = (t, x, y, z) ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡ ❡ xµ = (t, r, θ, φ) ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❢ís✐❝♦✳ ❖s ❞♦✐s s✐st❡♠❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s dqa = ea
µdxµ✱ ❝♦♠ ✐ss♦ ♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿
eaµ=
∂qa
∂xµ =
1 0 0 0
0 sinθ cosφ rcosθ cosφ −rsinθ sinφ
0 sinθ sinφ rcosθ sinφ rsinθ cosφ
0 cosθ −rsinθ 0
, ✭✷✳✶✷✮
✶✶
❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s q✉❡ s❡ r❡❧❛❝✐♦♥❛♠ ❛tr❛✈és ❞❡ ✉♠ ✏❜♦♦st✧❞❡ ▲♦r❡♥t③✳
◆♦ ❝❛s♦ ❣❡r❛❧ ♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ∂µqa✱ ❡♥tã♦ ❛s tétr❛❞❛s sã♦ ❝❤❛♠❛❞❛ ❞❡ ♥ã♦✲❤♦❧ô♥♦♠❛s✳ P❛r❛ ❡ss❛ ❝❛t❡❣♦r✐❛ ❞❡ tétr❛❞❛s t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ∂µeaν −∂νeaµ 6= 0✱ ❧♦❣♦ ♣❛r❛ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❝♦♠ t♦rçã♦✱ t❡♠♦s tétr❛❞❛s ♥ã♦✲❤♦❧ô♥♦♠❛s✳ ◆♦s s✐st❡♠❛s ❛❜♦r❞❛❞♦s ♥❡st❛ t❡s❡ ❡♠ q✉❡ ❢♦r ♥❡❝❡ssár✐♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✱ ✈❛♠♦s ✉s❛r ❛ ❡str✉t✉r❛ ❞❡ ✭✷✳✶✷✮ ♣❛r❛ ❝♦♥str✉ír♠♦s ♦ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡✳
P❛r❛ ♦ ❚❊●❘ ❛ ♠❛♥✐❢❡st❛çã♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ s❡ ❞á ❡♠ ✉♠ ❡s♣❛ç♦✲ t❡♠♣♦ ❞♦t❛❞♦ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ tétr❛❞❛s ♥ã♦✲❤♦❧♦♥ô♠✐❝♦✱ q✉❡ ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❣r❛❞✐❡♥t❡ ❞❡ ❢✉♥çõ❡sqa✳ ❆s q✉❛♥t✐❞❛❞❡s ❞❡ ✐♥t❡r❡ss❡ ❢ís✐❝♦ t❛✐s ❝♦♠♦ ♦ ✈❡t♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r s❡rã♦ ❞❡✜♥✐❞♦s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♥❣❡♥t❡✱ q✉❡ s❡rá ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ♦ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ r❡❢❡rê♥❝✐❛✳
✷✳✸ ❆ ❋♦r♠✉❧❛çã♦ ▲❛❣r❛♥❣❡❛♥❛
◆❛ ❢♦r♠✉❧❛çã♦ ▲❛❣r❛♥❣❡❛♥❛ ❞♦ ❚❊●❘ ✈❛♠♦s ✐♠♣♦r q✉❡ ❛ ❝♦♥❡①ã♦ ❞❡ s♣✐♥
ωµab s❡❥❛ ✐❣✉❛❧ ❛ ③❡r♦✳ ❈♦♠ ✐ss♦ ❛ ❡①♣r❡ssã♦ ✭✷✳✸✮ s❡ r❡❞✉③ ❛✿
eR(e)≡ −e(1 4T
abcT abc+
1 2T
abcT
bac−TaTa) + 2∂µ(eTµ). ✭✷✳✶✸✮ ❆❧é♠ ❞✐ss♦✱ ❛ t♦rçã♦ ❡♠ ✭✷✳✹✮ ❛ss✉♠❡ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
Taµν(e) =∂µeaν −∂νeaµ. ✭✷✳✶✹✮ ❙❡ ❞❡s♣r❡③❛r♠♦s ❛ ❞✐✈❡r❣ê♥❝✐❛ ❡♠ ✭✷✳✶✸✮✱ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ♣❛r❛ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ♥♦ ❚❊●❘ é ❞❛❞❛ ♣♦r✿
L(eaµ) = −k e(1 4T
abcT abc+
1 2T
abcT
bac−TaTa)−LM
✶✷
♦♥❞❡ k = 1/(16π) ❡ LM é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ♣❛r❛ ♦s ❝❛♠♣♦s ❞❡ ♠❛tér✐❛✳ ❖ t❡r♠♦ ❞❡ ❞✐✈❡r❣ê♥❝✐❛ ♥ã♦ é ♥❡❝❡ssár✐♦ q✉❛♥❞♦ ❝♦♥str✉✐♠♦s ❛ ✐♥t❡❣r❛❧ ❞❡ ❛çã♦ ♣❛r❛ ❡s♣❛ç♦s✲t❡♠♣♦s ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ♣❧❛♥♦s✱ ♣♦✐s ❛s ✐♥t❡❣r❛✐s ❞❡ s✉♣❡r❢í❝✐❡ q✉❡ s✉r❣❡♠ ♣♦r ✐♥t❡❣r❛çõ❡s ♣♦r ♣❛rt❡s s❡ ❛♥✉❧❛♠✳ ◆♦ ✈á❝✉♦✱ ♥♦t❛♠♦s q✉❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❣❡r❛✐s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ♣♦r tr❛♥s✲ ❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ❣❧♦❜❛✐s ❙❖✭✸✱✶✮ ❝♦♠♦ é ❡s♣❡r❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ ✐♠♣✉s❡♠♦s
ωµab = 0✳ ❖ t❡♥s♦r Σabc é ❞❡✜♥✐❞♦ ♣♦r✿
Σabc = 1 4(T
abc+Tbac
−Tcab) + 1 2(η
acTb
−ηabTc), ✭✷✳✶✻✮
❡Ta =Tb
ba✳ ❆s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ sã♦ ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❡ ✭✷✳✶✺✮✱ ♣♦r ♠❡✐♦ ❞❡ s✉❛ ✈❛r✐❛çã♦ ❢✉♥❝✐♦♥❛❧ ❡♠ r❡❧❛çã♦ ❛eaµ ❡ sã♦ ❞❛❞❛s ♣♦r✿
eaλebµ∂ν(eΣbλν)−e(ΣbνaTbνµ− 1
4eaµTbcdΣ
bcd) = 1
4keTaµ. ✭✷✳✶✼✮
❈♦♠♦ ΣabcT
abc é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛ r❡❧❛t✐✈♦ à ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ❞✐✈❡r❣ê♥❝✐❛ t♦t❛❧✱ ♣♦❞❡✲s❡ ♠♦str❛r✱ ♣♦r ❝á❧❝✉❧♦s ❡①♣❧í❝✐t♦s✱ q✉❡ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ ✭✷✳✶✼✮ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ t❡♥s♦r ❞❡ ❊✐♥st❡✐♥Gaµ =
eaνGνµ✳ ❖✉ s❡❥❛✿
eaλebµ∂ν(eΣbλν)−e(ΣbνaTbνµ− 1
4eaµTbcdΣ
bcd) = 1
2e[Raµ(e)− 1
2eaµR(e)], ✭✷✳✶✽✮ ❝♦♠ ✐ss♦✱ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ s❡ t♦r♥❛ ❝❧❛r❛✿
Raµ(e)−1
2eaµR(e) = 1
2kTaµ. ✭✷✳✶✾✮
✶✸
q✉❛❞r♦ é ♦♣♦st♦✱ ♠❛s ❛❜s♦❧✉t❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡✳ ❚❡♠✲s❡ ❛ ❝✉r✈❛t✉r❛ ❝♦♥str✉í❞❛ ❛ ♣❛rt✐r ❞❛ ❝♦♥❡①ã♦ ❞❡ ❈❛rt❛♥ ♥✉❧❛ ❡ ❛ t♦rçã♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳
❆s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ✭✷✳✶✼✮ ♣♦❞❡♠ s❡r r❡❡s❝r✐t❛s ♥❛ ❢♦r♠❛✿
∂ν(eΣaλν) = 1 4ke e
aµ(tλµ+Tλµ), ✭✷✳✷✵✮ ♦♥❞❡
tλµ =k(4ΣbcλTbcµ−gλµΣbcdTbcd), ✭✷✳✷✶✮ é ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ♦ t❡♥s♦r ❞❡ ❡♥❡r❣✐❛ ♠♦♠❡♥t♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❬✷✶❪✳ ❉❡♥✲ tr❡ ♦s ✈ár✐♦s ♠♦t✐✈♦s q✉❡ s✉♣♦rt❛♠ ❡ss❛ ✐♥t❡r♣r❡t❛çã♦ ❬✷✷❪✱ ✈❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ tλµ é ✉♠ t❡♥s♦r ✈❡r❞❛❞❡✐r♦ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❡♥tr❡t❛♥t♦ tλµ ♥ã♦ é s✐♠étr✐❝♦✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s ✉♠❛ ❧❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦ t❛♥t♦ ♣❛r❛ etaλ q✉❛♥t♦ ♣❛r❛ eTaλ✳ P❛r❛ ❡♥t❡♥❞❡r♠♦s ✐ss♦ ❜❛st❛ ♥♦t❛r q✉❡ Σaλν é ❛♥t✐✲s✐♠étr✐❝♦ ♥♦s ❞♦✐s ú❧t✐♠♦s í♥❞✐❝❡s✱ ❧❡♠❜r❛♥❞♦ q✉❡ ✉♠❛ ❝♦♥tr❛çã♦ ❡♥tr❡ ✉♠ t❡♥s♦r s✐♠étr✐❝♦ ❡ ♦✉tr♦ ❛♥t✐✲s✐♠étr✐❝♦ é ♥✉❧❛✱ t❡♠♦s ♦ s❡❣✉✐♥t❡✿
∂λ∂ν(eΣaλν)≡0. ✭✷✳✷✷✮ ❆ss✐♠✱ ✐♠❡❞✐❛t❛♠❡♥t❡ ❝❤❡❣❛♠♦s à ❡q✉❛çã♦✿
∂λ(etaλ+eTaλ) = 0, ✭✷✳✷✸✮
✶✹
✷✳✹ ❆ ❋♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛
◆❡st❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❛♣r❡s❡♥t❛çã♦ r❡s✉♠✐❞❛ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❡st❛❜❡❧❡❝✐❞♦s ♥❛s r❡❢❡rê♥❝✐❛s ❬✶✺❪ ❡ ❬✷✸❪✳ P❛r❛ ♦❜t❡r♠♦s ❛ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ❞♦ ❚❊●❘ t❡♠♦s q✉❡✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❡st❛❜❡❧❡❝❡r ♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❛ t❡♦r✐❛✳ ❈♦♠♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ♥ã♦ ❝♦♥té♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧ ❞❡ea0✱
❡ss❛ q✉❛♥t✐❞❛❞❡ s✉r❣❡ ❝♦♠♦ ✉♠ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ ▲❛❣r❛♥❣❡✳ ❖ ♠♦♠❡♥t♦ ❝❛♥♦♥✐✲ ❝❛♠❡♥t❡ ❝♦♥❥✉❣❛❞♦ ❛ eai é ❞❛❞♦ ♣♦r Πai = δL/δe˙ai✳ ❆ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ✭♥ã♦ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❝♦✈❛r✐❛♥t❡✮ é ♦❜t✐❞❛ r❡❡s❝r❡✈❡♥❞♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ♥❛ ❢♦r♠❛ L = pq˙−H0✱ ❡♠ t❡r♠♦s ❞❡ eai✱ Πai ❡ ❞♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✳
❊①❡❝✉t❛♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲❡❣❡♥❞r❡✱ ❝❤❡❣❛♠♦s à ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛ ❬✶✺❪ ♥❛ ❢♦r♠❛✿
H =H0+αikΓ′ik+βkΓk, ✭✷✳✷✹✮
♠❛✐s t❡r♠♦s ❞❡ s✉♣❡r❢í❝✐❡✳ αik ❡ βk sã♦ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✳ ❆tr❛✈és ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥s✐stê♥❝✐❛ δH0
δea0 = 0✱ t❡♠♦s ✉♠ ♥♦✈♦ ✈í♥❝✉❧♦ C
a✱ q✉❡ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠
H0 ♣♦r H0 =ea0Ca✳ ❈♦♠ ✐ss♦ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛✿
H =ea0Ca+αikΓ′ik+βkΓk. ✭✷✳✷✺✮
❆♣ós r❡s♦❧✈❡r♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ✐❞❡♥t✐✜❝❛♠♦sαik = 1/2(Ti0k+Tk0i) ❡βk =
T00k✳ Ca✱ Γ′ik ❡ Γk sã♦ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✱ ❣❛r❛♥t✐♥❞♦ q✉❡ ❛ ❡✈♦❧✉çã♦
t❡♠♣♦r❛❧ ❞❛ t❡♦r✐❛ é ❜❡♠ ❞❡✜♥✐❞❛✳
❖ ✈í♥❝✉❧♦ Ca é ❡s❝r✐t♦ ❝♦♠♦ Ca = −∂iΠai +pa✱ ♦♥❞❡ pa é ✉♠❛ ❡①♣r❡ssã♦ ♠✉✐t♦ ❝♦♠♣❧✐❝❛❞❛ ❞❛s ✈❛r✐á✈❡✐s ❞❡ ❝❛♠♣♦✱ ❡①♣❧✐❝✐t❛♠❡♥t❡ t❡♠♦s✿
pa = kenea0h− 1
4g00
gikgjlPijPkl− 1 2P
2+1
4g
imgnjTb
mnTbij+ + 1
2g njTi
mnTmij −gikTmmiTnnk
i
− 2g100gikgjlγaijPkl−
− 1
2gijγ
aijP− eaig0mgnjTb
✶✺
− 2g0kTmmkTnni−2gjkT0ijTnnk
o
, ✭✷✳✷✻✮
❝♦♠ γaij ❡ Pik ❞❡✜♥✐❞♦s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
γaij = − 1 2ke(e
aiΓj+eajΓi)
−eakhg00(gjmTikm+gimTjkm+ 2gijTmmk) + + g0m(g0jTimk+g0iTjmk)−2g0ig0jTmmk+
+ (gjmg0i+gimg0j −2gijg0m)T0
mk
i
✭✷✳✷✼✮
Pik = 1
keΠ
(ik)+g0m(gkjTi
mj +gijTkmj −2gikTjmj) +
+ (gkmg0i+gimg0k)Tjmj. ✭✷✳✷✽✮ ❆ ❢♦r♠❛ ✐♥t❡❣r❛❧ ❞❛s ❡q✉❛çõ❡s ❞❡ ✈í♥❝✉❧♦ Ca = 0 é ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡ ❡♥❡r❣✐❛ ❞♦ t✐♣♦H−E = 0❡ ♥♦s ♣❡r♠✐t❡ ❞❡✜♥✐r ♦ ✈❡t♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❣r❛✈✐t❛❝✐♦♥❛❧Pa ✿
Pa =−
Z
V
d3xpa, ✭✷✳✷✾✮
❝♦♠♦pa =∂iΠai ✭♣❡❧❛ ❡q✉❛çã♦ ❞❡ ✈í♥❝✉❧♦✮✱ t❡♠♦s ❬✶✱ ✷✹❪✿
Pa=−
Z
V
d3x∂iΠai, ✭✷✳✸✵✮
V é ✉♠ ✈♦❧✉♠❡ ❛r❜✐trár✐♦ ❞♦ ❡s♣❛ç♦ tr✐✲❞✐♠❡♥s✐♦♥❛❧✳ ❊ss❛ é ✉♠❛ ❞❡✜♥✐çã♦ ❝♦♥✲
s✐st❡♥t❡ ♣♦✐s ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ✐♥❞✐❝❛♠ q✉❡ ✭✷✳✸✵✮ r❡♣r❡s❡♥t❛ ♦ ♠♦♠❡♥t♦✲❡♥❡r❣✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ❝♦♥t✐❞♦ ❡♠ ✉♠ ✈♦❧✉♠❡ ❱ ❡♠ ❡s♣❛ç♦s ✈❛③✐♦s✳ P❛rt✐❝✉❧❛r♠❡♥t❡ ✭✷✳✸✵✮ ❣❡r❛ ❛ ❡♥❡r❣✐❛ ❞❡ ❆❉▼ ❬✹✶❪ q✉❛♥❞♦ ❛♣❧✐❝❛❞❛ ❛ t♦❞♦ ❡s♣❛ç♦ tr✐✲❞✐♠❡♥s✐♦♥❛❧✳ ◆♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s✱ t❡♠♦s✿
Πai=−4keΣa0i. ✭✷✳✸✶✮
✶✻
❊♠ t❡r♠♦s ❞❛ ❞❡✜♥✐çã♦ ✭✷✳✷✶✮✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿
d dt
Z
V
d3x e eaµ(t0µ+T0µ) = −
I
S
dSj
e eaµ(tjµ+Tjµ)
, ✭✷✳✸✷✮
q✉❡ r❡♣r❡s❡♥t❛ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ♣❛r❛ ♦ t❡♥s♦r ❞❡ ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ t♦t❛❧ tλµ +Tλµ✳ ❆ss✐♠✱ ❛❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r Pa ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❢❛♠✐❧✐❛r✿
Pa =
Z
V
d3x e eaµ(t0µ+T0µ). ✭✷✳✸✸✮
❊♥tr❡t❛♥t♦ ✉t✐❧✐③❛r❡♠♦s ❛ ❡①♣r❡ssã♦ ✭✷✳✸✵✮✱ ♣♦r ♠♦t✐✈♦s ♣rát✐❝♦s✱ ✉♠❛ ✈❡③ q✉❡ é ♠✉✐t♦ ♠❛✐s ❢á❝✐❧ ❧✐❞❛r ❝♦♠ ✭✷✳✸✵✮ ❞♦ q✉❡ ✭✷✳✸✸✮✳ ❱❡♠♦s q✉❡ ❛ ❞✐✈❡r❣ê♥❝✐❛ q✉❡ ❛♣❛r❡❝❡ ❡♠Ca♣♦❞❡ s❡r r❡❛❧♠❡♥t❡ t♦♠❛❞❛ ♣❛r❛ ❞❡✜♥✐r♠♦s ♦ ✈❡t♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❡♠ ✈✐st❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ q✉❡ r❡❧❛❝✐♦♥❛♠ tµν ❝♦♠ Πai✳ P♦rt❛♥t♦ ❛ ♥♦ss❛ ❞❡✜♥✐çã♦ ❞❡ ♠♦♠❡♥t♦ ❡♥❡r❣✐❛ ♥ã♦ é ❛r❜✐trár✐❛✱ ♦ s❡✉ s❡♥t✐❞♦ é ❡str✐t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦✳
❆ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ✈í♥❝✉❧♦s ❝♦♠♦ ❡q✉❛çõ❡s q✉❡ ❞❡✜♥❡♠ ❛ ❡♥❡r❣✐❛ ❡ ♦ ♠♦✲ ♠❡♥t♦ ❣r❛✈✐t❛❝✐♦♥❛✐s s❡ ❥✉st✐✜❝❛ q✉❛♥❞♦ ❧✐❞❛♠♦s ♣♦r ❡①❡♠♣❧♦ ❝♦♠ ❛ ✐♥t❡❣r❛❧ ❞❡ ❛çã♦ ❞❡ ❏❛❝♦❜✐ ❬✷✺❪✳ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ❞❡s❝r✐t♦ ♣♦r ❝♦♦r❞❡♥❛❞❛sxi ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ♥✲❞✐♠❡♥s✐♦♥❛❧✱ (i= 1,2, ...n)✳ ❆ ❛çã♦ ❞❡ ❏❛❝♦❜✐ é✿
S[x] =
Z p
mijdxidxj
p
2(E−V(x)), ✭✷✳✸✹✮
♦♥❞❡ mij é ❛ ♠étr✐❝❛ ◆❡✇t♦♥✐❛♥❛✳ ❙❡ ✐♥tr♦❞✉③✐r♠♦s ✉♠ ♣❛râ♠❡tr♦ σ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❝❛♠✐♥❤♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❝♦♠ ❡①tr❡♠✐❞❛❞❡s ✜①❛s ❡♥tã♦ ❛ ❛çã♦ ❞❡ ❏❛❝♦❜✐ s❡ ❡s❝r❡✈❡ ❝♦♠♦✿
S[x] =
Z σ′′
σ′
dσpmijx˙ix˙j
p
2(E−V(x)), ✭✷✳✸✺✮
♦♥❞❡x(σ′) = x′ ❡x(σ′′) = x′′ sã♦ ✜①❛❞♦s✳
✶✼
❝❛♠❡♥t❡ ♥✉❧♦✳ ❆ ❞❡✜♥✐çã♦ ❞♦ ♠♦♠❡♥t♦ ❝❛♥♦♥✐❝❛♠❡♥t❡ ❝♦♥❥✉❣❛❞♦ às ❝♦♦r❞❡♥❛❞❛s✱
pi = ˙xiq2(E−V(x)) ˙
x2 ✱ ❡st❛❜❡❧❡❝❡ ✉♠ ✈í♥❝✉❧♦ ❈ q✉❡ ♣♦❞❡ s❡r ✉s❛❞♦ ❝♦♠♦ ✉♠❛ ❡q✉❛çã♦
q✉❡ ❞❡✜♥❡ ❛ ❡♥❡r❣✐❛✳ ❉❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
C ≡ 1
2m ijp
ipj+V(x)−E = 0. ✭✷✳✸✻✮ ❈♦♠ ✐ss♦ ♣♦❞❡♠♦s ♥♦t❛r ❛ s✐♠✐❧❛r✐❞❛❞❡ q✉❡ ❡①✐st❡ ❡♥tr❡ ♦ ❡①♣♦st♦ ❛❝✐♠❛ ❡ ❛q✉✐❧♦ q✉❡ ✉s❛♠♦s ♥❡st❛ t❡s❡ ♣❛r❛ ❞❡✜♥✐r ♦ ✈❡t♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳
❖s ✈í♥❝✉❧♦s Γ′ik ❡Γk sã♦ ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❛ r❡❧❛çã♦ ✭✷✳✸✶✮ q✉❛♥❞♦ ❡s❝r❡✈❡✲
♠♦s ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛ ❡ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦✿
Γ′ik = Π[ik]+ke
{gimgkjT0mj + (gkmg0i−gimg0k)Tjmj}
Γk = Π0k+ 2ke(gkjg0iT0ij −g0kg0iTjij +g00gikTjij). ✭✷✳✸✼✮ ❊ss❛s q✉❛♥t✐❞❛❞❡s sã♦ ✈í♥❝✉❧♦s ♥♦ s❡♥t✐❞♦ ❡str✐t♦ ❞♦ t❡r♠♦✱ ♣♦✐s r❡♣r❡s❡♥t❛♠ ✉♠❛ r❡❧❛çã♦ ❛❧❣é❜r✐❝❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❞❡ ❝❛♠♣♦✱ ♦✉ s❡❥❛✱ ❛s tétr❛❞❛s✱ ❡ ♦s ♠♦♠❡♥t♦s ❝❛♥♦♥✐❝❛♠❡♥t❡ ❝♦♥❥✉❣❛❞♦s ❛ ❡❧❛s✳ ■ss♦ é ❡①♣❧✐❝✐t❛❞♦ q✉❛♥❞♦ ❡①❡❝✉t❛♠♦s ❛ tr❛♥s❢♦r✲ ♠❛çã♦ ❞❡ ▲❡❣❡♥❞r❡ s♦❜r❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ▲❛❣r❛♥❣❡❛♥❛ ✭✷✳✶✺✮✳
❖ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ ❡♥tr❡ ❞✉❛s q✉❛♥t✐❞❛❞❡s ❞❡ ❝❛♠♣♦ ❋ ❡ ● é ❞❛❞♦ ♣♦r✿
{F, G}=
Z
d3x δF δeai(x)
δG δΠai(x) −
δF δΠai(x)
δG δeai(x)
. ✭✷✳✸✽✮
❈❛❧❝✉❧❛♥❞♦ ♦ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ ❡♥tr❡ ♦s ✈í♥❝✉❧♦sΓ′ij(x) ❡Γ′kl(y)✱ ✈❡♠♦s q✉❡ ❡❧❡s s❛t✐s❢❛③❡♠ ❛ á❧❣❡❜r❛ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❬✶✺❪✱ ❡①♣❧✐❝✐t❛♠❡♥t❡ t❡♠♦s✿
{Γ′ij(x),Γ′kl(y)
}= 1 2
gilΓ′jk +gjkΓ′il
−gikΓ′jl
−gjlΓ′ikδ(x
−y), ✭✷✳✸✾✮
✐ss♦ ❥✉st✐✜❝❛ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ✉♠❛ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛ ❞♦ ✈í♥❝✉❧♦ ❝♦♠♦ ❞❡✜♥✐çã♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✱ t❛❧ q✉❛❧ é ❢❡✐t♦ ♣❛r❛ ♦ ✈❡t♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦✳
❊♠ ✈✐st❛ ❞✐ss♦ é ✐♠♣♦rt❛♥t❡ r❡❡s❝r❡✈❡r♠♦s ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛
✶✽
r❡❡s❝r❡✈❡♥❞♦✲♦s ❝♦♠♦ ✉♠ ú♥✐❝♦ ✈í♥❝✉❧♦Γab✳ ❆♥t❡s✱ ♣♦ré♠✱ ❞❡✈❡♠♦s ♥♦t❛r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
Σµ0ν −Σν0µ= 1 2[g
µmgνjT0
mj + (gνmg0µ−gµmg0ν)Tjmj], ✭✷✳✹✵✮ ❛ ♣❛rt✐r ❞✐ss♦ t♦♠❛♠♦s ♦s ❝♦♠♣♦♥❡♥t❡sµ= 0 ❡ ν=k✱ ♦❜t❡♥❞♦✿
Σ00k= 1 2[g
0mgkjT0
mj + (gkmg00−g0mg0k)Tjmj]. ✭✷✳✹✶✮ ❘❡❛❧✐③❛♥❞♦ ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛µ=i ❡ ν=k✱ t❡♠♦s✿
Σi0k−Σk0i = 1 2[g
imgkjT0
mj + (gkmg0i−gimg0k)Tjmj]. ✭✷✳✹✷✮ ❙❡ ❞❡✜♥✐r♠♦s ✉♠❛ q✉❛♥t✐❞❛❞❡ Mµν ♣♦r✿
Mik = 2Π[ik] =eaiΠak −eakΠai, ✭✷✳✹✸✮
M0k = Π0k =ea0Πak, ✭✷✳✹✹✮ ❡♥tã♦ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ♦ ✈í♥❝✉❧♦ s✐♠♣❧✐✜❝❛❞♦Γab é✿
Γab =Mab+ 4ke(Σa0b−Σb0a), ✭✷✳✹✺✮
❝♦♠ Mab = ea
µebνMµν = −Mba✳ ❖ ♥♦✈♦ ✈í♥❝✉❧♦ Γab = −Γba ❡♥❝❡rr❛ ❛♠❜♦s ♦s ❛♥t✐❣♦s ✈í♥❝✉❧♦sΓ′ik ❡ Γk ❛tr❛✈és ❞❛s r❡❧❛çõ❡s Γik = 2Γ′ik =e
aiebkΓab✱ Γk ≡Γ0k =
ea0ebkΓab✳
❈♦♠ ✐ss♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s q✉❛♥❞♦ ❝♦♠♣❛r❛❞❛ ❛ ✭✷✳✷✺✮✿
H=ea0Ca+
1 2λabΓ
ab, ✭✷✳✹✻✮
✶✾
❡ βk✱ ♥❛ ❡①♣r❡ssã♦ ✭✷✳✷✺✮✳ P❛r❛ ✐ss♦ ♦ s❡❣✉♥❞♦ t❡r♠♦ ❡♠ ✭✷✳✹✻✮ ❞❡✈❡ s❡r s❡♣❛r❛❞♦
❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
1 2λabΓ
ab = 1 2λµνΓ
µν = 1 2(λ00Γ
00+λ
0iΓ0i+λi0Γi0 +λijΓij) = λ0iΓ0i+
1 2λijΓ
ij
= λ0iΓi +λijΓ′ij, ✭✷✳✹✼✮
❧♦❣♦ ✐❞❡♥t✐✜❝❛♠♦s ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❝♦♠♦λik =αik❡λ0k=−λk0 =βk✳
➱ ❜♦♠ ❧❡♠❜r❛r q✉❡ ♥❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✉t✐❧✐③❛♠♦s ❛s r❡❧❛çõ❡s λµν = eaµebνλab ❡ Γµν =e
aµebνΓab✳
❈♦♠♦ ✈✐♠♦s✱ q✉❛♥❞♦ ❡s❝r❡✈❡♠♦s ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ❞❡ ❛çã♦ ❞❡ ❏❛❝♦❜✐✱ s✉r❣❡ ✉♠ ✈í♥❝✉❧♦✱ ❡①♣r❡ss♦ ♣♦r ✭✷✳✸✻✮✱ q✉❡ é ✉s❛❞♦ ❝♦♠♦ ❞❡✜♥✐çã♦ ❞❡ ❡♥❡r❣✐❛✳ ❈♦♠♣❛r❛♥❞♦ ❝♦♠ ♦ ♥♦ss♦ ♣r♦❝❡❞✐♠❡♥t♦✱ ✈❡♠♦s q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ✐♥t❡r♣r❡t❛çã♦ ❞♦ ✈í♥❝✉❧♦ Ca ❝♦♠♦ ❞❡✜♥✐çã♦ ❞♦ ✈❡t♦r ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ ❣r❛✈✐t❛❝✐♦♥❛❧ é ♦ ♠❡s♠♦✳ ❆ss✐♠✱ t❡♥❞♦ ♣♦r ❜❛s❡ ❛s ♠❡s♠❛s ✐❞é✐❛s✱ ❛ ❡st❡♥sã♦ ♠❛✐s ♥❛t✉r❛❧ ❞❡ss❡s ❝♦♥❝❡✐t♦s é ✐♥t❡r♣r❡t❛r ♦ ✈í♥❝✉❧♦Γab ❝♦♠♦ ❞❡✜♥✐çã♦ ❞❡ ❛❧❣✉♠❛ q✉❛♥✲ t✐❞❛❞❡ ❢ís✐❝❛✳ ❆♦ ❛♥❛❧✐s❛r♠♦s ❛s ❞✐♠❡♥sõ❡s ❞❡ss❡ ✈í♥❝✉❧♦ ❡ t❡♥❞♦ ♣♦r ❜❛s❡ ❛ r❡❧❛çã♦ ✭✷✳✸✾✮✱ ✈❡♠♦s q✉❡ ❡ss❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛ t❡♠ q✉❡ ❡st❛r r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✳ ■ss♦ s❡rá ❡①❡♠♣❧✐✜❝❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✹ q✉❛♥❞♦✱ ❡♠ ♥♦ss❛ ❛♥á❧✐s❡✱ r❡✐♥tr♦✲ ❞✉③✐r♠♦s ❛s ❝♦♥st❛♥t❡sc❡ G✳
P♦rt❛♥t♦ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ à ❞❡✜♥✐çã♦ ❞❡ Pa ❬✷✶❪✱ ❛ ❢♦r♠❛ ✐♥t❡❣r❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ ✈í♥❝✉❧♦ Γab = 0 ♠♦t✐✈❛ ❛ ❞❡✜♥✐çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞♦ ✹✲♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✿
Mab =−4ke(Σa0b −Σb0a). ✭✷✳✹✽✮
❡ q✉❡✱ ♣♦rt❛♥t♦✱ ❞❡✜♥❡
Lab =−
Z
V
