Aspectos quânticos da gravidade de ChernSimons nãocomutativa

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❋❮❙■❈❆ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖

❋r❛♥❝✐s❝♦ ❆❞❡✈❛❧❞♦ ●♦♥ç❛❧✈❡s ❞❛ ❙✐❧✈❡✐r❛

❆s♣❡❝t♦s q✉â♥t✐❝♦s ❞❛ ❣r❛✈✐t❛çã♦ ❞❡

❈❤❡r♥✲❙✐♠♦♥s ♥ã♦✲❝♦♠✉t❛t✐✈❛

❋r❛♥❝✐s❝♦ ❆❞❡✈❛❧❞♦ ●♦♥ç❛❧✈❡s ❞❛ ❙✐❧✈❡✐r❛

❆s♣❡❝t♦s q✉â♥t✐❝♦s ❞❛ ❣r❛✈✐t❛çã♦ ❞❡

❈❤❡r♥✲❙✐♠♦♥s ♥ã♦✲❝♦♠✉t❛t✐✈❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡✲ r❛❧ ❞♦ ❈❡❛rá ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ ❘♦❜❡rt♦ ❱✐♥❤❛❡s ▼❛❧✉❢ ❈❛✈❛❧❝❛♥t❡

❈♦✲♦r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❆❧❜❡rt♦ ❙❛♥t♦s ❞❡ ❆❧♠❡✐❞❛

▼❡str❛❞♦ ❡♠ ❋ís✐❝❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá

❉❡❞✐❝❛tór✐❛

❆❣r❛❞❡❝✐♠❡♥t♦s

◗✉❡r♦ ❛❣r❛❞❡❝❡r à t♦❞♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❡ss❛ ❞✐ss❡rt❛çã♦✳ ❊♠ ❡s♣❡❝✐❛❧✿ ✕ ❆ ❉❡✉s ❡ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣r✐♠❡✐r❛♠❡♥t❡❀

✕ ❆♦ Pr♦❢✳ ❘♦❜❡rt♦ ❱✐♥❤❛r❡s ▼❛❧✉❢ ❈❛✈❛❧❝❛♥t❡✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❝♦♥✜❛♥ç❛ ❡♠ ♠✐♠ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s❀

✕ ❆♦ Pr♦❢✳ ❈❛r❧♦s ❆❧❜❡rt♦ ❙❛♥t♦s ❞❡ ❆❧♠❡✐❞❛✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❞❡♣♦s✐t❛❞❛❀

✕ ❆♦s t♦❞♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ▲❆❙❙❈❖✱ ♣♦✐s t✐✈❡r❛♠ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛❀

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ✈❛♠♦s ✐♥✈❡st✐❣❛r q✉❛✐s ❛s ♠♦❞✐✜❝❛çõ❡s q✉❡ ♦ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧ ❝♦♠ t❡r♠♦ ❞❡ ❈❤❡r♥✲❙✐♠♦♥s s♦❢r❡ ❝♦♠ ❛ ❛❞✐çã♦ ❞❛ t❡♦r✐❛ ♥ã♦ ❝♦♠✉t❛t✐✈❛ ♥♦ ❡s♣❛ç♦✲ t❡♠♣♦✳ ❋❛r❡♠♦s ✐st♦ ❡♠ ❞♦✐s ❝❛s♦s✿ ♦ ♣r✐♠❡✐r♦ ✉t✐❧✐③❛♥❞♦ s♦♠❡♥t❡ ❛ t❡♦r✐❛ ❞❡ ❊✐♥st❡✐♥✲ ❍✐❧❜❡rt ❡ ♥♦ s❡❣✉♥❞♦ ❝❛s♦ ❛❝r❡s❝❡♥t❛♥❞♦ ♦ t❡r♠♦ ❞❡ ❣r❛✈✐❞❛❞❡ t♦♣♦❧ó❣✐❝❛ t✐♣♦ ❈❤❡r♥✲ ❙✐♠♦♥s✳ ❆s ♠♦❞✐✜❝❛çõ❡s q✉❡ ❡st❛♠♦s ✐♥✈❡st✐❣❛♥❞♦ ♦❝♦rr❡♠ ❡♠ ✉♠ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ❞♦✐s ❜ós♦♥s ✈❡t♦r✐❛✐s tr♦❝❛♥❞♦ ✉♠ ❣rá✈✐t♦♥✳ ❆té ♣♦❞❡r♠♦s ❝❤❡❣❛r ❛ ✉♠❛ ❝♦♥❝❧✉sã♦ ❞❡ ❝♦♠♦ ❛ ♥ã♦ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❛❧t❡r❛ ♦ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧✱ ✐r❡♠♦s ✐♥✐❝✐❛r ♥♦ss♦ ❡st✉❞♦ ❝♦♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ❣r❛✈✐❞❛❞❡ ❡♠ ❜❛✐①❛s ❞✐♠❡♥sõ❡s✳ ❆♣ós ❛♣r❡❡♥❞❡r ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥ ♣❛r❛ t❡♦r✐❛ q✉❛❞rát✐❝❛s ❞❛ ❣r❛✈✐❞❛❞❡✱ ❡①♣❛♥❞✐♠♦s ♦s ❝♦♥❝❡✐t♦s ♣❛r❛ ✉♠❛ ❣r❛✈✐❞❛❞❡ t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ♠❛ss✐✈❛✳ ❘❡✈✐s❛r❡♠♦s tó♣✐❝♦s ✐♠♣♦rt❛♥t❡s ❞❛ t❡♦r✐❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ P♦r ✜♠ ❛♥❛❧✐s❛♥❞♦ ❛ ✐♥t❡r❛çã♦ ❝♦♠ ❝❛♠♣♦ ❞♦ ❣rá✈✐t♦♥ ❝♦♠ ♠❛tér✐❛ ❡s❝r❡✈❡r❡♠♦s ♦ ✈ért✐❝❡ ❞❛ t❡♦r✐❛ ❡ ❡♥❝♦♥tr❛r❡♠♦s ❛s ♠♦❞✐✜❝❛çõ❡s ♦r✐✉♥❞❛s ❞❛ ♥ã♦ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❞♦✐s ❝❛s♦s ❝✐t❛❞♦s ❛❝✐♠❛✳ ❱❡r✐✜❝❛♠♦s q✉❡ ❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❛❧t❡r❛ ❛ ❢♦r♠❛ ❞♦ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧ t❛♥t♦ ♥❛ ♦r✐❣❡♠✱ ❞❡✐①❛♥❞♦✲♦ ❜❡♠ ❝♦♠♣♦rt❛❞♦✱ q✉❛♥t♦ ♥♦ ✐♥✜♥✐t♦✳

❆❜str❛❝t

■♥ t❤✐s ♣❛♣❡r ✇❡ ✐♥✈❡st✐❣❛t❡ ✇❤❛t ❝❤❛♥❣❡s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ♣♦t❡♥t✐❛❧ ✇✐t❤ ❈❤❡r♥✲ ❙✐♠♦♥s t❡r♠ s✉✛❡rs ❢r♦♠ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ t❤❡♦r② ✐♥ s♣❛❝❡✲t✐♠❡✳ ❲❡ ❞♦ t❤✐s ✐♥ t✇♦ ❝❛s❡s✿ t❤❡ ✜rst ✉s✐♥❣ ♦♥❧② t❤❡ t❤❡♦r② ♦❢ ❊✐♥st❡✐♥✲❍✐❧❜❡rt ❛♥❞ ✐♥ t❤❡ s❡❝♦♥❞ ❝❛s❡✱ ❛❞❞✐♥❣ t❤❡ t❡r♠ t♦♣♦❧♦❣✐❝❛❧ ❈❤❡r♥✲❙✐♠♦♥s ❣r❛✈✐t② t②♣❡✳ ❚❤❡ ❝❤❛♥❣❡s t❤❛t ♦❝❝✉r ❛r❡ ✐♥✈❡st✐❣❛t✐♥❣ ♦♥ ❛ s❝❛tt❡r✐♥❣ ♦❢ t✇♦ ✈❡❝t♦r ❜♦s♦♥s ❡①❝❤❛♥❣✐♥❣ ❛ ❣r❛✈✐t♦♥✳ ❯♥t✐❧ ✇❡ r❡❛❝❤ ❛ ❝♦♥❝❧✉s✐♦♥ ❛s t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈✐t② ❝❤❛♥❣❡s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ♣♦t❡♥t✐❛❧✱ ✇❡ ✇✐❧❧ ❜❡❣✐♥ ♦✉r st✉❞② ✇✐t❤ ❛ ❣r❛✈✐t② ♠♦❞❡❧ ✐♥ ❧♦✇ ❞✐♠❡♥s✐♦♥s✳ ❆❢t❡r ❧❡❛r♥✐♥❣ ❤♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❣r❛✈✐t♦♥ ♣r♦♣❛❣❛t♦r ❢♦r q✉❛❞r❛t✐❝ t❤❡♦r② ♦❢ ❣r❛✈✐t②✱ ✇❡ ❡①♣❛♥❞❡❞ t❤❡ ❝♦♥❝❡♣ts ❢♦r ❛ t♦♣♦❧♦❣✐❝❛❧❧② ♠❛ss✐✈❡ ❣r❛✈✐t②✳ ❲❡ ✇✐❧❧ r❡✈✐❡✇ ✐♠♣♦rt❛♥t t♦♣✐❝s ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ t❤❡♦r② ✐♥ s♣❛❝❡✲t✐♠❡✳ ❋✐♥❛❧❧② ❛♥❛❧②③✐♥❣ t❤❡ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ t❤❡ ❣r❛✈✐t♦♥ ✜❡❧❞ ✇✐t❤ ♠❛tt❡r✱ ✇r✐t❡ t❤❡ ✈❡rt❡① ♦❢ t❤❡ t❤❡♦r② ❛♥❞ ✜♥❞ t❤❡ ❝❤❛♥❣❡s ❛r✐s✐♥❣ ❢r♦♠ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈✐t② ♦❢ t❤❡ t✇♦ ❝❛s❡s ❝✐t❡❞ ❛❜♦✈❡✳ ❲❡ ❢♦✉♥❞ t❤❛t t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈✐t② ❛❧t❡rs t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ♣♦t❡♥t✐❛❧ ❜♦t❤ ✐♥ ♦r✐❣✐♥✱ ❧❡❛✈✐♥❣ ❤✐♠ ✇❡❧❧ ❜❡❤❛✈❡❞✱ ❛s ❛t ✐♥✜♥✐t②✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ♣✳ ✽

✶ ❚❡♦r✐❛ ❞❛ ❣r❛✈✐❞❛❞❡ q✉❛❞rát✐❝❛ ❡♠ ❉✲❞✐♠❡♥sõ❡s✳ ♣✳ ✶✶

✶✳✶ ❉❡t❡r♠✐♥❛çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ♣❛r❛ ❛ ❣r❛✈✐t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✶ ✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠ hµν ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✷

✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❞❡ ❛❧t❛s ❞❡r✐✈❛✲

❞❛s✳ ♣✳ ✶✾

✷✳✶ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦♠

❛❧t❛s ❞❡r✐✈❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✾ ✷✳✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s q✉❛✲

❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✸ ✷✳✷✳✶ ❙✐♠❡tr✐③❛çã♦ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✸

✸ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ♣❛r❛ t❡♦r✐❛ ❞❡ ❣r❛✈✐❞❛❞❡ t♦♣♦❧♦❣✐❝❛♠❡♥t❡

♠❛ss✐✈❛ ❞❡ ❈❤❡r♥✲❙✐♠♦♥s✳ ♣✳ ✷✾

✸✳✶ ❚❡♦r✐❛ ❞❡ ❈❤❡r♥✲❙✐♠♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✾ ✸✳✷ ▲✐♥❡❛r✐③❛çã♦ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ ❝♦♠ t❡r♠♦ ❞❡ ❈❤❡r♥✲❙✐♠♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶

✹ ❚❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❝❛♠♣♦s ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♣✳ ✸✺

✹✳✺✳✶ ❈á❧❝✉❧♦ ❞♦ ✈ért✐❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✹ ✹✳✺✳✷ ❱ért✐❝❡ ♥❛ t❡♦r✐❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✼ ✹✳✺✳✸ P♦t❡♥❝✐❛❧ ♥ã♦✲r❡❧❛t✐✈íst✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✾

✺ P♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧ ❞❛ t❡♦r✐❛ ❞❡ ❈❤❡r♥✲❙✐♠♦♥s ♥ã♦✲❝♦♠✉t❛t✐✈❛✳ ♣✳ ✺✺ ✺✳✶ ▲❛❣r❛♥❣✐❛♥❛ ❞❡ ❊✐♥st❡✐♥✲❍✐❧❜❡rt✲❈❤❡r♥✲❙✐♠♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✺ ✺✳✷ ▼❛tr✐③ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ M✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✻ ✺✳✸ ❉❡❢♦r♠❛çã♦ ❞♦ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✾

❈♦♥❝❧✉sã♦ ♣✳ ✻✷

✽

■♥tr♦❞✉çã♦

❆s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥ ♣❛r❛ ❛ ❣r❛✈✐t❛çã♦ ❢♦r❛♠ ❢♦r♠✉❧❛❞❛s ♣❛r❛ s❡r❡♠ ✉t✐❧✐③❛❞❛s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧✱ ❡❧❛s ♣♦❞❡♠ t❛♠❜é♠ s❡r ✉t✐❧✐③❛❞❛s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♥❛t✉r❡③❛ ❞❛ ❣r❛✈✐t❛çã♦ ❞✐❢❡r❡ ❞❛q✉❡❧❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧❬✶❪✳ ❆ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ é ❞✐♥❛♠✐❝❛♠❡♥t❡ tr✐✈✐❛❧ ❡♠ três ❞✐♠❡♥sõ❡s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✿ ❢♦r❛ ❞❛s ❢♦♥t❡s ♦ ❡s♣❛ç♦✲t❡♠♣♦ é ♣❧❛♥♦ ❬✷❪❀ t♦❞♦s ♦s ❡❢❡✐t♦s ❞❛s ❢♦♥t❡s ❧♦❝❛✲ ❧✐③❛❞❛s ♠❛♥✐❢❡st❛♠✲s❡ ♥❛ ❣❡♦♠❡tr✐❛ ❣❧♦❜❛❧❬✸❪✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♥ã♦ ❡①✐st❡♠ ❣rá✈✐t♦♥s✱ ❡ ❛s ❢♦rç❛s ♥ã♦ sã♦ ♠❡❞✐❛❞❛s ♣❡❧❛s tr♦❝❛s ❣r❛✈✐tô♥✐❝❛s❀ ♥❛ ✈❡r❞❛❞❡ ❡❧❛s sã♦ ❞❡ ♥❛t✉r❡③❛ ❣❡♦♠étr✐❝❛✴t♦♣♦❧ó❣✐❝❛✱ ❡ t❡♠ s✉❛ ♦r✐❣❡♠ ❡♠ ♣r♦♣r✐❡❞❛❞❡s ❣❧♦❜❛✐s ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✱ q✉❡ ♥ã♦ é ▼✐♥❦♦✇s❦✐❛♥♦ ❡♠ s✉❛ t♦t❛❧✐❞❛❞❡✱ ♠❡s♠♦ q✉❛♥❞♦ ❡❧❡ é ❧♦❝❛❧♠❡♥t❡ ♣❧❛♥♦❬✹✱ ✺❪✳

P♦❞❡✲s❡ r❡ss❛❧t❛r ♦✉tr❛ ❞ú✈✐❞❛ ❞❛ ❣r❛✈✐t❛çã♦ ❡♠ (2 + 1)D✱ ❛ ❞❡ ♥ã♦ ♣♦ss✉✐r ✉♠ ❧✐♠✐t❡ ◆❡✇t♦♥✐❛♥♦✿ ✐♠♣❧✐❝❛♥❞♦ ❤❛✈❡r ❡♠ ✉♠❛ q✉❡❜r❛ ♥❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ s❡ ❡s♣❡r❛✈❛ t❡r ❡♥tr❡ ❡❧❛ ❡ ❛ t❡♦r✐❛ ❞❡ ◆❡✇t♦♥ ❬✶❪✳ ❱❛r✐❛s ✐♥✈❡st✐❣❛çõ❡s ❡①❛✉st✐✈❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❣r❛✈✐t❛❝✐♦♥❛✐s ❡♠ ❜❛✐①❛ ❞✐♠❡♥❝✐♦♥❛❧✐❞❛❞❞❡ t❡♠ s✐❞♦ r❡❛❧✐③❛❞❛s ♥♦s ú❧t✐♠♦s ❛♥♦s❬✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻❪✱ ❞❡♠♦♥str❛♥❞♦ ✉♠ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❞♦s ❢ís✐❝♦s ❡♠ ♠❡❧❤♦r❛r à ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧ q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧ ✈✐❛ ♦ ❡st✉❞♦ ❞❡ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❡♠ ❞✐♠❡♥sõ❡s ♠❛✐s ❜❛✐①❛s✱ ❛s q✉❛✐s✱ ❡s♣❡r❛✲s❡ s❡r❡♠ r❡♥♦r♠❛❧✐③á✈❡✐s✳

❆ ✜♠ ❞❡ t❡r ✉♠❛ t❡♦r✐❛ ❞❡ ❣r❛✈✐t❛çã♦ ❡♠ ✭✷✰✶✮ ❞✐♠❡♥sõ❡s q✉❡ ♥ã♦ s❡❥❛ ❞✐♥❛♠✐❝❛♠❡♥t❡ tr✐✈✐❛❧✱ ❝♦♠ ♣♦t❡♥❝✐❛❧ r❡❧❛t✐✈íst✐❝♦ ❜❡♠ ❝♦♠♣♦rt❛❞♦ ❡ q✉❡ ♣♦ss❛ s❡r r❡♥♦♠❛❧✐③á✈❡❧✱ ♣♦❞❡✲ ♠♦s ✐♠❛❣✐♥❛r ❛ ✐♥❝❧✉sã♦ ❞❡ t❡r♠♦s ❝♦♠ ❞❡r✐✈❛❞❛s q✉❛❞rát✐❝❛sR R2

µν√gd3x❡ R

R2√_gd3_x

♥❛ ❛çã♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❊✐♥st❡✐♥❬✼❪✳

❖✉tr♦ ❣r❛♥❞❡ ♣r♦❜❧❡♠❛ q✉❡ ♣r❡♦❝✉♣❛ ♦s ❢ís✐❝♦s t❡ór✐❝♦s é ♦ ❢❛t♦ ❞❛ t❡♦r✐❛ ❞❛ r❡❧❛t✐✲ ✈✐❞❛❞❡ ❣❡r❛❧ ❞❡ ❊✐♥st❡✐♥ s❡r ♥ã♦✲ r❡♥♦r♠❛❧✐③á✈❡❧ ❡♠ q✉❛tr♦ ❞✐♠❡♥sõ❡s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❛♣r❡s❡♥t❛♥❞♦ ❞✐✈❡r❣ê♥❝✐❛s ✉❧tr❛✈✐♦❧❡t❛s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ❡❧✐♠✐♥❛❞❛s ♣❡❧♦s ♠❡❝❛♥✐s♠♦s ✉s✉❛✐s✳ ❚❛❧ ❢❛t♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❝❛♠♣♦s ♥♦ ♠❡s♠♦ ♣♦♥t♦✱ q✉❡ ❛ ♣r✐♥❝✐♣✐♦ ♥ã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❊♠ ✉♠❛ t❡♥t❛t✐✈❛ ❞❡ ❞r✐❜❧❛r ❡ss❡ ✐♠♣❛ss❡✱ ❍❡✐s❡♥❜❡r❣ ❬✾❪ ♣r♦♣ôs ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❝é❧✉❧❛ ♠í♥✐♠❛✱ q✉❡ ✐♠♣❧✐❝❛r✐❛ ♥✉♠ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❝❡rt❡③❛ ♣❛r❛ ♠❡❞✐❞❛s ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❡ ❡❧✐♠✐♥❛✈❛ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦♥t♦

■♥tr♦❞✉çã♦ ✾

❝♦♠✉t❛t✐✈♦✱ ♥♦ q✉❛❧ ❛s ❝♦♦r❞❡♥❛❞❛s ♣❛ss❛♠ ❛ ♦❜❡❞❡❝❡r à r❡❧❛çã♦ ❞❡ ❝♦♠✉t❛çã♦✿

[xµ, xν] = Θµν,

❡ ❢♦r❛♠ ✐♥✐❝✐❛❧♠❡♥t❡ ✉t✐❧✐③❛❞❛s ♣♦r ❙♥②❞❡r ❬✶✵❪ ❝♦♠♦ ❢♦r♠❛ ❞❡ s✉❛✈✐③❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ✉❧tr❛✈✐♦❧❡t❛ ❡♠ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❝❛♠♣♦s✳ ❚❛✐s ✐❞❡✐❛s ❢♦r❛♠ ❡sq✉❡❝✐❞❛s ♣♦r ✉♠ ❧♦♥❣♦ ♣❡rí♦❞♦ ❡♠ ✈✐rt✉❞❡ ❞♦ ❡♥♦r♠❡ s✉❝❡ss♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ r❡♥♦r♠❛❧✐③❛çã♦✳ ❖ ✐♥t❡r❡ss❡ ♣❡❧❛ ♥ã♦ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❢♦✐ r❡t♦♠❛❞♦ ♣♦r ✈♦❧t❛ ❞♦s ❛♥♦s ✾✵✱ ❞❡✈✐❞♦ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ q✉❡ ❛ t❡✲ ♦r✐❛ ❞❡ ❨❛♥❣✲▼✐❧❧s ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♣♦❞❡r✐❛ s❡r ♦❜t✐❞❛ ❝♦♠♦ ✉♠ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❞❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ♥❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡ ❢✉♥❞♦ ❬✽❪✳ ❖✉tr❛ r❛③ã♦ ♣❛r❛ ❝♦♥s✐❞❡r❛r♠♦s ❛ ✐❞❡✐❛ ❞❛ q✉❛♥t✐③❛çã♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ♣♦❞❡ s❡r ❝♦♥❝❡❜✐❞❛ ♣♦r ❛r❣✉♠❡♥✲ t♦s ❡♥✈♦❧✈❡♥❞♦ ❛ t❡♦r✐❛ ❞❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❞✐stâ♥❝✐❛s ♣ró①✐♠❛s ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ P❧❛♥❝❦ lp =

p

G~_/c3 ∼_{= 10}−33_{cm✱ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ s❡ t♦r♥❛}

tã♦ ✐♥t❡♥s♦ q✉❡ ♥❡♠ ❛ ❧✉③ ♦✉ ♦✉tr♦ s✐♥❛❧ sã♦ ❝❛♣❛③❡s ❞❡ tr❛♥s♠✐t✐r ✐♥❢♦r♠❛çã♦ ❞❡ ♠♦❞♦ q✉❡ ♠❡❞✐❞❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❡r❞❡♠ ♦ s✐❣♥✐✜❝❛❞♦ ❬✶✶❪✳ ❈♦♥t✉❞♦✱ ❛✐♥❞❛ q✉❡ t❡♥❤❛ ♦❝♦r✲ r✐❞♦ ❜♦♥s r❡s✉❧t❛❞♦s r❡❧❛❝✐♦♥❛❞♦s à ♥ã♦✲❝♦♠✐✉t❛t✐✈✐❞❛❞❡ ❡ à t❡♦r✐❛ ❞❡ ❝♦r❞❛s✱ s✉r❣✐r❛♠ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s✳ ❖ ♣r✐♠❡✐r♦ r❡❧❛❝✐♦♥❛❞♦ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ✉❧tr❛✈✐♦❧❡t❛ ❞♦s ❞✐❛❣r❛♠❛s ❞❡ ❋❡②♥♠❛♥ ♥❛ ♣r❡s❡♥ç❛ ❞❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❝❤❛♠❛r❛♠ ❛t❡♥çã♦✳ ❖ ❡①❡♠♣❧♦ ♠❛✐s ♥♦tá✈❡❧ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠✐st✉r❛ ❯❱✴■❘✱ q✉❡ ❝♦♥s✐st❡ ♥❛ ❝♦♥✈❡rsã♦ ❞❡ ♣❛rt❡ ❞❛s ❞✐✈❡r✲ ❣ê♥❝✐❛s ✉❧tr❛✈✐♦❧❡t❛s ✭❯❱✮ ❞❛ t❡♦r✐❛ ❝♦♠✉t❛t✐✈❛ ❡♠ s✐♥❣✉❧❛r✐❞❛❞❡s ✐♥❢r❛✈❡r♠❡❧❤❛s ✭■❘✮ q✉❡ ♣♦❞❡♠ ✐♠♣♦ss✐❜✐❧✐t❛r ♦ tr❛t❛♠❡♥t♦ ♣❡rt✉r❜❛t✐✈♦ ✉s✉❛❧ ❬✶✷✱ ✶✸✱ ✶✹❪✳ ❖✉tr♦ ♣♦♥t♦ ❞❡ ❞❡st❛q✉❡ é ❛ ✈✐♦❧❛çã♦ ❞❛ ✉♥✐t❛r✐❡❞❛❞❡ ❡♠ ♠♦❞❡❧♦s ❡♥✈♦❧✈❡♥❞♦ ❛ ♥ã♦ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♥tr❡ ❡s♣❛ç♦ ❡ t❡♠♣♦ ❬✶✺❪✳

❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦✱ ❛ ✜♠ ❞❡ ❜✉s❝❛r♠♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛✐s ♣r♦❢✉♥❞♦s s♦❜r❡ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❡♠ ❜❛✐①❛s ❞✐♠❡♥sõ❡s ❡ ❛♥❛❧✐s❛r ❛s ❝♦♥tr✐❜✉✐çõ❡s ♦r✐✉♥❞❛s ❞❡ ✉♠❛ t❡♦r✐❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❢♦✐ ❛♥❛❧✐s❛❞❛ q✉❛❧ ❛ ❢♦r♠❛ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ q✉❡ ❞❡✈❡r❡♠♦s ✉t✐❧✐③❛r ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❞❛ t❡♦r✐❛✳ ❖ s❡❣✉♥❞♦ ❝❛♣✐t✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ♣❛r❛ ♦ ❝ô♠♣✉t♦ ❞♦ ♣r♦♣❛❣❛❞♦r✱ ❣❡♥❡r❛❧✐③❛♥❞♦ ♦ ❝á❧❝✉❧♦ ♣❛r❛ ❉✲❞✐♠❡♥sõ❡s

■♥tr♦❞✉çã♦ ✶✵

✶✶

✶ ❚❡♦r✐❛ ❞❛ ❣r❛✈✐❞❛❞❡ q✉❛❞rát✐❝❛

❡♠ ❉✲❞✐♠❡♥sõ❡s✳

◆❡st❡ ❝❛♣ít✉❧♦ ❞❡s❡♥✈♦❧✈❡r❡♠♦s t♦❞♦s ♦s ♣❛ss♦s ♣❛r❛ ♦❜t❡r ❛ ❢♦r♠❛ ❛♣r♦♣r✐❛❞❛ ♣❛r❛ ❛çã♦ ❞❡ ❣r❛✈✐t❛çã♦ q✉❛❞rát✐❝❛ ❡♠D ❞✐♠❡♥sõ❡s✱ q✉❡ ♣♦st❡r✐♦r♠❡♥t❡ s❡rá ✉t✐❧✐③❛❞❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r r❡❢❡r❡♥t❡ à t❡♦r✐❛✳ ❊♠ ♣♦ss❡ ❞❡ ✉♠❛ ❛çã♦ ❣❡♥ér✐❝❛ ♣❛r❛ ❛ ❞✐♥â♠✐❝❛ ❞♦s ❝❛♠♣♦s ❣r❛✈✐t❛❝✐♦♥❛✐s s❡♠ t♦rçã♦ ❢❛r❡♠♦s ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❢r❛❝♦✱ ♠❛♥t❡♥❞♦ ❛♣❡♥❛s ♦s t❡r♠♦s ❞❛ ♣❡rt✉r❜❛çã♦ ❞❡ ♦r❞❡♠ q✉❛❞rát✐❝❛✳

✶✳✶ ❉❡t❡r♠✐♥❛çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ♣❛r❛ ❛ ❣r❛✈✐t❛çã♦

❆ ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ q✉❛❞rát✐❝❛ ❡♠ ❉✲❞✐♠❡♥sõ❡s ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧✱ é ❞❛❞❛ ♣♦r❀

I =

Z

dD_xp₍

−1)D−1_g

2R κ2 +

α

2R

2₊ β

2R

2

µν+

γ

2R

2

µνρσ+

δ

2R

. ✭✶✳✶✮

♦♥❞❡ α, β, γ ❡ δ sã♦ ♣❛râ♠❡tr♦s ❝♦♠ ❞✐♠❡♥sõ❡s ❞❡ LD−4_{✱ ❥á ♦ t❡r♠♦ κ}2 _{é ✉♠❛ ❝♦♥st❛♥t❡}

q✉❡ t❡♠ ❞✐♠❡♥sã♦ ❞❡LD−2_{✳ ❊♠ q✉❛tr♦ ❞✐♠❡♥sõ❡s t❡♠♦sκ= 32πG✱ s❡♥❞♦ G❛ ❝♦♥st❛♥t❡}

❞❡ ◆❡✇t♦♥✳ ❖ t❡r♠♦ ❡s❝r✐t♦ ❝♦♠♦ R2_{✱ r❡♣r❡s❡♥t❛ ∂}

ν∂νR2✱ ❡ ♣♦❞❡ s❡r ❞❡s❝❛rt❛❞♦ ❞❛

▲❛❣r❛♥❣✐❛♥❛ ✭✶✳✶✮✳ ❉❡ ❢❛t♦✱ ✉t✐❧✐③❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈❡r❣ê♥❝✐❛✿

Z

V

(_∇.v)dτ =

I

S

v.da, ✭✶✳✷✮

❛❞♦t❛♥❞♦ v=∂_νR2✱ ❡ ❧❡♠❜r❛♥❞♦ q✉❡ à ❛çã♦ é ✐♥t❡❣r❛❞❛ ❡♠ t♦❞♦ ♦ ❡s♣❛ç♦✳ ❊ ♥♦ ✐♥✜♥✐t♦

♦s ❝❛♠♣♦s ❞❡✈❡♠ s❡r ♥✉❧♦s✳ ❋✐❝❛♠♦s ❡♥tã♦ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❛çã♦✿

LEH = p

(₋1)D−1_g

2R κ2 +

α

2R

2 ₊β

2R

2

µν+

γ

2R

2

µνρσ

. ✭✶✳✸✮

❆♥❛❧✐s❛♥❞♦ ♦ ❡s♣❛ç♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ✭❉❂✶✮✱ ♦s t❡♥s♦r❡s R✱ Rµν ❡ R2µνρσ sã♦ ✐❞❡♥t✐✲

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✷

q✉❛❞rát✐❝❛ só t❡rá s❡♥t✐❞♦ ♣❛r❛ ❞✐♠❡♥sõ❡sD>2✳

❆♥❛❧✐s❛♥❞♦ ♦ ❝❛s♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ D = 2✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ ❡ ♦

t❡♥s♦r ❞❡ ❘✐❝❝✐ ❡♠ ❢✉♥çã♦ ❞♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿

Rµνρσ =

1

2R[gµσgνρ−gµρgνσ], ✭✶✳✹✮

Rµν =

1

2Rgµν. ✭✶✳✺✮

❙✉❜st✐t✉✐♥❞♦ ❛s ❞✉❛s ❡①♣r❡ssõ❡s ❛❝✐♠❛ ♥❛ ❡q✉❛çã♦ ✭✶✳✸✮✿

LEH = p

(₋1)D−1_g

2R κ2 +

α

2R

2₊β

2R 2 µν + γ 2R 2 µνρσ

= √₋g

2R κ2 +

α

2R

2₊β

2 R2 2 + γ 2R 2

= √₋g

2R κ2 +

α 2 + β 4 + γ 2 R2 . ✭✶✳✻✮

P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❝♦♥❝❡r♥❡♥t❡ ❞❛ ❣r❛✈✐t❛çã♦ q✉❛❞rát✐❝❛ ❡♠ ❉❂✷ ❛ ▲❛✲ ❣r❛♥❣✐❛♥❛ s❡ r❡❞✉③ ❛ ❢♦r♠❛✿

L =√₋g

2R κ + α 2R 2 ✭✶✳✼✮

◆ã♦ é ❞♦ ✐♥t✉✐t♦ ❞❡s❞❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛r ❛ t❡♦r✐❛ ❡♠ ✷❉✲❞✐♠❡♥sõ❡s✳ ❋✐❝❛r❡♠♦s tr❛✲ ❜❛❧❤❛♥❞♦ ❝♦♠ ❛ ▲❛❣r❛♥❣✐❛♥❛ ❞❛ ❡q✉❛çã♦ ✭✶✳✸✮✳

LEH = p

(₋1)D−1_g

2R κ2 +

α

2R

2 ₊β

2R 2 µν+ γ 2R 2 µνρσ .

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠

h

❖ ♣r✐♠❡✐r♦ ♣❛ss♦ q✉❡ ❞❡✈❡♠♦s ❞❛r ❛♥t❡s ❞♦ ❝á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥ é ❡s❝r❡✈❡r ❛ ▲❛❣r❛♥❣✐❛♥❛ ❡♠ ❢✉♥çã♦ ❞❡ ✉♠ ❝❛♠♣♦ h✱ ♣❛r❛ ✐ss♦ ✐r❡♠♦s ❞❡❝♦♠♣♦r ❛ ♠étr✐❝❛ ❡♠ t❡r♠♦s ❞❡✱

gµν =ηµν +khµν, ✭✶✳✽✮

❡ s✉❛ ✐♥✈❡rs❛✿

gµν _=ηµν

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✸

♦♥❞❡ ♦ t❡r♠♦ ηµν ❡ ❛ ♠étr✐❝❛ ❞❡ ▼✐♥❦♦✇s❦✐✱ κ é ❝♦♥st❛♥t❡ ❡ ♦ hµν é ❝❛♠♣♦ ❞♦ ❣rá✈✐t♦♥✳

❉❡✈❡r❡♠♦s ❝♦❧❡❝✐♦♥❛r ❛té s❡❣✉♥❞❛ ♦r❞❡♠ ♣❛r❛ ♦❜t❡r ♦ ♣r♦♣❛❣❛❞♦r ❞❛ t❡♦r✐❛ ♠❛✐s ❛❞✐❛♥t❡✳ ❆ ❡q✉❛çã♦ ✭✶✳✸✮✱ ❝♦♥té♠ ✉♠ ❢❛t♦r ♠✉❧t✐♣❧✐❝❛t✐✈♦ g✱ q✉❡ é ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠étr✐❝❛ ❞❡ ▼✐♥❦♦✇s❦✐✱ ❧♦❣♦ ❞❡✈❡rá s❡r ❡s❝r✐t♦ ❡♠ ❢✉♥çã♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✽✮✳ ❘❡❡s❝r❡✈❡♥❞♦ ♦ t❡r♠♦ q✉❡ ❝♦♥té♠g✿

p

(₋1)D−1_{g =}q₍₋₁₎D−1_detg

µν ✭✶✳✶✵✮

❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ✭✶✳✽✮✱ ✈❛♠♦s ✉t✐❧✐③❛r ♦ tr❛ç♦ ❞❛ ♠étr✐❝❛ ❝♦♠♦ ηµν = (+− · · ·−)✱

❞❡st❛ ❢♦r♠❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

p

(₋1)D−1_{g = (−1)}D−1_{det [η}

µν +khµν] 1

2

= (₋1)D−1_{det [η}

µα(δνα+kh α ν)]

1 2

= (₋1)D−1det(ηµα) det(δαν +kh α ν)

1 2

= (det(δαν +kh α ν))

1 2

∼

= p1 +khα

α ✭✶✳✶✶✮

❊①♣❛♥❞✐♥❞♦ ❛ ❢✉♥çã♦ ✶✳✶✶✱ ❡♠ t♦r♥♦ ❞♦ ❝❛♠♣♦ h✱ ❛té s❡❣✉♥❞❛ ♦r❞❡♠✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡❀

p

(₋1)D−1_{g = 1 +} 1

2kh

α

α ✭✶✳✶✷✮

◆♦t❡ q✉❡ ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ✐♥❞❡♣❡♥❞❡ ❞❛ ❞✐♠❡♥sã♦ q✉❡ ❡st❡❥❛ s❡♥❞♦ ❡st✉❞❛❞❛✱ ✐st♦ ♦❝♦rr❡ ♣♦r ❝♦♥t❛ ❞❛ ❢♦r♠❛ ❞♦ ηµν✱ q✉❡ t❡♠ s❡✉ ❞❡t❡r♠✐♥❛♥t❡ ♣♦s✐t✐✈♦ ❡♠ ❞✐♠❡♥sõ❡s

✐♠♣❛r❡s ❡ ♥❡❣❛t✐✈♦ ❡♠ ❞✐♠❡♥sõ❡s ♣❛r❡s✱ ♠❛✐s t❛❧ ♣r♦♣r✐❡❞❛❞❡ é ❜❛❧❛♥❝❡❛❞❛ ♣❡❧♦ t❡r♠♦

(₋1)D−1_{✱ q✉❡ q✉❛♥❞♦ ❛ ❞✐♠❡♥sã♦ é í♠♣❛r ♦ r❡s✉❧t❛❞♦ ♣♦s✐t✐✈♦✱ ❡ q✉❛♥❞♦ ❛ ❞✐♠❡♥sã♦ ❢♦r}

♣❛r ♦ r❡s✉❧t❛❞♦ é ♥❡❣❛t✐✈♦✳

❘❡st❛✲♥♦s ❛✐♥❞❛✱ ❛♥❛❧✐s❛r ♦s ♦✉tr♦s t❡r♠♦s ❢♦r♠❛❞♦r❡s ❞❛ ❡q✉❛çã♦ ✭✶✳✸✮✳ P❛r❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡❧❛ ♣❡rt✉r❜❛çã♦ ❞❛ ♠étr✐❝❛✱ ❞❡t❛❧❤❛r❡♠♦s ❡①♣❧✐❝✐t❛✲ ♠❡♥t❡ ❛❧❣✉♥s ♣❛ss♦s t♦♠❛❞♦s✳

❖s t❡r♠♦s R✱ Rµν ❡ Rµνρσ ❡s❝r✐t♦s ❡♠ ❢✉♥çã♦ ❞❛ ♠étr✐❝❛gµν✱ ❛♣r❡s❡♥t❛♠ ✉♠❛ ❡str✉✲

t✉r❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝♦♥❡①ã♦ ❛✜♠✱ ♦✉ sí♠❜♦❧♦ ❞❡ ❈❤r✐st♦✛❡❧✱ q✉❡ t❡♠ s✉❛ ❞❡✜♥✐çã♦✿

Γβ µν =

1 2g

αβ_[∂

µgνα+∂νgµα−∂αgµν]. ✭✶✳✶✸✮

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✹

❞♦ ❝❛♠♣♦h❀

Γβµν =

1 2 η

αβ

−khαβ[∂µ(ηνα+khνα) +∂ν(ηµα+khµα)−∂α(ηµν+khµν)]

= k 2 η

αβ

[∂µhνα+∂νhµα−∂αhµν]−khαβ[∂µhνα+∂νhµα−∂αhµν]

Γβ

µν =

k

2 η

αβ_∂

µhνα+ηαβ∂νhµα−ηαβ∂αhµν

−k

2

2 h

αβ_∂

µhνα+hαβ∂νhµα−hαβ∂αhµν

. ✭✶✳✶✹✮

P♦❞❡♠♦s ♥♦t❛r q✉❡ ❛ ❝♦♥❡①ã♦ ❛♣r❡s❡♥t❛ ❞♦✐s t❡r♠♦s✿ ✉♠ ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ❡♠ h✱ ❡ ♦ ♦✉tr♦ ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ q✉❛❞rát✐❝❛ ❡♠ h✳ ❊st❡ ❢❛t♦ s❡r✈✐rá ♣❛r❛ ♥♦s ❛✉①✐❧✐❛r ♠❛✐s ❛❞✐❛♥t❡ ♥♦ ❝á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r✱ ♣♦✐s ♣❛r❛ ♦ ❝ô♠♣✉t♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥ ✐r❡♠♦s ❝♦❧❡❝✐♦♥❛r t♦❞♦s ♦s t❡r♠♦s ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ t✐♣♦ h2_{✳ ❖ tr❛t❛♠❡♥t♦ ❢❡✐t♦ ❞❛ ❡q✉❛çã♦}

✭✶✳✶✸✮ ♣❛r❛ ✭✶✳✶✹✮ s❡rá ❢❡✐t♦ ❞✐✈❡rs❛s ✈❡③❡s✱ ❞❡✈✐❞♦ ❛ ❡str✉t✉r❛ ❞♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛✱ t❡♥s♦r ❞❡ ❘✐❝❝✐ ❡ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥✳

❉❛ ❞❡✜♥✐çã♦ ♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥✱ t❡♠✲s❡✿

Rµσνρ=∂νΓρµσ−∂σΓρµν

| {z }

R(1)µσνρ

+ Γλ σνΓ

ρ µλ−Γ

λ σλΓ

ρ µν

| {z }

R(2)µσνρ

✭✶✳✶✺✮

❋♦✐ s❡♣❛r❛❞♦ ♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ ❡♠ ❞♦✐s t❡r♠♦s✿ ✉♠ ❞❡♣❡♥❞❡♥t❡ ❡♠ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❞❡h ❡ ❛❧❣✉♠❛s ♣❛r❝❡❧❛s ❞❡♣❡♥❞❡♥t❡ ❞❡ h2_{✱ ❡ ♦ ♦✉tr♦ ❞❡♣❡♥❞❡ ❞❡ h}2_+O(h)3_{+...✳ P♦❞❡✲}

♠♦s ❛✜r♠❛r q✉❡ ❛ ♣❛rt❡ ❞❡♣❡♥❞❡♥t❡ ❞❡h2 _{♥♦ ♣r✐♠❡✐r♦ t❡r♠♦ ✐rá ❞❡s❛♣❛r❡❝❡r ❞❡✈✐❞♦ ❛♦s}

t❡r♠♦s ❞❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞♦ t❡♥s♦r✳ ❖❧❤❛♥❞♦ ♣❛r❛ ❛ ▲❛❣r❛♥❣✐❛♥❛ ✭✶✳✸✮✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♣r❡❝✐s❛♠♦s ❞♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥ ❡❧❡✈❛❞♦ ❛ s❡❣✉♥❞❛ ♣♦tê♥❝✐❛ (Rµσνρ)2✳ ❊♥tã♦ ♣❛r❛ ❡st❡

t❡r♠♦ s❡rá ❧❡✈❛❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ s♦♠❡♥t❡ ♦ t❡r♠♦ ♥♦♠❡❛❞♦R(1)µσνρ✱ ♣♦✐s ♦s ♦✉tr♦s t❡r♠♦s

❝♦♥tr✐❜✉✐rá ❝♦♠ t❡r♠♦s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r❡s ❡♠ hµν✳P♦❞❡♠♦s ❡st❡♥❞❡r t❛❧ ❛r❣✉♠❡♥t❛çã♦

♣❛r❛ ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✿

Rµν =∂ρΓρµν−∂νΓρµρ

| {z }

R(1)µν

+ Γλ ρλΓ

ρ µν −Γ

λ ρνΓ

ρ µλ

| {z }

R(2)µν

. ✭✶✳✶✻✮

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✺

❝✉r✈❛t✉r❛ t❡♠ s✉❛ ❞❡✜♥✐çã♦ ❝♦♠♦✿

R =gµνRµν. ✭✶✳✶✼✮

P♦❞❡ s❡r r❡❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

R = gµνRµν

= (ηµν

−khµν_)(R(1)

µν +R(2)µν)

= ηµν_R(1)

µν +η µν_R(2)

µν −kh µν_R(1)

µν −h µν_R(1)

µν. ✭✶✳✶✽✮

❈♦♥s❡r✈❛r❡♠♦s t♦❞♦s ♦s t❡r♠♦s ❞❡ ✭✶✳✶✽✮✱ ❡♠❜♦r❛ ♥ã♦ ❛♣r❡s❡♥t❡ ❞❡♣❡♥❞ê♥❝✐❛ q✉❛❞rá✲ t✐❝❛ ❡♠ h✳ ➱ ✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r q✉❡ t♦❞❛s ❛s ❡str✉t✉r❛s ❛❝✐♠❛ ♠❡♥❝✐♦♥❛❞❛s✱ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥✱ ❘✐❝❝✐ ❡ ♦ ♣ró♣r✐♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛ ❡stã♦ ♠✉❧t✐♣❧✐❝❛❞♦s ♣♦r √−g✳

❉❡ ♣♦ss❡ ❞❡ t♦❞❛s ❛s ❞❡✜♥✐çõ❡s ♣♦❞❡♠♦s ♦❜t❡r ♦s t❡r♠♦s q✉❡ ❢♦r♠❛♠ ❛ ▲❛❣r❛♥❣✐❛♥❛ ✭✶✳✸✮✳ ❊s❝r❡✈❡♥❞♦ ♣r✐♠❡✐r❛♠❡♥t❡ ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ ♣♦✐s ❡❧❡ ❝♦♥té♠ t❡r♠♦s ✉s❛❞♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛✳ ❆ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ ✜❝❛ ❞❡✜♥✐❞❛ ❝♦♠♦✿

Rµν =Rµν(1) =

k

2(hµν +∂µ∂νh

α

α−∂α∂νhαµ−∂α∂µhαν). ✭✶✳✶✾✮

❖ s❡❣✉♥❞♦ t❡r♠♦ q✉❡ ❝♦♥tr✐❜✉✐rá ❝♦♠ ♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛ ❞❡✈❡ ✜❝❛r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

R(2)_µν = k

2

4 [∂αh∂µh

α

ν +∂αh∂νhαµ−∂αh∂αhµν

− ∂αhβν∂µhαβ −∂αhνβ∂βhαµ+∂αhβν∂αhµβ

− ∂νhβα∂µhαβ −∂νhβα∂βhαµ+∂νhβα∂ α

hµβ

+ ∂βhνα∂µhαβ +∂ β

hνα∂βhαµ−∂ β

hνα∂αhµβ]. ✭✶✳✷✵✮

❯t✐❧✐③❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛❞♦ ❡♠ ✭✶✳✶✽✮ ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r √−g✱ s❡rã♦ ❢❡✐t❛s ❛❧❣✉♠❛s s✐♠♣❧✐✜❝❛çõ❡s✱ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱ ❡ ♦s t❡r♠♦s ❞❡ s✉♣❡r❢í❝✐❡ s❡rã♦ ✐❣♥♦r❛❞♦s✱ ♣♦✐s ♦s ❝❛♠♣♦s ✈ã♦ ❛ ③❡r♦ ♥♦ ✐♥✜♥✐t♦✳

L1 = √−g

2R k2

= 2

k2

ηµνR(2)µν −kh µν

R(1)µν +

k

2h

µ νη

µν

R(1)µν

L1 =

−1₂hµνhµν +

1 2h

µ µh

α α−h

µ

µ∂α∂βhαβ+hµν∂µ∂αhαν

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✻

❆ ✜♠ ❞❡ ❞❡✐①❛r ❛ ♥♦t❛çã♦ ♠❛✐s ❝♦♠♣❛❝t❛ ♣♦❞❡♠♦s ❞❡✜♥✐r Aµ

≡ ∂νhµν ❡ φ ≡ hαα✱

❞❡st❡ ♠♦❞♦ ❛ ❡q✉❛çã♦ ✶✳✷ ✜❝❛ ♥♦ s❡❣✉✐♥t❡ ❢♦r♠❛t♦✿

L1 =−

1 2

hµνhµν +A2ν+ (Aν −∂νφ)2

. ✭✶✳✷✷✮

❊st❛ ❢♦r♠❛ ♠❛✐s ❝♦♠♣❛❝t❛❞❛ ✐rá s❡r✈✐r ♠❛✐s à ❢r❡♥t❡ ♣❛r❛ r❡❛❧✐③❛r ❝♦♠ ♠❛✐♦r ❢❛❝✐❧✐❞❛❞❡ ✉♠❛ s✐♠♣❧✐✜❝❛çã♦ ♥♦s t❡r♠♦s ❞❛ ❡q✉❛çã♦ ✭✶✳✸✮✳

❆♥❛❧✐s❛♥❞♦ ♦ s❡❣✉♥❞♦ t❡r♠♦ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ ✭✶✳✸✮✱ ♦ ❡s❝❛❧❛r ❞❡ ❝✉r✈❛t✉r❛ ❛♦ q✉❛❞r❛❞♦✱ ❝❤❛♠❛r❡♠♦s ❡st❡ t❡r♠♦ ❞❡L2✿

L2 =

α

2R

2

= α 2k

2_(hµ

µ−∂µ∂νhµν)(hαα−∂α∂βhαβ)

= α 2k

2_(hµ µ2h

α α−2h

µ

µ∂α∂βhαβ +hµν∂µ∂ν∂α∂βhαβ). ✭✶✳✷✸✮

❯t✐❧✐③❛♥❞♦ ❛s ♠❡s♠❛s ❞❡✜♥✐çõ❡s ✉s❛❞❛s ♣❛r❛ ♦❜t❡r ❛ ❡q✉❛çã♦ ✭✶✳✷✷✮✱ ♣♦❞❡♠♦s r❡❡s✲ ❝r❡✈❡r ❛ ❡q✉❛çã♦ ✭✶✳✷✸✮✿

L2 =

αk2

2 (∂

α_A

α−φ)2. ✭✶✳✷✹✮

❖ ♣ró①✐♠♦ t❡r♠♦ ❛ s❡r ❛♥❛❧✐s❛❞♦ s❡rá ❞♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ ❛♦ q✉❛❞r❛❞♦✱ ❝❤❛♠❛♥❞♦ ❡st❛ ♣❛r❝❡❧❛ ❞❡ L3✳ ❖♥❞❡✱ ♣♦r ♠♦t✐✈♦s ❥á ❡①♣❧✐❝❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♥s✐❞❡r❛♠♦s ❛♣❡♥❛s ❛

♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ t❡♥s♦r✳ L3 =

β

2RµνR

µν ₌ β

2R

2

µν

= β 2

k2

4(h

µν

+∂µ∂νhαα−∂µ∂αhαν −∂ν∂αhαν)

×(hµν +∂µ∂νhβ_β−∂µ∂βhνβ−∂ν∂βhµβ)

L3 =

β

2

k2

4(h

µν2_h

µν+hµµ

2_hα

α−2h µ

µ∂α∂βhαβ−2hµβ∂µ∂αhαβ

+2hµν_∂

µ∂ν∂α∂βhαβ). ✭✶✳✷✺✮

❖❧❤❛♥❞♦ ❝♦♠ ❛t❡♥çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ✶✳✷✺✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♥❡❧❛ ❡①✐st❡♠ t❡r♠♦s s❡✲ ♠❡❧❤❛♥t❡s ❝♦♥t✐❞♦s ♥❛ ❡q✉❛çã♦ ✶✳✷✸✱ ❝♦♠ ✉♠❛ ❞✐❢❡r❡♥ç❛ ♥❛ ❝♦♥st❛t❡ ♠✉❧t✐♣❧✐❝❛t✐✈❛✳ ❊♥tã♦ s❡ ❛❞✐❝✐♦♥❛r♠♦s ❛s ❞❡✜♥✐çõ❡s b = βh₂2✱ Fµν = Aµ,ν −Aν,µ✱ ♦♥❞❡ Aµ,ν = ∂νAµ✳ P♦❞❡♠♦s

❢❛③❡r ✉♠ ❝á❧❝✉❧♦ ❛♥á❧♦❣♦✱ ❡♥❝♦♥tr❛♥❞♦ ❢❛❝✐❧♠❡♥t❡✿

L3 =

b

4(h

µν

2hµν −(A,µµ)2−Fµν2 + (∂ α

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✼

P♦r ✜♠ ✈❛♠♦s ❛♥❛❧✐s❛r ♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥✱ ❡st❡ ❝á❧❝✉❧♦ é ♠✉✐t♦ ♣❛r❡❝✐❞♦ ❝♦♠ ♦s ❝á❧❝✉❧♦s ❞♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ ♣❛r❛ ❡st❛ ❝♦♥tr✐❜✉✐çã♦ ❢♦✐ ♥♦♠❡❛❞❛ ❞❡L4✿

L4 =

γ

2RµβναR

µβνα

= γ 2R

2

µβνα

= γ 2

k2

4 (∂µ∂νh

α

β −∂ν∂αhµβ−∂β∂µhαν +∂β∂αhµν)

×(∂µ_∂ν_hβ α−∂

ν_∂

αhµβ−∂β∂µhνα+∂ β_∂

αhµν)

= γk

2

2 (h

µν2_h

µν −2hµβ∂α∂µhαβ +hµν∂µ∂ν∂α∂βhαβ). ✭✶✳✷✼✮

❘❡❛❧✐③❛♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s ❛❞♦t❛❞♦s ♥❛s ❡q✉❛çõ❡s ❛❝✐♠❛✱ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ ❢♦r♠❛ ❝♦♠♣❛❝t❛❞❛✱ ♦♥❞❡d= γk

2

2 ✱ ❛ ❡q✉❛çã♦ ✶✳✷✼ ✜❝❛ ♥❛ ❢♦r♠❛✿

L4 =d(hµν2hµν−(Aµ,µ)2−Fµν2 ). ✭✶✳✷✽✮

❊♠ ♣♦ss❡ ❞♦s r❡s✉❧t❛❞♦s ✭✶✳✷✷✮✱✭✶✳✷✹✮✱✭✶✳✷✻✮ ❡ ✭✶✳✷✽✮✱ s♦♠♦s ❝❛♣❛③❡s ❞❡ ❡s❝r❡✈❡r ❛ ▲❛❣r❛♥❣✐❛♥❛ ✭✶✳✸✮ ❝♦♠♣❧❡t❛✱ ❡♠ ❢✉♥çã♦ ❞♦s t❡r♠♦s ❜✐❧✐♥❡❛r❡s ❞❡ hµν✳ ❈❤❛♠❛r❡♠♦s ❛

▲❛❣r❛♥❣✐❛♥❛ ❝♦♠♣❧❡t❛ ❞❡L1,2,3,4 =L1+L2+L3+L4✱ ❢❛③❡♥❞♦ ✉♠ ♣♦✉❝♦ ❞❡ ♠❛♥✐♣✉❧❛çã♦

❛❧❣é❜r✐❝❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ❧❛❣r❛♥❣✐❛♥♦ ✭✶✳✸✮ ♥♦ s❡❣✉✐♥t❡ ❢♦r♠❛t♦✿

L1,2,3,4 =

b

4 +d h

µν

2hµν−(Aµ,µ)2−Fµν2 +

(b/4)(1 + 4c) (b/4) +d (∂

α

Aα−φ)2

−1₂hµνhµν+A2ν + (Aν −∂νφ)2

. ✭✶✳✷✾✮

❆♥t❡r✐♦r♠❡♥t❡ ❢♦✐ ♠❡♥❝✐♦♥❛❞♦ q✉❡ ♥❛ ❡q✉❛çã♦ ✭✶✳✷✼✮✱ ❡♥❝♦♥tr❛♠♦s t❡r♠♦s s❡♠❡❧❤❛♥✲ t❡s ❛♦ ❞❛ ❡①♣r❡ssã♦ ✭✶✳✷✺✮✱ ♣♦r ✐ss♦ ♣♦❞❡♠♦s ❢❛③❡r ✉♠❛ ♣❡q✉❡♥❛ ❛♥á❧✐s❡✱ ❡♠ ✈❡③ ❞❡ ❛❞♦t❛r ♦L1,2,3,4✱ ✈❛♠♦s ❡s❝r❡✈❡r ♦ t❡r♠♦ L1,2,3 =L1+L2+L3❀

L1,2,3 =

b

4[h

µν2_h

µν−(Aµ,µ)2−Fµν2 + (1 + 4c)(∂αAα−φ)2]

−1₂hµνhµν+A2ν + (Aν−∂νφ)2

, ✭✶✳✸✵✮

♥❛s ❞✉❛s ❡q✉❛çõ❡s ❛❝✐♠❛ ✜③❡♠♦sc=α/β✳

❆s ❡q✉❛çõ❡s ✭✶✳✷✾✮ ❡ ✭✶✳✸✵✮ sã♦ ❞✐❢❡r❡♥t❡s ❛♣❡♥❛s ❞❡✈✐❞♦ às ❝♦♥st❛♥t❡s ♠✉❧t✐♣❧✐❝❛t✐✈❛s✱ ❝♦♠♦ ❛s ❝♦♥st❛♥t❡s sã♦ ❛r❜✐tr❛r✐❛s ♣♦❞❡♠♦s ❛❞♦t❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥st❛♥t❡s ♦♥❞❡ ❛s ❞✉❛s ❡q✉❛çõ❡s ✜q✉❡♠ ✐❞ê♥t✐❝❛s✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❞❡s♣r❡③❛r ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ t❡r♠♦ L4✱

q✉❡ ❡q✉✐✈❛❧❡ ❛ ❞❡s♣r❡③❛r ♦ t❡r♠♦ R2

µβνα ♥❛ ▲❛❣r❛♥❣✐❛♥❛ ✐♥✐❝✐❛❧ ❞❛ t❡♦r✐❛✳

✶✳✷ ❖❜t❡♥çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❜✐❧✐♥❡❛r ❡♠h_µν ✶✽

❉❃✷✳

❊♥❝♦♥tr❛❞♦ ♦ ▲❛❣r❛♥❣✐❛♥♦ ❜✐❧✐♥❡❛r ✭✶✳✸✵✮✱ ❞❡✈❡ s❡r r❡ss❛❧t❛❞♦ q✉❡ ❡❧❡ é ✐♥✈❛r✐❛♥t❡ s♦❜ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s xµ

−→ xµ _{+ κξ}µ_{✱ ♦♥❞❡ ξ}µ _{é ✉♠}

❝❛♠♣♦ ✈❡t♦r✐❛❧ ❛r❜✐trár✐♦ ✐♥✜♥✐t❡s✐♠❛❧✱ ❡st❛ tr❛♥s❢♦r♠❛çã♦ ❞❡✈❡ s❡r ✐♥✜♥✐t❡s✐♠❛❧ ♣❛r❛ ❡✈✐t❛r ✐♥❝♦♥s✐stê♥❝✐❛ ❝♦♠ ❛ ❡①♣r❡ssã♦ ✭✶✳✽✮✳ ❯s❛♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ ❡♠ ✭✶✳✽✮ ♦❜t❡♠♦s✿

hµν(x)−→hµν(x)−ξµ,ν −ξν,µ. ✭✶✳✸✶✮

❆ ♣r❡s❡♥ç❛ ❞❡ ✉♠❛ s✐♠❡tr✐❛ ❧♦❝❛❧ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ ♥♦s ❡①✐❣❡ ❛❞✐❝✐♦♥❛r ✉♠ t❡r♠♦ ❞❡ ✜①❛çã♦ ❞❡ ❣❛✉❣❡ Lgf✱ ❛ ❡s❝♦❧❤❛ ❝♦♠✉♠ ❞❛ ❧✐t❡r❛t✉r❛✱ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ Aµ ❡

φ✳ ❱❛♠♦s ❛❞♦t❛r t❛♠❜é♠ ♦✉tr❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❡♥tr❡ Fµν ❡(Aµ,µ−φ)✱♣♦r ❝♦♥t❛ q✉❡

❡st❡s t❡r♠♦ ❛♣❛r❡❝❡♠ ♥♦ ❡①♣r❡ssã♦ ❞♦ ❧❛❣r❛♥❣✐❛♥♦ ✭✶✳✸✵✮✱ ❢❛③❡♠♦s ❛ss✐♠✱ ♣♦✐s ♦ ✜①❛❞♦r ❞❡ ❣❛✉❣❡ ✜❝❛rá ♦ ♠❛✐s ❣❡r❛❧ ♣♦ssí✈❡❧✿

Lgf =λ1(Aν −λ∂νφ)2+

b

4[λ2(A

µ

,µ−φ)

2

−λ3Fµν]. ✭✶✳✸✷✮

♦s ❢❛t♦r❡sλ✱λ1✱λ2 ❡ λ3 sã♦ ♣❛râ♠❡tr♦s ❛❥✉stá✈❡✐s ♣❛r❛ ❝❛❞❛ ❝❛s♦ ❡st✉❞❛❞♦✳

❆♥❛❧✐s❛♥❞♦ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s ♣♦❞❡♠♦s ❧✐st❛r três ✜①❛❞♦r❡s ❞❡ ❣❛✉❣❡ ♠❛✐s ✉t✐❧✐③❛❞♦s❬✼✱ ✶✻✱ ✶✼❪✱ ❛♠❜♦s ❢♦r♠❛❞♦s ♣♦r ❡str✉t✉r❛s ♦r✐✉♥❞❛s ❞❛ ❡q✉❛çã♦✳

P♦st❡r✐♦r♠❡♥t❡ ♣♦❞❡r❡♠♦s ❛❞♦t❛r ❛❧❣✉♠ ❣❛✉❣❡ ❡s♣❡❝í✜❝♦ ♣❛r❛ ♦ ♥♦ss♦ tr❛❜❛❧❤♦✳

❊①❡♠♣❧♦ ✶ ❏♦❧✈❡✲❚♦♥✐♥ ❣❛✉❣❡(λ= 0)

Lgf =λ1Aν +

b

4[λ2(A

µ

,µ−φ)

2

−λ3Fµν]. ✭✶✳✸✸✮

❊①❡♠♣❧♦ ✷ ❞❡ ❉♦♥❣❡r ❣❛✉❣❡ (λ2 =λ3 = 0, λ = 1₂)

Lgf =λ1(Aν −

1 2∂νφ)

2_{. ✭✶✳✸✹✮}

❊①❡♠♣❧♦ ✸ ❋❡②♥♠❛♥ ❣❛✉❣❡ (λ1 = 1, λ2 =λ3 = 0, λ= 1₂)

Lgf = (Aν−

1 2∂νφ)

✶✾

✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ♣❛r❛

t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❞❡ ❛❧t❛s

❞❡r✐✈❛❞❛s✳

❊♥❝♦♥tr❛❞♦ ❛ ❧❛❣r❛♥❣✐❛♥❛ ❛♣r♦♣r✐❛❞❛ ❡ ❛❞✐❝✐♦♥❛♥❞♦ ✉♠ ✜①❛❞♦r ❞❡ ❣❛✉❣❡✱ ♣♦❞❡♠♦s ❞❛r ❝♦♥t✐♥✉✐❞❛❞❡ à ❜✉s❝❛ ❞❛ ♦❜t❡♥çã♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦♠ t❡r♠♦s ❞❡ ❞❡r✐✈❛❞❛s s✉♣❡r✐♦r❡s✱ ♣❛r❛ ❞✐♠❡♥sã♦ D > 2✳ ❆ ❜✉s❝❛ ❞❡ ❡♥❝♦♥tr❛r ♦ ♣r♦♣❛❣❛❞♦r ❞❡

✉♠❛ t❡♦r✐❛ é ❜❛st❛♥t❡ r❡❧❡✈❛♥t❡✱ ♣♦✐s✱ ♣♦❞❡♠♦s ❛♥❛❧✐s❛r ❛ ❡str✉t✉r❛ ❞❛ t❡♦r✐❛✱ s❡ ❛ t❡♦r✐❛ é ✉♥✐tár✐❛ ❡ ❛ ♣r❡s❡♥ç❛ ♦✉ ♥ã♦ ❞❡ ♠♦❞♦s ❞❡ ♣r♦♣❛❣❛çã♦ ♥ã♦ ❢ís✐❝♦s✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ✉s❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥ ❞❡ ❇❛r♥❡s✲❘✐✈❡rs✱ ♣❛r❛ ❛✉①✐❧✐❛r ♥❛ ✐♥✈❡rsã♦ ❞♦ ♦♣❡r❛❞♦r ❞❛ t❡♦r✐❛ q✉❡ é ❡①tr❛í❞♦ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❧❛❣r❛♥❣✐❛♥❛✱ ❝♦♠♦ r❡s✉❧t❛❞♦ t❡rá ♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥✳

✷✳✶ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦✲

r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦♠ ❛❧t❛s ❞❡r✐✈❛❞❛s

❆❞♦t❛♥❞♦ L ❝♦♠♦ s❡♥❞♦ ✉♠ ❧❛❣r❛♥❣✐❛♥♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❣r❛✈✐t❛❝✐♦♥❛❧✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥ ❞❡✈❡r❡♠♦s ❡♥❝♦♥tr❛r ❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❞❡st❡ ❧❛❣r❛♥❣✐❛♥♦✱ ❛tr❛✈és ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❛ ♠étr✐❝❛ ✭✶✳✽✮✱ s✉❜st✐t✉✐♥❞♦ ♥♦ ❧❛❣r❛♥❣✐❛♥♦ ❡ ❝♦❧❡t❛♥❞♦ s♦♠❡♥t❡ ♦s t❡r♠♦s ❞❡ ♦r❞❡♠ q✉❛❞rát✐❝❛ ❡♠hµν✱ ❡♥❝♦♥tr❛r❡♠♦sL0✳ ❙❡ ❛ t❡♦r✐❛ ❡♥✈♦❧✈✐❞❛ ❢♦r ✐♥✈❛r✐❛♥t❡

❞❡ ❣❛✉❣❡✱ ❞❡✈❡rá s❡r ❛❞✐❝✐♦♥❛❞♦ ✉♠ ✜①❛❞♦r ❞❡ ❣❛✉❣❡Lgf✳ ❆ ❧❛❣r❛♥❣✐❛♥❛ r❡s✉❧t❛♥t❡ s❡rá

L=_Lt+Lf g✳ P♦❞❡rá s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

L= 1 2h

µν

Oµν,αβhαβ, ✭✷✳✶✮

♦♥❞❡ ♦ t❡r♠♦Oµν,αβ✱ é ✉♠ ♦♣❡r❛❞♦r t❡♥s♦r✐❛❧✳ ❆ ✈ír❣✉❧❛ ♥♦s í♥❞✐❝❡s ❞♦ ♦♣❡r❛❞♦r ✈❡t♦r✐❛❧

é ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ❡❧❡ é s✐♠étr✐❝♦ ♥♦s í♥❞✐❝❡s✳ ❈♦♠♦ ❞✐t♦ ♥♦ ✐♥✐❝✐♦ ❞❡st❡ ❝❛♣✐t✉❧♦ ♦ ♣r♦♣❛❣❛❞♦r s❡rá ♦❜t✐❞♦ ❞❡♣♦✐s ❞❛ ✐♥✈❡rsã♦ ❞❡st❡ ♦♣❡r❛❞♦r✳

✷✳✶ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦♠ ❛❧t❛s ❞❡r✐✈❛❞❛s ✷✵

s♦r✐❛✐s ❞❡ ❇❛r♥❡s✲❘✐✈❡rs✳ ❖ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ss❡s ♦♣❡r❛❞♦r❡s ❡♠ ❉✲❞✐♠❡♥sõ❡s é ❞❛❞♦ ❬✼❪✿

Pµν,αβ1 =

1

2(θµαωνβ+θµβωνα+θναωµβ+θνβωµα),

P_µν,αβ2 = 1

2(θµαθνβ+θµβθνα)− 1

D₋1θµνθαβ,

Pµν,αβ0 =

1

D₋1θµνθαβ,

P0µν,αβ =ωµνωαβ,

P0µν,αβ =θµνωαβ+ωµνθαβ.

❖s t❡r♠♦s θµν ❡ ωµν✱ sã♦ ❛s ♣r♦❥❡çõ❡s ❞♦s ♦♣❡r❛❞♦r❡s ✈❡t♦r✐❛✐s tr❛♥s✈❡rs❛❧ ❡ ❧♦♥❣✐t✉✲

❞✐♥❛❧ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

θµν ≡ηµν−

kµkν

k2 ,

ωµν ≡

kµkν

k2 .

♦ t❡r♠♦ kµ é ♦ ♠♦♠❡♥t✉♠ ❞♦ ❣rá✈✐t♦♥ ❡ k2 ≡kµkµ✳

❆s r❡❧❛çõ❡s ❡♥tr❡ ❛s ♣r♦❥❡çõ❡s ♣♦❞❡♠ s❡r ❧✐st❛❞❛s ❝♦♠♦✿

θµβθβν =θµν, ωµβωβν =ωµν, θµβωνβ = 0.

❖s ♦♣❡r❛❞♦r❡s ❞❡ ❇❛r♥❡s✲❘✐✈❡rs ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ❢❡❝❤❛❞❛ ❡ s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦ ❞❡ ❝♦♠♣❧❡t✉❞❡✿

[P2+P1+P0+P0]µν,αβ =

1

2(ηµαηνβ +ηµβηνα)≡Iµν,αβ,

♦♥❞❡Iµν,αβ é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ ♦s t❡♥s♦r❡s ❞❡ q✉❛tr♦ í♥❞✐❝❡s✳

❖s ♦♣❡r❛❞♦r❡s P2_{, P}1_{, P}0_{, P}0_{✱ r❡♣r❡s❡♥t❛♠ ❛s ♣r♦❥❡çõ❡s ❞♦s s♣✐♥✲✶✱ s♣✐♥✲✷ ❡ ❞♦✐s s♣✐♥✲}

✵✳ ❖ ♦♣❡r❛❞♦rP0✱ é ❛ s♦♠❛ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ tr❛♥s❢❡rê♥❝✐❛✳

P0µν,αβ ≡[P θω

+Pωθ]µν,αβ.

❯t✐❧✐③❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞♦s ♣r♦❥❡t♦r❡s tr❛♥s✈❡rs❛✐s θµν ❡ ωµν✱ ♣❛r❛ r❡❡s❝r❡✈❡r ♦ ♦♣❡✲

r❛❞♦rP0✱ t❡r❡♠♦s✿

Pθω

µν,αβ ≡θµνωαβ, Pµν,αβθω ≡ωµνθαβ.

✷✳✶ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦♠ ❛❧t❛s ❞❡r✐✈❛❞❛s ✷✶

q✉❡ ♣r♦❞✉t♦s ❞❡ ♣r♦❥❡çõ❡s ❞❡ s♣✐♥ ❞✐❢❡r❡♥t❡s s❡❥❛♠ ♥✉❧♦s✱ ❡♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

P0P1 =P0P1 =P2P0 =P2P0 =O.

❖✉tr♦ r❡s✉❧t❛❞♦ ❜❡♠ ❡s♣❡r❛❞♦ sã♦ ♦s ♣r♦❞✉t♦s ❞♦s ♣r♦❥❡t♦r❡s ❞❡ s♣✐♥ ✐❣✉❛✐s✿

P(1)µν,λγP(1) λγ

,αβ =P(1)µν,αβ,

P(2)µν,λγP(2) λγ

,αβ =P(2)µν,αβ,

P(0)µν,λγP(0) λγ

,αβ =P(0)µν,αβ,

P0µν,λγP

0λγ ,αβ =P

0

µν,αβ.

❆ ❞✐❢❡r❡♥ç❛ ❡stá ♥♦ t❡r♠♦P0✱ ❝♦♠♦ ❢♦✐ ♠❡♥❝✐♦♥❛❞♦✱ ❡❧❡ é ✉♠❛ s♦♠❛ ❞♦s ❞♦✐s ♣r♦❥❡t♦r❡s ❞❡ s♣✐♥✲✵

(P)2 = (D₋1)(P0+P0), P0P0 =P0P0 =Pθω, P0P0 =P0P0 =Pωθ.

❆♣ós ❞❡✐①❛r ♦ ❧❛❣r❛♥❣✐❛♥♦ ♥❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ✭✷✳✶✮✱ ♦ ♦♣❡r❛❞♦r Oµν,αβ ❞❡✈❡rá s❡r

❡①♣❛♥❞✐❞♦ ❡♠ t❡r♠♦s ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥ ❞❡ ❇❛r♥❡s✲❘✐✈❡rs P2_{, P}1_{, P}0_{, P}0_{, P}0_{✳ P❛r❛}

❛✉①✐❧✐❛r ♥❡st❡ tr❛❜❛❧❤♦✱ ♣♦❞❡✲s❡ ✉s❛r ❛❧❣✉♠❛s ✐❞❡♥t✐❞❛❞❡s t❡♥s♦r✐❛✐s✿

1

2(ηµαηνβ +ηµβηνα) = [P

2_+P1_+P0_+P0_]

µν,αβ,

ηµνηαβ = [(D−1)P0+P

0

]µν,αβ],

1

k2(ηµαkνkβ+ηµβkνkα+ηναkµkβ +ηνβkµkα) = [2P

1_{+ 4P}0_]

µν,αβ,

1

k2(ηµνkαkβ +ηαβkµkν) = [P 0

+ 2P0]µν,αβ, ✭✷✳✷✮

1

k4(kµkνkαkβ) =P 0

µν,αβ,

P2µν,αβ =

1

2(ηµαηνβ+ηµβηνα)− 1

D₋1ηµνηαβ −

P1+D−2

D₋1P

0

− _D1 −1P

0

µν,αβ

,

P = 1

D₋1ηµνηαβ − 1

D₋1

h

P0+P0i

µν,αβ,

❛s ✐❞❡♥t✐❞❛❞❡s ❛❝✐♠❛ ❧✐st❛❞❛s ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ ❝♦♥❢❡r✐❞❛s✱ ❜❛st❛ ❛♣❧✐❝❛r ❛ ❞❡✜♥✐çã♦ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❇❛r♥❡s✲❘✐✈❡rs✳

✷✳✶ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❝♦♠ ❛❧t❛s ❞❡r✐✈❛❞❛s ✷✷

❝♦♥s❡❣✉✐♠♦s ❡①♣❛♥❞✐r ♥❛ ❜❛s❡ ♦s P2_{, P}1_{, P}0_{, P}0_{, P}0_{✱ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧✱ ♦ ♦♣❡r❛❞♦r}

✜❝❛rá ♥❛ ❢♦r♠❛✿

O =x1P1+x2P2+x0P0+x0P 0

+x0P 0

. ✭✷✳✸✮

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❡♥❝♦♥tr❛r ♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥✱ ❡♥tã♦ ❡♠ ♣♦ss❡ ❞♦ ♦♣❡r❛❞♦r ❡①♣❛♥❞✐❞♦ ✭✷✳✷✮✱ ❞❡✈❡ s❡r ✐♥✈❡rt✐❞♦✱ ✉t✐❧✐③❛♥❞♦ ✐❞❡♥t✐❞❛❞❡ OO−1 _{= I✱}

♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ♦♣❡r❛❞♦r ✐♥✈❡rs♦ ❝♦♠♦✿

O−1 =y1P1+y2P2+y0P0 +y0P 0

+y₀P0.

❖❜s❡r✈❛♥❞♦ ❛ r❡❧❛çã♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡s❝r✐t❛ ❛❝✐♠❛✱ ❥✉♥t♦ ❝♦♠ ❛q✉❡❧❛ ❞♦ ✐♥✐❝✐♦ ❞♦ ❝❛♣✐t✉❧♦✿

[P2+P1+P0+P0]µν,αβ =Iµν,αβ,

❢❛❝✐❧♠❡♥t❡ ❝❤❡❣❛r❡♠♦s ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s✱ q✉❡ ❛♣ós r❡s♦❧✈❡r t❡r❡♠♦s ❛ ❢♦r♠❛ ❡①❛t❛ ❞♦ ♣r♦♣❛❣❛❞♦r✳

P❛r❛ ♥ã♦ ♥♦s ❛tr❛♣❛❧❤❛r♠♦s ❡s❝r❡✈❡r❡♠♦s t❡r♠♦ ❛ t❡r♠♦✱ ❥á q✉❡ ♦s t❡r♠♦s ❝r✉③❛❞♦s q✉❡ ♥ã♦ sã♦ ♥✉❧♦s s❡rã♦ s♦♠❡♥t❡ ❞❡ s♣✐♥✲✵✳

I = _OO−1

I = (x1P1+x2P2+x0P0+x0P 0

+x0P 0

)_×(y1P1+y2P2+y0P0 +y0P 0

+y₀P0)

I = x1y1(P1)2 +x2y2(P2)2+x0y0(P0)2+x0P0y0P 0

+x0y0(P 0

)2+x0y0P 0

P0 +x0P 0

y0P0+x0P 0

y₀P0+x0P 0

y₀P0

I = x1y1P1+x2y2P2+x0y0P0+x0y0Pωθ

+x0y0P 0

+x0y0Pθω+x0y0Pθω +x0y0Pωθ +x0y0(D−1)(P0+P 0

).

❈♦♠♣❛r❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r ❝♦♠ ❛ ✐❞❡♥t✐❞❛❞❡ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❇❡r♥❡s✲❘✐✈❡rs✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞✐r❡t❛♠❡♥t❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s s✐♠✉❧tâ♥❡❛s✿

x1y1 = 1,

x2y2 = 1,

x0y0+ (D−1)x0y0 = 1,

x0y0 + (D−1)x0y0 = 1,

✷✳✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s q✉❛❞rát✐❝❛ ✷✸

x0y0x0y0 = 0.

❉❡st❛ ❢♦r♠❛ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ ♣r♦♣❛❣❛❞♦r O−1_✿

O−1 = 1

x1

P1+ 1

x2

P2+ 1

x0x0−(D−1)x20

[x0P0+x0P 0

−x0P0]. ✭✷✳✹✮

❈♦♠ ❡st❡ r❡s✉❧t❛❞♦ s♦♠♦s ❝❛♣❛③❡s ❞❡ ❝❛❧❝✉❧❛r ❛ ❢♦r♠❛ ❞❡ ♦ ♣r♦♣❛❣❛❞♦r ♣❛rt✐r ❞❡ ✉♠❛ ❧❛❣r❛♥❣✐❛♥❛ q✉❛❧q✉❡r✱ t♦♠❛♥❞♦ ♦ ❝✉✐❞❛❞♦✱ ♣♦✐s ❡♠ ❞❡t❡r♠✐♥❛❞❛s ♦❝❛s✐õ❡s ❞❡✈❡✲s❡ ✜①❛r ♦ ❣❛✉❣❡✳

✷✳✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦✲

r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s q✉❛❞rát✐❝❛

◆❛ s❡çã♦ ❛♥t❡r✐♦r ✈✐♠♦s ♦s ♣❛ss♦s ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ♣r♦♣❛❣❛❞♦r ❞♦ ❣rá✈✐t♦♥ ❞❡ ✉♠❛ t❡♦r✐❛ q✉❛❧q✉❡r✳ ◆❡st❛ s❡çã♦ ✈❛♠♦s r❡❛❧✐③❛r ❜❛s✐❝❛♠❡♥t❡ ♦s ♣❛ss♦s ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ ♠❛s ❝♦♠ ✉♠ ❧❛❣r❛♥❣✐❛♥♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✸✵✮✳ Pr✐♠❡✐r❛♠❡♥t❡ ❞❡✈❡♠♦s s✐♠❡tr✐③❛r ♦ ❧❛❣r❛♥❣✐❛♥♦ ❛ ✜♠ ❞❡ ❞❡✐①❛r ❞❛ ❢♦r♠❛ ✭✷✳✶✮✳

✷✳✷✳✶ ❙✐♠❡tr✐③❛çã♦ ❞❛ ▲❛❣r❛♥❣✐❛♥❛

❖ ❧❛❣r❛♥❣✐❛♥♦ q✉❡ ✈❛♠♦s ❛♥❛❧✐s❛r ♥❡st❛ s❡çã♦ é ❡♥❝♦♥tr❛❞♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✸✵✮ ❡ ❛❞✐✲ ❝✐♦♥❛❞♦ ✉♠ ✜①❛❞♦r ❞❡ ❣❛✉❣❡✳ ❯s❛r❡♠♦s ♦ ♠❛✐s ❣❡r❛❧✳

❘❡❧❡♠❜r❛♥❞♦ q✉❡ ♦ ❧❛❣r❛♥❣✐❛♥♦ ❛❞♦t❛❞♦ é ❞❛ ❢♦r♠❛ L=_L1+Lgf+L2+L3✱ ✈❛♠♦s

❛♥❛❧✐s❛r s❡♣❛r❛❞❛♠❡♥t❡✱ ♣♦✐s ❥✉❧❣♦ s❡r ✉♠❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s s✐♠♣❧❡s ❡ ♠❛✐s ♦r❣❛♥✐③❛❞❛✱ ♣❛r❛ ♦ ❛❝♦♠♣❛♥❤❛♠❡♥t♦ ❞❛s s✐♠❡tr✐③❛çõ❡s ❞❡ ❝❛❞❛ ♣❛rt❡✳

❈♦♠❡ç❛♥❞♦ ♣❡❧❛ ❛ ❡q✉❛çã♦ ✭✶✳✷✶✮✱ ♠❛✐s ♦ t❡r♠♦ ❞❡ ✜①❛çã♦ ❞❡ ❣❛✉❣❡✱ ❡st❡ ú❧t✐♠♦ ✐r❡✐ ❡s❝r❡✈❡r s✉❜st✐t✉✐♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞❡Aµ ❡φ✿

L1 +Lgf = −

1 2h

µν

hµν−

1 2h

µ µh

α α+h

µ

µ∂α∂βhαβ −hµν∂µ∂αhαν

+1 2λ1(2h

µ

µ∂α∂βhαβ −2λhµν∂ν∂αhαµ−2λ

2_hµ µh

α α)

+b 4(λ2[h

µν_∂

µ∂ν∂α∂βhαβ −2λhµµ∂α∂βhαβ +λ2hµµ

2_hα α

✷✳✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s q✉❛❞rát✐❝❛ ✷✹

❘❡s❝r❡✈❡♥❞♦ ❛❣♦r❛ ♥♦ ❢♦r♠❛t♦ L1+Lgf = 1₂hµνOµν,αβhαβ✱ t❡♠♦s✿

L1+Lgf =

1 2h

µν

(ηµαηνβ−ηµνηαβ + 2ηµν∂α∂β−2∂µ∂αηνβ

+λ1[2ηµν∂α∂β−2λ∂ν∂αηµβ−2λ2ηµνηαβ]

+b

2(λ2[∂µ∂ν∂α∂β−2ληµν∂α∂β +λ

2_η

µνηαβ]

+λ3[2ηνβ∂µ∂α−2∂µ∂ν∂α∂β])hαβ. ✭✷✳✻✮

❉❡♣♦✐s q✉❡ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛❧❣✉♥s t❡r♠♦s ♥ã♦ ❛♣r❡s❡♥t❛♠ s✐♠❡tr✐❛ ♥♦s í♥❞✐❝❡s✱ tr♦❝❛ ❞❡µ_→ν ❡ α _→β✱ ♣♦✐s ✐st♦ é ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ q✉❡ ❞❡✈❡♠♦s ❜✉s❝❛r✱ ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ♣r♦♣❛❣❛❞♦r✳ ❆♣ós s✐♠❡tr✐③❛r ❛ ❡q✉❛çã♦ ✭✷✳✻✮✱ ❡♥❝♦♥tr❛r❡♠♦s✿

L1+Lgf =

1 2h

µν

([1

2(ηµαηνβ +ηµβηνα)−ηµνηαβ]−(∂α∂βηµν +∂µ∂νηαβ) +1

2[∂µ∂αηνβ +∂µ∂βηνα+∂ν∂αηµβ+∂ν∂βηµα]

+λ1[(∂α∂βηµν +∂µ∂νηαβ)−λ[∂µ∂αηνβ+∂µ∂βηνα+∂ν∂αηµβ

+∂ν∂βηµα]−2λ2ηµνηαβ]

+b

2(λ2[∂µ∂ν∂α∂β −

λ

2 (ηµν∂α∂β+ηα∂µ∂ν) +λ

22_η

µνηαβ]

+λ3[

1

2(∂µ∂αηνβ +∂µ∂βηνα+∂ν∂αηµβ+∂ν∂βηµα]

−2∂µ∂ν∂α∂β)hαβ. ✭✷✳✼✮

❯s❛♥❞♦ ❛s r❡❧❛çõ❡s t❡♥s♦r✐❛✐s ✭✷✳✷✮✱ ♣♦❞❡r❡♠♦s r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ✭✷✳✼✮✱ ❡♠ ❢✉♥çã♦ ❞♦s ♣r♦❥❡t♦r❡s ❞❡ ❇❛r♥❡s✲❘✐✈❡rs✿

L1+Lgf =

1 2h

µν

([bλ3k

4

2 ]P

1_{+ [k}2_]P2

+[2(D₋1)k2λ1λ2−k2(D−2) +

bλ2λ2k4[D−1]

2 ]P

0

+

2k2λ1−4k2λλ1+ 2k2λ1λ2+

bλ2k4

2 −2

bλ2λk4

2 +

bλ2λ2k4

2

P0

+[2k2λ1λ2−2k2λλ1−

bλ2λk4

2 + +

bλ2λ2k4

2 ]P

0

)hαβ_{. ✭✷✳✽✮}

✷✳✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s q✉❛❞rát✐❝❛ ✷✺

L2 =

α

2κ

2_(hµ µ2h

α α−2h

µ

µ∂α∂βhαβ +hµν∂µ∂ν∂α∂βhαβ)

= 1 2h

µν

(ακ2[2ηµνηαβ−2ηµν∂α∂β+∂µ∂ν∂α∂β])hαβ. ✭✷✳✾✮

❆♣ós s✐♠❡tr✐③❛r ✜❝❛♠♦s ❝♦♠✿

L2 =

1 2h

µν

(ακ2[2(ηµνηαβ)−(∂α∂βηµν+∂µ∂νηαβ)

+∂µ∂ν∂α∂β])hαβ. ✭✷✳✶✵✮

❘❡❡s❝r❡✈❡♥❞♦ ❡♠ ❢✉♥çã♦ ❞♦s ♣r♦❥❡t♦r❡s✿ L2 =

1 2h

µν_(ακ2_[k4_(D

−1)P0])hαβ_{. ✭✷✳✶✶✮}

❋✐♥❛❧♠❡♥t❡✱ ❝♦♠ ♦ ú❧t✐♠♦ ❢❛t♦r ♦ t❡♥s♦r ❞❡ ❘✐❡♠❛♥♥✿ L3 =

b

4(h

µν2_h

µν+hµµ

2_hα

α−2h µ

µ∂α∂βhαβ −2hµβ∂µ∂αhαβ

+2hµν_∂

µ∂ν∂α∂βhαβ) ✭✷✳✶✷✮

= 1 2h

µν

(b 2[

2_[η

µαηνβ +ηµνηαβ]−2[ηµν∂α∂β+ηνβ∂µ∂α]

+2[∂µ∂ν∂α∂β])hαβ. ✭✷✳✶✸✮

❙✐♠❡tr✐③❛♥❞♦✿ L3 =

1 2h

µνb

2(

2_[1

2(ηµαηνβ+ηµβηνα) +ηµνηαβ]−[(∂α∂βηµν+∂µ∂νηαβ) +1

2[∂µ∂αηνβ+∂µ∂βηνα+∂ν∂αηµβ+∂ν∂βηµα] + 2[∂µ∂ν∂α∂β])h

αβ_{. ✭✷✳✶✹✮}

❊s❝r❡✈❡♥❞♦ ❡♠ ❢✉♥çã♦ ❞♦s ♣r♦❥❡t♦r❡s✿ L3 =

1 2h

µνb

2k

4_(P2_+DP0_)hαβ_{. ✭✷✳✶✺✮}

✷✳✷ ❈á❧❝✉❧♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡♠ ❉✲❞✐♠❡♥sõ❡s ♣❛r❛ t❡♦r✐❛s ❣r❛✈✐t❛❝✐♦♥❛✐s q✉❛❞rát✐❝❛ ✷✻

❡♠ t❡r♠♦s ❞♦s ♣r♦❥❡t♦r❡s ❞❡ s♣✐♥✿

L = _L1+Lgf+L2+L3

= 1 2h

µν

{₂b[λ3k4 +

2λ1k2

b ]P

1₊ b

2[k

4₊ 2k2

b ]P

2

+b 2[Dk

4

− 2k

2_(D−2)

b + 4(D−1)k

4_{c+ (D}

−1)k4λ2λ2+

4k2_(D−1)λ 1λ2

b ]P

0

+b 2

k4λ2−2k4λλ2+

4k2_λ 1

b −

8k2_λλ

1+k4λ2λ2

b +

4k2_λ 1λ2

b

P0

+b 2[4k

2_λ

1λ2+k4λ2λ2−k4λ2λ−4k2λλ1]P 0

}hαβ. ✭✷✳✶✻✮

❈♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r ♦ ❧❛❣r❛♥❣✐❛♥♦ ❡♠ ❢✉♥çã♦ ❞♦s ♣r♦❥❡t♦r❡s ❡♠ ❉✲❞✐♠❡♥sõ❡s✱ ❛✐♥❞❛ ♥ã♦ t♦♠❛♠♦s ♥❡♥❤✉♠ ✜①❛❞♦r ❞❡ ❣❛✉❣❡ ❡s♣❡❝í✜❝♦✱ ♦ ❝❛❧❝✉❧♦ r❡❛❧✐③❛❞♦ ❛❝✐♠❛ ❢♦✐ r❡❛❧✐③❛❞♦ ❞❡ ♠❛♥❡✐r❛ q✉❡ ✜q✉❡ ♦ ♠❛✐s ❣❡r❛❧ ♣♦ssí✈❡❧✳ ❖ t❡r♠♦ ❡♥tr❡ ♣❛rê♥t❡s❡s ❞❛ ❡q✉❛çã♦ ✭✷✳✶✻✮ é ♦ ♦♣❡r❛❞♦r ✈❡t♦r✐❛❧ q✉❡ ❞❡✈❡♠♦s ✐♥✈❡rt❡r ♣❛r❛ ♦❜t❡r ♦ ♣r♦♣❛❣❛❞♦r✳ P❛r❛ ❝✉♠♣r✐r ❡st❛ t❛r❡❢❛✱ s❡❣✉✐r❡♠♦s ♦s ♣❛ss♦s ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❝♦♠❡ç❛♥❞♦ ♣♦r ❡s❝r❡✈❡r ♦ ♦♣❡r❛❞♦r ✈❡t♦r✐❛❧✿

O =x1P1+x2P2+x0P0+x0P 0

+x0P 0

❙✉❜st✐t✉✐♥❞♦ ❝♦♠ ♦s ✈❛❧♦r❡s q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❡♥❝♦♥tr❛r

O = b 2[λ3k

4₊2λ1k2

b ]P

1₊ b

2[k

4₊ 2k2

b ]P

2

+b 2[Dk

4

− 2k

2_(D−2)

b + 4(D−1)k

4_{c+ (D}

−1)k4λ2λ2+

4k2_(D−1)λ 1λ2

b ]P

0

+b 2

k4λ2−2k4λλ2+

4k2_λ 1

b −

8k2_λλ 1

b +k

4_λ 2λ2+

4k2_λ 1λ2

b

P0

+b 2[

4k2_λ 1λ2

b +k

4_λ

2λ2−k4λ2λ−

4k2_λλ 1

b ]P

0

. ✭✷✳✶✼✮

❊st❡ é ♦ ♦♣❡r❛❞♦r ✈❡t♦r✐❛❧ ❡♠ ❉✲❞✐♠❡♥sã♦ ❡ s❡♠ ✉♠ ❣❛✉❣❡ ❡s♣❡❝í✜❝♦✱ ❡❧❡ é ❛ ❜❛s❡ ♣❛r❛ ♦❜t❡r ♦s ♣r♦♣❛❣❛❞♦r❡s q✉❛❧q✉❡r ❞✐♠❡♥sã♦ ❝♦♠ D > 2✱ ❡ ❝♦♠ ✜①❛❞♦r ❞❡ ❣❛✉❣❡ ❞❛

❢♦r♠❛ ❞❡s❝r✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✿

Lgf =λ1(Aν −λ∂νφ)2+

b

4[λ2(A

µ

,µ−φ)2−λ3Fµν].