t rt r - PowerPoint PPT Presentation

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ❙✉♠♠❛r② ❚❤❡ Pr♦❜❧❡♠✱ ✐♥ ❝❧❛ss✐❝❛❧ ✜❡❧❞ t❤❡♦r②✳ ❍♦✇ t♦ r❡❢♦r♠✉❧❛t❡ ❊✐♥st❡✐♥✬s ♣❤②s✐❝❛❧ ❡q✉❛t✐♦♥ ❢♦r ❣r❛✈✐t② ✐♥ ❛♥ ❝❛✉s❛❧ ❢r❛♠❡ ✇✐t❤ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t♦ ❜❡ ❞❡s✐❣♥❛t❡❞ t②♣❡ ♦❢ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥✳ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ s②st❡♠s✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜rst t✐♠❡ ♦r❞❡r ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ❝♦♥str❛✐♥ts ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❤②♣❡r❜♦❧✐❝ ♣r♦❜❧❡♠ ✐♥❝❧✉❞❡s t✐♠❡❧✐❦❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ❤②♣❡rs✉r❢❛❝❡s s❡r✈✐♥❣ ❛s ❜♦✉♥❞❛r✐❡s t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦♠t✐♥❣ t♦ ❛✿ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ✲ ❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥ Pr♦❜❧❡♠✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ❙✉♠♠❛r② ❚❤❡ Pr♦❜❧❡♠✱ ✐♥ ❝❧❛ss✐❝❛❧ ✜❡❧❞ t❤❡♦r②✳ ❍♦✇ t♦ r❡❢♦r♠✉❧❛t❡ ❊✐♥st❡✐♥✬s ♣❤②s✐❝❛❧ ❡q✉❛t✐♦♥ ❢♦r ❣r❛✈✐t② ✐♥ ❛♥ ❝❛✉s❛❧ ❢r❛♠❡ ✇✐t❤ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t♦ ❜❡ ❞❡s✐❣♥❛t❡❞ t②♣❡ ♦❢ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥✳ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ s②st❡♠s✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜rst t✐♠❡ ♦r❞❡r ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ❝♦♥str❛✐♥ts ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❤②♣❡r❜♦❧✐❝ ♣r♦❜❧❡♠ ✐♥❝❧✉❞❡s t✐♠❡❧✐❦❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ❤②♣❡rs✉r❢❛❝❡s s❡r✈✐♥❣ ❛s ❜♦✉♥❞❛r✐❡s t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦♠t✐♥❣ t♦ ❛✿ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ✲ ❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥ Pr♦❜❧❡♠✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ❙✉♠♠❛r② ❚❤❡ Pr♦❜❧❡♠✱ ✐♥ ❝❧❛ss✐❝❛❧ ✜❡❧❞ t❤❡♦r②✳ ❍♦✇ t♦ r❡❢♦r♠✉❧❛t❡ ❊✐♥st❡✐♥✬s ♣❤②s✐❝❛❧ ❡q✉❛t✐♦♥ ❢♦r ❣r❛✈✐t② ✐♥ ❛♥ ❝❛✉s❛❧ ❢r❛♠❡ ✇✐t❤ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t♦ ❜❡ ❞❡s✐❣♥❛t❡❞ t②♣❡ ♦❢ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥✳ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ s②st❡♠s✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜rst t✐♠❡ ♦r❞❡r ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ❝♦♥str❛✐♥ts ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❤②♣❡r❜♦❧✐❝ ♣r♦❜❧❡♠ ✐♥❝❧✉❞❡s t✐♠❡❧✐❦❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ❤②♣❡rs✉r❢❛❝❡s s❡r✈✐♥❣ ❛s ❜♦✉♥❞❛r✐❡s t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦♠t✐♥❣ t♦ ❛✿ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ✲ ❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥ Pr♦❜❧❡♠✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ❙✉♠♠❛r② ❚❤❡ Pr♦❜❧❡♠✱ ✐♥ ❝❧❛ss✐❝❛❧ ✜❡❧❞ t❤❡♦r②✳ ❍♦✇ t♦ r❡❢♦r♠✉❧❛t❡ ❊✐♥st❡✐♥✬s ♣❤②s✐❝❛❧ ❡q✉❛t✐♦♥ ❢♦r ❣r❛✈✐t② ✐♥ ❛♥ ❝❛✉s❛❧ ❢r❛♠❡ ✇✐t❤ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t♦ ❜❡ ❞❡s✐❣♥❛t❡❞ t②♣❡ ♦❢ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥✳ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥❢♦r♠❛t✐♦♥ ❢♦r s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ s②st❡♠s✿ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✜rst t✐♠❡ ♦r❞❡r ♦❢ ✜❡❧❞s ✐♥✐t✐❛❧ ✈❛❧✉❡ ❝♦♥str❛✐♥ts ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ❤②♣❡r❜♦❧✐❝ ♣r♦❜❧❡♠ ✐♥❝❧✉❞❡s t✐♠❡❧✐❦❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ❤②♣❡rs✉r❢❛❝❡s s❡r✈✐♥❣ ❛s ❜♦✉♥❞❛r✐❡s t♦ t❤❡ s♦❧✉t✐♦♥✱ ♣r♦♠t✐♥❣ t♦ ❛✿ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ✲ ❇♦✉♥❞❛r② ❈♦♥❞✐t✐♦♥ Pr♦❜❧❡♠✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❖✉t❧✐♥❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ✶ Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❊✐♥st❡✐♥✬s ❚❤❡♦r② ♦❢ ●r❛✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ✷ Pr❡❧✐♠✐♥❛r✐❡s ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❡❝✐❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ❙♣❡❝✐❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐s♦♠❡tr✐❡s ♦❢ s♣❛❝❡t✐♠❡✳ s❡♥s❡ ♦❢ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ❊❧❡❝tr♦♠❛❣♥❡t✐s♠ ✐s s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ s❤✐❡❧❞❡❞ ❢r♦♠ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❡❝✐❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ❙♣❡❝✐❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐s♦♠❡tr✐❡s ♦❢ s♣❛❝❡t✐♠❡✳ s❡♥s❡ ♦❢ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ❊❧❡❝tr♦♠❛❣♥❡t✐s♠ ✐s s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ s❤✐❡❧❞❡❞ ❢r♦♠ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❡❝✐❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ❙♣❡❝✐❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐s♦♠❡tr✐❡s ♦❢ s♣❛❝❡t✐♠❡✳ s❡♥s❡ ♦❢ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ❊❧❡❝tr♦♠❛❣♥❡t✐s♠ ✐s s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ s❤✐❡❧❞❡❞ ❢r♦♠ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❡❝✐❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ❙♣❡❝✐❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐s♦♠❡tr✐❡s ♦❢ s♣❛❝❡t✐♠❡✳ s❡♥s❡ ♦❢ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ❊❧❡❝tr♦♠❛❣♥❡t✐s♠ ✐s s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ s❤✐❡❧❞❡❞ ❢r♦♠ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❡❝✐❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ❙♣❡❝✐❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐s♦♠❡tr✐❡s ♦❢ s♣❛❝❡t✐♠❡✳ s❡♥s❡ ♦❢ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ❊❧❡❝tr♦♠❛❣♥❡t✐s♠ ✐s s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦ ✐♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ s❤✐❡❧❞❡❞ ❢r♦♠ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ s♣❛❝❡t✐♠❡✳ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ✐s ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥t✳✳✳ ✭❛♥❞ ❧♦❝❛❧❧② s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦✮ ◆♦ s❡♥s❡ ♦❢ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✦ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ♥♦ ❦♥♦✇♥ ♠❡t❤♦❞ ❢♦r s❤✐❡❧❞✐♥❣ ❢r♦♠ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ s♣❛❝❡t✐♠❡✳ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ✐s ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥t✳✳✳ ✭❛♥❞ ❧♦❝❛❧❧② s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦✮ ◆♦ s❡♥s❡ ♦❢ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✦ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ♥♦ ❦♥♦✇♥ ♠❡t❤♦❞ ❢♦r s❤✐❡❧❞✐♥❣ ❢r♦♠ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ s♣❛❝❡t✐♠❡✳ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ✐s ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥t✳✳✳ ✭❛♥❞ ❧♦❝❛❧❧② s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦✮ ◆♦ s❡♥s❡ ♦❢ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✦ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ♥♦ ❦♥♦✇♥ ♠❡t❤♦❞ ❢♦r s❤✐❡❧❞✐♥❣ ❢r♦♠ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ s♣❛❝❡t✐♠❡✳ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ✐s ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥t✳✳✳ ✭❛♥❞ ❧♦❝❛❧❧② s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦✮ ◆♦ s❡♥s❡ ♦❢ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✦ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ♥♦ ❦♥♦✇♥ ♠❡t❤♦❞ ❢♦r s❤✐❡❧❞✐♥❣ ❢r♦♠ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ P❤②s✐❝s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞✐✛❡♦♠♦r♣❤✐s♠s ♦❢ s♣❛❝❡t✐♠❡✳ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ✐s ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥t✳✳✳ ✭❛♥❞ ❧♦❝❛❧❧② s♣❡❝✐❛❧ ❝♦✈❛r✐❛♥t✦✮ ◆♦ s❡♥s❡ ♦❢ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✦ ■♥❡rt✐❛❧ ♦❜s❡r✈❡r✿ ♥♦ ❦♥♦✇♥ ♠❡t❤♦❞ ❢♦r s❤✐❡❧❞✐♥❣ ❢r♦♠ ❣r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ ❚❤❡ t❤❡♠❡ ❤❡r❡ ✐s t❤❛t ✐♥❡rt✐❛❧ ♦❜s❡r✈❡rs ❝❛♥♥♦t ❜❡ ❞❡s✐❣♥❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❣r❛✈✐t②✳ ❊✐♥st❡✐♥ ♣r♦♣♦s❡❞ ❞❡s✐❣♥❛t✐♥❣ ❛❧❧ ♦❜s❡r✈❡rs ✐♥❡rt✐❛❧✿ ●r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞ ✈❛♥✐s❤❡s ✐♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✳ P❤❡♥♦♠❡♥♦♥s ❧✐♥❦❡❞ t♦ ❣r❛✈✐t② ❛r❡ ♥♦✇ ♣✉t t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✳ ❊♠❡r❣❡♥t ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡✱ ❣♦✐♥❣ ❜② t❤❡ ♥❛♠❡✿ ✏❊q✉✐✈❛❧❡♥❝❡ Pr✐♥❝✐♣❧❡✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ ❚❤❡ t❤❡♠❡ ❤❡r❡ ✐s t❤❛t ✐♥❡rt✐❛❧ ♦❜s❡r✈❡rs ❝❛♥♥♦t ❜❡ ❞❡s✐❣♥❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❣r❛✈✐t②✳ ❊✐♥st❡✐♥ ♣r♦♣♦s❡❞ ❞❡s✐❣♥❛t✐♥❣ ❛❧❧ ♦❜s❡r✈❡rs ✐♥❡rt✐❛❧✿ ●r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞ ✈❛♥✐s❤❡s ✐♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✳ P❤❡♥♦♠❡♥♦♥s ❧✐♥❦❡❞ t♦ ❣r❛✈✐t② ❛r❡ ♥♦✇ ♣✉t t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✳ ❊♠❡r❣❡♥t ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡✱ ❣♦✐♥❣ ❜② t❤❡ ♥❛♠❡✿ ✏❊q✉✐✈❛❧❡♥❝❡ Pr✐♥❝✐♣❧❡✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ ❚❤❡ t❤❡♠❡ ❤❡r❡ ✐s t❤❛t ✐♥❡rt✐❛❧ ♦❜s❡r✈❡rs ❝❛♥♥♦t ❜❡ ❞❡s✐❣♥❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❣r❛✈✐t②✳ ❊✐♥st❡✐♥ ♣r♦♣♦s❡❞ ❞❡s✐❣♥❛t✐♥❣ ❛❧❧ ♦❜s❡r✈❡rs ✐♥❡rt✐❛❧✿ ●r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞ ✈❛♥✐s❤❡s ✐♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✳ P❤❡♥♦♠❡♥♦♥s ❧✐♥❦❡❞ t♦ ❣r❛✈✐t② ❛r❡ ♥♦✇ ♣✉t t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✳ ❊♠❡r❣❡♥t ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡✱ ❣♦✐♥❣ ❜② t❤❡ ♥❛♠❡✿ ✏❊q✉✐✈❛❧❡♥❝❡ Pr✐♥❝✐♣❧❡✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ ❚❤❡ t❤❡♠❡ ❤❡r❡ ✐s t❤❛t ✐♥❡rt✐❛❧ ♦❜s❡r✈❡rs ❝❛♥♥♦t ❜❡ ❞❡s✐❣♥❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❣r❛✈✐t②✳ ❊✐♥st❡✐♥ ♣r♦♣♦s❡❞ ❞❡s✐❣♥❛t✐♥❣ ❛❧❧ ♦❜s❡r✈❡rs ✐♥❡rt✐❛❧✿ ●r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞ ✈❛♥✐s❤❡s ✐♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✳ P❤❡♥♦♠❡♥♦♥s ❧✐♥❦❡❞ t♦ ❣r❛✈✐t② ❛r❡ ♥♦✇ ♣✉t t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✳ ❊♠❡r❣❡♥t ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡✱ ❣♦✐♥❣ ❜② t❤❡ ♥❛♠❡✿ ✏❊q✉✐✈❛❧❡♥❝❡ Pr✐♥❝✐♣❧❡✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ ❚❤❡ t❤❡♠❡ ❤❡r❡ ✐s t❤❛t ✐♥❡rt✐❛❧ ♦❜s❡r✈❡rs ❝❛♥♥♦t ❜❡ ❞❡s✐❣♥❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❣r❛✈✐t②✳ ❊✐♥st❡✐♥ ♣r♦♣♦s❡❞ ❞❡s✐❣♥❛t✐♥❣ ❛❧❧ ♦❜s❡r✈❡rs ✐♥❡rt✐❛❧✿ ●r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞ ✈❛♥✐s❤❡s ✐♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✳ P❤❡♥♦♠❡♥♦♥s ❧✐♥❦❡❞ t♦ ❣r❛✈✐t② ❛r❡ ♥♦✇ ♣✉t t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✳ ❊♠❡r❣❡♥t ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡✱ ❣♦✐♥❣ ❜② t❤❡ ♥❛♠❡✿ ✏❊q✉✐✈❛❧❡♥❝❡ Pr✐♥❝✐♣❧❡✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ●❡♥❡r❛❧ ❈♦✈❛r✐❛♥❝❡✱ s♣❛✇✐♥❣ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✳ ❚❤❡ t❤❡♠❡ ❤❡r❡ ✐s t❤❛t ✐♥❡rt✐❛❧ ♦❜s❡r✈❡rs ❝❛♥♥♦t ❜❡ ❞❡s✐❣♥❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❣r❛✈✐t②✳ ❊✐♥st❡✐♥ ♣r♦♣♦s❡❞ ❞❡s✐❣♥❛t✐♥❣ ❛❧❧ ♦❜s❡r✈❡rs ✐♥❡rt✐❛❧✿ ●r❛✈✐t❛t✐♦♥❛❧ ✜❡❧❞ ✈❛♥✐s❤❡s ✐♥ t❤✐s ♣❡rs♣❡❝t✐✈❡✳ P❤❡♥♦♠❡♥♦♥s ❧✐♥❦❡❞ t♦ ❣r❛✈✐t② ❛r❡ ♥♦✇ ♣✉t t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✳ ❊♠❡r❣❡♥t ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡✱ ❣♦✐♥❣ ❜② t❤❡ ♥❛♠❡✿ ✏❊q✉✐✈❛❧❡♥❝❡ Pr✐♥❝✐♣❧❡✑ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❖✉t❧✐♥❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ✶ Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❊✐♥st❡✐♥✬s ❚❤❡♦r② ♦❢ ●r❛✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ✷ Pr❡❧✐♠✐♥❛r✐❡s ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❛❝❡t✐♠❡ ✐♥tr✐♥s✐❝ ♣r♦♣❡rt✐❡s ■♥t❡r♥❛❧ ❙tr✉❝t✉r❡ ♠❡tr✐❝ � ❴ | ❴ � ✿ g ab ▲❡✈✐✲❈✐✈✐t❛ ❝♦♥♥❡❝t✐♦♥ ∇ ✿ Γ c ab = ( 1 / 2 ) g cd ( ∂ a g bd + ∂ b g da − ∂ d g ab ) ❘✐❡♠❛♥♥ ❝✉r✈❛t✉r❡ t❡♥s♦r✿ R a bcd = ∂ d Γ a cb − ∂ c Γ a db + Γ a de Γ e cb − Γ a ce Γ e db ❘✐❝❝✐ t❡♥s♦r✿ R ab = g c e R e acb ❝✉r✈❛t✉r❡ s❝❛❧❛r✿ R = g ab R ab ❊✐♥st❡✐♥ t❡♥s♦r✿ G ab = R ab − ( 1 / 2 ) Rg ab ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❛❝❡t✐♠❡ ✐♥tr✐♥s✐❝ ♣r♦♣❡rt✐❡s ■♥t❡r♥❛❧ ❙tr✉❝t✉r❡ ♠❡tr✐❝ � ❴ | ❴ � ✿ g ab ▲❡✈✐✲❈✐✈✐t❛ ❝♦♥♥❡❝t✐♦♥ ∇ ✿ Γ c ab = ( 1 / 2 ) g cd ( ∂ a g bd + ∂ b g da − ∂ d g ab ) ❘✐❡♠❛♥♥ ❝✉r✈❛t✉r❡ t❡♥s♦r✿ R a bcd = ∂ d Γ a cb − ∂ c Γ a db + Γ a de Γ e cb − Γ a ce Γ e db ❘✐❝❝✐ t❡♥s♦r✿ R ab = g c e R e acb ❝✉r✈❛t✉r❡ s❝❛❧❛r✿ R = g ab R ab ❊✐♥st❡✐♥ t❡♥s♦r✿ G ab = R ab − ( 1 / 2 ) Rg ab ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❛❝❡t✐♠❡ ♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s str❡ss✲❡♥❡r❣②✲♠♦♠❡♥t✉♠ t❡♥s♦r T ab ❉❡❝♦♠♣♦s❡❞ ✐♥ ❡♥❡r❣② E ✱ ♠♦♠❡♥t✉♠ ✈❡❝t♦r p ❛♥❞ str❡ss t❡♥s♦r✿ T ab υ a υ b = E T ab υ a x b = p x T ab υ a y b = p y T ab υ a z b = p x   T ab x a x b = σ xx T ab x a y b = σ xy T ab x a z b = σ xz   T ab y a y b = σ yy T ab y a z b = σ yz     T ab z a z b = σ zz ❢♦r ❛♥ ♦rt❤♦♥♦r♠❛❧ ❧♦❝❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✇✐t❤ t✐♠❡❧✐❦❡ ❜❛s✐s ✈❡❝t♦r υ a ❛♥❞ s♣❛❝❡❧✐❦❡ ❜❛s✐s ✈❡❝t♦rs x a ✱ y a ❛♥❞ z a ✐s s②♠♠❡tr✐❝ s❛t✐s✜❡s t❤❡ ❡♥❡r❣② ❝♦♥❞✐t✐♦♥✿ T ab υ a υ b ≥ 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❙♣❛❝❡t✐♠❡ ♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s str❡ss✲❡♥❡r❣②✲♠♦♠❡♥t✉♠ t❡♥s♦r T ab ❉❡❝♦♠♣♦s❡❞ ✐♥ ❡♥❡r❣② E ✱ ♠♦♠❡♥t✉♠ ✈❡❝t♦r p ❛♥❞ str❡ss t❡♥s♦r✿ T ab υ a υ b = E T ab υ a x b = p x T ab υ a y b = p y T ab υ a z b = p x   T ab x a x b = σ xx T ab x a y b = σ xy T ab x a z b = σ xz   T ab y a y b = σ yy T ab y a z b = σ yz     T ab z a z b = σ zz ❢♦r ❛♥ ♦rt❤♦♥♦r♠❛❧ ❧♦❝❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✇✐t❤ t✐♠❡❧✐❦❡ ❜❛s✐s ✈❡❝t♦r υ a ❛♥❞ s♣❛❝❡❧✐❦❡ ❜❛s✐s ✈❡❝t♦rs x a ✱ y a ❛♥❞ z a ✐s s②♠♠❡tr✐❝ s❛t✐s✜❡s t❤❡ ❡♥❡r❣② ❝♦♥❞✐t✐♦♥✿ T ab υ a υ b ≥ 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❊✐♥st❡✐♥✬s ❋✐❡❧❞ ❊q✉❛t✐♦♥ ✐♥ ♠❛ss ✉♥✐ts ✭ c = G = 1 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ G ab = 8 π T ab ❚❤❡ ♠❡tr✐❝ ✐s ✐♠♣❧✐❝✐t ✐♥ T ab ❛s ✇❡❧❧ ❛s G ab ✦ ❧❡❛❞✐♥❣ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ t♦ ❝♦♠♣r✐s❡ ❛ ❝♦✉♣❧❡❞✱ ♥♦♥✲❧✐♥❡❛r✱ s❡❝♦♥❞ ♦r❞❡r P❉❊ s②st❡♠ ❢♦r t❤❡ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❇✐❛♥❝❤✐ ■❞❡♥t✐t② ❊q✉❛t✐♦♥ ♦❢ ▼♦t✐♦♥ ∇ a G ab = 0 ∇ a T ab = 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❊✐♥st❡✐♥✬s ❋✐❡❧❞ ❊q✉❛t✐♦♥ ✐♥ ♠❛ss ✉♥✐ts ✭ c = G = 1 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ G ab = 8 π T ab ❚❤❡ ♠❡tr✐❝ ✐s ✐♠♣❧✐❝✐t ✐♥ T ab ❛s ✇❡❧❧ ❛s G ab ✦ ❧❡❛❞✐♥❣ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ t♦ ❝♦♠♣r✐s❡ ❛ ❝♦✉♣❧❡❞✱ ♥♦♥✲❧✐♥❡❛r✱ s❡❝♦♥❞ ♦r❞❡r P❉❊ s②st❡♠ ❢♦r t❤❡ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❇✐❛♥❝❤✐ ■❞❡♥t✐t② ❊q✉❛t✐♦♥ ♦❢ ▼♦t✐♦♥ ∇ a G ab = 0 ∇ a T ab = 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❊✐♥st❡✐♥✬s ❋✐❡❧❞ ❊q✉❛t✐♦♥ ✐♥ ♠❛ss ✉♥✐ts ✭ c = G = 1 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ G ab = 8 π T ab ❚❤❡ ♠❡tr✐❝ ✐s ✐♠♣❧✐❝✐t ✐♥ T ab ❛s ✇❡❧❧ ❛s G ab ✦ ❧❡❛❞✐♥❣ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ t♦ ❝♦♠♣r✐s❡ ❛ ❝♦✉♣❧❡❞✱ ♥♦♥✲❧✐♥❡❛r✱ s❡❝♦♥❞ ♦r❞❡r P❉❊ s②st❡♠ ❢♦r t❤❡ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❇✐❛♥❝❤✐ ■❞❡♥t✐t② ❊q✉❛t✐♦♥ ♦❢ ▼♦t✐♦♥ ∇ a G ab = 0 ∇ a T ab = 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❊✐♥st❡✐♥✬s ❋✐❡❧❞ ❊q✉❛t✐♦♥ ✐♥ ♠❛ss ✉♥✐ts ✭ c = G = 1 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ G ab = 8 π T ab ❚❤❡ ♠❡tr✐❝ ✐s ✐♠♣❧✐❝✐t ✐♥ T ab ❛s ✇❡❧❧ ❛s G ab ✦ ❧❡❛❞✐♥❣ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ t♦ ❝♦♠♣r✐s❡ ❛ ❝♦✉♣❧❡❞✱ ♥♦♥✲❧✐♥❡❛r✱ s❡❝♦♥❞ ♦r❞❡r P❉❊ s②st❡♠ ❢♦r t❤❡ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❇✐❛♥❝❤✐ ■❞❡♥t✐t② ❊q✉❛t✐♦♥ ♦❢ ▼♦t✐♦♥ ∇ a G ab = 0 ∇ a T ab = 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❊✐♥st❡✐♥✬s ❋✐❡❧❞ ❊q✉❛t✐♦♥ ✐♥ ♠❛ss ✉♥✐ts ✭ c = G = 1 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ G ab = 8 π T ab ❚❤❡ ♠❡tr✐❝ ✐s ✐♠♣❧✐❝✐t ✐♥ T ab ❛s ✇❡❧❧ ❛s G ab ✦ ❧❡❛❞✐♥❣ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ t♦ ❝♦♠♣r✐s❡ ❛ ❝♦✉♣❧❡❞✱ ♥♦♥✲❧✐♥❡❛r✱ s❡❝♦♥❞ ♦r❞❡r P❉❊ s②st❡♠ ❢♦r t❤❡ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❇✐❛♥❝❤✐ ■❞❡♥t✐t② ❊q✉❛t✐♦♥ ♦❢ ▼♦t✐♦♥ ∇ a G ab = 0 ∇ a T ab = 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❙✉♠♠❛r② ❊✐♥st❡✐♥✬s ❋✐❡❧❞ ❊q✉❛t✐♦♥ ✐♥ ♠❛ss ✉♥✐ts ✭ c = G = 1 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ G ab = 8 π T ab ❚❤❡ ♠❡tr✐❝ ✐s ✐♠♣❧✐❝✐t ✐♥ T ab ❛s ✇❡❧❧ ❛s G ab ✦ ❧❡❛❞✐♥❣ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ t♦ ❝♦♠♣r✐s❡ ❛ ❝♦✉♣❧❡❞✱ ♥♦♥✲❧✐♥❡❛r✱ s❡❝♦♥❞ ♦r❞❡r P❉❊ s②st❡♠ ❢♦r t❤❡ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❇✐❛♥❝❤✐ ■❞❡♥t✐t② ❊q✉❛t✐♦♥ ♦❢ ▼♦t✐♦♥ ∇ a G ab = 0 ∇ a T ab = 0 ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❖✉t❧✐♥❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ✶ Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❊✐♥st❡✐♥✬s ❚❤❡♦r② ♦❢ ●r❛✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ✷ Pr❡❧✐♠✐♥❛r✐❡s ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼♦t✐✈❡s ❢♦r ❛♥ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥ ✐s ❛ s♣❛❝❡t✐♠❡ ❡q✉❛t✐♦♥✿ Pr❡❞✐❝t❛❜✐❧✐t② ✐s ✐♠♣❧✐❝✐t✳ ◆♦ ❡①♣❡r✐♠❡♥t ❝❛♥ ❜❡ s❡t ♣r✐♦r t♦ ❤❛✈✐♥❣ ❛ s♣❛❝❡t✐♠❡ s♦❧✉t✐♦♥✳ ❖❜s❡r✈❛t✐♦♥s ❛r❡ s♣❛❝❡❧✐❦❡ ✐♥st❛♥❝❡s✦ ■❢ ❛ s♣❛❝❡❧✐❦❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s s❡t✱ ❤♦✇ ✐s ✐ts ❡✈♦❧✉t✐♦♥ ❡①tr❛❝t❡❞ ❢r♦♠ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥❄ ❚❤❡ ❧❛st q✉❡st✐♦♥ ❞❡♠♦♥str❛t❡s t❤❡ ❛❧r❡❛❞② ❦♥♦✇♥ ❛♥❞ ❛❝❝❡♣t❡❞ ♣r♦♣❡rt② t❤❛t ❛❧❧ P❤②s✐❝❛❧ ❚❤❡♦r✐❡s ❤❛✈❡✿ ❛♥ ✏■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥✑ ✇❤✐❝❤ st❛♥❞s ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♥❛t✉r❡ ♦❢ ❛❧❧ t❤❡♦r✐❡s✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❉❡♣❧♦②♠❡♥t G αβ = R αβ − 1 2 Rg αβ = 1 g σρ ( ∂ σ ∂ ρ g αβ + ∂ α ∂ β g σρ − 2 ∂ ρ ∂ ( α g β ) σ ) 2 ∑ σ ∑ ρ − 1 g σρ g αβ ∑ g µν ( ∂ σ ∂ ρ g µν − ∂ ρ ∂ µ g νσ )+ ... 2 ∑ σ ∑ µ ∑ ρ ν ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❉❡♣❧♦②♠❡♥t G αβ = R αβ − 1 2 Rg αβ = 1 g σρ ( ∂ σ ∂ ρ g αβ + ∂ α ∂ β g σρ − 2 ∂ ρ ∂ ( α g β ) σ ) 2 ∑ σ ∑ ρ − 1 g σρ g αβ ∑ g µν ( ∂ σ ∂ ρ g µν − ∂ ρ ∂ µ g νσ )+ ... 2 ∑ σ ∑ µ ∑ ρ ν ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❚❤❡♦r❡♠s ❚❤❡♦r❡♠ ✭❈❛✉❝❤②✲❑♦✇❛❧❡✇s❦✐✮ ❆❧❧ s❡❝♦♥❞ t✐♠❡ ♦r❞❡r P❉❊ s②st❡♠s ∂ 2 φ i ∂ x µ ; ∂ 2 φ i ∂ 2 φ i � t , x µ ; φ i ; ∂φ i ∂ t , ∂φ i � ∂ t 2 = F i ∂ t ∂ x µ , ∂ x µ ∂ x ν ❡♥❞♦✇❡❞ ✇✐t❤ ❛r❜✐tr❛r② ❛♥❛❧②t✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡s � φ i ( 0 , x µ ) = f i ( x µ ) and ∂φ i � ∂ t ( 0 , x µ ) = g i ( x µ ) ∈ C ω [ R dim M − 1 | R ] ❝♦♥st✐t✉t❡ ❛ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ ❛♥❛❧②t✐❝ s♦❧✉t✐♦♥✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❚❤❡♦r❡♠s ❚❤❡♦r❡♠ ✭❈❛✉❝❤②✲❑♦✇❛❧❡✇s❦✐✮ ❆❧❧ s❡❝♦♥❞ t✐♠❡ ♦r❞❡r P❉❊ s②st❡♠s ∂ 2 φ i ∂ x µ ; ∂ 2 φ i ∂ 2 φ i � t , x µ ; φ i ; ∂φ i ∂ t , ∂φ i � ∂ t 2 = F i ∂ t ∂ x µ , ∂ x µ ∂ x ν ❡♥❞♦✇❡❞ ✇✐t❤ ❛r❜✐tr❛r② ❛♥❛❧②t✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡s � φ i ( 0 , x µ ) = f i ( x µ ) and ∂φ i � ∂ t ( 0 , x µ ) = g i ( x µ ) ∈ C ω [ R dim M − 1 | R ] ❝♦♥st✐t✉t❡ ❛ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ ❛♥❛❧②t✐❝ s♦❧✉t✐♦♥✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆ss✉♠♣t✐♦♥s✱ ❙♣❛❝❡t✐♠❡ ✐s ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝✿ ✐t ❛❞♠✐ts ❛ ♠♦♥♣❛r❛♠❡tr✐❝ ❢♦❧✐❛t✐♦♥ ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡s ❛❧❧ ♦❢ s♣❛❝❡t✐♠❡ ✐s ❡✐t❤❡r ❢✉t✉r❡ ♦r ♣❛st t✐♠❡✲❞❡♣❡♥❞❡❞ ♦♥ ❡✈❡♥ts ♦♥ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❛ ❈❛✉❝❤② ❤②♣❡rs✉r❢❛❝❡ ❝✉ts t❤r♦✉❣❤ s♣❛❝❡t✐♠❡ s❡♣❡r❛t✐♥❣ ✐♥ ✐♥ ❛ ♣❛st ❛♥❞ ❛ ❢✉t✉r❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❆♥❛❧②t✐❝ s♦❧✉t✐♦♥s ❞♦ ♥♦t ✇♦r❦ ♦♥ ❝❛✉s❛❧ s♣❛❝❡t✐♠❡s✳ ❣❧♦❜❛❧❧② ❤②♣❡r❜♦❧✐❝ s♣❛❝❡t✐♠❡s ❛r❡ st❛❜❧② ❝❛s✉❛❧ ❛ss✉♠✐♥❣ ❛t ♠♦st ❞✐✛❡r❡♥t✐❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ s♦❧✉t✐♦♥s ♥♦ ❣❡♥❡r✐❝ t❤❡♦r❡♠s ❢♦r ✐t✦ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❚❤❡♦r❡♠s ❚❤❡♦r❡♠ ❆❧❧ ❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ P❉❊ s②st❡♠s ♦♥ M g ab ∇ a ∇ b φ i + ∑ ( A i j ) a ∇ a φ j + ∑ B i j φ j + C i = 0 j j ❡♥❞♦✇❡❞ ✇✐t❤ ❛r❜✐tr❛r② s♠♦♦t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦♥ Σ ✱ φ i ❛♥❞ n a ∇ a φ i ❝♦♥st✐t✉t❡ ❛ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ s♠♦♦t❤ s♦❧✉t✐♦♥✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❚❤❡♦r❡♠s ❚❤❡♦r❡♠ ❆❧❧ ❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ P❉❊ s②st❡♠s ♦♥ M g ab ∇ a ∇ b φ i + ∑ ( A i j ) a ∇ a φ j + ∑ B i j φ j + C i = 0 j j ❡♥❞♦✇❡❞ ✇✐t❤ ❛r❜✐tr❛r② s♠♦♦t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦♥ Σ ✱ φ i ❛♥❞ n a ∇ a φ i ❝♦♥st✐t✉t❡ ❛ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ s♠♦♦t❤ s♦❧✉t✐♦♥✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❚❤❡♦r❡♠s ❆❧❧ q✉❛s✐ ✲❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ s②st❡♠s ♦♥ M g ab ( φ j | ∇ c φ j ) ∇ a ( φ j | ∇ c φ j ) ∇ b ( φ j | ∇ c φ j ) φ i = F i ( φ j | ∇ c φ j ) ❡♥❞♦✇❡❞ ✇✐t❤ s♠♦♦t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦♥ Σ ( φ i and n a ∇ a φ i ) ∈ C ∞ [ Σ | R n ] ❧♦❝❛❧❧② s✉✣❝✐❡♥t❧② ❝❧♦s❡ t♦ t❤♦s❡ ♦❢ ❛ ❜❛❝❦❣r♦✉♥❞ s♦❧✉t✐♦♥✱ ❝♦♥st✐t✉t❡ ❛ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ s♠♦♦t❤ s♦❧✉t✐♦♥✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❚❤❡♦r❡♠s ❆❧❧ q✉❛s✐ ✲❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ s②st❡♠s ♦♥ M g ab ( φ j | ∇ c φ j ) ∇ a ( φ j | ∇ c φ j ) ∇ b ( φ j | ∇ c φ j ) φ i = F i ( φ j | ∇ c φ j ) ❡♥❞♦✇❡❞ ✇✐t❤ s♠♦♦t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s ♦♥ Σ ( φ i and n a ∇ a φ i ) ∈ C ∞ [ Σ | R n ] ❧♦❝❛❧❧② s✉✣❝✐❡♥t❧② ❝❧♦s❡ t♦ t❤♦s❡ ♦❢ ❛ ❜❛❝❦❣r♦✉♥❞ s♦❧✉t✐♦♥✱ ❝♦♥st✐t✉t❡ ❛ ✇❡❧❧ ♣♦s❡❞ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ s♠♦♦t❤ s♦❧✉t✐♦♥✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❖✉t❧✐♥❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ✶ Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❊✐♥st❡✐♥✬s ❚❤❡♦r② ♦❢ ●r❛✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ✷ Pr❡❧✐♠✐♥❛r✐❡s ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ♦❢ s♣❛❝❡t✐♠❡ ♠❡tr✐❝ g ab ✐♥t♦ ❛ s♣❛t✐❛❧ ♠❡tr✐❝ h ab ❛♥❞ ♠♦r❡✳✳✳ ∀ υ a s✉❝❤✱ t❤❛t υ a ∇ a t = 1 ✿ ❝♦✈❛r✐❛♥t ❞❡❝♦♠♣♦s✐t✐♦♥❄ g 00 = h i j N i N j − NN h ab = g ab + n a n b N = − υ a n a = ( n a ∇ a t ) − 1 g i 0 = N i / g 0 j = N j N a = h ab υ b g i j = h i j ✐♥ ❛❞❛♣t❡❞ ❝♦♦r❞✐♥❛t❡s g tt = h i j N i N j − NN  g tx = N x g ty = N y g tz = N z  g xt = N x g xx = h xx g xy = h xy g xz = h xz     g yt = N y g yx = h yx g yy = h yy g yz = h yz   g zt = N z g zx = h zx g zy = h zy g zz = h zz ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ♦❢ s♣❛❝❡t✐♠❡ ♠❡tr✐❝ g ab ✐♥t♦ ❛ s♣❛t✐❛❧ ♠❡tr✐❝ h ab ❛♥❞ ♠♦r❡✳✳✳ ∀ υ a s✉❝❤✱ t❤❛t υ a ∇ a t = 1 ✿ ❝♦✈❛r✐❛♥t ❞❡❝♦♠♣♦s✐t✐♦♥❄ g 00 = h i j N i N j − NN h ab = g ab + n a n b N = − υ a n a = ( n a ∇ a t ) − 1 g i 0 = N i / g 0 j = N j N a = h ab υ b g i j = h i j ✐♥ ❛❞❛♣t❡❞ ❝♦♦r❞✐♥❛t❡s g tt = h i j N i N j − NN  g tx = N x g ty = N y g tz = N z  g xt = N x g xx = h xx g xy = h xy g xz = h xz     g yt = N y g yx = h yx g yy = h yy g yz = h yz   g zt = N z g zx = h zx g zy = h zy g zz = h zz ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ♦❢ s♣❛❝❡t✐♠❡ ♠❡tr✐❝ g ab ✐♥t♦ ❛ s♣❛t✐❛❧ ♠❡tr✐❝ h ab ❛♥❞ ♠♦r❡✳✳✳ ∀ υ a s✉❝❤✱ t❤❛t υ a ∇ a t = 1 ✿ ❝♦✈❛r✐❛♥t ❞❡❝♦♠♣♦s✐t✐♦♥❄ g 00 = h i j N i N j − NN h ab = g ab + n a n b N = − υ a n a = ( n a ∇ a t ) − 1 g i 0 = N i / g 0 j = N j N a = h ab υ b g i j = h i j ✐♥ ❛❞❛♣t❡❞ ❝♦♦r❞✐♥❛t❡s g tt = h i j N i N j − NN  g tx = N x g ty = N y g tz = N z  g xt = N x g xx = h xx g xy = h xy g xz = h xz     g yt = N y g yx = h yx g yy = h yy g yz = h yz   g zt = N z g zx = h zx g zy = h zy g zz = h zz ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ♦❢ s♣❛❝❡t✐♠❡ ♠❡tr✐❝ g ab ✐♥t♦ ❛ s♣❛t✐❛❧ ♠❡tr✐❝ h ab ❛♥❞ ♠♦r❡✳✳✳ ∀ υ a s✉❝❤✱ t❤❛t υ a ∇ a t = 1 ✿ ❝♦✈❛r✐❛♥t ❞❡❝♦♠♣♦s✐t✐♦♥❄ g 00 = h i j N i N j − NN h ab = g ab + n a n b N = − υ a n a = ( n a ∇ a t ) − 1 g i 0 = N i / g 0 j = N j N a = h ab υ b g i j = h i j ✐♥ ❛❞❛♣t❡❞ ❝♦♦r❞✐♥❛t❡s g tt = h i j N i N j − NN  g tx = N x g ty = N y g tz = N z  g xt = N x g xx = h xx g xy = h xy g xz = h xz     g yt = N y g yx = h yx g yy = h yy g yz = h yz   g zt = N z g zx = h zx g zy = h zy g zz = h zz ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡q✉❛t✐♦♥s a 0 ❡q✉❛t✐♦♥s ab G ab n b = 8 π T ab n b G ab = 8 π T ab ❡q✉❛t✐♦♥s i j ❡q✉❛t✐♦♥ 00 ❡q✉❛t✐♦♥s i 0 G ab n a n b = 8 πρ a G cb n b = 8 π J a h c h c a h d b G cd = 8 πσ ab ρ = T ab n a n b J a = h c a G cb n b σ ab = h c a h d b T cd ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡q✉❛t✐♦♥s a 0 ❡q✉❛t✐♦♥s ab G ab n b = 8 π T ab n b G ab = 8 π T ab ❡q✉❛t✐♦♥s i j ❡q✉❛t✐♦♥ 00 ❡q✉❛t✐♦♥s i 0 G ab n a n b = 8 πρ a G cb n b = 8 π J a h c h c a h d b G cd = 8 πσ ab ρ = T ab n a n b J a = h c a G cb n b σ ab = h c a h d b T cd ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡q✉❛t✐♦♥s a 0 ❡q✉❛t✐♦♥s ab G ab n b = 8 π T ab n b G ab = 8 π T ab ❡q✉❛t✐♦♥s i j ❡q✉❛t✐♦♥ 00 ❡q✉❛t✐♦♥s i 0 G ab n a n b = 8 πρ a G cb n b = 8 π J a h c h c a h d b G cd = 8 πσ ab ρ = T ab n a n b J a = h c a G cb n b σ ab = h c a h d b T cd ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡q✉❛t✐♦♥s a 0 ❡q✉❛t✐♦♥s ab G ab n b = 8 π T ab n b G ab = 8 π T ab ❡q✉❛t✐♦♥s i j ❡q✉❛t✐♦♥ 00 ❡q✉❛t✐♦♥s i 0 G ab n a n b = 8 πρ a G cb n b = 8 π J a h c h c a h d b G cd = 8 πσ ab ρ = T ab n a n b J a = h c a G cb n b σ ab = h c a h d b T cd ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❊✐♥st❡✐♥✬s ❊q✉❛t✐♦♥ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡q✉❛t✐♦♥s a 0 ❡q✉❛t✐♦♥s ab G ab n b = 8 π T ab n b G ab = 8 π T ab ❡q✉❛t✐♦♥s i j ❡q✉❛t✐♦♥ 00 ❡q✉❛t✐♦♥s i 0 G ab n a n b = 8 πρ a G cb n b = 8 π J a h c h c a h d b G cd = 8 πσ ab ρ = T ab n a n b J a = h c a G cb n b σ ab = h c a h d b T cd ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❖✉t❧✐♥❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ✶ Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❊✐♥st❡✐♥✬s ❚❤❡♦r② ♦❢ ●r❛✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ✷ Pr❡❧✐♠✐♥❛r✐❡s ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ♦♥ Σ 0 ♦❢ t❤❡ ❢♦❧✐❛t✐♦♥ ❛s ✐♥✐t✐❛❧ ✈❛❧✉❡ s♣❛❝❡✳ s♣❛t✐❛❧ ♠❡tr✐❝ ■♥✐t✐❛❧✐③❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡s ♦❢ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❡①t❡r✐♦r ❝✉r✈❛t✉r❡ K ab : = D a n b = 1 2 ➾ n h ab ❝♦✈❛r✐❛♥t t✐♠❡ ❞❡r✐✈❛t✐✈❡ 1 2 ➾ t h ab = NK ab + 1 2 ➾ N h ab ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ♦♥ Σ 0 ♦❢ t❤❡ ❢♦❧✐❛t✐♦♥ ❛s ✐♥✐t✐❛❧ ✈❛❧✉❡ s♣❛❝❡✳ s♣❛t✐❛❧ ♠❡tr✐❝ ■♥✐t✐❛❧✐③❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡s ♦❢ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❡①t❡r✐♦r ❝✉r✈❛t✉r❡ K ab : = D a n b = 1 2 ➾ n h ab ❝♦✈❛r✐❛♥t t✐♠❡ ❞❡r✐✈❛t✐✈❡ 1 2 ➾ t h ab = NK ab + 1 2 ➾ N h ab ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ♦♥ Σ 0 ♦❢ t❤❡ ❢♦❧✐❛t✐♦♥ ❛s ✐♥✐t✐❛❧ ✈❛❧✉❡ s♣❛❝❡✳ s♣❛t✐❛❧ ♠❡tr✐❝ ■♥✐t✐❛❧✐③❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡s ♦❢ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❡①t❡r✐♦r ❝✉r✈❛t✉r❡ K ab : = D a n b = 1 2 ➾ n h ab ❝♦✈❛r✐❛♥t t✐♠❡ ❞❡r✐✈❛t✐✈❡ 1 2 ➾ t h ab = NK ab + 1 2 ➾ N h ab ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ♦♥ Σ 0 ♦❢ t❤❡ ❢♦❧✐❛t✐♦♥ ❛s ✐♥✐t✐❛❧ ✈❛❧✉❡ s♣❛❝❡✳ s♣❛t✐❛❧ ♠❡tr✐❝ ■♥✐t✐❛❧✐③❛t✐♦♥ ♦♥ t❤❡ ❧✐♥❡s ♦❢ ❆❉▼ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❡①t❡r✐♦r ❝✉r✈❛t✉r❡ K ab : = D a n b = 1 2 ➾ n h ab ❝♦✈❛r✐❛♥t t✐♠❡ ❞❡r✐✈❛t✐✈❡ 1 2 ➾ t h ab = NK ab + 1 2 ➾ N h ab ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ❈♦♥str❛✐♥ts✳ ❝♦♥str❛✐♥t 0 G ab n a n b = 1 2 ( ( 3 ) R + KK − K ab K ab ) = 8 πρ ❝♦♥tr❛✐♥t i b G cd n d = D a ( K ab − Kh ab ) = 8 π J b h c ✹ ♠❡tr✐❝ ♥♦♥✲❞❡✈❡❧♦♣♠❡♥t ❡q✉❛t✐♦♥s ❛❧❧♦✇✐♥❣ ✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ r❡❧❡✈❛♥t t♦ ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡ ♦❢ s♦❧✉t✐♦♥✱ ❡♠♣❧♦② ❝♦♦r❞✐♥❛t❡❞ t♦ ✜① ❣✉❛❣❡✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ❈♦♥str❛✐♥ts✳ ❝♦♥str❛✐♥t 0 G ab n a n b = 1 2 ( ( 3 ) R + KK − K ab K ab ) = 8 πρ ❝♦♥tr❛✐♥t i b G cd n d = D a ( K ab − Kh ab ) = 8 π J b h c ✹ ♠❡tr✐❝ ♥♦♥✲❞❡✈❡❧♦♣♠❡♥t ❡q✉❛t✐♦♥s ❛❧❧♦✇✐♥❣ ✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ r❡❧❡✈❛♥t t♦ ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡ ♦❢ s♦❧✉t✐♦♥✱ ❡♠♣❧♦② ❝♦♦r❞✐♥❛t❡❞ t♦ ✜① ❣✉❛❣❡✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ❈♦♥str❛✐♥ts✳ ❝♦♥str❛✐♥t 0 G ab n a n b = 1 2 ( ( 3 ) R + KK − K ab K ab ) = 8 πρ ❝♦♥tr❛✐♥t i b G cd n d = D a ( K ab − Kh ab ) = 8 π J b h c ✹ ♠❡tr✐❝ ♥♦♥✲❞❡✈❡❧♦♣♠❡♥t ❡q✉❛t✐♦♥s ❛❧❧♦✇✐♥❣ ✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ r❡❧❡✈❛♥t t♦ ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡ ♦❢ s♦❧✉t✐♦♥✱ ❡♠♣❧♦② ❝♦♦r❞✐♥❛t❡❞ t♦ ✜① ❣✉❛❣❡✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ❈♦♥str❛✐♥ts✳ ❝♦♥str❛✐♥t 0 G ab n a n b = 1 2 ( ( 3 ) R + KK − K ab K ab ) = 8 πρ ❝♦♥tr❛✐♥t i b G cd n d = D a ( K ab − Kh ab ) = 8 π J b h c ✹ ♠❡tr✐❝ ♥♦♥✲❞❡✈❡❧♦♣♠❡♥t ❡q✉❛t✐♦♥s ❛❧❧♦✇✐♥❣ ✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ r❡❧❡✈❛♥t t♦ ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡ ♦❢ s♦❧✉t✐♦♥✱ ❡♠♣❧♦② ❝♦♦r❞✐♥❛t❡❞ t♦ ✜① ❣✉❛❣❡✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ■♥✐t✐❛❧ ❱❛❧✉❡s✱ ❈♦♥str❛✐♥ts✳ ❝♦♥str❛✐♥t 0 G ab n a n b = 1 2 ( ( 3 ) R + KK − K ab K ab ) = 8 πρ ❝♦♥tr❛✐♥t i b G cd n d = D a ( K ab − Kh ab ) = 8 π J b h c ✹ ♠❡tr✐❝ ♥♦♥✲❞❡✈❡❧♦♣♠❡♥t ❡q✉❛t✐♦♥s ❛❧❧♦✇✐♥❣ ✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ r❡❧❡✈❛♥t t♦ ❣❡♥❡r❛❧ ❝♦✈❛r✐❛♥❝❡ ♦❢ s♦❧✉t✐♦♥✱ ❡♠♣❧♦② ❝♦♦r❞✐♥❛t❡❞ t♦ ✜① ❣✉❛❣❡✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❖✉t❧✐♥❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ✶ Pr✐♥❝✐♣❧❡s ♦❢ ❊✐♥st❡✐♥✬s ❚❤❡♦r② ❊✐♥st❡✐♥✬s ❚❤❡♦r② ♦❢ ●r❛✈✐t② ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ✷ Pr❡❧✐♠✐♥❛r✐❡s ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ●❛✉❣❡ ✜①✐♥❣✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❣❛✉❣❡ ❢r❡❡❞♦♠ ✭✜①❡❞ ❜② ❡♠♣❧♦②✐♥❣ ❤❛r♠♦♥✐❝ ❝♦♦r❞✐♥❛t❡s✮ g νµ ∑ ∂ ν g νµ + 1 � x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ ν g αβ = 0 2 ∑ α ∑ ν ν β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♦♥ R µν = F µν + 1 g αβ ∂ α ∂ β g µν = 0 2 ∑ α ∑ β ❈♦♠♣❛t✐❜✐❧✐t② ✇✐t❤ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♦s❡❞ ❧♦❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ P❉❊ s②st❡♠ ❢♦r � x µ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ●❛✉❣❡ ✜①✐♥❣✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❣❛✉❣❡ ❢r❡❡❞♦♠ ✭✜①❡❞ ❜② ❡♠♣❧♦②✐♥❣ ❤❛r♠♦♥✐❝ ❝♦♦r❞✐♥❛t❡s✮ g νµ ∑ ∂ ν g νµ + 1 � x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ ν g αβ = 0 2 ∑ α ∑ ν ν β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♦♥ R µν = F µν + 1 g αβ ∂ α ∂ β g µν = 0 2 ∑ α ∑ β ❈♦♠♣❛t✐❜✐❧✐t② ✇✐t❤ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♦s❡❞ ❧♦❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ P❉❊ s②st❡♠ ❢♦r � x µ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ●❛✉❣❡ ✜①✐♥❣✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❣❛✉❣❡ ❢r❡❡❞♦♠ ✭✜①❡❞ ❜② ❡♠♣❧♦②✐♥❣ ❤❛r♠♦♥✐❝ ❝♦♦r❞✐♥❛t❡s✮ g νµ ∑ ∂ ν g νµ + 1 � x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ ν g αβ = 0 2 ∑ α ∑ ν ν β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♦♥ R µν = F µν + 1 g αβ ∂ α ∂ β g µν = 0 2 ∑ α ∑ β ❈♦♠♣❛t✐❜✐❧✐t② ✇✐t❤ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♦s❡❞ ❧♦❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ P❉❊ s②st❡♠ ❢♦r � x µ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ●❛✉❣❡ ✜①✐♥❣✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❣❛✉❣❡ ❢r❡❡❞♦♠ ✭✜①❡❞ ❜② ❡♠♣❧♦②✐♥❣ ❤❛r♠♦♥✐❝ ❝♦♦r❞✐♥❛t❡s✮ g νµ ∑ ∂ ν g νµ + 1 � x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ ν g αβ = 0 2 ∑ α ∑ ν ν β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♦♥ R µν = F µν + 1 g αβ ∂ α ∂ β g µν = 0 2 ∑ α ∑ β ❈♦♠♣❛t✐❜✐❧✐t② ✇✐t❤ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♦♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♦s❡❞ ❧♦❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ P❉❊ s②st❡♠ ❢♦r � x µ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❉❡✈❡❧♦♣♠❡♥t✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❚❤❡ r❡st ✻ ❡q✉❛t✐♦♥s ❛r❡ q✉❛s✐ ✲❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ ❢♦r t❤❡ ♣✉r❡❧② s♣❛t✐❛❧ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❚❛❦✐♥❣ t❤❡ ✢❛t ✭▼✐♥❦♦✇s❦✐✮ ♠❡tr✐❝ η ab ❛s ❜❛❝❦❣r♦✉♥❞ s♦❧✉t✐♦♥ ❧♦❝❛❧ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ✭♠♦❞✉❧♦ ❞✐✛❡♦♠♦r♣❤✐s♠s✮ ❤♦❧❞s✳ ▲♦❝❛❧ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ✏♣❛t❝❤❡❞✑ t♦ ❢♦r♠ ❣❧♦❜❛❧ s♦❧✉t✐♦♥s ✭❝♦♠✐♥❣ ✉♣✦✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❉❡✈❡❧♦♣♠❡♥t✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❚❤❡ r❡st ✻ ❡q✉❛t✐♦♥s ❛r❡ q✉❛s✐ ✲❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ ❢♦r t❤❡ ♣✉r❡❧② s♣❛t✐❛❧ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❚❛❦✐♥❣ t❤❡ ✢❛t ✭▼✐♥❦♦✇s❦✐✮ ♠❡tr✐❝ η ab ❛s ❜❛❝❦❣r♦✉♥❞ s♦❧✉t✐♦♥ ❧♦❝❛❧ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ✭♠♦❞✉❧♦ ❞✐✛❡♦♠♦r♣❤✐s♠s✮ ❤♦❧❞s✳ ▲♦❝❛❧ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ✏♣❛t❝❤❡❞✑ t♦ ❢♦r♠ ❣❧♦❜❛❧ s♦❧✉t✐♦♥s ✭❝♦♠✐♥❣ ✉♣✦✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❉❡✈❡❧♦♣♠❡♥t✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❚❤❡ r❡st ✻ ❡q✉❛t✐♦♥s ❛r❡ q✉❛s✐ ✲❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ ❢♦r t❤❡ ♣✉r❡❧② s♣❛t✐❛❧ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❚❛❦✐♥❣ t❤❡ ✢❛t ✭▼✐♥❦♦✇s❦✐✮ ♠❡tr✐❝ η ab ❛s ❜❛❝❦❣r♦✉♥❞ s♦❧✉t✐♦♥ ❧♦❝❛❧ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ✭♠♦❞✉❧♦ ❞✐✛❡♦♠♦r♣❤✐s♠s✮ ❤♦❧❞s✳ ▲♦❝❛❧ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ✏♣❛t❝❤❡❞✑ t♦ ❢♦r♠ ❣❧♦❜❛❧ s♦❧✉t✐♦♥s ✭❝♦♠✐♥❣ ✉♣✦✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ❉❡✈❡❧♦♣♠❡♥t✱ ❢♦r ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❚❤❡ r❡st ✻ ❡q✉❛t✐♦♥s ❛r❡ q✉❛s✐ ✲❧✐♥❡❛r✱ ❞✐❛❣♦♥❛❧✱ s❡❝♦♥❞ ♦r❞❡r ❤②♣❡r❜♦❧✐❝ ❢♦r t❤❡ ♣✉r❡❧② s♣❛t✐❛❧ ♠❡tr✐❝ ❝♦♠♣♦♥❡♥ts✳ ❚❛❦✐♥❣ t❤❡ ✢❛t ✭▼✐♥❦♦✇s❦✐✮ ♠❡tr✐❝ η ab ❛s ❜❛❝❦❣r♦✉♥❞ s♦❧✉t✐♦♥ ❧♦❝❛❧ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ✭♠♦❞✉❧♦ ❞✐✛❡♦♠♦r♣❤✐s♠s✮ ❤♦❧❞s✳ ▲♦❝❛❧ s♦❧✉t✐♦♥s ❝❛♥ ❜❡ ✏♣❛t❝❤❡❞✑ t♦ ❢♦r♠ ❣❧♦❜❛❧ s♦❧✉t✐♦♥s ✭❝♦♠✐♥❣ ✉♣✦✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ s♦❧✈✐♥❣ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s R ab = 0 ✳ ❙♦❧✈❡ ✈❛❝✉✉♠ ❡q✉❛t✐♦♥s ❧♦❝❛❧❧② ♦♥ ❛❧❧ ❡✈❡♥ts ♦♥ Σ ✳ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ s♦❧✈❡❞ ✜❧♠ ♣r♦①✐♠❛ t♦ ❡♥t✐r❡ Σ ✳ ❚❛❦❡ ❛❧❧ s✉❝❤ ✭❧♦❝❛❧❧②✮ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ❈♦♠♣❛r❡ ❛♥② ♣❛✐r ♦❢ ❝❧❛ss❡s ♦❢ ❞✐✛❡♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ ⊆ ✱ t❤✉s ♣❛rt✐❛❧❧② ♦r❞❡r✐♥❣ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ♦♥ ❡♥t✐r❡ Σ ✳ ⊆ ✲❝❤❛✐♥s ❛r❡ ❛❧✇❛②s ✉♣✲❜♦✉♥❞ ❜② t❤❡ ✉♥✐♦♥ ❝♦♥t❡♥t ♦❢ ❡❛❝❤✱ t❤✉s ❣❡♥❡r❛t✐♥❣ ❛ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥ ♦♥ ❡♥t✐r❡ Σ ✳ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ❘❡❧❛t✐✈✐t② ❉❡♣❧♦②✐♥❣ t❤❡ Pr♦❜❧❡♠ ❚❤❡ ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ■♥✐t✐❛❧ ❱❛❧✉❡s ❛♥❞ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t ❙✉♠♠❛r② ❉❡✈❡❧♦♣♠❡♥t ❊q✉❛t✐♦♥s ✭✐♥ ✈❛❝✉✉♠✱ R ab = 0 ✮ ▼❛①✐♠❛❧ ❈❛✉❝❤② ❉❡✈❡❧♦♣♠❡♥t✱ ♥♦t ❡♥♦✉❣❤❄ ❈❛✉❝❤② ❞❡✈❡❧♦♣♠❡♥t ❜r❡❛❦s ❞♦✇♥ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ s✐♥❣✉❧❛r✐t✐❡s✱ ♠❛①✐♠❛❧ s♦❧✉t✐♦♥s ❛r❡ ♥♦t ♥❡❝❡ss❛r✐❧② ❣❡♦❞❡s✐❝❛❧❧② ❝♦♠♣❧❡t❡✳ ❋♦r ❛s②♠♣t♦t✐❝❛❧❧② ✢❛t ✐♥✐t✐❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❡tr✐❝✱ ❈❛✉❝❤② ❞❡✈❡❧♦♣♠❡♥t ❝❛rr✐❡s ✉♥❝♦♥❞✐t✐♦♥❛❧❧② ❛s②♠♣r♦t✐❝❛❧❧②✳ ✭❈❤r✐st♦❞♦✉❧♦✉ ✫ ❖✬▼✉r❝❤❛❞❤❛✱ ✶✾✽✶✮ ❙tr❛t♦s ❈❤✳ P❛♣❛❞♦✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②