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t r t

t rt r - PowerPoint PPT Presentation

Sep 22, 2023 •153 likes •1.43k views

trt r r tt t rt r t

  1. ■♥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

  2. ■♥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

  3. ■♥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

  4. ■♥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

  5. ■♥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

  6. ■♥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

  7. ■♥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

  8. ■♥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

  9. ■♥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

  10. ■♥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

  11. ■♥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

  12. ■♥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

  13. ■♥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

  14. ■♥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

  15. ■♥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

  16. ■♥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

  17. ■♥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

  18. ■♥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

  19. ■♥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

  20. ■♥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

  21. ■♥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

  22. ■♥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

  23. ■♥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✐❝ ï¿œ ❎ | ❎ ï¿œ ✿ 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

  24. ■♥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✐❝ ï¿œ ❎ | ❎ ï¿œ ✿ 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

  25. ■♥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

  26. ■♥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

  27. ■♥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

  28. ■♥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

  29. ■♥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

  30. ■♥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

  31. ■♥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

  32. ■♥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

  33. ■♥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

  34. ■♥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

  35. ■♥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

  36. ■♥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

  37. ■♥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

  38. ■♥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

  39. ■♥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

  40. ■♥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

  41. ■♥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

  42. ■♥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 µΜ ( ∂ σ ∂ ρ g µΜ − ∂ ρ ∂ µ g Μσ )+ ... 2 ∑ σ ∑ µ ∑ ρ Îœ ❙tr❛t♩s ❈❀✳ P❛♣❛❞♊✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♩r♠✉❧❛t✐♊♥ ♊❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

  43. ■♥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 µΜ ( ∂ σ ∂ ρ g µΜ − ∂ ρ ∂ µ g Μσ )+ ... 2 ∑ σ ∑ µ ∑ ρ Îœ ❙tr❛t♩s ❈❀✳ P❛♣❛❞♊✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♩r♠✉❧❛t✐♊♥ ♊❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

  44. ■♥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 ï¿œ t , x µ ; φ i ; ∂φ i ∂ t , ∂φ i ï¿œ ∂ t 2 = F i ∂ t ∂ x µ , ∂ x µ ∂ x Îœ ❡♥❞♊✇❡❞ ✇✐t❀ ❛r❜✐tr❛r② ❛♥❛❧②t✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡s ï¿œ φ i ( 0 , x µ ) = f i ( x µ ) and ∂φ i ï¿œ ∂ 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

  45. ■♥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 ï¿œ t , x µ ; φ i ; ∂φ i ∂ t , ∂φ i ï¿œ ∂ t 2 = F i ∂ t ∂ x µ , ∂ x µ ∂ x Îœ ❡♥❞♊✇❡❞ ✇✐t❀ ❛r❜✐tr❛r② ❛♥❛❧②t✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡s ï¿œ φ i ( 0 , x µ ) = f i ( x µ ) and ∂φ i ï¿œ ∂ 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

  46. ■♥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

  47. ■♥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

  48. ■♥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

  49. ■♥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

  50. ■♥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

  51. ■♥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

  52. ■♥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

  53. ■♥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

  54. ■♥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

  55. ■♥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

  56. ■♥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

  57. ■♥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

  58. ■♥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

  59. ■♥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

  60. ■♥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

  61. ■♥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

  62. ■♥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

  63. ■♥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

  64. ■♥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

  65. ■♥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

  66. ■♥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

  67. ■♥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

  68. ■♥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

  69. ■♥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

  70. ■♥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

  71. ■♥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

  72. ■♥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

  73. ■♥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

  74. ■♥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

  75. ■♥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

  76. ■♥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

  77. ■♥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

  78. ■♥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 Μµ ∑ ∂ Îœ g Μµ + 1 ï¿œ x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ Îœ g αβ = 0 2 ∑ α ∑ Îœ Îœ β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♊♥ R µΜ = F µΜ + 1 g αβ ∂ α ∂ β g µΜ = 0 2 ∑ α ∑ β ❈♊♠♣❛t✐❜✐❧✐t② ✇✐t❀ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♊♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♊s❡❞ ❧♊❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♊♥❛❧✱ s❡❝♊♥❞ ♩r❞❡r ❀②♣❡r❜♩❧✐❝ P❉❊ s②st❡♠ ❢♊r ï¿œ x µ ✳ ❙tr❛t♩s ❈❀✳ P❛♣❛❞♊✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♩r♠✉❧❛t✐♊♥ ♊❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

  79. ■♥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 Μµ ∑ ∂ Îœ g Μµ + 1 ï¿œ x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ Îœ g αβ = 0 2 ∑ α ∑ Îœ Îœ β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♊♥ R µΜ = F µΜ + 1 g αβ ∂ α ∂ β g µΜ = 0 2 ∑ α ∑ β ❈♊♠♣❛t✐❜✐❧✐t② ✇✐t❀ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♊♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♊s❡❞ ❧♊❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♊♥❛❧✱ s❡❝♊♥❞ ♩r❞❡r ❀②♣❡r❜♩❧✐❝ P❉❊ s②st❡♠ ❢♊r ï¿œ x µ ✳ ❙tr❛t♩s ❈❀✳ P❛♣❛❞♊✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♩r♠✉❧❛t✐♊♥ ♊❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

  80. ■♥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 Μµ ∑ ∂ Îœ g Μµ + 1 ï¿œ x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ Îœ g αβ = 0 2 ∑ α ∑ Îœ Îœ β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♊♥ R µΜ = F µΜ + 1 g αβ ∂ α ∂ β g µΜ = 0 2 ∑ α ∑ β ❈♊♠♣❛t✐❜✐❧✐t② ✇✐t❀ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♊♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♊s❡❞ ❧♊❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♊♥❛❧✱ s❡❝♊♥❞ ♩r❞❡r ❀②♣❡r❜♩❧✐❝ P❉❊ s②st❡♠ ❢♊r ï¿œ x µ ✳ ❙tr❛t♩s ❈❀✳ P❛♣❛❞♊✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♩r♠✉❧❛t✐♊♥ ♊❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

  81. ■♥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 Μµ ∑ ∂ Îœ g Μµ + 1 ï¿œ x µ = g ab ∇ a ∇ b x µ = ∑ g αβ ∂ Îœ g αβ = 0 2 ∑ α ∑ Îœ Îœ β ❊✐♥st❡✐♥ ❘❡❞✉❝❡❞ ❊q✉❛t✐♊♥ R µΜ = F µΜ + 1 g αβ ∂ α ∂ β g µΜ = 0 2 ∑ α ∑ β ❈♊♠♣❛t✐❜✐❧✐t② ✇✐t❀ ❊✐♥st❡✐♥✬s ❡q✉❛t✐♊♥✱ ②✐❡❧❞s ❛ ✇❡❧❧ ♣♊s❡❞ ❧♊❝❛❧✱ ❧✐♥❡❛r✱ ❞✐❛❣♊♥❛❧✱ s❡❝♊♥❞ ♩r❞❡r ❀②♣❡r❜♩❧✐❝ P❉❊ s②st❡♠ ❢♊r ï¿œ x µ ✳ ❙tr❛t♩s ❈❀✳ P❛♣❛❞♊✉❞✐s ■♥✐t✐❛❧ ❱❛❧✉❡ ❋♩r♠✉❧❛t✐♊♥ ♊❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②

  82. ■♥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

  83. ■♥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

  84. ■♥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

  85. ■♥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

  86. ■♥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

  87. ■♥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

  88. ■♥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

  89. ■♥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

  90. ■♥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

  91. ■♥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

  92. ■♥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

  93. ■♥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

  94. ■♥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

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