(Luận án tiến sĩ) Lý thuyết tương đối tổng quát cải tiến f(R) đối xứng cầu và ứng dụng trong vũ trụ học

t s t ỵ P ýt ❍⑨◆ ▲❹▼ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❱■➏❚ ◆❆▼ ❍➴❈ ❱■➏◆ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ P❍❸▼ ❱❿◆ ❑Ý ị ì f (R) ✣➮■ ❳Ù◆● ❈❺❯ ❱⑨ Ù◆● ❉Ư◆● ❚❘❖◆● ❱Ơ ❚❘Ư ❍➴❈ t ỵ ỵ tt t ỵ t số ữớ ữợ P ý P ỗ t s t ỵ P ýt ỡ ổ t ỡ ữớ ữợ tổ P ý P ỗ ❱➙♥ ✤➣ ❧✉ỉ♥ õ♥❣ ❤ë✱ ❣✐ó♣ ✤ï tỉ✐ ✈➔ tr✉②➲♥ t❤ư ❦✐➳♥ t❤ù❝ ❝❤♦ tỉ✐✳ ❚ỉ✐ ①✐♥ ❝↔♠ ì♥ ✤➳♥ t ỵ ổ ◆❣❤➺ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ▲✉➟♥ →♥✳ ❈❤♦ tỉ✐ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ◗✉ÿ ♣❤→t tr✐➸♥ ❦❤♦❛ ❤å❝ ✈➔ ❝ỉ♥❣ ♥❣❤➺ ◗✉è❝ ❣✐❛ ✭◆❛❢♦st❡❞✮ ✈➔ q✉ÿ ❤å❝ ❜ê♥❣ ❖❞♦♥ ❱❛❧❧❡t t❤✉ë❝ ❚ê ❝❤ù❝ ●➦♣ ❣ï ❱✐➺t ◆❛♠ ✤➣ ❤é trñ ♠ët ♣❤➛♥ ❦✐♥❤ ♣❤➼ ❝❤♦ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❧➔♠ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤✳ ❈↔♠ ì♥ ❣✐❛ ✤➻♥❤ tỉ✐ ✈➔ ❜↕♥ ❜➧ tỉ✐ ✤➣ ❧✉æ♥ õ♥❣ ❤ë ✈➔ ✤ë♥❣ ✈✐➯♥ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❍➔ ◆ë✐✱ ♠ò❛ t❤✉ ✷✵✷✷✳ ✷ ▲✉➟♥ →♥ t s t ỵ P ýt ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ♥❤ú♥❣ ❦➳t q✉↔ ✤↕t ✤÷đ❝ tr♦♥❣ ▲✉➟♥ →♥ ❧➔ ❬✶✱ ✷✱ ✸❪✱ ♥❤ú♥❣ ❦➳t q✉↔ ♥➔② ự tổ ũ ợ ữớ ữợ ❞➝♥✳ ◆❤ú♥❣ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ♠ỵ✐✱ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝✳ ❍➔ ◆ë✐✱ ✶✵✴✷✵✷✷ ◆❈❙ P❤↕♠ ❱➠♥ ❑ý ✸ ▼ö❝ ❧ö❝ ▼ð ✣➛✉ ❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✶ ❚ê♥❣ q ởt số ỵ tt rở ỵ tt f (R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tt tsrổ ữợ ụ rử ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✶ ❱ô rử tsr ổ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✸ ❚r÷í♥❣ ✈ỉ ữợ ữ q ỵ tt f (R) ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹ ▼ët sè ỵ tt rở ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✺ ❑❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ❝â ✤ë ①♦➢♥✱ ❤➻♥❤ tự Pt ỵ tt f (T ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✻ ❍➻♥❤ t❤ù❝ ❧✉➟♥ ❱✐❡r❜❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✻✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ♣❤➙♥ tr♦♥❣ ❤➻♥❤ t❤ù❝ ❧✉➟♥ ❱✐❡r❜❡✐♥ ✸✻ ✶✳✻✳✸ ❈→❝ ✈❡❝t♦r ❱✐❡r❜❡✐♥ ❝õ❛ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ✤è✐ ①ù♥❣ ❝➛✉ ✳ ✳ ✳ ✳ ✸✾ ✶✳✻✳✹ ✣↕♦ ❤➔♠ ❤✐➺♣ ❜✐➳♥ ♠ð rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ Pữỡ tr tờ qt ỵ tt trữớ tr ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ❝♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✶✳✻✳✻ ❚r÷í♥❣ ❙♣✐♥♦r tr♦♥❣ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ❝♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổtớ ỵ t❤✉②➳t ❑❛❧✉③❛✲❑❧❡✐♥✱ ♠ët sü t❤è♥❣ ♥❤➜t ❝õ❛ tr÷í♥❣ ❤➜♣ ❞➝♥ ✈➔ tr÷í♥❣ ✤✐➺♥ tø ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷ ỵ tt f (R) tr tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ✈➔ ♠ët sè ù♥❣ ❞ư♥❣ ✻✵ ỵ tt f (R) tr♦♥❣ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ◆❣❤✐➺♠ tr♦♥❣ ❝❤➙♥ ❦❤æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ◆❣❤✐➺♠ ♥❤✐➵✉ ❧♦↕♥ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❈❤✉②➸♥ ✤ë♥❣ tr trữớ ố ự ỵ ttf (R) ✳ ✳ ✳ ✷✳✷✳✶ ❈❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❤➔♥❤ t✐♥❤ tr♦♥❣ trữớ ởt ổ s ỵ ttf (R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ✣÷í♥❣ tr✉②➲♥ ❝õ❛ t✐❛ s→♥❣ tr♦♥❣ tr÷í♥❣ ✤è✐ ①ù♥❣ ởt ổ s ỵ ttf (R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✻✷ ✻✻ ✼✶ ✼✸ ✳ ✼✼ ✳ t s t ỵ P ýt ✷✳✸ ▲í✐ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✸ tr ỵ tt f (R) rữớ ✤è✐ ①ù♥❣ ❝➛✉ t➽♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✸✳✶✳✶ ▼æ ❤➻♥❤ f (R) = R + λR2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✸✳✶✳✷ ▼æ ❤➻♥❤ f (R) = R + λR2 +∞ n=0 an R n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✸✳✶✳✸ ▼æ ❤➻♥❤ f (R) = ReλR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✹ ▼æ ❤➻♥❤ f (R) = αR1+ε ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ √ ✸✳✶✳✺ ▼æ ❤➻♥❤ f (R) = R + λ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❚r÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t t➽♥❤ ✸✳✷✳✶ ▼æ ❤➻♥❤ f (R) = R + λR2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ′ ✸✳✷✳✷ ▼æ ❤➻♥❤ f (R) = R + λR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ▲í✐ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ▼ët sè ✈➜♥ ✤➲ ❦❤→❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❧♦↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✾✻ ✾✼ ✾✽ ✶✵✵ ✶✵✶ ✶✵✶ ✶✵✸ ✹✳✶ ◆❣❤✐➺♠ ỵ tt f (R) tr ❱ơ trư ❋▲❘❲ ✶✵✸ ✹✳✶✳✶ ▼ët sè ♠ỉ ❤➻♥❤ ❝ư t❤➸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ✹✳✶✳✷ ▲í✐ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✹✳✷ ❙â♥❣ ❤➜♣ ❞➝♥ ❝õ❛ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ❝õ❛ ỵ tt f (R) ổ st ự ①↕ sâ♥❣ ❤➜♣ ❞➝♥ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ✳ ✶✶✸ ✹✳✷✳✷ ▼❡tr✐❝ ♥❤✐➵✉ ❧♦↕♥ ❝õ❛ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺ ✹✳✷✳✸ ❈æ♥❣ s✉➜t ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ ❝õ❛ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❧♦↕♥ ❝õ❛ ỵ tt f (R) ✳ ✶✶✼ ✹✳✷✳✹ ❙ü ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ tr♦♥❣ q✉→ tr➻♥❤ ❤➻♥❤ t❤➔♥❤ ♠ët ❤è ✤❡♥✶✶✼ ❆ ❈→❝ t❡♥s♦r tr str tr ỗ t ỵ tt tữỡ ố tờ qt ởt ỵ tt trữớ ữủ ổ ố ỵ tt ữủ st ỹ s ỵ tt ữ s❛✉ ♥â ✤➣ ✤÷đ❝ ♥❤✐➲✉ t❤➼ ♥❣❤✐➺♠ ❦✐➸♠ ❝❤ù♥❣✳ ◆❤÷ ú t t t ỵ tt sỹ tữỡ t→❝ ❣✐ú❛ ✈➟t ❝❤➜t ✈➔ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ❝â t❤➸ ❧➔♠ ❝❤♦ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ❦❤ỉ♥❣ ❝á♥ ❧➔ ♣❤➥♥❣ ♥ú❛ ♠➔ ❧➔ ❝♦♥❣✳ ❇↔♥ ❝❤➜t ❝õ❛ ❤➜♣ ❞➝♥ t❤ü❝ r❛ ❝❤➼♥❤ ❧➔ t➼♥❤ ❝❤➜t ❤➻♥❤ ❤å❝ ❝õ❛ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥✳ ▼ët ✈➟t ❝❤✉②➸♥ ữợ t ỹ t q ✤✐➸♠ ❝õ❛ ◆❡✇t♦♥✱ ❝â t❤➸ ✤÷đ❝ ❝♦✐ ❧➔ ❝❤✉②➸♥ ✤ë♥❣ tỹ ổ ữợ t ỹ ♥❤÷♥❣ tr♦♥❣ ♠ët ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ❝♦♥❣ t❤❡♦ q✉❛♥ ✤✐➸♠ ❝õ❛ ❊✐♥st❡✐♥✳ ❚❤✉②➳t t÷ì♥❣ ✤è✐ sû ❞ư♥❣ ❤➻♥❤ ❤å❝ ❘✐❡♠❛♥♥ ✤➸ t tữủ t ỵ tr ổtớ ❝♦♥❣ ✈➔ ♥â ✤÷đ❝ t❤➸ ❤✐➺♥ q✉❛ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ❬✹✱ ✺❪ 8πG Rµν − Rgµν = − Tà c Pữỡ tr õ t t ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ❧➜② ❜✐➳♥ ♣❤➙♥ t→❝ t→❝ ❞ư♥❣ ▲❛❣r❛♥❣✐❛♥ ❝õ❛ tr÷í♥❣ ❤➜♣ ❞➝♥ ❧➔ LG = R✳ P❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈ỵ✐ ✈➳ tr→✐ ❧➔ t❡♥s♦r ✤ë ❝♦♥❣ ❝õ❛ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥✱ ✈➳ ♣❤↔✐ ❧➔ t❡♥s♦r ♥➠♥❣ ①✉♥❣ ❧÷đ♥❣ ❝õ❛ ❝❤➜t ✈➔ ♥❤÷ ✈➟② ❝â ♥❣❤➽❛ ❧➔ ❤➻♥❤ ❤å❝ ❝õ❛ ❦❤ỉ♥❣ t❤í✐ ❣✐❛♥ ✤÷đ❝ q✉②➳t ✤à♥❤ ❝❤➼♥❤ ✈➟t ❝❤➜t ❝❤ù❛ ❜➯♥ tr♦♥❣ ♥â✱ ✈➟t ❝❤➜t ❜➫ ❝♦♥❣ ❦❤ỉ♥❣ t❤í✐ ❣✐❛♥✳ ✭✶✮ ợ t Pữỡ tr st tọ r rt ũ ủ t tữủ ố ợ ❝→❝ ✈➟t t❤➸ t❤ỉ♥❣ t❤÷í♥❣ ✈➔ ❤➺ ♠➦t trí✐✳ ❚✉② tữợ ù t ụ trử ỏ tữủ ỵ tt ổ t t ữủ ởt tr ỳ ợ t ♥❣➔② ♥❛② ✤â ❧➔ ❣✐↔✐ t❤➼❝❤ sü ❣✐➣♥ ♥ð t➠♥❣ tè❝ ❝õ❛ ✈ơ trư ✭✈➜♥ ✤➲ ♥➠♥❣ ❧÷đ♥❣ tè✐✮✳ ❈â ♥❤✐➲✉ q✉❛♥ s→t ❝❤♦ t❤➜② r➡♥❣ ✈ơ trư ✤❛♥❣ ❣✐➣♥ ♥ð t➠♥❣ tè❝ ❬✻✱ ✼❪✳ ❙ü ❣✐➣♥ ♥ð t➠♥❣ tè❝ ❝õ❛ ✈ơ trư ♥❣❤➽❛ ❧➔ ✈ơ trư ❣✐➣♥ ♥ð ✈ỵ✐ ✈➟♥ tè❝ ♥❣➔② ❝➔♥❣ t➠♥❣✱ tr♦♥❣ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ❝â ♥❣❤✐➺♠ ❝❤♦ ♠ët ✈ơ trư ❣✐➣♥ ♥ð ♥❤÷♥❣ ✈➟♥ tè❝ ❣✐➣♥ ♥ð ♥❣➔② ❝➔♥❣ ❣✐↔♠ ✭tị② t❤✉ë❝ ✈➔♦ ♠➟t ✤ë ✈➟t ❝❤➜t s♦ ✈ỵ✐ ♠➟t ✤ë tỵ✐ ❤↕♥ ♠➔ ✈ơ trư ❝â t❤➸ ❧➔ ✈ơ trư ✤â♥❣✱ ✈ơ trư ♠ð ❤❛② ✈ơ trư ♣❤➥♥❣✮✱ ♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ st ố ợ ụ trử ữủ ỳ ữỡ tr➻♥❤ ❋r✐❡❞♠❛♥♥✳ ✣➸ ❣✐↔✐ t❤➼❝❤ sü t➠♥❣ tè❝ ❝õ❛ ✈ô trư✱ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➔ t❛ ❝â t❤➸ t❤➯♠ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ♠ët ❤➡♥❣ sè✱ ✻ ▲✉➟♥ t s t ỵ P ýt ❤➡♥❣ sè ✈ơ trư Λ✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ s➩ ❝â ❞↕♥❣ 8πG Rµν − Rgµν + Λgµν = Tà c Pữỡ tr ♥➔② ❝â t❤➸ t❤✉ ✤÷đ❝ tø ❜✐➳♥ ♣❤➙♥ t→❝ ❞ư♥❣ ợ r trữớ LG = R2 ỵ tt ữủ ỵ tt tữỡ ố tờ q✉→t ✭●❘✮✳ ❚❤ü❝ r❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤➣ ✤÷đ❝ ❊✐♥st❡✐♥ ữ r tứ rt sợ t r ởt ụ trö t➽♥❤ ❧➦♥❣ ✤➸ tr✐➺t t✐➯✉ ✤✐ sü ❜✐➳♥ ✤ê✐ ❝õ❛ ✈ơ trư ♠➔ ❋r✐❡❞♠❛♥♥ ✤➣ ❝❤♦ t❤➜② r➡♥❣ ✈ơ trô ❝â ❦❤↔ ♥➠♥❣ ❣✐➣♥ ♥ð ❤♦➦❝ ❝♦ ❧↕✐✳ ❱➻ ❧ó❝ ✤â ❊✐♥st❡✐♥ ♠♦♥❣ ♠✉è♥ ♠ët ✈ơ trư t➽♥❤ ❧➦♥❣ ❝ơ♥❣ ♥❤÷ ❜❛♦ s✉② ♥❣❤➽ ❦❤→❝ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❝ị♥❣ t❤í✐✳ ◆❤÷♥❣ s❛✉ ❦❤✐ ❍✉❜❜❧❡ ❝ỉ♥❣ ❜è ♣❤→t ❤✐➺♥ ❝õ❛ ♠➻♥❤ r➡♥❣ ✈ơ trư ✤❛♥❣ ❣✐➣♥ ♥ð✱ ❊✐♥st❡✐♥ ✤➣ ❝❤➜♣ ♥❤➟♥ s❛✐ ❧➛♠ ✈➔ ♥â✐ ♠ët ❝➙✉ ✤↕✐ ❧♦↕✐ ♥❤÷ ✈✐➺❝ ❝❤♦ ✈➔♦ ❤➡♥❣ sè Λ ❧➔ s❛✐ ❧➛♠ ❧ỵ♥ ♥❤➜t ❝✉ë❝ ✤í✐ ỉ♥❣✳ ❚✉② ♥❤✐➯♥✱ ❝→✐ ♠➔ ❊✐♥st❡✐♥ ❝♦✐ ❧➔ s❛✐ ❧➛♠ ❧ỵ♥ ♥❤➜t ❝✉ë❝ ✤í✐ t❤➻ s tr t ởt ỵ tữ tt ♥â ❧➔ ❝ù✉ ❝→♥❤ ❝❤♦ ❝❤➼♥❤ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ỉ♥❣ ✤➸ ❣✐↔✐ t❤➼❝❤ sü ❣✐➣♥ ♥ð t➠♥❣ tè❝ ❝õ❛ ✈ô trử ợ tr số ữủ ❦❤→❝ ✤✐ s♦ ✈ỵ✐ ❣✐→ trà ❧ó❝ ✤➛✉ ♠➔ ❊✐♥st❡✐♥ ✤➣ ❞ị♥❣ ✤➸ t↕♦ r❛ ♠ët ✈ơ trư t➽♥❤ ❧➦♥❣✱ t❛ s➩ ❣✐↔✐ t❤➼❝❤ ✤÷đ❝ sü t➠♥❣ tè❝ ❝õ❛ ✈ơ trư✳ ◆❤÷ ✤➣ ♥â✐ t❤❡♦ q✉❛♥ s→t ❝õ❛ ❍✉❜❜❧❡ ✭❦➼♥❤ t❤✐➯♥ ✈➠♥ ❦❤ỉ♥❣ ❣✐❛♥✮✱ ✈ơ trư ✤❛♥❣ ❣✐➣♥ ♥ð t➠♥❣ tè❝✱ ❝→❝ t❤✐➯♥ ❤➔ ✤❛♥❣ ❧ò✐ r❛ ①❛ ♥❤❛✉✳ ❚❤❡♦ q✉❛♥ ✤✐➸♠ ❝õ❛ ♥❤➔ ✈ơ trư ❤å❝ ✈➔ ❧✐♥❤ ♠ư❝ ●❡♦r❣❡s ▲❡♠❛✐tr❡✱ ➢t ❤➥♥ t➜t ❝↔ ❝→❝ t❤✐➯♥ ❤➔ ❜❛♥ ✤➛✉ ♣❤↔✐ ð r➜t ❣➛♥ ♥❤❛✉✱ ❣➛♥ ♥❤÷ ♠ët ✤✐➸♠ ✈ỵ✐ ❣✐↔ t❤✉②➳t ✈➲ ✧♥❣✉②➯♥ tû ♥❣✉②➯♥ t❤õ②✧✳ ❈❤➼♥❤ ✤✐➲✉ ♥➔② ♠➔ s❛✉ ♥➔② ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✤➣ ♣❤→t tr t ỵ tt ợ t ữủ t ợ ỵ tt ụ trử õ tữợ rt ọ ❣➛♥ ♥❤÷ ♠ët ✤✐➸♠✱ ♠➟t ✤ë ✈ỉ ❝ị♥❣ ✤➟♠ ✤➦❝ ✈➔ ♥❤✐➺t ✤ë ❝ü❝ ❝❛♦✱ ✤✐➸♠ ❜❛♥ ✤➛✉ ♥➔② ❤✐➺♥ ữ õ ởt ỵ tt t ỵ ♠ỉ t↔✱ ♥â ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ P❧❛♥❝❦✳ ❙❛✉ ✤â ✈ơ trư ❣✐➣♥ ♥ð ✈➔ tr↔✐ q✉❛ ❜❛ ❣✐❛✐ ✤♦↕♥ ✤â ❧➔ ❦✛ ♥❣✉②➯♥ ❧↕♠ ♣❤→t✱ ❦✛ ♥❣✉②➯♥ ❜ù❝ ①↕ ✈➔ ❝✉è✐ ❝ị♥❣ ♥❤÷ ♥❣➔② ❤ỉ♠ ♥❛② ✤â ❧➔ ❦✛ t t ữủ tố ự ỵ t❤✉②➳t ❇✐❣ ❇❛♥❣ ✤➣ ✤÷đ❝ t❤➼ ♥❣❤✐➺♠ ❦✐➸♠ ❝❤ù♥❣ tø sü ♣❤→t ❤✐➺♥ r❛ t➔♥ t➼❝❤ ❝õ❛ ♥â ✤â ❧➔ ❜ù❝ ①↕ ♥➲♥ ✭❈▼❇✮✳ ❇ù❝ ①↕ ✤â ✤÷đ❝ ♣❤→t ❤✐➺♥ ✈➔♦ ♥➠♠ ✶✾✻✹ ❜ð✐ ❆r♥♦ P❡♥③✐❛s ✈➔ ❘♦❜❡rt ❲✐❧s♦♥✱ ❝❤➼♥❤ ♥❤í ♣❤→t ❤✐➺♥ ♥➔② ♠➔ ❤❛✐ ỉ♥❣ ✤➣ ✤÷đ❝ ♥❤➟♥ t ỵ ỵ tt ❇❛♥❣ ♥❣➔② ♥❛② ✤➣ ✤÷đ❝ ❤➛✉ ❤➳t ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❝ỉ♥❣ ♥❤➟♥✱ ✤❛♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♣❤→t tr✐➸♥✳ ❈â ♥❤✐➲✉ ❜➡♥❣ ❝❤ù♥❣ ❝❤➾ r❛ r➡♥❣ ✈ơ trư ✤➣ tr↔✐ q✉❛ ❤❛✐ ❦✛ ♥❣✉②➯♥ ❣✐➣♥ ♥ð t➠♥❣ tè❝✳ ❚❤í✐ ❦ý t➠♥❣ tè❝ ✤➛✉ t✐➯♥ ❧➔ ð ❦✛ ♥❣✉②➯♥ ❧↕♠ ♣❤→t ❬✽✱ ✾❪✳ ❚❤í✐ ❦ý t➠♥❣ tè❝ t❤ù ❤❛✐ ❧➔ ð ❦✛ ♥❣✉②➯♥ ❝õ❛ ✈➟t ❝❤➜t ✭♥❣➔② ♥❛②✮✱ sü t➠♥❣ tè❝ tr♦♥❣ ❦✛ ♥❣✉②➯♥ ♥➔② ✤÷đ❝ ❝❤♦ ❝â ♥❣✉②➯♥ ♥❤➙♥ tø ♥➠♥❣ ❧÷đ♥❣ tè✐ ❬✶✵✱ ✶✶❪✳ ❱ơ trư ♥❣➔② ♥❛② ❝â ❦❤♦↔♥❣ ✻✸✱✽% ♥➠♥❣ ❧÷đ♥❣ tè✐✱ ✷✻✱✽% ✈➟t ❝❤➜t tè✐✱ ❝❤➾ ❝â ❦❤♦↔♥❣ ✹✱✾% ✈➟t ❝❤➜t t❤ỉ♥❣ t❤÷í♥❣ ❬✶✷❪✳ ✼ t s t ỵ P ýt t q✉↔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ●❘ ❝❤♦ t❤➜② ✈ơ trư ✤❛♥❣ t tố t ỵ tt ỏ ♥❤✐➲✉ ♥❤÷đ❝ ✤✐➸♠ ✈➔ ♥❤✐➲✉ ❞➜✉ ❤ä✐✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ ỗ ố số ỏ ởt ởt ỵ tữ r õ õ ỗ ố tứ ởt ữủ tố ợ ữỡ tr tr↕♥❣ t❤→✐ P = −ρ ✭→♣ s✉➜t ❜➡♥❣ ➙♠ ♠➟t ✤ë ♥➠♥❣ ❧÷đ♥❣✮✳ ◆➳✉ ❝❤♦ r➡♥❣ ♥➠♥❣ ❧÷đ♥❣ tè✐ ❝â ỗ ố tứ ổ t ỵ t t ỵ ữủ tỷ r ổ ổ trè♥❣ ré♥❣ ♠➔ ♥â ✈➝♥ ♠❛♥❣ ♥➠♥❣ ❧÷đ♥❣ t❤ỉ♥❣ q✉❛ ❝→❝ t❤➠♥❣ ❣✐→♥❣ ❧÷đ♥❣ tû✮✱ t❤➻ ♥➠♥❣ ❧÷đ♥❣ tè✐ t❤❡♦ t t t ỵ ữủ tỷ q ợ s♦ ✈ỵ✐ ❣✐→ trà Λ ❝➛♥ ❝â ✤➸ ✈ơ trơ t tố ợ tố ữủ q st ữ ♥❛②✱ ♥â ❧ỵ♥ ❤ỵ♥ ❦❤♦↔♥❣ 10120 ❧➛♥ ❬✶✸✱ ✶✹❪✳ ❍↕♥ ❝❤➳ t❤ù ❤❛✐ ❝õ❛ ●❘ ❧➔✱ s❛✉ ❦➾ ♥❣✉②➯♥ ❧↕♠ ♣❤→t ♣❤↔✐ ✤÷đ❝ ♥è✐ t✐➳♣ ❜ð✐ ❦➾ ♥❣✉②➯♥ ❜ù❝ ①↕ ✈➔ s❛✉ ✤â ♥è✐ t✐➳♣ ❜ð✐ ❦➾ ♥❣✉②➯♥ ✈➟t ❝❤➜t ữủ tố ỵ tt ợ ởt số Λ ♥❤÷ ✈➟② s➩ ❦❤ỉ♥❣ t❤➸ ❝❤♦ ❧í✐ ❣✐↔✐ ♣❤ị ❤ñ♣ ❝õ❛ sü ♥è✐ t✐➳♣ ♥➔② ❬✶✺❪✳ ❍↕♥ ❝❤➳ t❤ù ỵ tt ụ ổ t t ✤÷đ❝ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈➟t ❝❤➜t tè✐✳ ▼ët ❧♦↕✐ ✈➟t ❝❤➜t ❞♦ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❞ü tỗ t ữ ữ ữủ t t❤➼❝❤ ♠ët tr♦♥❣ ♥❤ú♥❣ ❤✐➺♥ t÷đ♥❣ ✤â ❧➔✱ tè❝ ✤ë q✉❛② ❜➜t t❤÷í♥❣ ❝õ❛ r➻❛ ❝→❝ t❤✐➯♥ ❤➔ ❬✶✻✱ ✶✼❪✳ ❱➔ ❝✉è✐ ❝ị♥❣✱ ❝❤÷❛ ❝â ♠ët t❤➼ ♥❣❤✐➺♠ ♥➔♦ ♣❤→t ❤✐➺♥ t❤➜② ♠ët ❝→❝❤ trü❝ t✐➳♣ sü ❝â ♠➦t ❝õ❛ ♥➠♥❣ ❧÷đ♥❣ tè✐✳ ◆â ❝❤➾ ✤÷đ❝ ✤÷❛ ✈➔♦ ✤➸ ❣✐↔✐ t❤➼❝❤ ❝→❝ ❤✐➺♥ t÷đ♥❣ tr♦♥❣ ✤â ❝â ✈➜♥ ✤➲ t➠♥❣ tè❝ ❝õ❛ ✈ơ trư✳ ❱ỵ✐ t➜t ❝↔ ♥❤ú♥❣ ✈➜♥ ✤➲ ✤➣ t❤↔♦ ❧✉➟♥✱ rã r➔♥❣ r➡♥❣ ❝➛♥ ♣❤↔✐ t➻♠ ♠ët ỵ tt t t ỵ tt st ú t õ ỳ ỵ tt sỷ ỳ ỵ tt s t ỵ tt tữỡ ✤è✐ ♥❤➡♠ sû❛ ❝❤ú❛ ♥❤ú♥❣ ❦❤✉②➳t ✤✐➸♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ st ỳ ỵ tt õ t t ữủ sü t➠♥❣ tè❝ ❝õ❛ ✈ơ trư ♠ët ❝→❝❤ r➜t tü ♥❤✐➯♥ tø ❝❤➼♥❤ ❤➻♥❤ ❤å❝ ❝õ❛ sü ❝♦♥❣ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ổ ữ ữủ tố ỳ ỵ t❤✉②➳t ❤➜♣ ❞➝♥ sû❛ ✤ê✐ ❝á♥ ✤➸ ❣✐↔✐ t❤➼❝❤ r➜t ♥❤✐➲✉ ❤✐➺♥ t÷đ♥❣ ❦❤→❝ ♥❤÷✿ ❈→❝ ✈➜♥ ✤➲ tr♦♥❣ ❦➾ t t t ữợ ự ổ ụ trử tr ỵ tt ữủ tỷ t ỵ tr sõ ỵ tt sỷ ✤❛♥❣ trð t❤➔♥❤ ✈➜♥ ✤➲ t❤í✐ sü ❤➜♣ ❞➝♥✱ t❤✉ ❤ót r➜t ♥❤✐➲✉ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ t❤❛♠ ❣✐❛ ♥❣❤✐➯♥ ự t tr ởt tr số ỳ ỵ tt ❤➜♣ ❞➝♥✲sû❛ ✤ê✐ ✭♠ð rë♥❣✮ ✤ì♥ ❣✐↔♥ ♥❤➜t✱ ✤÷đ❝ r❛ tứ rt sợ ữủ ự r➜t sỉ✐ ♥ê✐✱ trð t❤➔♥❤ ✈➜♥ ✤➲ t❤í✐ sü r➜t õ ỵ tt f (R) ợ ▲❛❣r❛♥❣✐❛♥ ❝õ❛ tr÷í♥❣ ❤➜♣ ❞➝♥ LG = f (R)✱ ❦❤✐ õ ữỡ tr 8G f (R)Rà f (R)gµν + ∇µ ∇ν f ′ (R) − gµν □f ′ (R) = − Tµν c t s t ỵ P ❑ý✭♣❤❛♠❦②✈❛t❧②❅❣♠❛✐❧✳❝♦♠✮ (R) ❚r♦♥❣ ✤â f ′ (R) = dfdR ✱ ∇µ ❧➔ ✤↕♦ ❤➔♠ ❤✐➺♣ ❜✐➳♥ ✈➔ □ ≡ ∇α ữỡ tr ỵ tt ❤➜♣ ❞➝♥✲f (R) ❬✶✺✱ ✶✾✱ ✷✵❪✳ ❚❛ ❝ô♥❣ ❞➵ t❤➜② ữỡ tr st trữớ ủ r ỵ tt ợ f (R) = R ỵ t❤✉②➳t ❤➜♣ ❞➝♥✲f (R) r❛ ✤í✐ tø r➜t sỵ♠ s❛✉ st ữ r ỵ tt ỵ tt ổ ữủ ú ỵ ữ tứ ❝â ♥❤✐➲✉ q✉❛♥ s→t ❝❤♦ t❤➜② ✈ơ trư ✤❛♥❣ ❣✐➣♥ ♥ð t➠♥❣ tè❝✱ ❝→✐ ♠➔ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ❦❤ỉ♥❣ t❤➸ t ữủ t ỵ ttf (R) ữ ♠ët ù♥❣ ❝û ✈✐➯♥ s→♥❣ ❣✐→ ✤➸ ❣✐↔✐ t❤➼❝❤ ✈➜♥ t tố ụ trử ỵ tt t❤➼❝❤ sü t➠♥❣ tè❝ ❝õ❛ ✈ơ trư tø ❝❤➼♥❤ ❤➻♥❤ ❤å❝ ❝õ❛ ❦❤ỉ♥❣✲t❤í✐ ❣✐❛♥ ♠➔ ❦❤ỉ♥❣ ❝➛♥ ✤÷❛ ✈➔♦ ♥➠♥❣ ữủ tố õ tr ỵ tt ♥➠♥❣ ❧÷đ♥❣ tè✐ ❝â ❜↔♥ ❝❤➜t tø ❝❤➼♥❤ ❤➻♥❤ ❤å❝ ổtớ ỵ tt f (R) ụ õ t❤➸ ❣✐↔✐ t❤➼❝❤ ✤÷đ❝ ❝→❝ ✈➜♥ ✤➲ ❝õ❛ sü q✉❛② ❜➜t t❤÷í♥❣ ❝õ❛ r➻❛ ❝õ❛ ♠ët sè t❤✐➯♥ ❤➔ ❝→✐ ♠➔ ✤÷đ❝ ❝❤♦ ❧➔ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈➟t ❝❤➜t tè✐✳ r ỵ tt ụ õ t t ữủ ♠ët sè ❤✐➺♥ t÷đ♥❣ ❦❤→❝ ❝õ❛ ✈ơ trư ♠➔ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ❦❤ỉ♥❣ t❤➸ ❣✐↔✐ t❤➼❝❤ ✤÷đ❝✳ ❍➔♠ f (R) ✤÷đ❝ ❝❤å♥ ♥❤ú♥❣ ❞↕♥❣ ❝ư t❤➸ tị② ✈➔♦ ♠ư❝ ✤➼❝❤ ✤➸ ❣✐↔✐ t❤➼❝❤ ♠ët ✭♠ët sè✮ ❤✐➺♥ t÷đ♥❣ ♥➔♦ ✤â ❝õ❛ ✈ơ trư✱ ✈➼ ❞ư ♥❤÷ λ f (R) = R + λR2 ✱ f (R) = R + R ✱ ❡t❝✳ ✣➣ ❝â ♥❤✐➲✉ ♠ỉ ❤➻♥❤ ✤÷đ❝ ❝❤å♥✱ ♠é✐ ♠ỉ ❤➻♥❤ ❝â t❤➸ ❣✐↔✐ t❤➼❝❤ ✤÷đ❝ ♠ët sè ❤✐➺♥ t÷đ♥❣ tr♦♥❣ ✈ơ trư✱ ♥❤÷♥❣ ❝❤÷❛ ♠ỉ ❤➻♥❤ ♥➔♦ ❤♦➔♥ ỵ tt ữủ t✐➳♥✳ ❈â r➜t ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ t❤❛♠ ❣✐❛ ♥❣❤✐➯♥ ❝ù✉ ♣❤→t tr✐➸♥ ✈➔ ♥â ✤❛♥❣ trð t❤➔♥❤ ✈➜♥ ✤➲ t❤í✐ sü ♥❣➔② ♥❛②✳ ▲✉➟♥ →♥ s➩ t➟♣ tr✉♥❣ ❝❤õ ỵ tt f (R) Pữỡ tr ỵ tt f (R) õ t ữủ ❞➵ ❞➔♥❣ tr♦♥❣ ❝→❝ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t➽♥❤✳ ◆❣❤✐➺♠ ỵ tt f (R) tr trữớ ✤è✐ ①ù♥❣ ❝➛✉ t➽♥❤ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❬✻✼✱ ✻✽✱ ố ợ trữớ ố ự ❦❤æ♥❣ t➽♥❤✱ s➩ ❧➔ r➜t ❦❤â ❦❤➠♥ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✱ ✈➻ t❤➳ ✤➣ ❝â ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤÷đ❝ ✤➲ ①✉➜t ❬✼✸✱ ✼✹✱ ✼✺❪✳ ❍✐➺♥ ♥❛②✱ t❤❡♦ ❤✐➸✉ ❜✐➳t ❝õ❛ ❝❤ó♥❣ tỉ✐ t❤➻ ❝❤÷❛ ❝â ❛✐ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➜♣ ❞➝♥✲f (R) ❝❤♦ tr÷í♥❣ ❦❤ỉ♥❣ t➽♥❤✳ ❱➻ ✈➟② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤❡♦ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ❜➡♥❣ ❝→❝❤ →♣ ✤➦t ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â ❧➔ ởt tr ỳ ỹ ủ ỵ ú tổ t ỵ tt f (R) tờ qt tr tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ tê♥❣ q✉→t ✭❦❤ỉ♥❣ ♥❤➜t t❤✐➳t t➽♥❤✱ ❝❤♦ ❝↔ ❝❤➙♥ ❦❤ỉ♥❣ ✈➔ tr♦♥❣ ✈➟t ❝❤➜t✮ ❜➡♥❣ ♣❤÷ì♥❣ q tr ỵ tt ❞➝♥ ❊✐♥st❡✐♥✳ ❙❛✉ ✤â →♣ ❞ö♥❣ ❞ö♥❣ ❝❤♦ ♠ët sè ổ f (R) t ì ữỡ ♣❤→♣ ♥➔② ❧➔ ❝→❝ ♥❣❤✐➺♠ t❤✉ ✤÷đ❝ s➩ ❦❤ỉ♥❣ ①✉➜t ❤✐➺♥ t❤➯♠ ❜➜t ❦➻ ♠ët t❤❛♠ sè ♠ỵ✐ ♥➔♦ ✭✈➼ ❞ư ♥❤÷ ❝→❝ ❤➡♥❣ sè t➼❝❤ ♣❤➙♥✮✱ t❤❛♠ sè ❞✉② ♥❤➜t ①✉➜t ❤✐➺♥ ❧➔ t❤❛♠ sè ♥❤✐➵✉ ❧♦↕♥ λ ♠➔ ð ✤â ✤➣ ✤÷đ❝ ✤÷❛ ✈➔♦ ♥❣❛② tø ✤➛✉✳ ❱➻ ✈➟② t❛ ❝â t❤➸ →♣ ❞ư♥❣ ♥❤ú♥❣ ♥❣❤✐➺♠ t❤✉ ✤÷đ❝ ✈➔♦ ❝→❝ ❜➔✐ t♦→♥ ❝ư t❤➸ ✤➸ ✤↕t ✤÷đ❝ ♥❤ú♥❣ t q t ró r tr ỵ t❤✉②➳t ❤➜♣ ❞➝♥✲f (R) ✭♥❤÷ ❝→❝ ❜➔✐ t♦→♥ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ❤➔♥❤ t✐♥❤✱ →♥❤ s→♥❣✱ ✈➔ ❝→❝ ❜➔✐ ✾ t s t ỵ P ýt t ❦❤→❝ s❛✉ ♥➔②✱✳✳✳✮✳ ❙❛✉ ✤â q✉ÿ ✤↕♦ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ❤➔♥❤ t✐♥❤ ❝ơ♥❣ ♥❤÷ ✤÷í♥❣ tr✉②➲♥ ❝õ❛ →♥❤ s→♥❣ ữủ t t ởt tt ỵ t❤✉②➳t ❤➜♣ ❞➝♥✲f (R)✳ ❑➳t q✉↔ ❧➔ ❝→❝ ❣â❝ q✉❛② t✐♥❤ s❛✐ ❤❛② ❣â❝ ❧➺❝❤ ❝õ❛ t✐❛ s→♥❣ ❝â ♠ët sỹ ỵ tt st trữớ ố ①ù♥❣ ❝➛✉ t➽♥❤ t❤➻ ❝→❝ ✤↕✐ ❧÷đ♥❣ ❝❤➾ ❧➔ sü ỵ tt st ợ ữ ❦❤✐ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ❦❤ỉ♥❣ t➽♥❤✱ ♥❣♦➔✐ sü ❤✐➺✉ ❝❤➾♥❤ ❝❤♦ ❣✐→ trà ❊✐♥st❡✐♥ ✈➲ ✤ë ❧ỵ♥ ♥â ❝á♥ ởt số ự ợ ỵ tt st ổ t❤➸ ❝â ✤â ❧➔ ❝→❝ ✤↕✐ ❧÷đ♥❣ s➩ t❤❛② ✤ê✐ t❤❡♦ t❤í✐ ❣✐❛♥✱ ✈➼ ❞ư ♥❤÷ ❣â❝ q✉❛② t✐♥❤ s❛✐ ❝õ❛ ❤➔♥❤ t✐♥❤ tr♦♥❣ ❝→❝ ❝❤✉ ❦➻ ❦❤→❝ ♥❤❛✉ s➩ ❦❤→❝ ♥❤❛✉✱ ❝→❝ trư❝ ❝õ❛ ❊❧✐♣ ❝ơ♥❣ t❤❛② ✤ê✐ ✤ë t tớ ố ợ ỵ tt st tt ❝↔ ❝→❝ ✤↕✐ ❧÷đ♥❣ ♥❤÷ ❣â❝ q✉❛② t✐♥❤ s❛✐✱ ✤ë ❞➔✐ trư❝ ❊❧✐♣✱ ❣â❝ ❧➺❝❤ ❝õ❛ t✐❛ s→♥❣✱ ✈✳✈✱ ❝❤ó♥❣ ❧✉ỉ♥ ❧➔ ❤➡♥❣ sè✮✳ ◆❣✉②➯♥ ♥❤➙♥ ❝õ❛ ❝→❝ ❤✐➺♥ t÷đ♥❣ ♥➔② ❧➔✱ ❦❤✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ t❤➻ ♥❣❤✐➺♠ ✤↕t ✤÷đ❝ ❧➔ ❝→❝ t❡♥s♦r ♠❡tr✐❝ ❧✉ỉ♥ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ✭❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥✮✱ ❝á♥ ❦❤✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ f (R) ♥❣❤✐➺♠ t❤✉ ✤÷đ❝ ❧➔ ❝→❝ t❡♥s♦r ♠❡tr✐❝ s➩ ❦❤ỉ♥❣ ♣❤↔✐ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ♠➔ ♥â ♣❤ư tở tớ ứ t t ỵ r õ t ù tr ỵ tt ❞➝♥✲f (R)✳ ✣✐➲✉ ♥➔② ❝â t❤➸ ❣➙② r❛ ❝→❝ ❤✐➺✉ ù♥❣ t❤ó ✈à ❤ì♥✱ ✈➼ ❞ư ♥❤÷ ❦❤✐ ❜→♥ ❦➼♥❤ ởt ỗ r t❤❡♦ t❤í✐ ❣✐❛♥ t❤➻ ♥â ❝â ❦❤↔ ♥➠♥❣ ❜ù❝ ①↕ r sõ t ỵ tt f (R) ự ổ t õ tr ỵ tt st tr ỵ tt st ởt trữớ ố ự t↕✐ ❝❤➙♥ ❦❤æ♥❣ ❧✉æ♥ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ✈➔ ❦❤æ♥❣ t❤➸ ❜ù❝ ①↕ r❛ sâ♥❣ ❤➜♣ ❞➝♥✳ ❙â♥❣ ❤➜♣ ❞➝♥ ởt trữớ ố ự ỵ tt ❞➝♥✲f (R) ❝❤ó♥❣ tỉ✐ s➩ t❤↔♦ ❧✉➟♥ ❝❤✐ t✐➳t ❤ì♥ ð ❝❤÷ì♥❣ s❛✉ ❝õ❛ ❧✉➟♥ →♥ ♥➔②✳ ❈✉è✐ ❝ị♥❣ ❧➔ →♣ ❞ư♥❣ ♥❤ú♥❣ ♥❣❤✐➺♠ ✤➣ ✤↕t ✤÷đ❝ ✈➔♦ ❤➺ t❤è♥❣ ❙❣r❆✯✕❙✷ ð t➙♠ ❚❤✐➯♥ ❤➔ ❝õ❛ ❝❤ó♥❣ t❛ ✭❉↔✐ ◆❣➙♥ ❍➔✮✳ ◆❤ú♥❣ ❦➳t q✉↔ ♥➔② ♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦ ❝→❝ t❤➼ ♥❣❤✐➺♠ ❦✐➸♠ tr❛ tr♦♥❣ t÷ì♥❣ ❧❛✐ ✤➸ ✤→♥❤ ❣✐→ t ú ỵ tt f (R) →♥ ✤÷đ❝ tr➻♥❤ ❜➔② ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✱ tê♥❣ q✉❛♥✱ q ởt số ỵ tt sỷ rë♥❣✮✳ ❈❤÷ì♥❣ ✷ ✈➔ ❝❤÷ì♥❣ ✸ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr t t ỵ ✸❪✳ ❈❤÷ì♥❣ ✹ tâ♠ t➢t ❝→❝ ❦➳t q✉↔ ♠➔ ❝❤ó♥❣ tæ✐ ✤❛♥❣ t❤ü❝ ❤✐➺♥ ✈➔ ♣❤→t tr✐➸♥ ♥è✐ t✐➳♣ ❝õ❛ ❝❤÷ì♥❣ ✷ ✈➔ ✸✱ ✤➸ t❤➸ ❤✐➺♥ tr✐➸♥ ✈å♥❣ ♣❤→t tr✐➸♥ ❝õ❛ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ →♥✳ ▲✉➟♥ →♥ sỷ q ữợ s tr s s ữủ ữợ +, , , tự ds2 = ηµν dxµ dxν = dx0 dx0 + dxi dxi = c2 dt2 − dx2 − dy − dz , ✈ỵ✐ ❝❤➾ sè ▲❛✲t✐♥❤ i, j, k, l = 1, 2, 3❀ ❝→❝ ❝❤➾ sè ❍②✲❧↕♣ µ, ν, = 0, 1, 2, 3✳ ✶✵ ▲✉➟♥ t s t ỵ P ýt ữủ ❝↔ ❤➺ ❝ơ♥❣ ❝❤➼♥❤ ❜➡♥❣ t❡♥s♦r ♥➠♥❣✲①✉♥❣ ❧÷đ♥❣ ❝õ❛ r✐➯♥❣ tr÷í♥❣ ❤➜♣ ❞➝♥✳ ❚ø ✭✹✳✼✵✮ ✈➔ ✭✹✳✼✶✮ t❛ ❝â t❤➸ t ữợ tà = c4 Qαµν , 8πG ✭✹✳✼✾✮ tr♦♥❣ ✤â Qαµν = αν ρβ µσ η η η − η µν η ρβ η ασ − η αν η σβ η µρ + η µν η σβ η ρα +η µβ η νρ η ασ − η αβ η νρ η µσ ∂σ gρβ ✭✹✳✽✵✮ ❉➵ t❤➜② t➼♥❤ ❝❤➜t ♣❤↔♥ ①ù♥❣ Qαµν = −Qµαν ✈➔ ✤è✐ ①ù♥❣ ∂α Qαµν = ∂α Qà ổ st ự ữủ sõ q t S ỗ ♥➡♠ ð t➙♠ ♠➦t ❝➛✉✮ ✤÷đ❝ t➼❝❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ợ ct0i t ỏ ữủ P =− dE = dt ✭✹✳✽✶✮ ct0i dS i S ❚r♦♥❣ ✤â P ❧➔ ❝æ♥❣ s✉➜t✱ sü ❜✐➳♥ t❤✐➯♥ ♥➠♥❣ ❧÷đ♥❣ E ❜➯♥ tr♦♥❣ ♠➦t ❝➛✉ tr♦♥❣ ♠ët ✤ì♥ ✈à t❤í✐ ❣✐❛♥ ❝❤➼♥❤ ❜➡♥❣ ❝ỉ♥❣ s✉➜t ❜ù❝ ①↕ r❛ ❦❤ä✐ t ú ỵ Q00 = ❞♦ t➼♥❤ ♣❤↔♥ ①ù♥❣✮ t❛ ❝â P =− c5 8πG ✭✹✳✽✷✮ ∂j Qj0i dS i S ❚r♦♥❣ ✤â i, j = 1, 2, ✈➔ dS i = ni r2 dΩ ✈ỵ✐ dΩ = sin θdθdφ✱ ni = xi r✳ ❚ø ✭✹✳✽✵✮ t❛ ❝â Qj0i = (δij ∂0 gkk − δij ∂k g0k + ∂j gi0 − ∂0 gij ) ✭✹✳✽✸✮ ✹✳✷✳✷ ▼❡tr✐❝ ♥❤✐➵✉ ❧♦↕♥ ❝õ❛ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ tỉ✐ tâ♠ ❧÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ tr♦♥❣ ❬✶❪ ♠➔ s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❜➔✐ ♥➔②✳ ◆➳✉ ▲❛❣r❛♥❣✐❛♥ ❝õ❛ tr÷í♥❣ ❤➜♣ ❞➝♥ ữủ ữợ LG = R t s t ữủ ữỡ tr st ỏ ữủ ữợ tê♥❣ q✉→t ❤ì♥ ❧➔ LG = f (R) t❤➻ ♣❤÷ì♥❣ tr t ữủ s ữỡ tr ỵ tt ❤➜♣ ❞➝♥✲f (R)✮ ❬✶✺✱ ✶✾✱ ✷✵❪ f ′ (R)Rµν − gµν □f ′ (R) + ∇µ ∇ν f ′ (R) − f (R)gµν = −kTµν ✭✹✳✽✹✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✹✳✽✹✮ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❧♦↕♥ ❜➡♥❣ ❝→❝❤ ✤➦t f (R) = R + λh(R), ✶✶✺ ✭✹✳✽✺✮ ▲✉➟♥ t s t ỵ P ýt ❤➡♥❣ sè λ ✈➔ ❤➔♠ ❜➜t ❦➻ h(R) ♣❤↔✐ t❤ä❛ ♠➣♥ λh(R) ≪ R✳ ❈❤ó♥❣ t❛ s➩ t❤✉ ✤÷đ❝ ♠❡tr✐❝ ♥❤✐➵✉ ❧♦↕♥ ❝õ❛ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t↕✐ ❝❤➙♥ ❦❤ỉ♥❣ ❧➔ ✭♠❡tr✐❝ ❞↕♥❣ ❙❝❤✇❛r③s❝❤✐❧❞✮ ds2 = − 2GMf (t) dr2 02 − dx − r2 (dθ2 + sin2 θdφ2 ) 2GMf (t) c2 r − c2 r ✭✹✳✽✻✮ ❍❡r❡ ✭✹✳✽✼✮ Mf (t) = M − λM1 (t) − λM2 (t) ✈ỵ✐ G ❧➔ ❤➡♥❣ sè ❤➜♣ ❞➝♥✱ M ✈➔ R0 (t) ❧➛♥ ❧÷đt ❧➔ ❦❤è✐ ❧÷đ♥❣ ✈➔ ❜→♥ ỗ 2[Ro (t)]3 h(kT 00 ) + kT 00 h′ (kT 00 ) , 3kc2 4π ′′ ∂ M α(t), M2 (t) = h (kT ) kc ∂t [Ro (t)]3 ✭✹✳✽✽✮ M1 (t) = T 00 = α(t) = M c2 π[R0 (t)] ✭✹✳✽✾✮ ✭✹✳✾✵✮ , 3k c2 R0 (t) 256π [ξ(t)]4 arcsin[ξ(t)R0 (t)] ξ(t)R0 (t) − + 2[ξ(t)R0 (t)]2 × − [ξ(t)R0 (t)]2 ξ (t) = − [ξ(t)R0 (t)]2 −3/2 ✭✹✳✾✶✮ , 2GM o (t)] ✭✹✳✾✷✮ c2 [R ❈❤♦ ♠æ ❤➻♥❤ ✭✹✳✾✸✮ f (R) = R + λRb ✭b > 0✮✱ M1 (t) ✈➔ M2 (t) ❝â ❞↕♥❣ M1 (t) = 4π (b + 1)c2b (kM )b , kc2 31−b 22b+1 π b [Ro (t)]3b−3 M 2b−4 (3kM )b−2 ∂ ∂t [Ro (t)]3 4π b(b − 1)c M2 (t) = kc (4π)b−2 [Ro (t)]3b−6 ✭✹✳✾✹✮ α(t) ✭✹✳✾✺✮ ❱➔ ❝❤♦ ♠æ ❤➻♥❤ f (R) = R1+ε t s t ỵ P ❑ý✭♣❤❛♠❦②✈❛t❧②❅❣♠❛✐❧✳❝♦♠✮ ✭|ε| ≪ 1✮✱ ❝❤ó♥❣ trð t❤➔♥❤ λM1 (t) = −M + 4π (ε + 2)6ε (kc2 M )ε+1 , kc2 (8π)(ε+1) [Ro (t)]3ε M ε−1 ∂ ∂t [Ro (t)]3 4π ε(ε + 1)(3kc M ) λM2 (t) = kc (4π)ε−1 [Ro (t)]3ε−3 ✭✹✳✾✼✮ α(t) ✭✹✳✾✽✮ ú ỵ R0 (t) ỗ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥✱ tr÷í♥❣ ❤đ♣ ♥➔② ①↔② r❛ ỗ ởt ổ s t tr✐➸♥ ✈ỵ✐ ❜→♥ ❦➼♥❤ ❝â t❤➸ ❝♦ ❧↕✐ ❤❛② ♥ð r❛ t❤❡♦ t❤í✐ ❣✐❛♥✱ ♥❤÷ ✈➟② ❝→❝ ♠❡tr✐❝ t↕✐ ❝❤➙♥ ổ trữớ ố ự tr ỵ tt ❞➝♥ f (R) ❝â t❤➸ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥ ỵ ởt trữớ ố ①ù♥❣ ❝➛✉ ✈➝♥ ❝â t❤➸ ❜ù❝ ①↕ r❛ sâ♥❣ ❤➜♣ ❞➝♥✳ ❍✐➺✉ ù♥❣ ♥➔② ❦❤ỉ♥❣ t❤➸ ❝â tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ✈➻ ♠❡tr✐❝ t↕✐ ❝❤➙♥ ❦❤ỉ♥❣ ❝õ❛ ♠ët tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ❧✉æ♥ ð tr↕♥❣ t❤→✐ ❞ø♥❣✳ ✹✳✷✳✸ ❈æ♥❣ s✉➜t ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ ❝õ❛ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t ữỡ ỵ tt f (R) ❚â♠ t➢t ❦➳t q✉↔ ❝❤ó♥❣ tỉ✐ ✤↕t ✤÷đ❝ ♥❤÷ s❛✉✿ ❈æ♥❣ s✉➜t ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ ❝õ❛ ♠ët tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ ❧➔ P = −M˙ f (t)c2 ố ợ t t tổ tữớ t t❛ ❝â t❤➸ ❝♦✐ M˙ (t) = ✭❦❤è✐ ❧÷đ♥❣ t❤❛② ✤ê✐ r➜t ❝❤➟♠✮✱ ✈➻ ✈➟② sü ❜ù❝ ①↕ ♥➠♥❣ ❧÷đ♥❣ sâ♥❣ ❤➜♣ ❞➝♥ ❝â t❤➸ ❧➜② P = λ M˙ (t) + M˙ (t) c2 ✭✹✳✶✵✵✮ ◆➠♥❣ ❧÷đ♥❣ ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❝â t❤➸ ❝♦✐ ❧➔ sü t❤❛② ✤ê✐ ❝õ❛ t❤➳ tữỡ t ỳ ỗ ❤➜♣ ❞➝♥ ✭❝ơ♥❣ ♥❤÷ tr♦♥❣ ♠ët ❤➺ s❛♦ ✤ỉ✐ q✉❛② ①✉♥❣ q✉❛♥❤ ♥❤❛✉ t❤➻ t❛ ❝â t❤➸ ❝♦✐ ❦❤è✐ ❧÷đ♥❣ ❝õ❛ tø♥❣ ♥❣æ✐ s❛♦ ❧➔ ❦❤æ♥❣ ✤ê✐ ✈➔ sü ❜ù❝ ①↕ ♥➠♥❣ ❧÷đ♥❣ sâ♥❣ ❤➜♣ ❞➝♥ ❧➔ ❞♦ sü t❤❛② ✤ê✐ t❤➳ ♥➠♥❣ t÷ì♥❣ t→❝ ❣✐ú❛ ❤❛✐ ♥❣ỉ✐ s❛♦ ✈➔ ❧➔♠ ❝❤♦ ❝❤ó♥❣ ♥❣➔② ❝➔♥❣ t✐➳♥ ❧↕✐ ❣➛♥ ♥❤❛✉✮✳ ✹✳✷✳✹ ❙ü ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ tr♦♥❣ q✉→ tr➻♥❤ ❤➻♥❤ t❤➔♥❤ ♠ët ❤è ✤❡♥ ❚â♠ t➢t ❦➳t q✉↔ ❝❤ó♥❣ tỉ✐ ✤↕t ✤÷đ❝ ♥❤÷ s❛✉✿ ●✐û sû ❜→♥ ❦➼♥❤ ❝❤➙♥ trí✐ sü ❦✐➺♥ ❧➔ A1 = 2GM c2 ✭❤➻♥❤ ✹✳✶✮ ✈ỵ✐ M ❧➔ ❦❤è✐ ❧÷đ♥❣ ❝õ❛ ❤è ✤❡♥ ❦❤✐ ❜→♥ ❦➼♥❤ ✈➟t ❝❤➜t ❝♦ ❧↕✐ ❜➡♥❣ A1 ✳ ❇→♥ ❦➼♥❤ ✈➟t ❝❤➜t t✐➳♣ tư❝ ❝♦ ❧↕✐ ✤➳♥ ❜→♥ ❦➼♥❤ ❝✉è✐ ❝ị♥❣ t s t ỵ P ýt ❧➔ A2 ✈ỵ✐ A2 − A1 = −ψA1 ✱ t❤➻ ❦❤è✐ ❧÷đ♥❣ ❜à s✉② ❣✐↔♠ ♠ët ❧÷đ♥❣ ∆M = −3ψM ✳ ◆❤÷ ✈➟② ♥➠♥❣ ❧÷đ♥❣ sâ♥❣ ❤➜♣ ❞➝♥ ❜ù❝ ①↕ r❛ ❧➔ 3ψM c2 ✳ ◆❣÷đ❝ ❧↕✐ ♥➳✉ ✤♦ ❝ỉ♥❣ s✉➜t ❜ù❝ ①↕ r❛ ❝õ❛ ❤è ✤❡♥ tr♦♥❣ t❤í✐ ❣✐❛♥ ♥➔② s➩ t➼♥❤ ✤÷đ❝ ψ ✱ ❞♦ ✤â s➩ ❜✐➳t ✤÷đ❝ ❜→♥ ❦➼♥❤ ✈➟t ❝❤➜t ❝õ❛ ❤è ✤❡♥ ❧➔ A2 ❜➡♥❣ ❜❛♦ ♥❤✐➯✉✳ ❍➻♥❤ ✹✳✶✿ ❍➻♥❤ ❞↕♥❣ ❤è ✤❡♥ ✈ỵ✐ A1 = 2GM ❧➔ ❜→♥ ❦➼♥❤ ❝❤➙♥ trí✐ sü ❦✐➺♥✱ c2 A2 ❧➔ ❜→♥ ❦➼♥❤ ✈➟t ❝❤➜t✳ ◆❤÷ ✈➟② tr♦♥❣ ỵ tt f (R) ởt ổ s õ ✤õ ✤✐➲✉ ❦✐➺♥ s❛✉ ❦❤✐ sö♣ ✤ê ✤➸ trð t❤➔♥❤ ởt ố t õ s ự r ỗ ữủ ữợ sõ Pữỡ tr st ❦❤ỉ♥❣ t❤➸ ❝❤♦ ❤✐➺✉ ù♥❣ ♥➔② ✈➻ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t↕✐ ❝❤➙♥ ❦❤æ♥❣ ❧✉æ♥ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ✈➔ ♥❣➠♥ ❝➜♠ ❦❤è✐ ❧÷đ♥❣ ❜✐➳♥ ✤ê✐ ✤➸ ❜ù❝ ①↕ r❛ sâ♥❣ ❤➜♣ ❞➝♥✳ ❘ã r➔♥❣ ❦❤✐ ♠ët ♣❤➛♥ ❦❤è✐ ❧÷đ♥❣ t sõ t ởt ỗ sâ♥❣ ❤➜♣ ❞➝♥ ✤õ ♠↕♥❤ ✈➔ ❝â ❤✐ ✈å♥❣ ✤➸ t r õ t ỵ tt tữỡ ố tờ qt st ởt ỵ tt ①✉➜t s➢❝✱ ✈➔ ✤➣ ✤÷đ❝ r➜t ♥❤✐➲✉ t❤➼ ♥❣❤✐➺♠ ❦✐➸♠ ❝❤ù♥❣ t❤➔♥❤ ❝ỉ♥❣ rü❝ rï✳ ❚✉② ♥❤✐➯♥✱ ♥❤÷ ✤➣ ♥â✐✱ ♥â ❝á♥ ♥❤✐➲✉ ✈➜♥ ✤➲ ♥❤÷ ❧➔ sü ❣✐➣♥ ♥ð t➠♥❣ tè❝ ❝õ❛ ✈ơ trư ✭✈➜♥ ✤➲ ♥➠♥❣ ❧÷đ♥❣ tè✐✮✱ sü ❧↕♠ ♣❤→t ❝õ❛ ✈ơ trư✱ ♠ët sè ✈➜♥ ✤➲ tr ỵ tt ữủ tỷ ỵ tt ❤➜♣ ❞➝♥✲f (R) ✤➣ ✤÷đ❝ ✤➲ ①✉➜t ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ✤â✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ ✭✷✳✷✮ ữủ t t ữỡ tr ỵ tt ❤➜♣ ❞➝♥✲f (R)✳ P❤÷ì♥❣ tr➻♥❤ ♥➔② ❝❤➾ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ❞➵ ❞➔♥❣ tr♦♥❣ ❝→❝ tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t➽♥❤✱ ố ợ trữớ ố ự ổ t ♥â ❧➔ r➜t ❦❤â ❦❤➠♥✳ ❍✐➺♥ ♥❛② t❤❡♦ ♥❤÷ ❤✐➸✉ ❜✐➳t ❝õ❛ ❝❤ó♥❣ tỉ✐ t❤➻ ❝❤÷❛ ❝â ❜➔✐ ❜→♦ ♥➔♦ õ t t ỵ tt ❞➝♥✲f (R) ❝❤♦ tr÷í♥❣ ❦❤ỉ♥❣ t➽♥❤✳ ❱➻ ✈➟② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ♣❤↔✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤❡♦ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ❜➡♥❣ ❝→❝❤ →♣ ✤➦t ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✳ ❈❤ó♥❣ tỉ✐ ✤➣ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ t❤❡♦ ữỡ q ữỡ tr st ì ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ ❝→❝ ♥❣❤✐➺♠ t❤✉ ✤÷đ❝ s➩ ❦❤ỉ♥❣ ①✉➜t ❤✐➺♥ t❤➯♠ ❜➜t ❦➻ ♠ët t❤❛♠ sè ♠ỵ✐ ♥➔♦ ✭✈➼ ❞ư ♥❤÷ ❝→❝ ❤➡♥❣ sè t➼❝❤ ♣❤➙♥✮✱ t❤❛♠ sè ❞✉② ♥❤➜t ①✉➜t ❤✐➺♥ ❧➔ t❤❛♠ sè ♥❤✐➵✉ ❧♦↕♥ λ ♠➔ ð ✤â ✤➣ ✤÷đ❝ ✤÷❛ ✈➔♦ ♥❣❛② tø ✤➛✉✳ ❱➻ ✈➟② t❛ ❝â t❤➸ →♣ ❞ö♥❣ ♥❤ú♥❣ ♥❣❤✐➺♠ t❤✉ ✤÷đ❝ ✈➔♦ ❝→❝ ❜➔✐ t♦→♥ ❝ư t❤➸ ✤➸ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ ❝ư t❤➸ ✭rã r➔♥❣✮ ♥❤➡♠ ❦✐➸♠ tr ỵ tt f (R) ữ t ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ❤➔♥❤ t✐♥❤✱ →♥❤ s→♥❣✱ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❦❤→❝ s❛✉ ♥➔②✱✳✳✳✮✳ ❼ ❑➳t q✉↔ ♥❣❤✐➺♠ ♥❤✐➵✉ ❧♦↕♥ ❝õ❛ ❤➔♠ f (R) tê♥❣ q✉→t ❝❤♦ ♠ët tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ tê♥❣ q✉→t ✭❦❤æ♥❣ ♥❤➜t t❤✐➳t t➽♥❤✱ ❦❤æ♥❣ ❝❤➾ ❝❤♦ ❝❤➙♥ ❦❤æ♥❣✮ ❧➔ ✭✷✳✼✶✮ ✕ ✭✷✳✼✹✮✱ tr♦♥❣ ❦❤✐ ♥❣❤✐➺♠ t↕✐ ❝❤➙♥ ❦❤æ♥❣ s➩ ❧➔ ✭✷✳✸✷✮ ✕ ✭✷✳✸✺✮✳ ❼ t s ợ ỗ ❞➝♥ t❛ s➩ ✤÷đ❝ ♥❣❤✐➺♠ ✭✷✳✹✵✮ ✕ ✭✷✳✹✸✮✱ tø ✤➙② ♠ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ✤➣ ✤÷đ❝ t➼♥❤ t♦→♥ ❝ư t❤➸ ✈ỵ✐ ❝→❝ ♠ỉ ❤➻♥❤ f (R) ✭❝→❝ ❤➔♠ f (R)✮ ❦❤→❝ ♥❤❛✉✳ ❼ ❙❛✉ ✤â q✉ÿ ✤↕♦ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ❤➔♥❤ t✐♥❤ ❝ơ♥❣ ♥❤÷ ✤÷í♥❣ ❝❤✉②➲♥ ❝õ❛ →♥❤ s→♥❣ ✤➣ ✤÷đ❝ t➼♥❤ t♦→♥ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ỵ tt f (R) t q ❣â❝ q✉❛② t✐♥❤ s❛✐ ❤❛② ❣â❝ ❧➺❝❤ ❝õ❛ t✐❛ s→♥❣ õ ởt sỹ ỵ tt st tr÷í♥❣ ✤è✐ ①ù♥❣ ❝➛✉ t➽♥❤ t❤➻ ❝→❝ ✤↕✐ ❧÷đ♥❣ ✶✶✾ t s t ỵ P ýt sỹ ỵ tt st ợ ữ trữớ ố ự ổ t sü ❤✐➺✉ ❝❤➾♥❤ ❝❤♦ ❣✐→ trà ❊✐♥st❡✐♥ ✈➲ ✤ë ❧ỵ♥ õ ỏ ởt số ự ợ ỵ tt ❊✐♥st❡✐♥ ❦❤ỉ♥❣ t❤➸ ❝â ✤â ❧➔ ❝→❝ ✤↕✐ ❧÷đ♥❣ s➩ t❤❛② ✤ê✐ t❤❡♦ t❤í✐ ❣✐❛♥✱ ✈➼ ❞ư ♥❤÷ ❣â❝ q✉❛② t✐♥❤ s❛✐ ❝õ❛ ❤➔♥❤ t✐♥❤ tr♦♥❣ ❝→❝ ❝❤✉ ❦➻ ❦❤→❝ ♥❤❛✉ s➩ ❦❤→❝ ♥❤❛✉✱ ❝→❝ trư❝ ❝õ❛ ❊❧✐♣ ❝ơ♥❣ t❤❛② t tớ ố ợ ỵ tt ❊✐♥st❡✐♥ t➜t ❝↔ ❝→❝ ✤↕✐ ❧÷đ♥❣ ♥❤÷ ❣â❝ q✉❛② t✐♥❤ s❛✐✱ ✤ë ❞➔✐ trö❝ ❊❧✐♣✱ ❣â❝ ❧➺❝❤ ❝õ❛ t✐❛ s→♥❣✱ ✈✳✈✱ ❝❤ó♥❣ ❧✉ỉ♥ ❧➔ ❤➡♥❣ sè✮✳ ❼ ⑩♣ ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ✈➔♦ t➼♥❤ t♦→♥ ❝❤♦ ♠ët tr÷í♥❣ ❤➜♣ ❞➝♥ ♠↕♥❤ ❤ì♥✱ ✤â ❧➔ ♥❣❤✐➯♥ ❝ù✉ ♥❣ỉ✐ s❛♦ ❙✷ q✉❛② q✉❛♥❤ ♠ët s✐➯✉ ❤è ✤❡♥ ✭❝â ❦❤è✐ ❧÷đ♥❣ ❜➡♥❣ ✹ tr✐➺✉ ❧➛♥ ▼➦t ❚rí✐✮ ð t➙♠ ❝õ❛ ❉↔✐ ◆❣➙♥ ❍➔ ✭❙❣r ❆✯✮✳ ❑➳t q✉↔ t❤✉ ✤÷đ❝ ❧➔ ❝→❝ ữủ õ sỹ ỡ ỵ t❤✉②➳t ❝õ❛ ❊✐♥st❡✐♥✳ ❼ ❑❤✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❊✐♥st❡✐♥ t❤➻ ♥❣❤✐➺♠ ✤↕t ✤÷đ❝ ❧➔ ❝→❝ t❡♥s♦r ♠❡tr✐❝ ❧✉ỉ♥ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ✭❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥✮✱ ❝á♥ ❦❤✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ f (R) ♥❣❤✐➺♠ t❤✉ ✤÷đ❝ ❧➔ ❝→❝ t❡♥s♦r ♠❡tr✐❝ s➩ ❦❤æ♥❣ ♣❤↔✐ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ♠➔ ♥â ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥✳ ❚ø ✤➙② t❛ t❤➜② ỵ r õ t ù tr ỵ t❤✉②➳t ❤➜♣ ❞➝♥✲f (R)✳ ✣✐➲✉ ♥➔② ❝â t❤➸ ❣➙② r❛ ❝→❝ ❤✐➺✉ ù♥❣ t❤ó ✈à ❤ì♥✱ ✈➼ ❞ư ♥❤÷ ❦❤✐ ởt ỗ r ❝♦ ❧↕✐ t❤❡♦ t❤í✐ ❣✐❛♥ t❤➻ ♥â ❝â ❦❤↔ ♥➠♥❣ ự r sõ t ỵ tt f (R) ự ổ t õ tr ỵ tt st tr ỵ tt st ởt trữớ ố ①ù♥❣ ❝➛✉ t↕✐ ❝❤➙♥ ❦❤æ♥❣ ❧✉æ♥ ð tr↕♥❣ t❤→✐ ❞ø♥❣ ✈➔ ❦❤æ♥❣ t❤➸ ❜ù❝ ①↕ r❛ sâ♥❣ ❤➜♣ ❞➝♥✳ ❈æ♥❣ s✉➜t ❜ù❝ ①↕ sâ♥❣ ❤➜♣ ❞➝♥ tr♦♥❣ q✉→ tr➻♥❤ sö♣ ✤ê ❝õ❛ ♠ët ♥❣æ✐ s❛♦ ✭❤❛② ♠ët ❤è ✤❡♥✮ ✤è✐ ①ù♥❣ ❝➛✉ ❝ơ♥❣ ✤➣ t❤✉ ✤÷đ❝✳ ❼ ❑❤✐ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❧♦↕♥ ✈➔♦ ✤➸ ♠✐➯✉ t↔ ❱ơ trư✳ t q t ỵ tt f (R) t ỵ tt st ữ ❚❤❛♠ sè ❍✉❜❜❧❡ ❍ s➩ t❤❛② ✤ê✐ t❤❡♦ t❤í✐ ❣✐❛♥ ✤➸ ♣❤ị ❤đ♣ ✈ỵ✐ sü t✐➳♥ tr✐➸♥ ❝õ❛ ✈ơ trư✳ ◆❣❤✐➺♠ ♥❤✐➵✉ ❧♦↕♥ t❤✉ ✤÷đ❝ ❝ơ♥❣ ♠✐➯✉ t↔ ✤÷đ❝ sü ♥è✐ t✐➳♣ ♥❤❛✉ ❝õ❛ ❜❛ ❦➾ ♥❣✉②➯♥ ✈ơ trư ✭✤â ❧➔ ❦➾ ♥❣✉②➯♥ ❧↕♠ ♣❤→t✱ ❦➾ ♥❣✉②➯♥ ❜ù❝ ①↕ ✈➔ ❦➾ ♥❣✉②➯♥ ♥❣➔② ♥❛② ✭✈➜♥ ✤➲ ♥➠♥❣ ❧÷đ♥❣ tè✐✮✮✳ ❚ø õ t ỵ tt f (R) õ t❤➸ ❧➔ ♠ët t❤û ♥❣❤✐➺♠ tèt ✤➸ ♠✐➯✉ t↔ ✈ô trö✳ ◆❤ú♥❣ ❦➳t q✉↔ ♥➔② ♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦ ❝→❝ t❤➼ ♥❣❤✐➺♠ ❦✐➸♠ tr❛ tr♦♥❣ t÷ì♥❣ ❧❛✐ ✤➸ ✤→♥❤ ❣✐→ t ú ỵ tt f (R) Pử tsr tr str tr ỗ ❤➜♣ ❞➝♥ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝ỉ♥❣ t❤ù❝ ✭✷✳✸✼✮✳ ❚ø ✭✷✳✼✶✮ ✈➔ ✭✷✳✼✷✮ tr♦♥❣ ❣✐ỵ✐ ❤↕♥ ❊✐♥st❡✐♥ ✭❧➜② λh = 0✮ ❝❤ó♥❣ t❛ ❝â g11 (r, t) = 1− r r −1 , kT 00 (r′ , t)r′2 dr′ r g00 (r, t) = ❡①♣ r′ g11 (r′ , t) − ∞ 1 + kT 11 (r′ , t) − ′ ′2 r r ✭❆✳✶✮ dr. T = ỗ ❤➜♣ ❞➝♥ ❬❜➯♥ ♥❣♦➔✐ ❜→♥ ❦➼♥❤ Ro (t)❪ ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ r r′ g11 (r′ , t) − g00 (r, t) = exp Ro (t) Ro (t) × exp ∞ 1 + kT 11 (r′ , t) − ′ ′2 r r dr′ −g11 (r′ , t) − ′ dr r′ ✭❆✳✸✮ ▼➦t ❦❤→❝✱ ♥❤÷ t❛ t❤➜② ỗ t ổ g11 = ❜ð✐ t❤➳✱ g00 (r, t) = − 2GM exp c2 Ro (t) r r′ g11 (r′ , t) − Ro (t) 1 ′ + kT (r , t) − r′2 r′ −1 1− 2GM c2 r ✱ dr′ ✭❆✳✹✮ ✶✷✶ ▲✉➟♥ →♥ t✐➳♥ sÿ ✈➟t ỵ P ýt ứ s r 2GM g00 (r, t) = − exp c Ro (t) r k r′2 Ro (t) r′ ′′2 ′′ r T (r , t)dr′′ − kr′ T 11 (r′ , t) ′ dr r′ − rk′ r′′2 T 00 (r′′ , t)dr′′ ✭❆✳✺✮ ▼➦t ❦❤→❝ ♥❤÷ ✭✷✳✸✻✮ ❝❤ó♥❣ t❛ ✤❛♥❣ ①➨t T 00 ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥ t ✭❝♦✐ ♠➟t ✤ë ỗ ỗ t t 2GM exp g00 (r, t) = − c Ro (t) r k ′ ′ ′ r T (t) − kr T (r , t) ′ dr − k3 r′2 T 00 (t) Ro (t) ✭❆✳✻✮ ◆➳✉ ❜ä q✉❛ T 11 ❧➔ r➜t ọ s ợ T 00 ú t t ữủ g00 (r, t) = − r k ′ r T (t) dr′ , k ′2 Ro (t) − r T (t) 2GM exp c2 Ro (t) ✭❆✳✼✮ ❜ð✐ t❤➳✱ g00 (r, t) = − − k3 [Ro (t)]2 T 00 (t) 2GM c2 Ro (t) − k3 r2 T 00 (t) ✭❆✳✽✮ ◆➳✉ T 00 ✤÷đ❝ ①❡♠ ❧➔ ỗ t t ữủ ỡ g11 (r, t) = −1 1− k r T (t) ✭❆✳✾✮ ❚❤❛② t❤➳ ✭✷✳✷✻✮ ✈➔♦ tr♦♥❣ ✭❆✳✽✮ ✈➔ ✭❆✳✾✮✱ ❝❤ó♥❣ t❛ t❤✉ ✤÷đ❝ g11 (r, t) = 1− −1 1− kM c2 4πRo (t) kM c2 r2 4π[Ro (t)]3 1− kM c2 g00 (r, t) = − 4πRo (t) kM c2 r2 4π[Ro (t)]3 ✭❆✳✶✵✮ , ✭❆✳✶✶✮ , t❤❛② t❤➳ ✭❆✳✶✵✮ ✈➔ ✭❆✳✶✶✮ ✈➔♦ tr♦♥❣ ✭✷✳✸✻✮✱ s✉② r❛ ′ ∼ ∇i ∇E i h (kT ) = = k 8π k 8π 1− kM c2 4πRo (t) 1− kM c2 4πRo (t) ∂h′ (kT 00 ) 3/2 ∂t r2 1− kM c2 r2 4π[Ro (t)]3 ∂(kT 00 ) ′′ h (kT 00 ) 3/2 ∂t r2 1− ∂ M ∂t [Ro (t)]3 kM c2 r2 4π[Ro (t)]3 ∂ M ∂t [Ro (t)]3 t s t ỵ P ❑ý✭♣❤❛♠❦②✈❛t❧②❅❣♠❛✐❧✳❝♦♠✮ ❙❛✉ ✤â✱ tø ✭✷✳✷✻✮ ✈➔ ✭❆✳✶✷✮✱ t❛ ✤÷đ❝ ′ ∼ ∇i ∇E i h (kT ) = 3k c2 32π h′′ (kT 00 ) kM c2 4πRo (t) 1− r2 3/2 ∂ M ∂t [Ro (t)]3 kM c2 r2 4π[Ro (t)]3 1− , ✭❆✳✶✸✮ ❜ð✐ t❤➳✱ Ro (t) ∇ i ′ ∇E i h (kT )r dr ∼ = 3k2 c2 32π o ∂ M ∂t [Ro (t)]3 1− h′′ (kT 00 ) 3/2 kM c2 Ro (t) r4 1− o 4πRo (t) dr kM c2 r2 4π[Ro (t)]3 ✭❆✳✶✹✮ ❦➼ ❤✐➺✉ ξ (t) = kM c2 4π[Ro (t)]3 ✭❆✳✶✺✮ ❝❤ó♥❣ t❛ ✈✐➳t ❧↕✐ ✭❆✳✶✹✮ ❧➔ Ro (t) ∇ i ′ ∇E i h (kT )r dr ∼ = 3k2 c2 32π o ∂ M ∂t [Ro (t)]3 h′′ (kT 00 ) 3/2 {1 − [ξ(t)Ro (t)]2 } Ro (t) r4 − ξ (t)r2 o dr ✭❆✳✶✻✮ ❈✉è✐ ❝ò♥❣✱ ❞➵ ❞➔♥❣ t❤➜② r➡♥❣ r4 − ξ r2 = ∂ 3❛r❝s✐♥(ξr) − ξr(3 + 2ξ r2 ) 8ξ ∂r − ξ r2 , ✭❆✳✶✼✮ ♥❤÷ ✈➟②✱ Ro (t) ∇ i ′ ∇E i h (kT )r dr ∼ = h′′ (kT 00 ) o ∂ M ∂t [Ro (t)]3 α(t), ✭❆✳✶✽✮ ✈ỵ✐ α(t) = 3k c2 R0 (t) 256π [ξ(t)]4 arcsin[ξ(t)R0 (t)] ξ(t)R0 (t) − + 2[ξ(t)R0 (t)]2 × − [ξ(t)R0 (t)]2 ✶✷✸ − [ξ(t)R0 (t)]2 3/2 t s t ỵ P❤↕♠ ❱➠♥ ❑ý✭♣❤❛♠❦②✈❛t❧②❅❣♠❛✐❧✳❝♦♠✮ ❈→❝ ❝æ♥❣ ❜è ❦❤♦❛ ❤å❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧✉➟♥ →♥ ✶✳ ◆✳ ❆✳ ❑②✱ P✳ ❱✳ ❑② ❛♥❞ ◆✳ ❚✳ ❍✳ ❱❛♥✱ ✏P❡rt✉r❜❛t✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ f (R)✲t❤❡♦r② ♦❢ ❣r❛✈✐t② ✐♥ ❛ ❝❡♥tr❛❧ ❣r❛✈✐t❛t✐♦♥❛❧ ❢✐❡❧❞ ❛♥❞ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s✱✑ ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✽✱ ♥♦✳ ✼✱ ✺✸✾ ✭✷✵✶✽✮ ❬❛r❳✐✈✿✶✽✵✼✳✵✹✻✷✽ ❬❣r✲q❝❪❪✳ ✷✳ ◆✳ ❆✳ ❑②✱ P✳ ❱✳ ❑② ❛♥❞ ◆✳ ❚✳ ❍✳ ❱❛♥✱ ✏❚❡st✐♥❣ t❤❡ f (R)✲t❤❡♦r② ♦❢ ❣r❛✈✐t②✱✑ ❈♦♠♠✉♥✳ P❤②s✳ ✷✾✱ ♥♦✳✶✱ ✸✺✲✹✻ ✭✷✵✶✾✮ ❬❛r❳✐✈✿✶✾✵✹✳✵✹✵✶✸ ❬♣❤②s✐❝s✳❣❡♥✲♣❤❪❪✳ ✸✳ P✳ ❱✳ ❑②✱ ◆✳ ❚✳ ❍✳ ❱❛♥ ❛♥❞ ◆✳ ❆✳ ❑②✱ ❝❡♥tr❛❧ ❣r❛✈✐t❛t✐♦♥❛❧ ❢✐❡❧❞✱✑ ✏◆❡✇ ♣❤❡♥♦♠❡♥❛ ✇✐t❤ t❤❡ ❢✭❘✮✲t❤❡♦r② ♦❢ ❣r❛✈✐t❛t✐♦♥ ✐♥ ❛ ❏✳ P❤②s✳ ❈♦♥❢✳ ❙❡r✳ ✶✺✵✻✱ ♥♦✳✶✱ ✵✶✷✵✵✶ ✭✷✵✷✵✮ ✶✷✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆✳ ❆✳ ❑②✱ P✳ ❱✳ ❑② ❛♥❞ ◆✳ ❚✳ ❍✳ ❱❛♥✱ ✏P❡rt✉r❜❛t✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ✐♥ ❛ ❝❡♥tr❛❧ ❣r❛✈✐t❛t✐♦♥❛❧ ❢✐❡❧❞ ❛♥❞ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s✱✑ f (R)✲t❤❡♦r② ♦❢ ❣r❛✈✐t② ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✽✱ ♥♦✳ ✼✱ ✺✸✾ ✭✷✵✶✽✮ ❬❛r❳✐✈✿✶✽✵✼✳✵✹✻✷✽ ❬❣r✲q❝❪❪✳ ❬✷❪ ◆✳ ❆✳ ❑②✱ P✳ ❱✳ ❑② ❛♥❞ ◆✳ ❚✳ ❍✳ ❱❛♥✱ ✏❚❡st✐♥❣ t❤❡ f (R)✲t❤❡♦r② ♦❢ ❣r❛✈✐t②✱✑ ❈♦♠♠✉♥✳ P❤②s✳ ✷✾✱ ♥♦✳✶✱ ✸✺✲✹✻ ✭✷✵✶✾✮ ❬❛r❳✐✈✿✶✾✵✹✳✵✹✵✶✸ ❬♣❤②s✐❝s✳❣❡♥✲♣❤❪❪✳ ❬✸❪ P✳ ❱✳ ❑②✱ ◆✳ ❚✳ ❍✳ ❱❛♥ ❛♥❞ ◆✳ ❆✳ ❑②✱ ❝❡♥tr❛❧ ❣r❛✈✐t❛t✐♦♥❛❧ ❢✐❡❧❞✱✑ ❏✳ P❤②s✳ ❈♦♥❢✳ ❙❡r✳ ✶✺✵✻✱ ♥♦✳✶✱ ✵✶✷✵✵✶ ✭✷✵✷✵✮ ❬✹❪ ▲✳ ❉✳ ▲❛♥❞❛✉ ❛♥❞ ❊✳ ▼✳ ▲✐❢s❤✐t③✱ ✏ ❚❤❡ ❬✺❪ ❙✳ ❲❡✐♥❜❡r❣✱ ✏◆❡✇ ♣❤❡♥♦♠❡♥❛ ✇✐t❤ t❤❡ ❢✭❘✮✲t❤❡♦r② ♦❢ ❣r❛✈✐t❛t✐♦♥ ✐♥ ❛ ❈❧❛ss✐❝❛❧ ❚❤❡♦r② ♦❢ ❋✐❡❧❞s✱✧ ✈♦❧✳ ✷✱ ❊❧s❡✈✐❡r✱ ❖①❢♦r❞✱ ✶✾✾✹✳ ✏●r❛✈✐t❛t✐♦♥ ❛♥❞ ❝♦s♠♦❧♦❣②✿ Pr✐♥❝✐♣❧❡ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②✱✑ ❏♦❤♥ ❲✐❧❡② & ❙♦♥✱ ◆❡✇ ❨♦r❦✱ ✶✾✼✷✳ ❬✻❪ ❆✳ ●✳ ❘✐❡ss ❡t ❛❧✳ ❬❙✉♣❡r♥♦✈❛ ❙❡❛r❝❤ ❚❡❛♠❪✱ ✏❖❜s❡r✈❛t✐♦♥❛❧ ❡✈✐❞❡♥❝❡ ❢r♦♠ s✉♣❡r♥♦✈❛❡ ❢♦r ❛♥ ❛❝✲ ❝❡❧❡r❛t✐♥❣ ✉♥✐✈❡rs❡ ❛♥❞ ❛ ❝♦s♠♦❧♦❣✐❝❛❧ ❝♦♥st❛♥t✱✑ ❬✼❪ ❙✳ P❡r❧♠✉tt❡r ❡t ❛❧✳ ❬❙✉♣❡r♥♦✈❛ ❈♦s♠♦❧♦❣② Pr♦❥❡❝t ❈♦❧❧❛❜♦r❛t✐♦♥❪✱ ▲❛♠❜❞❛ ❢r♦♠ ✹✷ ❤✐❣❤ r❡❞s❤✐❢t s✉♣❡r♥♦✈❛❡✱✑ ❬✽❪ ❆✳ ❆✳ ❙t❛r♦❜✐♥s❦②✱ ❆str♦♥✳ ❏✳ ✶✶✻✱ ✶✵✵✾ ✭✶✾✾✽✮ ❬❛str♦✲♣❤✴✾✽✵✺✷✵✶❪✳ ✏▼❡❛s✉r❡♠❡♥ts ♦❢ ❖♠❡❣❛ ❛♥❞ ❆str♦♣❤②s✳ ❏✳ ✺✶✼✱ ✺✻✺ ✭✶✾✾✾✮ ❬❛str♦✲♣❤✴✾✽✶✷✶✸✸❪✳ ✏❆ ◆❡✇ ❚②♣❡ ♦❢ ■s♦tr♦♣✐❝ ❈♦s♠♦❧♦❣✐❝❛❧ ▼♦❞❡❧s ❲✐t❤♦✉t ❙✐♥❣✉❧❛r✐t②✱✑ P❤②s✳ ▲❡tt✳ ✾✶❇✱ ✾✾ ✭✶✾✽✵✮✳ ❬✾❪ ❉✳ ❑❛③❛♥❛s✱ ✏❉②♥❛♠✐❝s ♦❢ t❤❡ ❯♥✐✈❡rs❡ ❛♥❞ ❙♣♦♥t❛♥❡♦✉s ❙②♠♠❡tr② ❇r❡❛❦✐♥❣✱✑ ❆str♦♣❤②s✳ ❏✳ ✷✹✶✱ ▲✺✾ ✭✶✾✽✵✮✳ ❬✶✵❪ ❉✳ ❍✉t❡r❡r ❛♥❞ ▼✳ ❙✳ ❚✉r♥❡r✱ ♠❡❛s✉r❡♠❡♥ts✱✑ ✏Pr♦s♣❡❝ts ❢♦r ♣r♦❜✐♥❣ t❤❡ ❞❛r❦ ❡♥❡r❣② ✈✐❛ s✉♣❡r♥♦✈❛ ❞✐st❛♥❝❡ P❤②s✳ ❘❡✈✳ ❉ ✻✵✱ ✵✽✶✸✵✶ ✭✶✾✾✾✮ ❬❛str♦✲♣❤✴✾✽✵✽✶✸✸❪✳ ❬✶✶❪ ❱✳ ❙❛❤♥✐ ❛♥❞ ❆✳ ❆✳ ❙t❛r♦❜✐♥s❦②✱ ✏❚❤❡ ❈❛s❡ ❢♦r ❛ ♣♦s✐t✐✈❡ ❝♦s♠♦❧♦❣✐❝❛❧ ▲❛♠❜❞❛ t❡r♠✱✑ ■♥t✳ ❏✳ ▼♦❞✳ P❤②s✳ ❉ ✾✱ ✸✼✸ ✭✷✵✵✵✮ ❬❛str♦✲♣❤✴✾✾✵✹✸✾✽❪✳ ❬✶✷❪ ▼✳ ❙✳ ❚✉r♥❡r✱ ✏❉❛r❦ ♠❛tt❡r ❛♥❞ ❡♥❡r❣② ✐♥ t❤❡ ✉♥✐✈❡rs❡✱✑ P❤②s✳ ❙❝r✐♣t❛ ❚ ✽✺✱ ✷✶✵ ✭✷✵✵✵✮ ❬❛str♦✲ ♣❤✴✾✾✵✶✶✵✾❪✳ ❬✶✸❪ ❙✳ ❲❡✐♥❜❡r❣✱ ❬✶✹❪ ❘✳ ❇♦✉ss♦✱ ❚❤❡♦r②✱✑ ✏❚❤❡ ❈♦s♠♦❧♦❣✐❝❛❧ ❈♦♥st❛♥t Pr♦❜❧❡♠✱✑ ❘❡✈✳ ▼♦❞✳ P❤②s✳ ✻✶✱ ✶ ✭✶✾✽✾✮✳ ✏❚❤❡ ❈♦s♠♦❧♦❣✐❝❛❧ ❈♦♥st❛♥t Pr♦❜❧❡♠✱ ❉❛r❦ ❊♥❡r❣②✱ ❛♥❞ t❤❡ ▲❛♥❞s❝❛♣❡ ♦❢ ❙tr✐♥❣ P♦♥t✐❢✳ ❆❝❛❞✳ ❙❝✐✳ ❙❝r✳ ❱❛r✐❛ ✶✶✾✱ ✶✷✾ ✭✷✵✶✶✮ ❬❛r❳✐✈✿✶✷✵✸✳✵✸✵✼ ❬❛str♦✲♣❤✳❈❖❪❪✳ ❬✶✺❪ ❆✳ ❉❡ ❋❡❧✐❝❡ ❛♥❞ ❙✳ ❚s✉❥✐❦❛✇❛✱ ✏❢✭❘✮ t❤❡♦r✐❡s✱✑ ❬❣r✲q❝❪❪✳ ✶✷✺ ▲✐✈✐♥❣ ❘❡✈✳ ❘❡❧✳ ✶✸✱ ✸ ✭✷✵✶✵✮ r t s t ỵ P ýt ❬✶✻❪ ❘✳ ❋♦♦t✱ ✏❉✐ss✐♣❛t✐✈❡ ❞❛r❦ ♠❛tt❡r ❛♥❞ t❤❡ r♦t❛t✐♦♥ ❝✉r✈❡s ♦❢ ❞✇❛r❢ ❣❛❧❛①✐❡s✱✑ ❏❈❆P ✶✻✵✼✱ ♥♦✳ ✵✼✱ ✵✶✶ ✭✷✵✶✻✮ ❬❛r❳✐✈✿✶✺✵✻✳✵✶✹✺✶ ❬❛str♦✲♣❤✳●❆❪❪✳ ❬✶✼❪ ❑✳ ❆✳ ❖♠❛♥✱ ❏✳ ❋✳ ◆❛✈❛rr♦✱ ▲✳ ❱✳ ❙❛❧❡s✱ ❆✳ ❋❛tt❛❤✐✱ ❈✳ ❙✳ ❋r❡♥❦✱ ❚✳ ❙❛✇❛❧❛✱ ▼✳ ❙❝❤❛❧❧❡r ❛♥❞ ❙✳ ❉✳ ▼✳ ❲❤✐t❡✱ ✏▼✐ss✐♥❣ ❞❛r❦ ♠❛tt❡r ✐♥ ❞✇❛r❢ ❣❛❧❛①✐❡s❄✱✑ ▼♦♥✳ ◆♦t✳ ❘♦②✳ ❆str♦♥✳ ❙♦❝✳ ✹✻✵✱ ♥♦✳ ✹✱ ✸✻✶✵ ✭✷✵✶✻✮ ❬❛r❳✐✈✿✶✻✵✶✳✵✶✵✷✻ ❬❛str♦✲♣❤✳●❆❪❪✳ ❬✶✽❪ ❆✳ ●✉❛r♥✐③♦✱ ▲✳ ❈❛st❛♥❡❞❛ ❛♥❞ ❏✳ ▼✳ ❚❡❥❡✐r♦✱ ✏❇♦✉♥❞❛r② ❚❡r♠ ✐♥ ▼❡tr✐❝ ❢✭❘✮ ●r❛✈✐t②✿ ❋✐❡❧❞ ●❡♥✳ ❘❡❧✳ ●r❛✈✳ ✹✷✱ ✷✼✶✸ ✭✷✵✶✵✮ ❬❛r❳✐✈✿✶✵✵✷✳✵✻✶✼ ❬❣r✲q❝❪❪✳ ❊q✉❛t✐♦♥s ✐♥ t❤❡ ▼❡tr✐❝ ❋♦r♠❛❧✐s♠✱✑ ❬✶✾❪ ❚❤♦♠❛s P✳ ❙♦t✐r✐♦✉ ❛♥❞ ❱❛❧❡r✐♦ ❋❛r❛♦♥✐✱ ✏ f (R) t❤❡♦r✐❡s ♦❢ ❣r❛✈✐t②✱✧ ❘❡✈✳ ▼♦❞✳ P❤②s✳ ✽✷✱ ✹✺✶ ✕ P✉❜❧✐s❤❡❞ ✶ ▼❛r❝❤ ✷✵✶✵✳ ❬✷✵❪ ❙✳ ❈❛♣♦③③✐❡❧❧♦ ❛♥❞ ▼✳ ❉❡ ▲❛✉r❡♥t✐s✱ ✏ ❊①t❡♥❞❡❞ ❚❤❡♦r✐❡s ♦❢ ●r❛✈✐t②✱✑ P❤②s✳ ❘❡♣t✳ ✺✵✾✱ ✶✻✼ ✭✷✵✶✶✮ ❞♦✐✿✶✵✳✶✵✶✻✴❥✳♣❤②sr❡♣✳✷✵✶✶✳✵✾✳✵✵✸ ❬❛r❳✐✈✿✶✶✵✽✳✻✷✻✻ ❬❣r✲q❝❪❪✳ ❬✷✶❪ ▲➯ ❚❤à ◆❣❛✱ ✏ ▲✉➟♥ ❱➠♥ ❚❤↕❝ ❙ÿ✱✑ ❍å❝ ❱✐➺♥ ❑❤♦❛ ❍å❝ ❱➔ ❈æ♥❣ ◆❣❤➺✱ ✷✵✷✵✳ ❬✷✷❪ ❚✳ ❈❧✐❢t♦♥✱ P✳ ●✳ ❋❡rr❡✐r❛✱ ❆✳ P❛❞✐❧❧❛ ❛♥❞ ❈✳ ❙❦♦r❞✐s✱ ✏▼♦❞✐❢✐❡❞ ●r❛✈✐t② ❛♥❞ ❈♦s♠♦❧♦❣②✱✑ P❤②s✳ ❘❡♣t✳ ✺✶✸✱ ✶ ✭✷✵✶✷✮ ❬❛r❳✐✈✿✶✶✵✻✳✷✹✼✻ ❬❛str♦✲♣❤✳❈❖❪❪✳ ❬✷✸❪ ❙✳ ❈❛♣♦③③✐❡❧❧♦✱ ❘✳ ❉❡ ❘✐t✐s✱ ❈✳ ❘✉❜❛♥♦ ❛♥❞ P✳ ❙❝✉❞❡❧❧❛r♦✱ ✏ ❊①❛❝t ■♥t✳ ❏✳ ▼♦❞✳ P❤②s✳ ❉ ✺✱ ✽✺ ✭✶✾✾✻✮✳ ❝♦s♠♦❧♦❣✐❡s✱✑ ❬✷✹❪ ❙✳ ❈❛♣♦③③✐❡❧❧♦ ❛♥❞ ▼✳ ❘♦s❤❛♥✱ ✏ ❊①❛❝t t✐❡s✱✑ s♦❧✉t✐♦♥s ✐♥ ❇r❛♥s✲❉✐❝❦❡ ♠❛tt❡r ❝♦s♠♦❧♦❣✐❝❛❧ s♦❧✉t✐♦♥s ❢r♦♠ ❍♦❥♠❛♥ ❝♦♥s❡r✈❛t✐♦♥ q✉❛♥t✐✲ P❤②s✳ ▲❡tt✳ ❇ ✼✷✻ ✭✷✵✶✸✮ ✹✼✶ ❬❛r❳✐✈✿✶✸✵✽✳✸✾✶✵ ❬❣r✲q❝❪❪✳ ❬✷✺❪ ▼✳ P❛♦❧❡❧❧❛ ❛♥❞ ❙✳ ❈❛♣♦③③✐❡❧❧♦✱ ✏ ❍♦❥♠❛♥ s②♠♠❡tr② ❛♣♣r♦❛❝❤ ❢♦r s❝❛❧❛r✕t❡♥s♦r ❝♦s♠♦❧♦❣②✱✑ P❤②s✳ ▲❡tt✳ ❆ ✸✼✾✱ ✶✸✵✹ ✭✷✵✶✺✮ ❬❛r❳✐✈✿✶✺✵✸✳✵✵✵✾✽ ❬❣r✲q❝❪❪✳ ❬✷✻❪ ❙✳ ❑❛♠✐❧②❛ ❛♥❞ ❇✳ ▼♦❞❛❦✱ ✏ ◆♦❡t❤❡r s②♠♠❡tr② st✉❞② ✐♥ ❣❡♥❡r❛❧ s❝❛❧❛r t❡♥s♦r t❤❡♦r②✱✑ ●❡♥✳ ❘❡❧✳ ●r❛✈✳ ✸✻✱ ✻✼✸ ✭✷✵✵✹✮✳ ❬✷✼❪ ❙✳ ❲❛ts♦♥✱ ✏❆♥ ❊①♣♦s✐t✐♦♥ ♦♥ ✐♥❢❧❛t✐♦♥❛r② ❝♦s♠♦❧♦❣②✱✑ ❛str♦✲♣❤✴✵✵✵✺✵✵✸✳ ❬✷✽❪ ❘✳ ❊♠❛♠✐✱ ❍✳ ❋✐r♦✉③❥❛❤✐✱ ❙✳ ▼✳ ❙❛❞❡❣❤ ▼♦✈❛❤❡❞ ❛♥❞ ▼✳ ❩❛r❡✐✱ ❈❤❛r❣❡❞ ❙❝❛❧❛r ❋✐❡❧❞s✱✑ ❬✷✾❪ ❆✳ ❉✳ ▲✐♥❞❡✱ ✏❆♥✐s♦tr♦♣✐❝ ■♥❢❧❛t✐♦♥ ❢r♦♠ ❏❈❆P ✶✶✵✷✱ ✵✵✺ ✭✷✵✶✶✮ ❬❛r❳✐✈✿✶✵✶✵✳✺✹✾✺ ❬❛str♦✲♣❤✳❈❖❪❪✳ ✏❚❤❡ ■♥❢❧❛t✐♦♥❛r② ❯♥✐✈❡rs❡✱✑ ❬✸✵❪ ▲✳ ❆rt✉r♦ ❯r❡♥❛✲▲â♣❡③✱ ✏❙❝❛❧❛r ❘❡♣t✳ Pr♦❣✳ P❤②s✳ ✹✼✱ ✾✷✺ ✭✶✾✽✹✮✳ ❢✐❡❧❞s ❈♦s♠♦❧♦❣②✿ ❞❛r❦ ♠❛tt❡r ❛♥❞ ✐♥❢❧❛t✐♦♥✱✑ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✿ ❈♦♥❢❡r❡♥❝❡ ❙❡r✐❡s ✼✻✶ ✭✷✵✶✻✮ ✵✶✷✵✼✻✳ ❬✸✶❪ ■✳ ▲❡❛♥✐③❜❛rr✉t✐❛✱ ❆✳ ❘♦③❛s✲❋❡r♥→♥❞❡③ ❛♥❞ ■✳ ❚❡r❡♥♦✱ ✏❈♦s♠♦❧♦❣✐❝❛❧ ♠❛tt❡r✲❡♥❡r❣② s❝❛❧❛r ❢✐❡❧❞ ♠♦❞❡❧ ✇✐t❤ ❢❛st tr❛♥s✐t✐♦♥✱✑ ❝♦♥str❛✐♥ts ♦♥ ❛ ✉♥✐❢✐❡❞ ❞❛r❦ P❤②s✳ ❘❡✈✳ ❉ ✾✻✱ ♥♦✳ ✷✱ ✵✷✸✺✵✸ ✭✷✵✶✼✮ ❬❛r❳✐✈✿✶✼✵✻✳✵✶✼✵✻ ❬❛str♦✲♣❤✳❈❖❪❪✳ ❬✸✷❪ ❙✳ ▼✳ ❈❛rr♦❧❧✱ ✏◗✉✐♥t❡ss❡♥❝❡ ❛♥❞ t❤❡ r❡st ♦❢ t❤❡ ✇♦r❧❞✱✑ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✽✶✱ ✸✵✻✼ ✭✶✾✾✽✮ ❬❛str♦✲ ♣❤✴✾✽✵✻✵✾✾❪✳ ❬✸✸❪ ❘✳ ❞❡ ❘✐t✐s✱ ●✳ ▼❛r♠♦✱ ●✳ P❧❛t❛♥✐❛✱ ❈✳ ❘✉❜❛♥♦✱ P✳ ❙❝✉❞❡❧❧❛r♦ ❛♥❞ ❈✳ ❙t♦r♥❛✐♦❧♦✱ ✏ ◆❡✇ t♦ ❢✐♥❞ ❡①❛❝t s♦❧✉t✐♦♥s ❢♦r ❝♦s♠♦❧♦❣✐❝❛❧ ♠♦❞❡❧s ✇✐t❤ ❛ s❝❛❧❛r ❢✐❡❧❞✱✑ ✶✷✻ ❛♣♣r♦❛❝❤ P❤②s✳ ❘❡✈✳ ❉ ✹✷✱ ✶✵✾✶ ✭✶✾✾✵✮✳ ▲✉➟♥ →♥ t✐➳♥ sÿ t ỵ P ýt tr s②♠♠❡tr② ✐♥ ❢✭❘✮ ❝♦s♠♦❧♦❣②✱✑ P❤②s✳ ▲❡tt✳ ❇ ✻✻✹ ✭✷✵✵✽✮ ✶✻ ❬❛r❳✐✈✿✵✽✵✹✳✸✹✹✾ ❬❣r✲q❝❪❪✳ ❬✸✺❪ ❆✳ ◆❛r✉❦♦✱ ❉✳ ❨♦s❤✐❞❛ ❛♥❞ ❙✳ ▼✉❦♦❤②❛♠❛✱ ✏ ●r❛✈✐t❛t✐♦♥❛❧ s❝❛❧❛r✕t❡♥s♦r t❤❡♦r②✱✑ ❈❧❛ss✳ ◗✉❛♥t✳ ●r❛✈✳ ✸✸✱ ♥♦✳ ✾✱ ✵✾▲❚✵✶ ✭✷✵✶✻✮ ❬❛r❳✐✈✿✶✺✶✷✳✵✻✾✼✼ ❬❣r✲q❝❪❪✳ ❬✸✻❪ ❚✳ ❍❛r❦♦✱ ❋✳ ❙✳ ◆✳ ▲♦❜♦✱ ❙✳ ◆♦❥✐r✐ ❛♥❞ ❙✳ ❉✳ ❖❞✐♥ts♦✈✱ ✏ f (R, T ) ❣r❛✈✐t②✱✑ P❤②s✳ ❘❡✈✳ ❉ ✽✹✱ ✵✷✹✵✷✵ ✭✷✵✶✶✮ ❬❛r❳✐✈✿✶✶✵✹✳✷✻✻✾ ❬❣r✲q❝❪❪✳ ❬✸✼❪ ❈✳ P✳ ❙✐♥❣❤ ❛♥❞ P✳ ❑✉♠❛r✱ t❤❡♦r②✱✑ ✏❋r✐❡❞♠❛♥♥ ♠♦❞❡❧ ✇✐t❤ ✈✐s❝♦✉s ❝♦s♠♦❧♦❣② ✐♥ ♠♦❞✐❢✐❡❞ f (R, T ) ❣r❛✈✐t② ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✹✱ ✸✵✼✵ ✭✷✵✶✹✮ ❬❛r❳✐✈✿✶✹✵✻✳✹✷✺✽ ❬❣r✲q❝❪❪✳ ❬✸✽❪ ❚✳ ❍❛r❦♦ ❛♥❞ ❋✳ ❙✳ ◆✳ ▲♦❜♦✱ ✏❢✭❘✱Lm ✮ ❣r❛✈✐t②✱✑ ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✵✱ ✸✼✸ ✭✷✵✶✵✮ ❬❛r❳✐✈✿✶✵✵✽✳✹✶✾✸ ❬❣r✲q❝❪❪✳ ❬✸✾❪ ❚✳ ❍❛r❦♦ ❛♥❞ ❋✳ ❙✳ ◆✳ ▲♦❜♦✱ ✏●❡♥❡r❛❧✐③❡❞ ❞❛r❦ ❣r❛✈✐t②✱✑ ■♥t✳ ❏✳ ▼♦❞✳ P❤②s✳ ❉ ✷✶✱ ✶✷✹✷✵✶✾ ✭✷✵✶✷✮ ❬❛r❳✐✈✿✶✷✵✺✳✸✷✽✹ ❬❣r✲q❝❪❪✳ ❬✹✵❪ ❚✳ ❍❛r❦♦ ❛♥❞ ▼✳ ❏✳ ▲❛❦❡✱ ✏❈♦s♠✐❝ str✐♥❣s ✐♥ f (R, Lm ) ❣r❛✈✐t②✱✑ ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✺✱ ♥♦✳ ✷✱ ✻✵ ✭✷✵✶✺✮ ❬❛r❳✐✈✿✶✹✵✾✳✽✹✺✹ ❬❣r✲q❝❪❪✳ ❬✹✶❪ ❳✳ ❉♦✉ ❛♥❞ ❳✳ ❍✳ ▼❡♥❣✱ ✏❇✉❧❦ ✈✐s❝♦✉s ❝♦s♠♦❧♦❣②✿ ✉♥✐❢✐❡❞ ❞❛r❦ ♠❛tt❡r✱✑ ❆❞✈✳ ❆str♦♥✳ ✷✵✶✶✱ ✽✷✾✸✹✵ ✭✷✵✶✶✮ ❬❛r❳✐✈✿✶✵✶✷✳✸✵✹✺ ❬❛str♦✲♣❤✳❈❖❪❪✳ ❬✹✷❪ ❳✳ ❍✳ ▼❡♥❣ ❛♥❞ ❳✳ ❉♦✉✱ ❡♥❡r❣②✱✑ ✏❋r✐❡❞♠❛♥♥ ❝♦s♠♦❧♦❣② ✇✐t❤ ❜✉❧❦ ✈✐s❝♦s✐t②✿ ❛ ❝♦♥❝r❡t❡ ♠♦❞❡❧ ❢♦r ❞❛r❦ ❈♦♠♠✉♥✳ ❚❤❡♦r✳ P❤②s✳ ✺✷✱ ✸✼✼ ✭✷✵✵✾✮ ❬❛r❳✐✈✿✵✽✶✷✳✹✾✵✹ ❬❛str♦✲♣❤❪❪✳ ❬✹✸❪ ■✳ ❇r❡✈✐❦✱ ❱✳ ❱✳ ❖❜✉❦❤♦✈ ❛♥❞ ❆✳ ❱✳ ❚✐♠♦s❤❦✐♥✱ ❱✐s❝♦✉s ❋❧✉✐❞ ❈♦s♠♦❧♦❣②✱✑ ❆str♦♣❤②s✳ ❙♣❛❝❡ ❙❝✐✳ ✸✺✺✱ ✸✾✾ ✭✷✵✶✺✮ ❬❛r❳✐✈✿✶✹✶✵✳✷✼✺✵ ❬❣r✲q❝❪❪✳ ❬✹✹❪ ▼✳ ❇♦r✉♥❞❛✱ ❇✳ ❏❛♥ss❡♥ ❛♥❞ ▼✳ ❇❛st❡r♦✲●✐❧✱ ✈❛t✉r❡ ❣r❛✈✐t②✱✑ ❬✹✺❪ ❘✳ ❋❡rr❛r♦✱ ✏❉❛r❦ ❊♥❡r❣② ❈♦✉♣❧❡❞ ✇✐t❤ ❉❛r❦ ▼❛tt❡r ✐♥ ✏P❛❧❛t✐♥✐ ✈❡rs✉s ♠❡tr✐❝ ❢♦r♠✉❧❛t✐♦♥ ✐♥ ❤✐❣❤❡r ❝✉r✲ ❏❈❆P ✵✽✶✶✱ ✵✵✽ ✭✷✵✵✽✮ ❬❛r❳✐✈✿✵✽✵✹✳✹✹✹✵ ❬❤❡♣✲t❤❪❪✳ ✏❢✭❘✮ ❛♥❞ ❢✭❚✮ t❤❡♦r✐❡s ♦❢ ♠♦❞✐❢✐❡❞ ❣r❛✈✐t②✱✑ ❆■P ❈♦♥❢✳ Pr♦❝✳ ✶✹✼✶✱ ✶✵✸ ✭✷✵✶✷✮ ❬❛r❳✐✈✿✶✷✵✹✳✻✷✼✸ ❬❣r✲q❝❪❪✳ ❬✹✻❪ ❙✳ ❉❡s❡r ❛♥❞ ❈✳ ❏✳ ■s❤❛♠✱ ✏❈❛♥♦♥✐❝❛❧ ❱✐❡r❜❡✐♥ ❋♦r♠ ♦❢ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t②✱✑ P❤②s✳ ❘❡✈✳ ❉ ✶✹✱ ✷✺✵✺ ✭✶✾✼✻✮✳ ❬✹✼❪ ❏✳ ❨❡♣❡③✱ ❬✹✽❪ ❚✳ ▼❡✐✱ ✏❊✐♥st❡✐♥✬s ✈✐❡r❜❡✐♥ ❢✐❡❧❞ t❤❡♦r② ♦❢ ❝✉r✈❡❞ s♣❛❝❡✱✑ ❛r❳✐✈✿✶✶✵✻✳✷✵✸✼ ❬❣r✲q❝❪✳ ✏❖♥ t❤❡ ✈✐❡r❜❡✐♥ ❢♦r♠❛❧✐s♠ ♦❢ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②✱✑ ●❡♥✳ ❘❡❧✳ ●r❛✈✳ ✹✵✱ ✶✾✶✸ ✭✷✵✵✽✮ ❬❛r❳✐✈✿✵✼✵✼✳✷✻✸✾ rq ự Pũ ỏ ỵ tt tữỡ ố rở ợ ổtớ ❳✉➜t ❇↔♥ ❑❤♦❛ ❍å❝ ✈➔ ❑➽ ❚❤✉➟t✱ ✷✵✶✺✳ ❬✺✵❪ ▲✳ ❊✳ P❛r❦❡r ❛♥❞ ❉✳ ❏✳ ❚♦♠s✱ ✏◗✉❛♥t✉♠ ❢✐❡❧❞ t❤❡♦r② ✐♥ ❝✉r✈❡❞ s♣❛❝❡t✐♠❡✑✱ ❈❛♠❜r✐❞❣❡ ✉♥✐✈❡rs✐t② ♣r❡ss✱ ✷✵✵✾✳ ❬✺✶❪ ▼✳ ❊✳ P❡s❦✐♥ ❛♥❞ ❉✳ ❱✳ ❙❝❤r♦❡❞❡r✱ ✏ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ q✉❛♥t✉♠ ❢✐❡❧❞ t❤❡♦r②✱✑ P✉❜❧✐s❤✐♥❣✱ ▲✳▲✳❈✱ ✶✾✾✺✳ ✶✷✼ P❡rs❡✉s ❇♦♦❦s ▲✉➟♥ →♥ t✐➳♥ s t ỵ P ýt P ▲✐♥❣✲❋♦♥❣ ▲✐✱ ✏●❛✉❣❡ t❤❡♦r② ♦❢ ❡❧❡♠❡♥t❛r② ♣❛rt✐❝❧❡ ♣❤②s✐❝s✱✑ ❈❧❛r❡♥❞♦♥ Pr❡ss✳❖①❢♦r❞✱ ✷✵✵✵✳ ❬✺✸❪ ◗✉❛♥❣ ❍♦✲❑✐♠ ❛♥❞ ❳✉❛♥✲❨❡♠ P❤❛♠✱ ✏❊❧❡♠❡♥t❛r② ♣❛rt✐❝❧❡s ❛♥❞ t❤❡✐r ✐♥t❡r❛❝t✐♦♥s✱✑ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✱ ✶✾✾✽✳ ❬✺✹❪ ❏✳ ▼✳ ❖✈❡r❞✉✐♥ ❛♥❞ P✳ ❙✳ ❲❡ss♦♥✱ ✏❑❛❧✉③❛✲❑❧❡✐♥ ❣r❛✈✐t②✱✑ P❤②s✳ ❘❡♣t✳ ✷✽✸✱ ✸✵✸ ✭✶✾✾✼✮ ❬❣r✲ q❝✴✾✽✵✺✵✶✽❪✳ ❬✺✺❪ ▲✳ ▲✳ ❲✐❧❧✐❛♠s✱ ✏❋✐❡❧❞ ❊q✉❛t✐♦♥s ❛♥❞ ▲❛❣r❛♥❣✐❛♥ ❢♦r t❤❡ ❑❛❧✉③❛ ▼❡tr✐❝ ❊✈❛❧✉❛t❡❞ ✇✐t❤ ❚❡♥s♦r ❆❧❣❡❜r❛ ❙♦❢t✇❛r❡✱✑ ❬✺✻❪ ❙✳ ❲❡✐♥❜❡r❣✱ r❡❧❛t✐✈✐t②✧✱ ❏✳ ●r❛✈✳ ✷✵✶✺✱ ✾✵✶✽✼✵ ✭✷✵✶✺✮ ✏●r❛✈✐t❛t✐♦♥ ❛♥❞ ❝♦s♠♦❧♦❣②✿ Pr✐♥❝✐♣❧❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❣❡♥❡r❛❧ t❤❡♦r② ♦❢ ❏♦❤♥ ❲✐❧❡② & ❙♦♥✱ ◆❡✇ ❨♦r❦✱ ✶✾✼✷✳ ❬✺✼❪ ❇✳ P✳ ❆❜❜♦tt ❡t ❛❧✳ ❬▲■●❖ ❙❝✐❡♥t✐❢✐❝ ❛♥❞ ❱✐r❣♦ ❈♦❧❧❛❜♦r❛t✐♦♥s❪✱ ✏ ❖❜s❡r✈❛t✐♦♥ ✇❛✈❡s ❢r♦♠ ❛ ❜✐♥❛r② ❜❧❛❝❦ ❤♦❧❡ ♠❡r❣❡r ✑✱ ♦❢ ❣r❛✈✐t❛t✐♦♥❛❧ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✶✻✱ ✵✻✶✶✵✷ ✭✷✵✶✻✮ ❬❛r❳✐✈✿✶✻✵✷✳✵✸✽✸✼ ❬❣r✲q❝❪❪✳ ❬✺✽❪ ❇✳ P✳ ❆❜❜♦tt ❡t ❛❧✳ ❬▲■●❖ ❙❝✐❡♥t✐❢✐❝ ❛♥❞ ❱✐r❣♦ ❈♦❧❧❛❜♦r❛t✐♦♥s❪✱ ✏ ●❲✶✼✵✽✶✼✿ ❖❜s❡r✈❛t✐♦♥ ♦❢ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✶✾✱ ✶✻✶✶✵✶ ✭✷✵✶✼✮ ❣r❛✈✐t❛t✐♦♥❛❧ ✇❛✈❡s ❢r♦♠ ❛ ❜✐♥❛r② ♥❡✉tr♦♥ st❛r ✐♥s♣✐r❛❧ ✑✱ ❬❛r❳✐✈✿✶✼✶✵✳✵✺✽✸✷ ❬❣r✲q❝❪❪✳ ❬✺✾❪ P✳ ❏✳ ❊✳ P❡❡❜❧❡s✱ ✏ Pr✐♥❝✐♣❧❡s ♦❢ ♣❤②s✐❝❛❧ ❝♦s♠♦❧♦❣② ✧✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡②✱ ✶✾✾✸✳ ❬✻✵❪ ❙✳ ❲❡✐♥❜❡r❣✱ ✏❚❤❡ ❝♦s♠♦❧♦❣✐❝❛❧ ❝♦♥st❛♥t ♣r♦❜❧❡♠✑ ✱ ❬✻✶❪ ▲✳ ❆♠❡♥❞♦❧❛✱ ❉✳ P♦❧❛rs❦✐ ❛♥❞ ❙✳ ❚s✉❥✐❦❛✇❛✱ ✏ ❆r❡ ❘❡✈✳ ▼♦❞✳ P❤②s✳ ✻✶✱ ✶✲✷✸✱ ✭✶✾✽✾✮✳ ❢✭❘✮ ❞❛r❦ ❡♥❡r❣② ♠♦❞❡❧s ❝♦s♠♦❧♦❣✐❝❛❧❧② ✈✐❛❜❧❡ ❄✑✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✽✱ ✶✸✶✸✵✷ ✭✷✵✵✼✮ ❬❛str♦✲♣❤✴✵✻✵✸✼✵✸❪✳ ❬✻✷❪ ❍✳ ❲❡✐✱ ❍✳ ❨✳ ▲✐ ❛♥❞ ❳✳ ❇✳ ❩♦✉✱ ✏ ❊①❛❝t s②♠♠❡tr② ✑✱ ❝♦s♠♦❧♦❣✐❝❛❧ s♦❧✉t✐♦♥s ♦❢ ◆✉❝❧✳ P❤②s✳ ❇ ✾✵✸✱ ✶✸✷ ✭✷✵✶✻✮ ✏ ❆r❡ f (R) t❤❡♦r✐❡s ✈✐❛ ❍♦❥♠❛♥ ❢✭❘✮ ❞❛r❦ ❡♥❡r❣② ♠♦❞❡❧s ❝♦s♠♦❧♦❣✐❝❛❧❧② ✈✐❛❜❧❡ ❄✑✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✽✱ ✶✸✶✸✵✷ ✭✷✵✵✼✮ ❬❛str♦✲♣❤✴✵✻✵✸✼✵✸❪✳ ❬❛r❳✐✈✿✶✺✶✶✳✵✵✸✼✻ ❬❣r✲q❝❪❪✳ ❬✻✸❪ ❙✳ ◆♦❥✐r✐ ❛♥❞ ❙✳ ❉✳ ❖❞✐♥ts♦✈✱ ✏ ▼♦❞✐❢✐❡❞ ❣r❛✈✐t② ✇✐t❤ ♥❡❣❛t✐✈❡ ❛♥❞ ♣♦s✐t✐✈❡ ♣♦✇❡rs ♦❢ t❤❡ ❝✉r✈❛t✉r❡✿ ❯♥✐❢✐❝❛t✐♦♥ ♦❢ t❤❡ ✐♥❢❧❛t✐♦♥ ❛♥❞ ♦❢ t❤❡ ❝♦s♠✐❝ ❛❝❝❡❧❡r❛t✐♦♥ ✑✱ P❤②s✳ ❘❡✈✳ ❉ ✻✽✱ ✶✷✸✺✶✷ ✭✷✵✵✸✮ ❬❤❡♣✲ t❤✴✵✸✵✼✷✽✽❪✳ ❬✻✹❪ ❍✳ ▲✐✉✱ ❳✳ ❲❛♥❣✱ ❍✳ ▲✐ ❛♥❞ ❨✳ ▼❛✱ ❣r❛✈✐t❛t✐♦♥❛❧ ❧❡♥s✐♥❣ ❡❢❢❡❝t✑ ✱ ✏❉✐st✐♥❣✉✐s❤✐♥❣ ❢✭❘✮ t❤❡♦r✐❡s ❢r♦♠ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t② ❜② ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✼✱ ♥♦✳ ✶✶✱ ✼✷✸ ✭✷✵✶✼✮ ❬❛r❳✐✈✿✶✺✵✽✳✵✷✻✹✼ ❬❣r✲q❝❪❪✳ ❬✻✺❪ ❩✳ ❆♠✐r❛❜✐✱ ▼✳ ❍❛❧✐❧s♦② ❛♥❞ ❙✳ ❍❛❜✐❜ ▼❛③❤❛r✐♠♦✉s❛✈✐✱ ♠❡tr✐❝s ✐♥ ❢✭❘✮ ❣r❛✈✐t②✑ ✱ ✏●❡♥❡r❛t✐♦♥ ♦❢ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✻✱ ♥♦✳ ✻✱ ✸✸✽ ✭✷✵✶✻✮ ❬❛r❳✐✈✿✶✺✵✾✳✵✻✾✻✼ ❬❣r✲q❝❪❪✳ ❬✻✻❪ ❉✳ ▼☎ ✉❧❧❡r✱ ❱✳ ❈✳ ❞❡ ❆♥❞r❛❞❡✱ ❈✳ ▼❛✐❛✱ ▼✳ ❏✳ ❘❡❜♦✉☛❝❛s ❛♥❞ ❆✳ ❋✳ ❋✳ ❚❡✐①❡✐r❛✱ ✐♥ ❢✭❘✮ t❤❡♦r✐❡s✑ ✱ ✏❋✉t✉r❡ ❞②♥❛♠✐❝s ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✺✱ ♥♦✳ ✶✱ ✶✸ ✭✷✵✶✺✮ ❬❛r❳✐✈✿✶✹✵✺✳✵✼✻✽ ❬❛str♦✲♣❤✳❈❖❪❪✳ ❬✻✼❪ ❚✳ ▼✉❧t❛♠❛❦✐ ❛♥❞ ■✳ ❱✐❧❥❛✱ ✏ ❙♣❤❡r✐❝❛❧❧② t❤❡♦r✐❡s ♦❢ ❣r❛✈✐t② ✑✱ s②♠♠❡tr✐❝ s♦❧✉t✐♦♥s ♦❢ ♠♦❞✐❢✐❡❞ ❢✐❡❧❞ ❡q✉❛t✐♦♥s ✐♥ ❢✭❘✮ P❤②s✳ ❘❡✈✳ ❉ ✼✹✱ ✵✻✹✵✷✷ ✭✷✵✵✻✮ ❬❛str♦✲♣❤✴✵✻✵✻✸✼✸❪✳ ❬✻✽❪ ❑✳ ❑❛✐♥✉❧❛✐♥❡♥✱ ❏✳ P✐✐❧♦♥❡♥✱ ❱✳ ❘❡✐❥♦♥❡♥ ❛♥❞ ❉✳ ❙✉♥❤❡❞❡✱ ✏ ❙♣❤❡r✐❝❛❧❧② ❢✭❘✮ ❣r❛✈✐t② t❤❡♦r✐❡s ✑✱ s②♠♠❡tr✐❝ s♣❛❝❡t✐♠❡s ✐♥ P❤②s✳ ❘❡✈✳ ❉ ✼✻✱ ✵✷✹✵✷✵ ✭✷✵✵✼✮ ❬❛r❳✐✈✿✵✼✵✹✳✷✼✷✾ ❬❣r✲q❝❪❪✳ ✶✷✽ ▲✉➟♥ t s t ỵ P ýt ❙❤♦❥❛✐ ❛♥❞ ❋✳ ❙❤♦❥❛✐✱ ✏ ❙♦♠❡ st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ✐♥t❡r✐♦r s♦❧✉t✐♦♥s ♦❢ f (R) ❣r❛✈✐t② ✑✱ ●❡♥✳ ❘❡❧✳ ●r❛✈✳ ✹✹✱ ✷✶✶ ✭✷✵✶✷✮ ❬❛r❳✐✈✿✶✶✵✾✳✷✶✾✵ ❬❣r✲q❝❪❪✳ ❬✼✵❪ ▼✳ ❙❤❛r✐❢ ❛♥❞ ❍✳ ❘✳ ❑❛✉s❛r✱ ✏ ❉✉st st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥ ✐♥ f (R) ❣r❛✈✐t② ✑✱ ❏✳ P❤②s✳ ❙♦❝✳ ❏❛♣✳ ✽✵✱ ✵✹✹✵✵✹ ✭✷✵✶✶✮ ❬❛r❳✐✈✿✶✶✵✷✳✹✶✷✹ ❬♣❤②s✐❝s✳❣❡♥✲♣❤❪❪✳ ❬✼✶❪ ▲✳ ❙❡❜❛st✐❛♥✐ ❛♥❞ ❙✳ ❩❡r❜✐♥✐✱ ✏ ❙t❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥s ✐♥ ❋✭❘✮ ❣r❛✈✐t② ✑✱ ❊✉r✳ P❤②s✳ ❏✳ ❈ ✼✶✱ ✶✺✾✶ ✭✷✵✶✶✮ ❬❛r❳✐✈✿✶✵✶✷✳✺✷✸✵ ❬❣r✲q❝❪❪✳ ❬✼✷❪ ❆✳ ▲✳ ❊r✐❝❦❝❡❦✱ ❚✳ ▲✳ ❙♠✐t❤ ❛♥❞ ▼✳ ❑❛♠✐♦♥❦♦✇s❦✐✱ ✏ ❙♦❧❛r ❙②st❡♠ t❡sts ❞♦ r✉❧❡ ♦✉t ✶✴❘ ❣r❛✈✐t② ✑✱ P❤②s✳ ❘❡✈✳ ❉ ✼✹✱ ✶✷✶✺✵✶ ✭✷✵✵✻✮ ❬❛str♦✲♣❤✴✵✻✶✵✹✽✸❪✳ ❬✼✸❪ ❊✳ ❱✳ ❆r❜✉③♦✈❛✱ ❆✳ ❉✳ ❉♦❧❣♦✈ ❛♥❞ ▲✳ ❘❡✈❡r❜❡r✐✱ ✏ ❙♣❤❡r✐❝❛❧❧② ❙②♠♠❡tr✐❝ s♦❧✉t✐♦♥s ✐♥ ❋✭❘✮ ❣r❛✈✐t② ❛♥❞ ❣r❛✈✐t❛t✐♦♥❛❧ r❡♣✉❧s✐♦♥ ✑✱ ❬✼✹❪ ❆✳ ❙t❛❜✐❧❡✱ ✏ ❚❤❡ ❆str♦♣❛rt✳ P❤②s✳ ✺✹✱ ✹✹ ✭✷✵✶✹✮ ❬❛r❳✐✈✿✶✸✵✻✳✺✻✾✹ ❬❣r✲q❝❪❪✳ P♦st✲◆❡✇t♦♥✐❛♥ ❧✐♠✐t ♦❢ ❢✭❘✮✲❣r❛✈✐t② ✐♥ t❤❡ ❤❛r♠♦♥✐❝ ❣❛✉❣❡ ✑✱ P❤②s✳ ❘❡✈✳ ❉ ✽✷✱ ✵✻✹✵✷✶ ✭✷✵✶✵✮ ❬❛r❳✐✈✿✶✵✵✹✳✶✾✼✸ ❬❣r✲q❝❪❪✳ ❬✼✺❪ ❙✳ ❈❛♣♦③③✐❡❧❧♦✱ ❆✳ ❙t❛❜✐❧❡ ❛♥❞ ❆✳ ❚r♦✐s✐✱ ✏ ❆ ❣r❛✈✐t② ✑✱ ●❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐♥ t❤❡ ◆❡✇t♦♥✐❛♥ ❧✐♠✐t ♦❢ ❢✭❘✮✲ ▼♦❞✳ P❤②s✳ ▲❡tt✳ ❆ ✷✹✱ ✻✺✾ ✭✷✵✵✾✮ ❬❛r❳✐✈✿✵✾✵✶✳✵✹✹✽ ❬❣r✲q❝❪❪✳ ❬✼✻❪ ❙✳ ❈❛♣♦③③✐❡❧❧♦✱ ❱✳ ❋✳ ❈❛r❞♦♥❡✱ ❙✳ ❈❛r❧♦♥✐ ❛♥❞ ❆✳ ❚r♦✐s✐✱ ♦❜s❡r✈❛t✐♦♥❛❧ ❞❛t❛✱✑ ✏❈✉r✈❛t✉r❡ q✉✐♥t❡ss❡♥❝❡ ♠❛t❝❤❡❞ ✇✐t❤ ■♥t✳ ❏✳ ▼♦❞✳ P❤②s✳ ❉ ✶✷✱ ✶✾✻✾✲✶✾✽✷ ✭✷✵✵✸✮ ❬❛r❳✐✈✿❛str♦✲♣❤✴✵✸✵✼✵✶✽ ❬❛str♦✲ ♣❤❪❪✳ ❬✼✼❪ ▲✳ ❉✳ ▲❛♥❞❛✉ ❛♥❞ ❊✳ ▼✳ ▲✐❢s❤✐t③✱ ✏ ▼❡❝❤❛♥✐❝s✱✧ ✈♦❧✳ ✶✱ ❊❧s❡✈✐❡r✱ ❖①❢♦r❞✱ ✶✾✾✹✳ ❬✼✽❪ ❇✳ ▼❛❥✉♠❞❡r✱ ✏❚❤❡ ♣❡r✐❤❡❧✐♦♥ ♣r❡❝❡ss✐♦♥ ♦❢ ▼❡r❝✉r② ❛♥❞ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ✉♥❝❡rt❛✐♥t② ♣r✐♥❝✐♣❧❡✑ ✱ ❛r❳✐✈✿ ✶✶✵✺✳✷✹✷✽ ❬❣r✲q❝❪ ❛♥❞ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✳ ❬✼✾❪ ▼✳ ❍❛❜❡rr❡✐t❡r✱ ❲✳ ❙❝❤♠✉t③ ❛♥❞ ❆✳ ●✳ ❑♦s♦✈✐❝❤❡✈✱ ❛♥❞ ♣❤♦t♦s♣❤❡r✐❝ s♦❧❛r r❛❞✐✉s✱✑ ✏❙♦❧✈✐♥❣ t❤❡ ❞✐s❝r❡♣❛♥❝② ❜❡t✇❡❡♥ t❤❡ s❡✐s♠✐❝ P❤②s✳ ❘❡✈✳ ❆ ✼✼✱ ✵✷✷✶✵✼ ✭✷✵✵✽✮ ❬❛r❳✐✈✿✵✼✶✶✳✷✸✾✷ ❬❛str♦✲♣❤❪❪✳ ❬✽✵❪ ❙✳ ●✐❧❧❡ss❡♥✱ ❋✳ ❊✐s❡♥❤❛✉❡r✱ ❙✳ ❚r✐♣♣❡✱ ❚✳ ❆❧❡①❛♥❞❡r✱ ❘✳ ●❡♥③❡❧✱ ❋✳ ▼❛rt✐♥s ❛♥❞ ❚✳ ❖tt✱ ✐t♦r✐♥❣ st❡❧❧❛r ♦r❜✐ts ❛r♦✉♥❞ t❤❡ ♠❛ss✐✈❡ ❜❧❛❝❦ ❤♦❧❡ ✐♥ t❤❡ ●❛❧❛❝t✐❝ ❝❡♥t❡r✑ ✏▼♦♥✲ ❆str♦♣❤②s✳ ❏✳ ✻✾✷✱ ✶✵✼✺ ✭✷✵✵✾✮ ❬❛r❳✐✈✿✵✽✶✵✳✹✻✼✹ ❬❛str♦✲♣❤❪❪✳ ❬✽✶❪ ❉✳ Ps❛❧t✐s✱ ❉✳ P❡rr♦❞✐♥✱ ❑✳ ❘✳ ❉✐❡♥❡s ❛♥❞ ■✳ ▼♦❝✐♦✐✉✱ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②✑✱ ✏❑❡rr ❜❧❛❝❦ ❤♦❧❡s ❛r❡ ♥♦t ✉♥✐q✉❡ t♦ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✵✵✱ ✵✾✶✶✵✶ ✭✷✵✵✽✮ ❬P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✵✵✱ ✶✶✾✾✵✷ ✭✷✵✵✽✮❪ ❬❛r❳✐✈✿✵✼✶✵✳✹✺✻✹ ❬❛str♦✲♣❤❪❪✳ ❬✽✷❪ ❈✳ P✳ ▲✳ ❇❡rr② ❛♥❞ ❏✳ ❘✳ ●❛✐r✱ s②st❡♠ t❡sts✑✱ ✏▲✐♥❡❛r✐③❡❞ ❢✭❘✮ ❣r❛✈✐t②✿ ❣r❛✈✐t❛t✐♦♥❛❧ r❛❞✐❛t✐♦♥ ❛♥❞ s♦❧❛r P❤②s✳ ❘❡✈✳ ❉ ✽✸✱ ✶✵✹✵✷✷ ✭✷✵✶✶✮ ❊rr❛t✉♠✿ ❬P❤②s✳ ❘❡✈✳ ❉ ✽✺✱ ✵✽✾✾✵✻ ✭✷✵✶✷✮❪ ❬❛r❳✐✈✿✶✶✵✹✳✵✽✶✾ ❬❣r✲q❝❪❪✳ ❬✽✸❪ ❊✳ ❱✳ ▲✐♥❞❡r✱ ✏❊①♣♦♥❡♥t✐❛❧ ❣r❛✈✐t②✱✑ P❤②s✳ ❘❡✈✳ ❉ ✽✵✱ ✶✷✸✺✷✽ ✭✷✵✵✾✮ ❬❛r❳✐✈✿✵✾✵✺✳✷✾✻✷ ❬❛str♦✲ ♣❤✳❈❖❪❪✳ ❬✽✹❪ ❑✐♣♣❡♥❤❛❤♥ ❘ ❛♥❞ ❲❡✐❣❡rt ❆ ❛♥❞ ❲❡✐ss ❆✱ ✏ ❙t❡❧❧❛r ✷✵✶✷✳ ✶✷✾ str✉❝t✉r❡ ❛♥❞ ❡✈♦❧✉t✐♦♥ ✑✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱