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r t ssr - - PowerPoint PPT Presentation

Jun 28, 2023 122 likes •740 views

r t ssr rstr tss t rtt

slide-1
SLIDE 1

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡

❜②

❆❜❤✐❥✐t ▼❛♥❞❛❧

❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❏❛❞❛✈♣✉r ❯♥✐✈❡rs✐t②✱ ❑♦❧❦❛t❛✱ ■♥❞✐❛ ▼P❍❨❙✶✵

✶✵t❤ ❙❡♣t❡♠❜❡r✱ ✷✵✶✾

slide-2
SLIDE 2

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡

❜②

❆❜❤✐❥✐t ▼❛♥❞❛❧

❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❏❛❞❛✈♣✉r ❯♥✐✈❡rs✐t②✱ ❑♦❧❦❛t❛✱ ■♥❞✐❛ ▼P❍❨❙✶✵

✶✵t❤ ❙❡♣t❡♠❜❡r✱ ✷✵✶✾

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✷

slide-3
SLIDE 3

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❖✈❡r✈✐❡✇

✶ ■♥tr♦❞✉❝t✐♦♥

❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

✷ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ✸ ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ✹ ❉✐s❝✉ss✐♦♥s

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✸

slide-4
SLIDE 4

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-5
SLIDE 5

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-6
SLIDE 6

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-7
SLIDE 7

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-8
SLIDE 8

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-9
SLIDE 9

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-10
SLIDE 10

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-11
SLIDE 11

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❲❤❛t ❛ ❇❧❛❝❦ ❍♦❧❡ ✐s

❍✐❣❤❧② ❞❡♥s❡ ❝♦❧❧❛♣s❡❞ ♠❛ss✐✈❡ st❛r ❢r♦♠ ✇❤✐❝❤ ❣r❛✈✐t② ♣r❡✈❡♥ts ❛♥②t❤✐♥❣ ✐♥❝❧✉❞✐♥❣ ❧✐❣❤t✱ ❢r♦♠ ❡s❝❛♣✐♥❣✳

  • ■♥ ◆❡✇t♦♥✐❛♥ ♣❤②s✐❝s
  • t❤❡ ❡s❝❛♣❡ ✈❡❧♦❝✐t②✭vesc =
  • 2GM/R✮❀ ✐❢ vesc ❡①❝❡❡❞s t❤❡

s♣❡❡❞ ♦❢ ❧✐❣❤t t❤❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦❜❥❡❝t ❜❡❝♦♠❡s ❇❍

  • ■♥ ❣❡♥❡r❛❧ r❡❧❛t✐✈✐t②
  • ❇❧❛❝❦ ❤♦❧❡ ✐s ❛ r❡❣✐♦♥ ✇r❛♣♣❡❞ ❜② ❡✈❡♥t ❤♦r✐③♦♥
  • ❊①❛♠♣❧❡ ✿❙❝❤✇❛r③❝❤✐❧❞ ▼❡tr✐❝

ds2 = −(1 − 2GM

c2r )dt2 + 1 (1− 2GM

c2r )dr2 + r2(dθ2 + sin2θdφ2)

❲❤❡r❡✱ rh = 2GM

c2

✐s t❤❡ ❡✈❡♥t ❤♦r✐③♦♥✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✹

slide-12
SLIDE 12

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✿❳r❛② ❜✐♥❛r✐❡s✱ ❈❡♥tr❡ ♦❢ ❣❛❧❛①✐❡s✱ ❡s♣❡❝✐❛❧❧②✱ ❛❝t✐✈❡ ❣❛❧❛①✐❡s ♠❛② ❤♦st ❇❍s✳ ❆♥❞r♦♠❡❞❛ ❣❛❧❛①② ▼✸✶ ❛t ✐ts ❝❡♥tr❡ ✐s ❤♦st✐♥❣ ❛ s✉♣❡r ♠❛ss✐✈❡ ❇❍✳ ❖❏✷✽✼ ♠❛② ❜❡ ❝♦♥t❛✐♥✐♥❣ t✇♦ s✉♣❡r♠❛s✐✈❡ ❇❍s r♦t❛t✐♥❣ ❡❛❝❤ ♦t❤❡r ❡t❝✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✺

slide-13
SLIDE 13

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✿❳r❛② ❜✐♥❛r✐❡s✱ ❈❡♥tr❡ ♦❢ ❣❛❧❛①✐❡s✱ ❡s♣❡❝✐❛❧❧②✱ ❛❝t✐✈❡ ❣❛❧❛①✐❡s ♠❛② ❤♦st ❇❍s✳ ❆♥❞r♦♠❡❞❛ ❣❛❧❛①② ▼✸✶ ❛t ✐ts ❝❡♥tr❡ ✐s ❤♦st✐♥❣ ❛ s✉♣❡r ♠❛ss✐✈❡ ❇❍✳ ❖❏✷✽✼ ♠❛② ❜❡ ❝♦♥t❛✐♥✐♥❣ t✇♦ s✉♣❡r♠❛s✐✈❡ ❇❍s r♦t❛t✐♥❣ ❡❛❝❤ ♦t❤❡r ❡t❝✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✺

slide-14
SLIDE 14

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✿❳r❛② ❜✐♥❛r✐❡s✱ ❈❡♥tr❡ ♦❢ ❣❛❧❛①✐❡s✱ ❡s♣❡❝✐❛❧❧②✱ ❛❝t✐✈❡ ❣❛❧❛①✐❡s ♠❛② ❤♦st ❇❍s✳ ❆♥❞r♦♠❡❞❛ ❣❛❧❛①② ▼✸✶ ❛t ✐ts ❝❡♥tr❡ ✐s ❤♦st✐♥❣ ❛ s✉♣❡r ♠❛ss✐✈❡ ❇❍✳ ❖❏✷✽✼ ♠❛② ❜❡ ❝♦♥t❛✐♥✐♥❣ t✇♦ s✉♣❡r♠❛s✐✈❡ ❇❍s r♦t❛t✐♥❣ ❡❛❝❤ ♦t❤❡r ❡t❝✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✺

slide-15
SLIDE 15

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✿❳r❛② ❜✐♥❛r✐❡s✱ ❈❡♥tr❡ ♦❢ ❣❛❧❛①✐❡s✱ ❡s♣❡❝✐❛❧❧②✱ ❛❝t✐✈❡ ❣❛❧❛①✐❡s ♠❛② ❤♦st ❇❍s✳ ❆♥❞r♦♠❡❞❛ ❣❛❧❛①② ▼✸✶ ❛t ✐ts ❝❡♥tr❡ ✐s ❤♦st✐♥❣ ❛ s✉♣❡r ♠❛ss✐✈❡ ❇❍✳ ❖❏✷✽✼ ♠❛② ❜❡ ❝♦♥t❛✐♥✐♥❣ t✇♦ s✉♣❡r♠❛s✐✈❡ ❇❍s r♦t❛t✐♥❣ ❡❛❝❤ ♦t❤❡r ❡t❝✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✺

slide-16
SLIDE 16

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✿❳r❛② ❜✐♥❛r✐❡s✱ ❈❡♥tr❡ ♦❢ ❣❛❧❛①✐❡s✱ ❡s♣❡❝✐❛❧❧②✱ ❛❝t✐✈❡ ❣❛❧❛①✐❡s ♠❛② ❤♦st ❇❍s✳ ❆♥❞r♦♠❡❞❛ ❣❛❧❛①② ▼✸✶ ❛t ✐ts ❝❡♥tr❡ ✐s ❤♦st✐♥❣ ❛ s✉♣❡r ♠❛ss✐✈❡ ❇❍✳ ❖❏✷✽✼ ♠❛② ❜❡ ❝♦♥t❛✐♥✐♥❣ t✇♦ s✉♣❡r♠❛s✐✈❡ ❇❍s r♦t❛t✐♥❣ ❡❛❝❤ ♦t❤❡r ❡t❝✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✺

slide-17
SLIDE 17

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ❇❧❛❝❦ ❍♦❧❡ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✿❳r❛② ❜✐♥❛r✐❡s✱ ❈❡♥tr❡ ♦❢ ❣❛❧❛①✐❡s✱ ❡s♣❡❝✐❛❧❧②✱ ❛❝t✐✈❡ ❣❛❧❛①✐❡s ♠❛② ❤♦st ❇❍s✳ ❆♥❞r♦♠❡❞❛ ❣❛❧❛①② ▼✸✶ ❛t ✐ts ❝❡♥tr❡ ✐s ❤♦st✐♥❣ ❛ s✉♣❡r ♠❛ss✐✈❡ ❇❍✳ ❖❏✷✽✼ ♠❛② ❜❡ ❝♦♥t❛✐♥✐♥❣ t✇♦ s✉♣❡r♠❛s✐✈❡ ❇❍s r♦t❛t✐♥❣ ❡❛❝❤ ♦t❤❡r ❡t❝✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✺

slide-18
SLIDE 18

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡ ❆ st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥ ❢♦r ❊✐♥st❡✐♥✬s ✜❡❧❞ ❡q✉❛t✐♦♥s✱ s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✱ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ds2 = −f(r)dt2 + 1 f(r)dr2 + r2(dθ2 + sin2θdφ2), ✇❤❡r❡✱ f(r) = 1 − 2M r + Q2 r2 − c r(3ωq+1) . ❚❤❡ r❛♥❣❡ ♦❢ q✉✐♥t❡ss❡♥t✐❛❧ st❛t❡ ♣❛r❛♠❡t❡r ✐s −1 < ωq < − 1

3 ✳

ρq = − c 2 3ωq r3(1+ωq) ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✻

slide-19
SLIDE 19

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡ ❆ st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥ ❢♦r ❊✐♥st❡✐♥✬s ✜❡❧❞ ❡q✉❛t✐♦♥s✱ s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✱ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ds2 = −f(r)dt2 + 1 f(r)dr2 + r2(dθ2 + sin2θdφ2), ✇❤❡r❡✱ f(r) = 1 − 2M r + Q2 r2 − c r(3ωq+1) . ❚❤❡ r❛♥❣❡ ♦❢ q✉✐♥t❡ss❡♥t✐❛❧ st❛t❡ ♣❛r❛♠❡t❡r ✐s −1 < ωq < − 1

3 ✳

ρq = − c 2 3ωq r3(1+ωq) ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✻

slide-20
SLIDE 20

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡ ❆ st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥ ❢♦r ❊✐♥st❡✐♥✬s ✜❡❧❞ ❡q✉❛t✐♦♥s✱ s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✱ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ds2 = −f(r)dt2 + 1 f(r)dr2 + r2(dθ2 + sin2θdφ2), ✇❤❡r❡✱ f(r) = 1 − 2M r + Q2 r2 − c r(3ωq+1) . ❚❤❡ r❛♥❣❡ ♦❢ q✉✐♥t❡ss❡♥t✐❛❧ st❛t❡ ♣❛r❛♠❡t❡r ✐s −1 < ωq < − 1

3 ✳

ρq = − c 2 3ωq r3(1+ωq) ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✻

slide-21
SLIDE 21

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡ ❆ st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥ ❢♦r ❊✐♥st❡✐♥✬s ✜❡❧❞ ❡q✉❛t✐♦♥s✱ s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✱ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ds2 = −f(r)dt2 + 1 f(r)dr2 + r2(dθ2 + sin2θdφ2), ✇❤❡r❡✱ f(r) = 1 − 2M r + Q2 r2 − c r(3ωq+1) . ❚❤❡ r❛♥❣❡ ♦❢ q✉✐♥t❡ss❡♥t✐❛❧ st❛t❡ ♣❛r❛♠❡t❡r ✐s −1 < ωq < − 1

3 ✳

ρq = − c 2 3ωq r3(1+ωq) ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✻

slide-22
SLIDE 22

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡ ❆ st❛t✐❝ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ s♦❧✉t✐♦♥ ❢♦r ❊✐♥st❡✐♥✬s ✜❡❧❞ ❡q✉❛t✐♦♥s✱ s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✱ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ds2 = −f(r)dt2 + 1 f(r)dr2 + r2(dθ2 + sin2θdφ2), ✇❤❡r❡✱ f(r) = 1 − 2M r + Q2 r2 − c r(3ωq+1) . ❚❤❡ r❛♥❣❡ ♦❢ q✉✐♥t❡ss❡♥t✐❛❧ st❛t❡ ♣❛r❛♠❡t❡r ✐s −1 < ωq < − 1

3 ✳

ρq = − c 2 3ωq r3(1+ωq) ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✻

slide-23
SLIDE 23

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ♠❛ss✱ ✇❡ s❡t f(r) = 0✱ ✇❤✐❝❤ ②✐❡❧❞s M = r+

2

  • 1 + Q2

r2

+ −

c r

(3ωq+1) +

  • ✳

❚❤❡ ❡♥tr♦♣② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ S = πr2 ✳ ❚❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s Φ =

  • ∂M

∂Q

  • S =
  • ∂M

∂Q

  • r+ = Q

r+ ✳

❚❤❡ ❍❛✇❦✐♥❣ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ✐s ❣✐✈❡♥ ❜②✱ TH = f ′(r)

4π

  • r=r+

=

1 4π

  • 1

r+ − Q2 r3

+ +

3cωq r

(3ωq+2) +

  • ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✼

slide-24
SLIDE 24

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ♠❛ss✱ ✇❡ s❡t f(r) = 0✱ ✇❤✐❝❤ ②✐❡❧❞s M = r+

2

  • 1 + Q2

r2

+ −

c r

(3ωq+1) +

  • ✳

❚❤❡ ❡♥tr♦♣② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ S = πr2 ✳ ❚❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s Φ =

  • ∂M

∂Q

  • S =
  • ∂M

∂Q

  • r+ = Q

r+ ✳

❚❤❡ ❍❛✇❦✐♥❣ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ✐s ❣✐✈❡♥ ❜②✱ TH = f ′(r)

4π

  • r=r+

=

1 4π

  • 1

r+ − Q2 r3

+ +

3cωq r

(3ωq+2) +

  • ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✼

slide-25
SLIDE 25

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ♠❛ss✱ ✇❡ s❡t f(r) = 0✱ ✇❤✐❝❤ ②✐❡❧❞s M = r+

2

  • 1 + Q2

r2

+ −

c r

(3ωq+1) +

  • ✳

❚❤❡ ❡♥tr♦♣② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ S = πr2 ✳ ❚❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s Φ =

  • ∂M

∂Q

  • S =
  • ∂M

∂Q

  • r+ = Q

r+ ✳

❚❤❡ ❍❛✇❦✐♥❣ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ✐s ❣✐✈❡♥ ❜②✱ TH = f ′(r)

4π

  • r=r+

=

1 4π

  • 1

r+ − Q2 r3

+ +

3cωq r

(3ωq+2) +

  • ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✼

slide-26
SLIDE 26

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ♠❛ss✱ ✇❡ s❡t f(r) = 0✱ ✇❤✐❝❤ ②✐❡❧❞s M = r+

2

  • 1 + Q2

r2

+ −

c r

(3ωq+1) +

  • ✳

❚❤❡ ❡♥tr♦♣② ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠ S = πr2 ✳ ❚❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s Φ =

  • ∂M

∂Q

  • S =
  • ∂M

∂Q

  • r+ = Q

r+ ✳

❚❤❡ ❍❛✇❦✐♥❣ t❡♠♣❡r❛t✉r❡ ♦❢ t❤❡ ❜❧❛❝❦ ❤♦❧❡ ✐s ❣✐✈❡♥ ❜②✱ TH = f ′(r)

4π

  • r=r+

=

1 4π

  • 1

r+ − Q2 r3

+ +

3cωq r

(3ωq+2) +

  • ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✼

slide-27
SLIDE 27

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ◆♦✇ t❤❡ s♣❡❝✐✜❝ ❤❡❛t ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠✱ CQ = T ∂S

∂T

  • Q =

2πr2

+(r2 +−Q2+3ωqcr −(3ωq−1) +

) 3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

✳ ■♥ ♦r❞❡r t♦ ✜♥❞ ❛ ❞✐✈❡r❣❡♥❝❡ ✐♥ s♣❡❝✐✜❝ ❤❡❛t ♦♥❡ ♠✉st s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥✱ 3Q2 − r2

+ − 3ωq(3ωq + 2)cr−(3ωq−1) +

= 0 ✳

❋✐❣✳✶❛ ❋✐❣✳✶❜

10 20 30 40

r

8106 6106 4106 2106 2106 4106 6106

CQ

1 2 3 4 5 6

r

4000 2000 2000 4000 6000 8000

CQ

❋✐❣✿ ✭1a✮ ❛♥❞ ✭1b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s♣❡❝✐✜❝ ❤❡❛t ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 2

3 ✱ Q = 2 ❛♥❞

c = 0.8 ✳ ❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✽

slide-28
SLIDE 28

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ◆♦✇ t❤❡ s♣❡❝✐✜❝ ❤❡❛t ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠✱ CQ = T ∂S

∂T

  • Q =

2πr2

+(r2 +−Q2+3ωqcr −(3ωq−1) +

) 3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

✳ ■♥ ♦r❞❡r t♦ ✜♥❞ ❛ ❞✐✈❡r❣❡♥❝❡ ✐♥ s♣❡❝✐✜❝ ❤❡❛t ♦♥❡ ♠✉st s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥✱ 3Q2 − r2

+ − 3ωq(3ωq + 2)cr−(3ωq−1) +

= 0 ✳

❋✐❣✳✶❛ ❋✐❣✳✶❜

10 20 30 40

r

8106 6106 4106 2106 2106 4106 6106

CQ

1 2 3 4 5 6

r

4000 2000 2000 4000 6000 8000

CQ

❋✐❣✿ ✭1a✮ ❛♥❞ ✭1b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s♣❡❝✐✜❝ ❤❡❛t ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 2

3 ✱ Q = 2 ❛♥❞

c = 0.8 ✳ ❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✽

slide-29
SLIDE 29

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ◆♦✇ t❤❡ s♣❡❝✐✜❝ ❤❡❛t ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠✱ CQ = T ∂S

∂T

  • Q =

2πr2

+(r2 +−Q2+3ωqcr −(3ωq−1) +

) 3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

✳ ■♥ ♦r❞❡r t♦ ✜♥❞ ❛ ❞✐✈❡r❣❡♥❝❡ ✐♥ s♣❡❝✐✜❝ ❤❡❛t ♦♥❡ ♠✉st s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥✱ 3Q2 − r2

+ − 3ωq(3ωq + 2)cr−(3ωq−1) +

= 0 ✳

❋✐❣✳✶❛ ❋✐❣✳✶❜

10 20 30 40

r

8106 6106 4106 2106 2106 4106 6106

CQ

1 2 3 4 5 6

r

4000 2000 2000 4000 6000 8000

CQ

❋✐❣✿ ✭1a✮ ❛♥❞ ✭1b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s♣❡❝✐✜❝ ❤❡❛t ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 2

3 ✱ Q = 2 ❛♥❞

c = 0.8 ✳ ❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✽

slide-30
SLIDE 30

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ◆♦✇ t❤❡ s♣❡❝✐✜❝ ❤❡❛t ✇✐❧❧ t❛❦❡ t❤❡ ❢♦r♠✱ CQ = T ∂S

∂T

  • Q =

2πr2

+(r2 +−Q2+3ωqcr −(3ωq−1) +

) 3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

✳ ■♥ ♦r❞❡r t♦ ✜♥❞ ❛ ❞✐✈❡r❣❡♥❝❡ ✐♥ s♣❡❝✐✜❝ ❤❡❛t ♦♥❡ ♠✉st s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥✱ 3Q2 − r2

+ − 3ωq(3ωq + 2)cr−(3ωq−1) +

= 0 ✳

❋✐❣✳✶❛ ❋✐❣✳✶❜

10 20 30 40

r

8106 6106 4106 2106 2106 4106 6106

CQ

1 2 3 4 5 6

r

4000 2000 2000 4000 6000 8000

CQ

❋✐❣✿ ✭1a✮ ❛♥❞ ✭1b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ s♣❡❝✐✜❝ ❤❡❛t ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 2

3 ✱ Q = 2 ❛♥❞

c = 0.8 ✳ ❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✽

slide-31
SLIDE 31

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐s♦t❤❡r♠❛❧ ❝♦♠♣r❡s✐❜✐❧✐t② ✐s ❣✐✈❡♥ ❜②✱ K−1

T

= Q

  • ∂φ

∂Q

  • T = Q

r+

  • Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

  • ✳

❋✐❣✳✷❛ ❋✐❣✳✷❜

2 4 6 8 10

r

1 1 2

KT

1 2 4 6 8

r

1 1 2

KT

1

❋✐❣✿ ✭2a✮ ❛♥❞ ✭2b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐s♦t❤❡r♠❛❧ ❝♦♠♣r❡s✐❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 1

3 ❛♥❞ ωq = − 2 3 ✱ ✇❤❡r❡ Q = 2 ❛♥❞ c = 0.8 ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✾

slide-32
SLIDE 32

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐s♦t❤❡r♠❛❧ ❝♦♠♣r❡s✐❜✐❧✐t② ✐s ❣✐✈❡♥ ❜②✱ K−1

T

= Q

  • ∂φ

∂Q

  • T = Q

r+

  • Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

  • ✳

❋✐❣✳✷❛ ❋✐❣✳✷❜

2 4 6 8 10

r

1 1 2

KT

1 2 4 6 8

r

1 1 2

KT

1

❋✐❣✿ ✭2a✮ ❛♥❞ ✭2b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐s♦t❤❡r♠❛❧ ❝♦♠♣r❡s✐❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 1

3 ❛♥❞ ωq = − 2 3 ✱ ✇❤❡r❡ Q = 2 ❛♥❞ c = 0.8 ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✾

slide-33
SLIDE 33

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s ▼❡tr✐❝

❚❤❡r♠♦❞②♥❛♠✐❝s ❚❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐s♦t❤❡r♠❛❧ ❝♦♠♣r❡s✐❜✐❧✐t② ✐s ❣✐✈❡♥ ❜②✱ K−1

T

= Q

  • ∂φ

∂Q

  • T = Q

r+

  • Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

3Q2−r2

+−3ωq(3ωq+2)cr −(3ωq−1) +

  • ✳

❋✐❣✳✷❛ ❋✐❣✳✷❜

2 4 6 8 10

r

1 1 2

KT

1 2 4 6 8

r

1 1 2

KT

1

❋✐❣✿ ✭2a✮ ❛♥❞ ✭2b✮ r❡♣r❡s❡♥t t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ✐♥✈❡rs❡ ♦❢ t❤❡ ✐s♦t❤❡r♠❛❧ ❝♦♠♣r❡s✐❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ r+ ❢♦r ωq = − 1

3 ❛♥❞ ωq = − 2 3 ✱ ✇❤❡r❡ Q = 2 ❛♥❞ c = 0.8 ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✾

slide-34
SLIDE 34

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❖✈❡r t❤❡ ♣❛st ❢❡✇ ❞❡❝❛❞❡s✱ t❤❡ st✉❞② ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ❤❛s ❝♦♠❡ t♦ ❝♦♥❝❡♥tr❛t❡ ♠♦r❡ ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡t ♦❢ ✐♥❞✐❝❡s (α, β, γ, δ, ϕ, ψ, ν, η)✱ ❦♥♦✇♥ ❛s ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ t♦ ❞❡s❝r✐❜❡ t❤❡ s✐♥❣✉❧❛r ❜❡❤❛✈✐♦r ♦❢ ✈❛r✐♦✉s t❤❡r♠♦❞②♥❛♠✐❝ q✉❛♥t✐t✐❡s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳ ❚❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❛r❡✱ CQ ∼ |T − Tc|−α✱ K−1

T

∼ |T − Tc|−γ✱ Φ(r) − Φ(rc) ∼ |T − Tc|β✱ Φ(r) − Φ(rc) ∼ |Q − Qc|

1 δ ✱

CQ ∼ |Q − Qc|−ϕ✱ S(r) − S(rc) ∼ |Q − Qc|ψ ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✵

slide-35
SLIDE 35

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❖✈❡r t❤❡ ♣❛st ❢❡✇ ❞❡❝❛❞❡s✱ t❤❡ st✉❞② ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ❤❛s ❝♦♠❡ t♦ ❝♦♥❝❡♥tr❛t❡ ♠♦r❡ ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡t ♦❢ ✐♥❞✐❝❡s (α, β, γ, δ, ϕ, ψ, ν, η)✱ ❦♥♦✇♥ ❛s ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ t♦ ❞❡s❝r✐❜❡ t❤❡ s✐♥❣✉❧❛r ❜❡❤❛✈✐♦r ♦❢ ✈❛r✐♦✉s t❤❡r♠♦❞②♥❛♠✐❝ q✉❛♥t✐t✐❡s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳ ❚❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❛r❡✱ CQ ∼ |T − Tc|−α✱ K−1

T

∼ |T − Tc|−γ✱ Φ(r) − Φ(rc) ∼ |T − Tc|β✱ Φ(r) − Φ(rc) ∼ |Q − Qc|

1 δ ✱

CQ ∼ |Q − Qc|−ϕ✱ S(r) − S(rc) ∼ |Q − Qc|ψ ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✵

slide-36
SLIDE 36

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❖✈❡r t❤❡ ♣❛st ❢❡✇ ❞❡❝❛❞❡s✱ t❤❡ st✉❞② ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ❤❛s ❝♦♠❡ t♦ ❝♦♥❝❡♥tr❛t❡ ♠♦r❡ ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ ❛ s❡t ♦❢ ✐♥❞✐❝❡s (α, β, γ, δ, ϕ, ψ, ν, η)✱ ❦♥♦✇♥ ❛s ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ t♦ ❞❡s❝r✐❜❡ t❤❡ s✐♥❣✉❧❛r ❜❡❤❛✈✐♦r ♦❢ ✈❛r✐♦✉s t❤❡r♠♦❞②♥❛♠✐❝ q✉❛♥t✐t✐❡s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳ ❚❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❛r❡✱ CQ ∼ |T − Tc|−α✱ K−1

T

∼ |T − Tc|−γ✱ Φ(r) − Φ(rc) ∼ |T − Tc|β✱ Φ(r) − Φ(rc) ∼ |Q − Qc|

1 δ ✱

CQ ∼ |Q − Qc|−ϕ✱ S(r) − S(rc) ∼ |Q − Qc|ψ ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✵

slide-37
SLIDE 37

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ r❡✲❡①♣r❡ss ♣❤②s✐❝❛❧ q✉❛♥t✐t✐❡s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s r = rc(1 + ∆)✱ T(r) = T(rc)(1 + ǫ)✱ Q(r) = Q(rc)(1 + Π) ✱ ✇❤❡r❡✱ ∆ << 1✱ ǫ << 1 ❛♥❞ Π << 1✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✶

slide-38
SLIDE 38

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ r❡✲❡①♣r❡ss ♣❤②s✐❝❛❧ q✉❛♥t✐t✐❡s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s r = rc(1 + ∆)✱ T(r) = T(rc)(1 + ǫ)✱ Q(r) = Q(rc)(1 + Π) ✱ ✇❤❡r❡✱ ∆ << 1✱ ǫ << 1 ❛♥❞ Π << 1✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✶

slide-39
SLIDE 39

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ r❡✲❡①♣r❡ss ♣❤②s✐❝❛❧ q✉❛♥t✐t✐❡s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s r = rc(1 + ∆)✱ T(r) = T(rc)(1 + ǫ)✱ Q(r) = Q(rc)(1 + Π) ✱ ✇❤❡r❡✱ ∆ << 1✱ ǫ << 1 ❛♥❞ Π << 1✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✶

slide-40
SLIDE 40

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

P❡r❢♦r♠✐♥❣ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ T(r+) ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❝❤❛r❣❡ ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ rc✱ ✇❡ ♦❜t❛✐♥ T = T(r+c)+

  • ∂T

∂r+

  • Q=Qc
  • r+=r+c

(r+ −r+c)+ 1

2

  • ∂2T

∂r2

+

  • Q=Qc
  • r+=r+c

(r+ − r+c)2 + ❤✐❣❤❡r ♦r❞❡r t❡r♠s . ◆♦✇✱ ♥❡❣❧❡❝t✐♥❣ t❤❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s✱ ǫTc = 1 2 ∂2T ∂r2

  • Q
  • r+=rc

r2

c△2

❯s✐♥❣ t❤❡ r❡✲❡①♣r❡ss❡❞ q✉❛♥t✐t✐❡s ✇❡ ❣❡t✱ △ = 1 rc

  • 2ǫTc

D , ✇❤❡r❡, D = ∂2T ∂r2

  • Q=Qc
  • r+=rc

.

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✷

slide-41
SLIDE 41

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

P❡r❢♦r♠✐♥❣ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ T(r+) ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❝❤❛r❣❡ ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ rc✱ ✇❡ ♦❜t❛✐♥ T = T(r+c)+

  • ∂T

∂r+

  • Q=Qc
  • r+=r+c

(r+ −r+c)+ 1

2

  • ∂2T

∂r2

+

  • Q=Qc
  • r+=r+c

(r+ − r+c)2 + ❤✐❣❤❡r ♦r❞❡r t❡r♠s . ◆♦✇✱ ♥❡❣❧❡❝t✐♥❣ t❤❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s✱ ǫTc = 1 2 ∂2T ∂r2

  • Q
  • r+=rc

r2

c△2

❯s✐♥❣ t❤❡ r❡✲❡①♣r❡ss❡❞ q✉❛♥t✐t✐❡s ✇❡ ❣❡t✱ △ = 1 rc

  • 2ǫTc

D , ✇❤❡r❡, D = ∂2T ∂r2

  • Q=Qc
  • r+=rc

.

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✷

slide-42
SLIDE 42

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

P❡r❢♦r♠✐♥❣ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ T(r+) ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❝❤❛r❣❡ ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ rc✱ ✇❡ ♦❜t❛✐♥ T = T(r+c)+

  • ∂T

∂r+

  • Q=Qc
  • r+=r+c

(r+ −r+c)+ 1

2

  • ∂2T

∂r2

+

  • Q=Qc
  • r+=r+c

(r+ − r+c)2 + ❤✐❣❤❡r ♦r❞❡r t❡r♠s . ◆♦✇✱ ♥❡❣❧❡❝t✐♥❣ t❤❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s✱ ǫTc = 1 2 ∂2T ∂r2

  • Q
  • r+=rc

r2

c△2

❯s✐♥❣ t❤❡ r❡✲❡①♣r❡ss❡❞ q✉❛♥t✐t✐❡s ✇❡ ❣❡t✱ △ = 1 rc

  • 2ǫTc

D , ✇❤❡r❡, D = ∂2T ∂r2

  • Q=Qc
  • r+=rc

.

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✷

slide-43
SLIDE 43

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆♦✇✱ ✇❡ ❝❛♥ ❡①♣❛♥❞✐♥❣ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ CQ ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s CQ = 2πr2

+(r2 + − Q2 + 3ωqcr−(3ωq−1) +

) △

  • −2r2

+ + 3ωq(3ωq − 1)(3ωq + 2)r−(3ωq−1) +

, ✇❤✐❝❤ ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦✱ CQ = π √ 2Dr2

+

  • r2

+ − Q2 + 3ωqcr−(3ωq−1) +

  • −2r+ + 3ωq(3ωq − 1)(3ωq + 2)r−3ωq

+

  • (T − Tc)1/2 .

❈♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❣❡t α = 1/2✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✸

slide-44
SLIDE 44

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆♦✇✱ ✇❡ ❝❛♥ ❡①♣❛♥❞✐♥❣ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ CQ ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s CQ = 2πr2

+(r2 + − Q2 + 3ωqcr−(3ωq−1) +

) △

  • −2r2

+ + 3ωq(3ωq − 1)(3ωq + 2)r−(3ωq−1) +

, ✇❤✐❝❤ ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦✱ CQ = π √ 2Dr2

+

  • r2

+ − Q2 + 3ωqcr−(3ωq−1) +

  • −2r+ + 3ωq(3ωq − 1)(3ωq + 2)r−3ωq

+

  • (T − Tc)1/2 .

❈♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❣❡t α = 1/2✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✸

slide-45
SLIDE 45

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆♦✇✱ ✇❡ ❝❛♥ ❡①♣❛♥❞✐♥❣ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ CQ ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s CQ = 2πr2

+(r2 + − Q2 + 3ωqcr−(3ωq−1) +

) △

  • −2r2

+ + 3ωq(3ωq − 1)(3ωq + 2)r−(3ωq−1) +

, ✇❤✐❝❤ ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦✱ CQ = π √ 2Dr2

+

  • r2

+ − Q2 + 3ωqcr−(3ωq−1) +

  • −2r+ + 3ωq(3ωq − 1)(3ωq + 2)r−3ωq

+

  • (T − Tc)1/2 .

❈♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❣❡t α = 1/2✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✸

slide-46
SLIDE 46

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆❡①t✱ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t β ✇❤✐❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ✭Φ✮ ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ ❝❤❛r❣❡ Q ❛♥❞ ❞❡✜♥❡❞ t❤r♦✉❣❤ t❤❡ r❡❧❛t✐♦♥ ❛s Φ(r+) − Φ(rc) ∼ |T − Tc|β . ▲❡t ✉s ✜rst✱ ✇❡ ✉s❡ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ Φ(r+) ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ rc✱ Φ(r+) = Φ(rc)+ ∂Φ ∂r+

  • Q=Qc
  • r+=rc

(r+ −rc)+ ❤✐❣❤❡r ♦r❞❡r t❡r♠s .

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✹

slide-47
SLIDE 47

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆❡①t✱ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t β ✇❤✐❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ✭Φ✮ ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ ❝❤❛r❣❡ Q ❛♥❞ ❞❡✜♥❡❞ t❤r♦✉❣❤ t❤❡ r❡❧❛t✐♦♥ ❛s Φ(r+) − Φ(rc) ∼ |T − Tc|β . ▲❡t ✉s ✜rst✱ ✇❡ ✉s❡ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ Φ(r+) ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ rc✱ Φ(r+) = Φ(rc)+ ∂Φ ∂r+

  • Q=Qc
  • r+=rc

(r+ −rc)+ ❤✐❣❤❡r ♦r❞❡r t❡r♠s .

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✹

slide-48
SLIDE 48

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆❡①t✱ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t β ✇❤✐❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❡❧❡❝tr♦st❛t✐❝ ♣♦t❡♥t✐❛❧ ✭Φ✮ ❢♦r ❛ ✜①❡❞ ✈❛❧✉❡ ♦❢ ❝❤❛r❣❡ Q ❛♥❞ ❞❡✜♥❡❞ t❤r♦✉❣❤ t❤❡ r❡❧❛t✐♦♥ ❛s Φ(r+) − Φ(rc) ∼ |T − Tc|β . ▲❡t ✉s ✜rst✱ ✇❡ ✉s❡ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ Φ(r+) ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ rc✱ Φ(r+) = Φ(rc)+ ∂Φ ∂r+

  • Q=Qc
  • r+=rc

(r+ −rc)+ ❤✐❣❤❡r ♦r❞❡r t❡r♠s .

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✹

slide-49
SLIDE 49

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆♦✇✱ ♥❡❣❧❡❝t✐♥❣ t❤❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s ❛♥❞ t❤❡♥ ✉s✐♥❣ r❡✲❡①♣r❡ss❡❞ t❡r♠ ♦❢ r❛❞✐✉s ♦❢ ❡✈❡♥t ❤♦r✐③♦♥ ♥❡❛r ❝r✐t✐❝❛❧ ♣♦✐♥t✱ ✇❡ ❣❡t Φ(r+) − Φ(rc) = −Qc r2

c

  • 2

D(T − Tc)1/2. ❈♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❣❡t β = 1/2✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✺

slide-50
SLIDE 50

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆♦✇✱ ♥❡❣❧❡❝t✐♥❣ t❤❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s ❛♥❞ t❤❡♥ ✉s✐♥❣ r❡✲❡①♣r❡ss❡❞ t❡r♠ ♦❢ r❛❞✐✉s ♦❢ ❡✈❡♥t ❤♦r✐③♦♥ ♥❡❛r ❝r✐t✐❝❛❧ ♣♦✐♥t✱ ✇❡ ❣❡t Φ(r+) − Φ(rc) = −Qc r2

c

  • 2

D(T − Tc)1/2. ❈♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❣❡t β = 1/2✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✺

slide-51
SLIDE 51

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

◆♦✇✱ ♥❡❣❧❡❝t✐♥❣ t❤❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s ❛♥❞ t❤❡♥ ✉s✐♥❣ r❡✲❡①♣r❡ss❡❞ t❡r♠ ♦❢ r❛❞✐✉s ♦❢ ❡✈❡♥t ❤♦r✐③♦♥ ♥❡❛r ❝r✐t✐❝❛❧ ♣♦✐♥t✱ ✇❡ ❣❡t Φ(r+) − Φ(rc) = −Qc r2

c

  • 2

D(T − Tc)1/2. ❈♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❞❡✜♥✐t✐♦♥✱ ✇❡ ❣❡t β = 1/2✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✺

slide-52
SLIDE 52

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❙❝❛❧✐♥❣ ▲❛✇ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts α β γ δ ϕ ψ ❱❛❧✉❡s ✶✴✷ ✶✴✷ ✶✴✷ ✷ ✶✴✷ ✶✴✷ ◆♦✇ ✇❡ ❞✐s❝✉ss ❛❜♦✉t t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ ❧❛✇ ❢♦r ♦✉r ♣r❡s❡♥t ✇♦r❦✳ ❚❤❡s❡ r❡❧❛t✐♦♥s✭❧❛✇s✮ ❛r❡ st❛t❡❞ ❜❡❧♦✇✿ α + 2β + γ = 2✱ α + β(δ + 1) = 2, (2 − α)(δψ − 1) + 1 = (1 − α)δ✱ γ(δ + 1) = (2 − α)(δ − 1)✱ βδ = β + γ, δ = 2−α+γ

2−α−γ ✱

ϕβδ = α ❛♥❞ ψβδ = 1 − α ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✻

slide-53
SLIDE 53

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❙❝❛❧✐♥❣ ▲❛✇ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts α β γ δ ϕ ψ ❱❛❧✉❡s ✶✴✷ ✶✴✷ ✶✴✷ ✷ ✶✴✷ ✶✴✷ ◆♦✇ ✇❡ ❞✐s❝✉ss ❛❜♦✉t t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ ❧❛✇ ❢♦r ♦✉r ♣r❡s❡♥t ✇♦r❦✳ ❚❤❡s❡ r❡❧❛t✐♦♥s✭❧❛✇s✮ ❛r❡ st❛t❡❞ ❜❡❧♦✇✿ α + 2β + γ = 2✱ α + β(δ + 1) = 2, (2 − α)(δψ − 1) + 1 = (1 − α)δ✱ γ(δ + 1) = (2 − α)(δ − 1)✱ βδ = β + γ, δ = 2−α+γ

2−α−γ ✱

ϕβδ = α ❛♥❞ ψβδ = 1 − α ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✻

slide-54
SLIDE 54

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❙❝❛❧✐♥❣ ▲❛✇ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts α β γ δ ϕ ψ ❱❛❧✉❡s ✶✴✷ ✶✴✷ ✶✴✷ ✷ ✶✴✷ ✶✴✷ ◆♦✇ ✇❡ ❞✐s❝✉ss ❛❜♦✉t t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ ❧❛✇ ❢♦r ♦✉r ♣r❡s❡♥t ✇♦r❦✳ ❚❤❡s❡ r❡❧❛t✐♦♥s✭❧❛✇s✮ ❛r❡ st❛t❡❞ ❜❡❧♦✇✿ α + 2β + γ = 2✱ α + β(δ + 1) = 2, (2 − α)(δψ − 1) + 1 = (1 − α)δ✱ γ(δ + 1) = (2 − α)(δ − 1)✱ βδ = β + γ, δ = 2−α+γ

2−α−γ ✱

ϕβδ = α ❛♥❞ ψβδ = 1 − α ✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✻

slide-55
SLIDE 55

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❉✐s❝✉ss✐♦♥s ❇❛s❡❞ ♦♥ st❛♥❞❛r❞ t❤❡r♠♦❞②♥❛♠✐❝ ❛♣♣r♦❛❝❤✱ ✇❡ ❤❛✈❡ ♣r♦✈✐❞❡❞ ❛ ❣❡♥❡r❛❧ s❝❤❡♠❡ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ❡♠♣❧♦②❡❞ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✐♥ ❘✲◆ ❜❧❛❝❦ ❤♦❧❡s✳ ❯s✐♥❣ ❛ ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ✇❡ ❤❛✈❡ ❡①♣❧♦✐t❡❞ t❤✐s s❝❤❡♠❡ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡s s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✳ ❚❤❡ ♦❜❧✐❣❛t♦r② s❦❡t❝❤ ♦✉t ♦❢ ❝❤❛r❣❡❞ ❇❍ t❤❡r♠♦❞②♥❛♠✐❝s ✇✐t❤ t❤❡ ❝❡♥tr❛❧ ❡♥❣✐♥❡ ❛♠♣❧✐✜❡❞ ✐♥t♦ ❛ q✉✐♥t❡ss❡♥❝❡ ✜❡❧❞ ✐s ❙t❛❜❧❡ s♠❛❧❧ ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❯♥st❛❜❧❡ s♠❛❧❧✴✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❋✐rst ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❙t❛❜❧❡ ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ✉♥st❛❜❧❡ s✉♣❡r ♠❛ss✐✈❡ ❇❍s✳ ❇❛s❡❞ ♦♥ t❤✐s ♥♦✈❡❧ ❛♣♣r♦❛❝❤ ✇❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ❛❧❧ t❤❡ st❛t✐❝ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ s❛t✐s❢② t❤❡ s♦ ❝❛❧❧❡❞ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ r❡❧❛t✐♦♥s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✼

slide-56
SLIDE 56

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❉✐s❝✉ss✐♦♥s ❇❛s❡❞ ♦♥ st❛♥❞❛r❞ t❤❡r♠♦❞②♥❛♠✐❝ ❛♣♣r♦❛❝❤✱ ✇❡ ❤❛✈❡ ♣r♦✈✐❞❡❞ ❛ ❣❡♥❡r❛❧ s❝❤❡♠❡ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ❡♠♣❧♦②❡❞ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✐♥ ❘✲◆ ❜❧❛❝❦ ❤♦❧❡s✳ ❯s✐♥❣ ❛ ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ✇❡ ❤❛✈❡ ❡①♣❧♦✐t❡❞ t❤✐s s❝❤❡♠❡ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡s s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✳ ❚❤❡ ♦❜❧✐❣❛t♦r② s❦❡t❝❤ ♦✉t ♦❢ ❝❤❛r❣❡❞ ❇❍ t❤❡r♠♦❞②♥❛♠✐❝s ✇✐t❤ t❤❡ ❝❡♥tr❛❧ ❡♥❣✐♥❡ ❛♠♣❧✐✜❡❞ ✐♥t♦ ❛ q✉✐♥t❡ss❡♥❝❡ ✜❡❧❞ ✐s ❙t❛❜❧❡ s♠❛❧❧ ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❯♥st❛❜❧❡ s♠❛❧❧✴✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❋✐rst ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❙t❛❜❧❡ ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ✉♥st❛❜❧❡ s✉♣❡r ♠❛ss✐✈❡ ❇❍s✳ ❇❛s❡❞ ♦♥ t❤✐s ♥♦✈❡❧ ❛♣♣r♦❛❝❤ ✇❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ❛❧❧ t❤❡ st❛t✐❝ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ s❛t✐s❢② t❤❡ s♦ ❝❛❧❧❡❞ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ r❡❧❛t✐♦♥s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✼

slide-57
SLIDE 57

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❉✐s❝✉ss✐♦♥s ❇❛s❡❞ ♦♥ st❛♥❞❛r❞ t❤❡r♠♦❞②♥❛♠✐❝ ❛♣♣r♦❛❝❤✱ ✇❡ ❤❛✈❡ ♣r♦✈✐❞❡❞ ❛ ❣❡♥❡r❛❧ s❝❤❡♠❡ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ❡♠♣❧♦②❡❞ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✐♥ ❘✲◆ ❜❧❛❝❦ ❤♦❧❡s✳ ❯s✐♥❣ ❛ ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ✇❡ ❤❛✈❡ ❡①♣❧♦✐t❡❞ t❤✐s s❝❤❡♠❡ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡s s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✳ ❚❤❡ ♦❜❧✐❣❛t♦r② s❦❡t❝❤ ♦✉t ♦❢ ❝❤❛r❣❡❞ ❇❍ t❤❡r♠♦❞②♥❛♠✐❝s ✇✐t❤ t❤❡ ❝❡♥tr❛❧ ❡♥❣✐♥❡ ❛♠♣❧✐✜❡❞ ✐♥t♦ ❛ q✉✐♥t❡ss❡♥❝❡ ✜❡❧❞ ✐s ❙t❛❜❧❡ s♠❛❧❧ ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❯♥st❛❜❧❡ s♠❛❧❧✴✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❋✐rst ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❙t❛❜❧❡ ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ✉♥st❛❜❧❡ s✉♣❡r ♠❛ss✐✈❡ ❇❍s✳ ❇❛s❡❞ ♦♥ t❤✐s ♥♦✈❡❧ ❛♣♣r♦❛❝❤ ✇❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ❛❧❧ t❤❡ st❛t✐❝ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ s❛t✐s❢② t❤❡ s♦ ❝❛❧❧❡❞ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ r❡❧❛t✐♦♥s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✼

slide-58
SLIDE 58

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❉✐s❝✉ss✐♦♥s ❇❛s❡❞ ♦♥ st❛♥❞❛r❞ t❤❡r♠♦❞②♥❛♠✐❝ ❛♣♣r♦❛❝❤✱ ✇❡ ❤❛✈❡ ♣r♦✈✐❞❡❞ ❛ ❣❡♥❡r❛❧ s❝❤❡♠❡ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ❡♠♣❧♦②❡❞ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✐♥ ❘✲◆ ❜❧❛❝❦ ❤♦❧❡s✳ ❯s✐♥❣ ❛ ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ✇❡ ❤❛✈❡ ❡①♣❧♦✐t❡❞ t❤✐s s❝❤❡♠❡ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡s s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✳ ❚❤❡ ♦❜❧✐❣❛t♦r② s❦❡t❝❤ ♦✉t ♦❢ ❝❤❛r❣❡❞ ❇❍ t❤❡r♠♦❞②♥❛♠✐❝s ✇✐t❤ t❤❡ ❝❡♥tr❛❧ ❡♥❣✐♥❡ ❛♠♣❧✐✜❡❞ ✐♥t♦ ❛ q✉✐♥t❡ss❡♥❝❡ ✜❡❧❞ ✐s ❙t❛❜❧❡ s♠❛❧❧ ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❯♥st❛❜❧❡ s♠❛❧❧✴✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❋✐rst ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❙t❛❜❧❡ ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ✉♥st❛❜❧❡ s✉♣❡r ♠❛ss✐✈❡ ❇❍s✳ ❇❛s❡❞ ♦♥ t❤✐s ♥♦✈❡❧ ❛♣♣r♦❛❝❤ ✇❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ❛❧❧ t❤❡ st❛t✐❝ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ s❛t✐s❢② t❤❡ s♦ ❝❛❧❧❡❞ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ r❡❧❛t✐♦♥s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✼

slide-59
SLIDE 59

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❉✐s❝✉ss✐♦♥s ❇❛s❡❞ ♦♥ st❛♥❞❛r❞ t❤❡r♠♦❞②♥❛♠✐❝ ❛♣♣r♦❛❝❤✱ ✇❡ ❤❛✈❡ ♣r♦✈✐❞❡❞ ❛ ❣❡♥❡r❛❧ s❝❤❡♠❡ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ❡♠♣❧♦②❡❞ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✐♥ ❘✲◆ ❜❧❛❝❦ ❤♦❧❡s✳ ❯s✐♥❣ ❛ ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ✇❡ ❤❛✈❡ ❡①♣❧♦✐t❡❞ t❤✐s s❝❤❡♠❡ t♦ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❘❡✐ss♥❡r✲◆♦r❞strö♠ ❇❧❛❝❦ ❍♦❧❡s s✉rr♦✉♥❞❡❞ ❜② q✉✐♥t❡ss❡♥❝❡✳ ❚❤❡ ♦❜❧✐❣❛t♦r② s❦❡t❝❤ ♦✉t ♦❢ ❝❤❛r❣❡❞ ❇❍ t❤❡r♠♦❞②♥❛♠✐❝s ✇✐t❤ t❤❡ ❝❡♥tr❛❧ ❡♥❣✐♥❡ ❛♠♣❧✐✜❡❞ ✐♥t♦ ❛ q✉✐♥t❡ss❡♥❝❡ ✜❡❧❞ ✐s ❙t❛❜❧❡ s♠❛❧❧ ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❯♥st❛❜❧❡ s♠❛❧❧✴✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❋✐rst ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ❙t❛❜❧❡ ✐♥t❡r♠❡❞✐❛t❡ ♠❛ss ❇❍ → ❙❡❝♦♥❞ ♦r❞❡r ♣❤❛s❡ tr❛♥s✐t✐♦♥ → ✉♥st❛❜❧❡ s✉♣❡r ♠❛ss✐✈❡ ❇❍s✳ ❇❛s❡❞ ♦♥ t❤✐s ♥♦✈❡❧ ❛♣♣r♦❛❝❤ ✇❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ❛❧❧ t❤❡ st❛t✐❝ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✇❤✐❝❤ s❛t✐s❢② t❤❡ s♦ ❝❛❧❧❡❞ t❤❡r♠♦❞②♥❛♠✐❝ s❝❛❧✐♥❣ r❡❧❛t✐♦♥s ♥❡❛r t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t✳

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✼

slide-60
SLIDE 60

■♥tr♦❞✉❝t✐♦♥ ❚❤❡r♠♦❞②♥❛♠✐❝ ◗✉❛♥t✐t✐❡s ❈r✐t✐❝❛❧ P❤❡♥♦♠❡♥❛ ❉✐s❝✉ss✐♦♥s

❚❍❆◆❑❙

❚❤❡r♠♦❞②♥❛♠✐❝ ❙t✉❞② ♦❢ ❘❡✐ss♥❡r−◆♦r❞strö♠ ◗✉✐♥t❡ss❡♥❝❡ ❇❧❛❝❦ ❍♦❧❡ ✶✽