η = shear viscosity η s 1 s = entropy density 4π ( = k B = 1)

Transcription

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2 s 1 = shear viscosity s = entropy density 4π ( = k B = 1)

3 = shear viscosity s 1 4π s

4 = shear viscosity

5 s water π 1 4π

6 s Liquid Helium 9 1 4π 1 4π

7 Experimental Data KSS Quark-Gluon Plasma Kovtun Son Starinets!!! s Quark Gluon (>?) Plasma 1 4π

8 ɛ τ m.f.t. s k B n s 1 ɛ k B n τ m.f.t. τ m.f.t. = Mean Free Time ɛ = energy density of (quasi)particles n = density of (quasi)particles

9 s 1 ɛ k B n τ m.f.t. τ m.f.t. s 1 4π

10 s 1 ɛ k B n τ m.f.t. τ m.f.t. = Mean Free Time ɛ n = average energy per (quasi)particle ɛ n τ m.f.t. s k B = 1

11 = shear viscosity G R 12,12(t, x) iθ(t) < [T 12 (t, x), T 12 (0, 0)] > 1 = lim ω 0 ω ImGR 12,12(ω, p = 0) 1 k B T

12 s ( 1 g 2 Y M N c ) 2 1 4π s 1 ɛ k B n τ m.f.t. τ m.f.t.

13

14

15 N = 4 SU(N c ) { g } Y 2 M N c N c Λ = 6 L 2 1 S = d 5 x g(r 2Λ) + S GH 16πG N

16 r = = Area 4G N =

17 r = h 1 2 T 2 1

18 r = h 1 2 T 2 1

19 r = h 1 2 T 2 1

20 G F,R,A [ G R 12,12(t, x) iθ(t) < [T 12 (t, x), T 12 (0, 0)] > 1 k B T ] [ 1 = lim ω 0 ω ImGR 12,12(ω, p = 0) ]

21 s = 1 4π s k B = 1 s 1 4π

22 s 1 4π s 1 4π s = 1 4π

23 α S = 1 16πG N d 5 x g(r 2Λ + α 3 R 4 ) + S GH s = 1 4π = 1 4π ( ( ζ(3) ζ(3) 8 α 3 L 6 ) + O 1 (g 2 Y M N c) 3 2 ( ) α 4 ) L 8 + O ( 1 (g 2 Y M N c) 2 ) α s

24 s 1 4π s = 1 s 1 4π 4π

25 1 S = 16πG N d 5 x [ 12 g L 2 + R +L 2 ( c 1 R 2 + c 2 R µν R µν + c 3 R µνρσ R µνρσ) ] + S GH c i α L 2 << 1 s = 1 4π (1 8c 3) + O(c 2 i ) s 1 4π

26 s 1 4π {

27 S = 1 16πG N d 5 x [ 12 g L 2 + R + λ GB 2 L2 ( R 2 4R µν R µν + R µνρσ R µνρσ) ] + S GH λ GB << 1 { λ GB O(1)

28 0 = R µν 1 2 g µν + λ GB ( 2 L2 2RR µν 4R µρ Rν ρ [ R 2Λ + λ GB ( 2 L2 R 2 4R ρσ R ρσ + R ρσλγ R ρσλγ) ] 4R ρσ R ρ µ σ ν + 2R µρσλ Rν ρσλ ) µ R ρσ µ ν R ρσ g µν + h µν 3 h 4 h 2 h + h + h = 0

29 S GH λ GB S = 1 16πG N d 5 x [ 12 g L 2 + R + λ GB 2 L2 ( R 2 4R µν R µν + R µνρσ R µνρσ) ] + S GH λ GB O(1)

30 ds 2 black brane = f(r)n 2 dt f(r) dr2 + r2 f(r) = r2 L 2 1 2λ GB ( 1 L 2 ( 3 i=1 dx 2 i 1 4λ GB + 4λ GB ( r+ r ) ) 4 ) (λ GB 14 not to have naked singularity ) ( N 2 2λ GB 1 1 4λ GB so that f(r)n 2 r2 L 2 + O ( ) 1 r ) ı.e., c = 1

31 s λ GB << 1 λ GB O(1)

32 0 = R µν 1 2 g µν + λ GB ( 2 L2 2RR µν 4R µρ Rν ρ f(r) = r2 L 2 1 2λ GB [ R 2Λ + λ GB ( 2 L2 R 2 4R ρσ R ρσ + R ρσλγ R ρσλγ) ] ( 1 4R ρσ R ρ µ σ ν + 2R µρσλ Rν ρσλ 1 4λ GB + 4λ GB ( r+ r ) ) 4 )

33 λ GB O(1)

34

35 [ { ( )}] (r+ ) 2 0 = [r r (r r g 1 )] f 1 λ GB r+ 2 r f r r ( ) ( ) + [r r g 1 ] r+ 2 r f ( r+ ) f λgb r r (r2 + r 2 f) ( r+ ) 2 ( 2 f + r 4 r f r + + r+ 2 r ( ) [ + r+ 2 r 2 f + r+ 2 r f ( r+ ) 2 4 f r r ( ) ( ) λ GB { 2(r+ 2 r 2 f) ( r+ ) 2 f + 2(r 2 r + r 2 f)r + 2 r f 2r+ 2 r f (r+ ) 2 ] f} r r r r f r ) (r+ ) 2 f r g 1 (r) f 1 2λ GB ( r r + ) 2 ( 1 1 4λ GB + 4λ GB ( r+ r ) 4 )

36

37 g 1 (r) = ( λ GB + 4λ GB ( r+ r 2 ) 4 ) 2 + arbitrary constant g 1 (r) = ln ( 1 1 4λ GB + 4λ GB ( r+ r ) 4 )

38 s = 1 4π (1 4λ GB) s 1 4π λ GB (λ GB 14 not to have naked singularity )

39 s = 1 4π (1 4λ GB) s 0 as λ GB 1 4 (λ GB 14 not to have naked singularity )

40 λ GB > 9 t 100 x 3

41 g µν g µν { }

42 ( f(r)n 2 dt f(r) dr2 + r2 L 2 0 = R µν 1 2 g µν h 1 2 ( 3 ) dx 2 i + 2h 1 2dx 1 dx 2 + λ GB ( 2 L2 2RR µν 4R µρ Rν ρ i=1 [ R 2Λ + λ GB ( 2 L2 R 2 4R ρσ R ρσ + R ρσλγ R ρσλγ) ] 4R ρσ R ρ µ σ ν + 2R µρσλ Rν ρσλ ) ) ds 2 eff = f(r)n 2 c 2 g(r) [ c 2 g (r)dt 2 + dx 2 3] + 1 f(r) dr2 h 1 2(t, r, x 3 ) x 1 x 2

43 c 2 g(r) c 2 g(r) λ GB λ GB > 9 100

44 ds 2 eff = f(r)n 2 c 2 g(r) [ c 2 g (r)dt 2 + dx 2 3] 1 + f(r) dr2 } c 2 g(r) ( dr ds ) 2 = ( E Q ) 2 c 2 g(r)

45 λ GB > c 2 g,max ( ) 2 E Q r max ( dr ds ) 2 = ( E Q ) 2 c 2 g(r)

46 λ GB > 9 As E Q c g,max, hovering near r max 100 r max ( Indeed, x 3 t c g,max > 1 t = Elapsed Time along graviton propagation x 3 = Elapsed Distance along graviton propagation )

47 λ GB > 9 As E Q c g,max, hovering near r max 100 r max ( Indeed, x 3 t c g,max > 1 t = Elapsed Time along graviton propagation x 3 = Elapsed Distance along graviton propagation )

48 λ GB > 9 As E Q c g,max, hovering near r max 100 r max ( Indeed, x 3 t c g,max > 1 t = Elapsed Time along graviton propagation x 3 = Elapsed Distance along graviton propagation )

49 x 3 t c g,max > 1 λ GB > 9 100

50 s = 1 4π (1 4λ GB) λ GB > 9 100

51 s 1 = shear viscosity s = entropy density 4π ( = k B = 1)

52 s ( 1 4π ) = shear viscosity s = entropy density ( = k B = 1)

53 ( 1 ) 1 4π > s π

54 E( n) 0 T ij a c λ GB 9 100

55 ( 4 3 ) 2 1 4π s ( 4 5 ) 2 1 4π

56 s = 1 4π ( 1 1 2N ) + O(smaller)

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