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1 Abstract Black Hole Black Hole Λ 1

2 1 Introduction Prelude : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : History : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2 Black Hole Schwarzschild : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Kerr : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Kerr Black Hole : : : : : : : : : : : : : : : : : : : Schawarzschild : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Black Hole : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Λ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : de Sitter : : : : : : : : : : : : : : : : : : : : : Rindler : : : : : : : : : : : : : : : : : : : : : dynamics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Einstein-Hilbert : : : : : : : : : : : : : : : : : : : : : : : : : : 33 2

3 5 GUP Paradox : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : GUP : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Black Hole : : : : : : : : : : : : : : : : : : : : : : : : : : : A 43 A.1 GUP : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

4 1 Introduction 1.1 Prelude Black Hole Black Hole Black Hole Black Hole Black Hole Black Hole Black Hole Black Hole Washington James M Bardeen Medon Brandon Carter S. W. Hawking [1] Black Hole Black Hole Black Hole 4

5 1 Introduction Planck h = 1: [ kgm 2 =s] [m] Black Hole Black Hole Black Hole Bekenstein Black Hole 4 Black Hole 1.2 History Penrose Kerr Black Hole Penrose Floyd Black Hole [2; 3] Christodulou Kerr Black Hole Black Hole [4; 5; 6] Black Hole Penrose Floyd Kerr Black Hole Kerr-Newman Black Hole Hawking Black Hole [7] Black Hole Hawking Black Hole Black Hole Black Hole Black Hole Black Hole Greif [8] Black Hole Black Hole 5

6 1 Introduction Carter Black Hole Kerr Black Hole [9] Black Hole argument 1970 Bekenstein Black Hole Black Hole [10] Black Hole Bekenstein 10 Black Hole 10 7 Black Hole [m 2 ] π Black Hole [m 2 ] 1 3 Black Hole 1 Black Hole 1.47 [ km ] 4.41 [ km ] π 4πr 2 g = 9:8 108 [m 2 ] 1 Black Hole Bekenstein 3 Black Hole 10 6 [K] Black Hole t Hoot Black Hole brick wall [11] brick wall cutoff Black Hole Black Hole 6

7 1 Introduction R.J.E.Clausius τρoπ η T Q S = Q T S [12] Black Hole A B Q=T B A S B S A Black Hole Black Hole Hawkin Black Hole Black Hole 1 2 S 1 S 2 S = S 1 + S 2 Black Hole Black Hole Black Hole Black Hole S U Black Hole Black Hole Black Hole Hawking 0 J 2 =M 4 + Q 2 =M 2 = 1 0 7

8 1 Introduction Black Hole Bekenstein Black Hole Black Hole Black Hole Black Hole Black Hole Bekenstein-Hawking 2 Black Hole Black Hole GUP Generalized Uncertainty Principle 6 8

9 2 Black Hole 2.1 Schwarzschild Black Hole Black Hole Schwarzschild Black Hole Black Hole ε de Broglie λ = hc ε h Planck ( ' 6: [Js]) c ( ' 2: [m/s]) Black Hole de Broglie Schwarzschild r g (= 2GM=c 2 ) ε hc = λ hc = hc2 r g 2GM (2.1) G ( ' 6: [Nm 2 =kg 2 ]) M Black Hole Black Hole (2:1) N = Mc2 ε ο 2GM2 hc N! Black Hole S = k B ln(n!) ' k B (N lnn N) ο 4πk BGM 2 hc (2.2) 9

10 2 Black Hole k B Boltzmann ( ' 1: [J/K]) h Dirac h(= h=2π) Black Hole 2GM h r hc c 2 = Mc =) m pl = G Planck ( ' 10 8 [kg] ) Planck `pl = h r G h m pl c = c 3 Black Hole Planck ( ' [m]) Boltzmann Black Hole 2 2GM 2 A = 4πr g = 4π c 2 (2.3) S = k Bc 3 4G h A (2.4) (c = h = G = k B = 1) S = A Black Hole Black Hole Schwarzschild Black Hole Scharzschild 10

11 2 Black Hole δt r g 2GM = c = c 3 ω E hδω = δe de Broglie δe = h δt = hc3 2GM (2.5) δωδt = 1 Black Hole (2:5) Black Hole T = δe 4πk = hc 3 8πk B GM 1 M (2.6) Hawking [13] Black Hole δ Black Hole Black Hole Kerr-Newman Black Hole Black Hole Black Hole M E = Mc 2 Black Hole (2:6) S =Z d(mc 2 ) T = 8πk BG hc Z MdM = 4πk BGM 2 hc 11

12 2 Black Hole (2:2) 2.2 Kerr Black Hole Schwarzschild Black Hole Black Hole Black Hole Black Hole Kerr Black Hole Kerr Black Hole A =ZZ pgθφ dθdφ (2.7) g θφ Kerr (θ ;φ) Kerr M J ds 2 = 1 r gr Σ ψ c 2 dt 2 + Σ dr2 + Σdθ 2 + ψ! 2car g r sin 2 θ dtdφ Σ r 2 + a 2 + r gra 2 sin 2 θ Σ! sin 2 θdφ 2 r g = 2GM c 2 a = J cm Σ = r 2 + a 2 cos 2 θ = r 2 r g r + a 2 [15] 12

13 2 Black Hole ( cosθ = 0) r =0 g rr = = 0 r 2 r g r + a 2 = 0 (2.8) r = r g 2 ± r rg 2 2 a 2 (2.9) r = r + ((2:9) ) dr = 0 dt = 0 ψ! ds 2 =(r a2 cos 2 θ )dθ 2 + r 2 r gr a 2 sin 2 θ a2 + r a2 cos 2 sin 2 θdφ 2 θ ψ! =(r a2 cos 2 θ )dθ 2 + r g r r gr + a 2 sin 2 θ + + r a2 cos 2 sin 2 θdφ 2 θ =(r a2 cos 2 θ )dθ 2 rg r r gr + a 2 r a2 cos 2 sin 2 θdφ 2 θ r =(r a2 cos 2 θ )dθ r g r + a2 + r a2 cos 2 sin 2 θdφ 2 θ =(r a2 cos 2 θ )dθ 2 + r 2 +r 2 g r a2 cos 2 θ sin2 θdφ (2:8) (2:7) A = ZZ 0 s 1 2a 2 r g r + sinθdθdφ = 4πr g r + = 2πrg + 1 A (2.10) r g Kerr Black Hole J = 0 a = 0 (2:10) A = 4πr 2 g 13

14 2 Black Hole (2:3) Black Hole Schwarzschild Black Hole Kerr Black Hole Black Hole Black Hole d(mc 2 )=TdS+ ΩdJ (2.11) [15] T Kerr Black Hole Ω Kerr Black Hole (2:10) 0 s A = 2πr + 1 2a r g 1 2 A dm dj da (2:4) S = kb c 3 4G h A (2:11) T = Ω 4πJ = MA 4 hg c 6 A 2 c 3 k B MA 2 32πG 2 2πJ2 J = 0 T = hc 3 8πk B GM (2:6) Hawking 14

15 3 3.1 Schawarzschild Black Hole [17; 18; 20] Black Hole Black Hole M r g = 2GM=c 2 0 E E ' pc ' hc 2 x ' k BT (3.1) E ' pc x p ' h=2 E ' k B T x ' c 1 r g T (3:1) T = c 1 = 2π hc hc 3 2c 1 k B ( 2GM ' (3.2) c 2 ) 4c 1 GMk B T = hc 3 8πGMk B (= T BH ) (3.3) (2:6) Hawking [13] S U = Mc 2 (3:3) T S = Z ds =Z du T = Z 8πGMkB c 2 dm hc 3 = 4πGM2 k B hc 15

16 3 Black Hole A = 4πr 2 g S k B = Ac3 4G h = A 4`2pl ' M M fi 2 (3.4) 1=4 M fi Black Hole Boltzmann S = k B logw W Black Hole N S=k B ο N ο (M=M fi ) Black Hole (3:4) S A M T ε = ãt 4 V = Mc 2 =ε s = 4ãT 3 =3 S = sv = 4Mc 2 =3T (3:2) A S c 1 A 1 c 1 A = = k B 3π `2pl 4 3π =4 `2pl (3.5) c 1 = 3π =4 S=k B = A=(4`2pl ) Black Hole T T Black Hole [19] Black Hole Black Hole Black Hole T Hawking [13] V = Mc 2 =ε(= V BH ) Black Hole V? ο 4πrg 3 =3 16

17 3 V BH V = V 4? 3 πr3 g = 15 (3πG) 4 M 5 π 2 hc π 2GM c 2 = 45 4π 3 G hc (3π) 4 M2 2 3 = M 2 π 32 m pl M fl m pl V BH fl V? Black Hole V = V BH Black Hole M(r) (r;r) (1 2GM(r)=rc 2 ) 1 S = sv S S = 4Mc 2 =3T T M 1 S A L( M) T 1=L s T 3 1=L 3 S sv L 2 V ο L GM 2 =r ' εv =3 M ' εv =c 2 r ' r g r M T (= T? ) r ' (3Mc 2 =4πε) 1=3 T? 1 k B T? 45 m pl c 2 ' 2 mpl 1 2 (3.6) 32π M ã = kb 4 = h3 c 3 T? M 1=2 S = 4Mc 2 =3T? S ' ψ! A M ' π `2pl M fi 17

18 3 (S A) Black Hole Black Hole ο (A=`2pl )1=4 ' (M=M fi ) 1=2 r fl r g r fl r g Black Hole Black Hole ο (A=`pl) 1=4 S ' Mc 2 =T T (3:2) T BH m pl c 2 (m pl =M) (3:6) T? m pl c 2 (m pl =M) 1=2 Mc 2 εv T 4V?? TBH 4 V BH T BH T? (V? =V BH ) 1=4 T? (m pl =M) 1=2 Black Hole T (1=V ) 1=3 (U T 4 V = const) T = V U U V T. U 1 = p T T V c V V p = T T V 4V T (1=V ) 1=4 p c V Black Hole Black Hole Black Hole [17; 18; 20] V? Mc 2 T γ Mc 2 = ãt 4 γ V? V? M 3 T γ M 1=2 T γ V? S γ = 4ãT 3 γ V? =3 S γ M 3=2 S BH =k B = A=4`2pl S BH S γ = π 4 M 2 m pl Tγ = 8 T BH S BH T BH ο S γ T γ ο U (ο Mc 2 ) Black Hole T γ =T BH ( M 1=2 ) Black Hole 18

19 ο ct ρ = 3=32πGt 2 M ' ρ(ct) 3 ο T 2 S ' 4M=3T ο M 3=2 M ' ρ(ct) 3 ' c 3 t =G ct ' GM=c 2 Schwarzschild Black Hole (S M 2 ) Black Hole 3.2 Λ WMAP Λ [21] Λ de Sitter ds 2 Λ3 = 1 r2 c 2 dt Λ r2 1 dr 2 + r 2 dθ 2 + sin 2 θdφ 2 3 [22] `Λ = p 3=Λ `Λ E (3:1) x = c 2`Λ T E ' pc ' hc 2 x ' k BT (3.7) k B T = hc 2c 2`Λ ' hc 2c 2 p 3 p Λ (3.8) c 2 = π Gibbons Hawking 1977 [22] k B T = hcλ 1=2 =p 12π ρ Λ ρ Λ = Λc2 8πG (3.9) (3:9) n Λ n Λ E c 2 = ρ Λ 19

20 3 (3:7) (3:8) (3:9) n ρ Λc 2 Λ = E = Λc4 2c 3`Λ 3c 3 = 8πG hc 4πG V V = 4 3 π `3Λ c 3 h`λ N Λ = n Λ V = 3c 2 4πG c 3 h`λ 4 c 3 π `3Λ 3 = c 2 G h `2Λ `2Λ = c 2 `2pl A = 4π `2Λ S ο N Λ ο c 2 k B 4π c 2 = π Gibbons Hawking S=k B = A=4`2pl [22] A ( N S=k B ο 0:2776)[23] S ' N 2 k B 0:2776 ' 4c `Λ 2 (3.10) S k B ο c 2 π c 2 = π =4 S=k B = A=4`2pl A p ρ p = ρc 2 (3:10) 0:2776 A `2pl A `2pl `pl 20

21 3 T ε γ = ãt 4 ρ Λ ρ Λ ρ Λ = ε ρ γ = Λc2 γ 8πG c 2 c 2 ãt 4 ' Λc2 8πG 15 h 3 c 3 π 2 c 2 hc 2c 2 p 3 p Λ 4 = c3 15 8πG h π 2 Λ p 4 2c c `2Λ = 8π 3 `2pl 3 ' 90 π 3 c4 2 2 `Λ `pl (3.11) `Λ fl `pl ρ Λ fl ρ γ ρ Λ T ρ γ ρ pl (ο [kg=m 3 ]) ρ pl ' m pl `3pl ' h `4pl c [24; 25] ρ Λ ρ Λ ρ pl = (3:11) ρ Λ ρ Λ ' 90 ρ γ ρ pl π 3 c4 2 Λc 2 8πG 3 = h 8π `4pl c `Λ `pl 2 `pl `Λ 2 3 8π 2 `pl ' O(1) `Λ ρ Λ ' p ρ γ ρ pl (3.12) `Λ T ρ γ planck planck ρ pl ρ Λ Dark Energy ρ Λ (3:12) 21

22 3 3.3 κ k B T = hκ =2πc Unruh Minkowski [26] κ `κ = c 2 =κ x = c 3`κ E E ' pc ' hc 2 x ' κ hc 2c 3`κ k B T ' hκ 2c 3 c ' hκ 2c 3 c ' k BT c 3 = π Unruh [27] S 1 = Ak B 4`2pl (3.13) [28] de Sitter [29] T κ M V M = ãt 4 c 2 V κ κ = GM V 2 3 = GãT 4 c 2 V V 2 3 = Gπ2 15 kb 4 T 4 V h 3 c c 2 k B T = hc=(2c 3`κ ); κ = c 2 =`κ c 2 `κ = π c 4 3 G hc c 2 V 1 3 `4κ 22

23 3 V 1 3 = 15 16c4 3 π 2 S = 4ãT 3 V =3 S=V `κ `2pl S k B V π 2 kb 3 T 3 = 3 15 h 3 c V 1 4 π c 4 π = c 3 3`3κ `3κ `2pl = 8 3 c 3 1 `2pl (3:13) c 3 = 3=32 A = V 2 3 S 1 k B A = 4`2pl 1=4`2pl T S=k B A = 1=4`2pl Black Hole (4+n) (3+n) (4+n) Schwarzschild ds 2 = h(r)c 2 dt 2 + h(r) 1 dr 2 + r 2 dω 2 2+n [30] dω 2 2+n (3+n) h(r) 23

24 3 rh h(r) =1 r A n+2 n+1 = 1 16π G(n) M BH 1 n + 2 A n+2 c 2 r A n+2 = n+3 2π 2 Γ( n+3 2 ) [30] n = 0 Γ(3=2)= p π =2 A 2 = 4π G(n) (4+n) (4+n) planck planck planck m pl (n) = `pl(n) = t pl (n) = h n+1 c 1 n G(n) G(n) h c n 1 2+n 1 G(n) h 2+n c 5+n r h MBH H = p π m pl (n) c m pl (n) ψ n+1 n+3 8Γ( 2 ) n + 2! 1 n+1 [31] Black Hole E ο pc ο hc r ο k BT =) k B T ο hc r H (4+n) ε(t ) s(t ) ε(t A n+3 n+2 )= (2π) n+3 k kb T BT (n + 2) Γ(n + 4)ζ (n + 4) ο (k B T ) n+4 hc s(t n )= + 4 n + 3 (n A n+3 + 2) n+2 (2π) 3+n k kb T B Γ(n + 4)ζ (n + 4) ο (k B T ) n+3 hc 24

25 3 V = Mc2 ε Mc 2 k B T kb T hc n+3 S = sv = k B kb T hc n+3 Mc 2 k B T n+3 kb T hc ο Mc2 T ο k Br H Mc 2 ο k n+1 Br H rh rh m plc ο k B hc hc `pl(n) `pl n+2 S (4+n) (3+n) (2+n) n = 0 (3:5) Black Hole de Sitter (4+n) de Sitter ds 2 Λ3 = 1 r2 c 2 dt Λ r2 1 dr 2 + dω 2 n+2 3 `Λ = p 3=Λ T E ο pc ο hc ο k B T Λ `Λ n Λ Λc2 ρ(n) = 8πG(n) n Λ E c 2 ο ρ Λ ο Λc2 8πG(n) 25

26 3 V ο `3+n Λ N Λ = n Λ V ο Λc4 8πG(n) S N Λ S `Λ `3+n hc Λ ο 2+n `Λ `pl 2+n `Λ `pl Rindler κ `κ = c 2 =κ E ο pc ο hc 2`κ ο k B T M V = L 3+n M = ε c 2V ο k BT c 2 kb T hc 3+n L 3+n κ c 2 `κ ο κ = G(n)M L 2+n ο G(n) 3+n k B T kb T 1 4+n L 2+n c 2 L 3+n h ο G(n) hc `κ cl (3.14) L (3:14) L = c3 `3+n κ = G(n) h 2+n `κ `κ S = sv = sl 3+n S 1 2+n L 2+n ο sl ο k B `κ `κ `κ `pl `pl 2+n ο k B 1 `pl 2+n 26

27 3 L `pl S 2+n ο k B (= constant) 27

28 4 4.1 dynamics g µν I = Z d 4 x p gl (g; g) Z d 4 x p gl (g;γ) Γ Christoffel Lagrangian L (q; q) Lagrangian L 0 q; q; 2 q L 0 = L d dt q L q [32] L 0 q q L= q δi 0 = = Z P2 P 1 Z P2 P 1 dt» L L δq + q q δ q δ q L fififi P 2 q P 1» L dt q d dt L q fi fi P 2 δq qδ pfi P 1 L 0 δ p = 0 L δq = 0 Euler-Lagrange L = L (q; q) q L= q q d=dt (q L= q) q L 0 L q q L 0 q L 0 28

29 4 δ p = 0 Lagrangian L 0 [32] I 0 = Z Z I Z I Z d 4 x p gl d 4 x λ " d 4 x λ p gv λ d 4 x λ P λ g µν p # gl λ g µν (4.1) V µ g µν Γ λ µν V µ Lagrangian L Γ Γ Lorentz I I 0 V µ P tensor Γ V µ V µ = c 1 (g)g µν Γ λ νλ + c 2 (g)g ρν Γ µ ρν c 1 (g) c 2 (g) Γ λ νλ = p ν ln g p gg ρν Γ µ ρν = λ p gg λµ P µ p gv µ P µ = c 3 (g)g µν ν p g + c4 (g) p g ν g νµ (4.2) 29

30 4 c 3 c 1 c 2 c 4 c 2 Lagrangian L g µν x 0 g 0µ = 0 P x = constant a i = (0;a) ;a = ln p g 00 Rindler Rindler T =j a j =2π ds 1 = (4.3) da? A pl A pl canonical ensemble Euclid Euclid canonical ensemble (4:1) S = A Euclid [33] S T Rindler Rindler I 0 Rindler I (4:1) Rindler A = Z V d 4 x µ P µ = Z β 0 dt Z V Z d 3 x P = β d 2 x? ˆn P (4.4) V β = 2π =jaj Rindler Euclid β x 1 x Rindler Rindler 30

31 4 ds 2 =(1 + 2al)dt 2 =[1 + 2al (x)]dt 2 dl al dy 2 + dz 2 l al (x) dx2 dy 2 + dz 2 (4.5) l (x) l 0 dl=dx x (t ;l;y;z) Rindler l x (4:5) (t ;l;y;z) (t;x;y;z) (4:2) P µ = 2ac 4 (g) [1 + 2al (x)] l00 l 02 [c 3 (g) 2c 4 (g)] 0 (4:4) Z I = βp µ d 2 x? = βp µ A? = S A? (y z) Euclid (4:3) ds da? = 2aβc 4 (g)+β (c 3 2c 4 )(1 + 2al) l00 l 02 = 1 A pl x l(x) x c 3 (g) =2c 4 (g) aβ = 2π c 4 (g) =1=4πA pl g P µ 31

32 4 P µ = 1 4πA pl 2g µν ν p g + p g ν g νµ = p g g µλ Γ ρ 4πA λρ gτλ Γ µ τλ pl = 1 4πA pl 1 p g ν (gg νµ ) (4:2) P µ = F 1 (g) ν [F 2 (g)g µν ] F 1 F 2 g Rindler (4:5) (4:3) F 1 = 1=p g F2 = g (4:2) P µ c 3 c 4 Rindler Euclid (0;β )=(0;2π =a) P µ a 1=a P µ Lagrangian p gl g ρν g ρν;µ = P µ = 1 4πA pl 2g µν ν p g + p g ν g µν (4.6) p 1 h p gg i gl µν = Γ τ 4πA µργ ρ ντ ΓµνΓ ρ ρτ τ pl (4.7) Langrangian Dirac-Schrodinger Lagrangian Γ 2 -Lagrangian tensor 32

33 4 (4:1) Lagrangian Einstein-Hilbert Lagrangian p p glgrav = gl Cµ R p g x µ = (4.8) 4πA pl Lagrangian Einstein-Hilbert Lagrangian Einstein Lagrangian (4:6) g ρν;µ Lagrangian g (4:7) L G g R (4:8) 4.2 Einstein-Hilbert u i Σ g ij = h ij u i u j h ij Σ Σ vector u j j u i ;u j i u j ;u i j u j a i = u j j u i u j Σ K = j u j trace (4:1) B i a i u i K [34; 35] R = 3 R + K ab K ab Ka a Kb b 2 i Ku i + a i L 2 i Ku i + a i (4.9) L ADM Lagrangian 33

34 4 R = Rg ab u a u b = 2 (G ab R ab )u a u b 2G ab u a u b = 3 R K ab K ab + K a a K b b R abcd u d =( a b u c b a u c ) R abcd u b u d = g ac u b u d R abcd = u b a b u a u b b a u a = a u b b u a a u b ( b u a ) b u b a u a + b u b 2 = i Ku i + a i K ab K ab + K a a Kb b Σ 1 ;Σ 2 S 1 ;S 2 V R=16π (4:9) h ab = g ab +u a u b γ ab = g ab n a n b σ ab = h ab n a n b = g ab + u a u b n a n b Q (4:9) V I EH = 1 16π = 1 16π Z Z V V R p gd 4 x Z p Σ2 L gd 4 x K p hd 3 x 1 Σ 1 8π Z S2 S 1 ai n i p σd 2 xndt (4.10) g 00 = N 2 K = 0 β S 1 N a i n i κ βκ = 2π 34

35 4 κ 8π Z β 0 Z dt d 2 x p σ = 1 4 A A Euclid βe E ADM Hamiltonian I Euclid EH 1 = A βe = S βe 4 Euclid β Euclid Hamiltonian Σ S Q K ab ;Θ ab ;q ab Einstein-Hilbert trace Θ ab = q ab + u a u b ni a i + 2σ i (u a u b )n j K ij Θ = q n i a i ;Θ Θ a a ;q qa a (4.11) (4:10) a i n i (4:11) I EH + 1 8π = 1 16π Z Σ2 Z V Σ 1 K p hd 3 x 1 8π p L gd 4 x 1 8π Z S2 S 1 Z S2 S 1 Θ p σd 2 xndt q p σd 2 xndt ADM Lagrangian L tensor 3 R 35

36 4 Einstein-Hilbert (4:10) a i n i Lagrangian 3 R q Θ q Einstein-Hilbert 36

37 5 GUP 5.1 Paradox Black Hole Black Hole Black Hole Hawking Black Hole Black Hole 3.1 Hawking Black Hole Hawking 5.2 GUP GUP Generalized Uncertainty Principle p x h p + `2pl p h (5.1) [36] String Theory Heisenberg p p G p=c 3 M x E = pc x! x + x 37

38 5 GUP x = 2G M c 2 = 2G( pc) c 4 x h 2G p p + c 3 ο h p + `2pl p h (5.2) 5.3 Black Hole Black Hole Schartzschild p ο h h x = = hc2 r g 2GM BH (5:1) p= h GUP 2 s 3 p h ο x 4`2pl 41 ± 1 5 2`2pl ( x) 2 (5.3) (5:3) Black Hole T GUP ο pc 4πk B ο M BHc 2 4πk B 2 41 ± s 1 mpl M BH x ο r g Hawking 38

39 5 GUP Black Hole Planck Schwartzschild 2`pl Black Hole Planck S GUP οz d(mc 2 ) T GUP οz 4πkB M BH c s 1 mpl M BH dm BH = 4πk B Z MBH m pl d MBH m pl 2 2 MBH = 2πk B + 2πk B 4 M BH m pl m pl Z s 2 MBH + 4πk B 1d m pl s MBH 2 m pl MBH m pl fi fi 1 lnfi M BH m pl + s MBH 2 m pl 3 fi fi 1fi5 A(= 16πG 2 M 2 BH =c4 ) s S GUP ο A k B c 2 A k B c 2 8 G h G h 2 1 mpl M BH 2 2 s fi fi 2πk B lnfi M BH m pl mpl M BH fi fi fi Black Hole Planck S GUP ο A k B c 2 8 G h Black Hole 8 1 Black Hole Planck Black Hole Planck 1 Black 39

40 5 GUP Hole Hawking Planck Dark Matter 40

41 6 Black Hole Λ Λ ρ Λ = 3Λ=(8πG) ρ Λ = p ρ γ ρ pl (4+n) 3.1 Black Hole Black Hole Black Hole Black Hole Black Hole Black Hole S A 3=4 black Hole Black Hole Λ 2 Black Hole 3.1 Black Hole T 41

42 6 Black Hole 2 T Black Hole tensor Einstein Lagrangian Black Hole Black Hole Black Hole GUP GUP Planck Hawking Hawking Planck Black Hole Wald Black Hole [37] Page Hawking Hawking [38] Kiefer[39] Ashtekar[40] 42

43 A A.1 GUP 5 GUP (5:1) Heisenberg x p h 2 (5:1) x h 2 p + `2pl p h p= h 2 s p h ο x `2pl 5 T GUP ο M BHc 2 4πk B 2 41 s `pl x (A.1) (A.2) mpl M BH (A.3) 5 Black Hole Planck 2 1 Planck 5 2 MBH S GUP ο 8πk B m pl +8πk B 2 4 M BH m pl s MBH m pl ln fi fifi M BH m pl + s MBH 2 m pl 3 1 fi fi fi5 4 A(= 16πG 2 M 2 BH =c4 ) Black Hole Planck S GUP ο A (1 + ln2) 8 43

44 A Black Hole Black Hole 44

45 [1] the Latest Theoritical Cosmology by Hawking. [2] R. Penrose, Riv. Nuovo. Cimento 1(1969)252. [3] R. Penrose and R.M. Floyd, Nature 229(1971)177. [4] D. Christodoulou, Phys. Rev. Letters 25 (1970) 177. [5] D. Christodoulou, Ph.D. thesis, Princeton University, 1971(unpublished). [6] D. Christodoulou and R. Ruffini, Phys. Rev. D4 (1971) [7] S. W. Hawking, Phys. Rev. Letters 26 (1971) 1344; contribution to Black Holes, edited by B. DeWitt and C. DeWitt ( Gordon and Breach, New York, 1973). [8] J. M. Greif, Junior thesis, Prenceton University(1969). [9] B. Carter, Nature 238 (1972) 71. [10] J. D. Bekenstein, Ph.D. thesis, Princeton University, 1972(unpublished). [11] G. t Hoot, On the Quantum Structure of a Black Hole, Nucl. Phys. B256 (1985) 727. [12],,, [13] S. W. Hawking, Particle creation by black hole, Commun. Math. Phys. 43 (1975) 199. [14] John Archibald Wheeler[ ] [ ],, Scientific American library. 10. [15],, [16] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curred Space, Cambridge University Press (1982). [17] S. W. Hawking, Mon. Not. R. astr. Soc. 152 (1971)

46 A [18] B. J. Carr and S. W. Hawking, Mon. Not. R. astr. Soc. 168 (1974) 399. [19] J. D. Bekenstein,Black Holes and Entropy, Phys. Rev. D7 (1979) [20] H. K. Lee, Phys. Rev. D66 (2002) [21] D. N. Spergel et al., Ap. J. S. 148 (2003) 175. [22] G. W. Gibbons and S. W. Haking,Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D15 (1977) [23] L. D. Landau and E. M. Lifshitz, Statistical Physics, Reed Edu. and Pro. Pub.Ltd. Oxford (1980). [24] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1. [25] T. Padmanabhan, Classical and Quantum Thermodynamics of horizons in spherically symmetric spacetimes, Class. Quant. Grav. 19 (2002) [26] G. E. A. Matsas and D. A. T. Vanzella,The Fulling-Davies-Unruh Effect is Mandatory: The Proton s Testimony, Int. J. Mod. Phys. D11(2002) [27] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D14 (1976) 870. [28] T. Padmanabhan, Gravity from Space-time Thermodynamics, Astrophys. Space Sci. 285 (2003) 407. [29] T. Jacobson and R. Parentani, Horizon Entropy, Found. Phys. 33 (2003) [30] R.C. Myers and M.J. Perry, Black Holes in Higher Dimensional Space-Times, Ann. Phys. 172 (1986) [31] C.M. Harris and P. Kanti, Hawking Radiation from a (4+n)-dimensional Black Hole: Exact Results for the Schwarzschild Phase, hep-ph/ [32] D. Lynden-Bell and T. Padmanabhan, (1994), unpublished T. Padmanabhan, Cosmology and Astrophysics - through problems, (Cambridge university press, 1996). [33] T. Padmanabhan, Thermodynamics and/or Horizons: A comparison of Schwarzshild, Rindler and de Sitter spacetimes, Mod. Phys. Lett. A17 (2002) [34] C. W. Misner, K. S. Throne and J. A. Wheeler, Gravitation, Freeman and co.,

47 A [35] J. W. York, Between Quantum and Cosmos,1987 eds W. H. Zurek et al (Princeton University Press, Princeton, 1998). [36] Pisin Chen, Generalized Uncertainty Principle and Dark Matter, astroph/ [37] R. M. Wald, The Thermodynamics of Black Hole, Living Rev. Rel. 4(2001), 6. [38] Don N. Page, Hawking Radiation and Black Hole Thermodynamics, hepth/ [39] C. Kiffer, Quantum Aspects of Black Holes, astro-ph/ [40] A. Ashtekar, Gravity and the Quantum, gr-qc/

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