GR_LN2001.dvi

Transcription

1 2001

2 Riemann Lie Killing Weyl Petrov Birkhoff Schwarzschild BH Minkowski Killing Cauchy Einstein Israel Israel Hodge

3 Weyl Schwarzschild Israel-Kahn C-metric Ernst Ernst Kerr-TS class Kerr-Newman

4 Riemann : M X T X T X X i) φ X φ = dφ(x) =X µ µ φ. ii) T,S a, b iii) X, Y X (at + bs) =a X T + b X S. X+Y T = X T + Y T. iv) f fx T = f X T. iv) T,S X (T S) =( X T ) S + T X S. X, Y X Y : Riemann Riemann R(X, Y )Z =( X Y Y X [X,Y ] )Z, (1.1)

5 1 5 Riemann : Riemann (M,g) Riemann i) Θ(X, Y ):= X Y Y X [X, Y ]=0. ii) g =0. : e a θ a X e a = e b ω b a(x), X θ a = ω a b(x)θ b (1.2) ω a b Θ a = dθ a + ω a b θ b (1.3) Riemann : Riemann g g ab g Riemann : g =0, (1.4) Torsion free : Θ(X, Y )= X Y Y X [X, Y ]=0, (1.5) dg ab = ω ab + ω ba, (1.6) dθ a = ω a b θ b. (1.7) : Riemann R a b = 1 2 Ra bcdθ c θ d. (1.8) R a b = dω a b + ω a c ω c b (1.9) Bianchi DR a b := dr a b + ω a c R c b R a c ω c b =0. (1.10)

6 Lie (φ, V µ,t µ,ν, ) U, U x µ,x µ U U Φ:U U x µ (Φ(p)) = x µ (p) p U Φ x µ = x µ, (1.1) Φ µ = µ, (1.2) Φ dx µ = dx µ (1.3) :(Φ φ)(φ(p)) φ(p) =φ(φ(p)), (1.4) :(Φ V ) Φ(p) =(Φ V µ )(Φ(p))(Φ µ ) Φ(p) = V µ (p)( µ) Φ(p) = V µ (Φ(p))( µ) Φ(p) = V Φ(p), (1.5) 1 :(Φ ω) Φ(p) =(Φ ω µ )(Φ(p))(Φ dx µ ) Φ(p) = ω µ (p)(dx µ ) Φ(p) = ω µ(φ(p))(dx µ ) Φ(p) = ω Φ(p). (1.6) U U Φ φ = φ, Φ V = V, Φ ω = ω, (1.7) (1 ) 1. U Φ λ : U M ( λ <a) Φ λ U i) (λ, p) Φ λ (p) ( a, a) U M ii) Φ λ U Φ λ (U )

7 1 7 iii) λ 1, λ 2, λ 1 + λ 2 <a p U Φ λ2 (p) U Φ λ1 (Φ λ2 (p)) = Φ λ1 +λ 2 (p) iv) Φ 0 =id. 2. Φ λ ( <λ< ) λ Φ λ M Φ λ M 1 3. U 1 Φ λ X µ (p) = d dλ xµ (Φ λ (p)) λ=0 (1.8) 1 Φ λ () M X p γ(λ), d dλ xµ (γ(λ)) = X µ (γ(λ)), (1.9) λ Φ λ M 1Φ λ {Φ λ (p) λ R} X 2. M X p U X Φ λ : U M ( λ <a) X Φ λ M (Lie )

8 1 8 T X M p X Φ λ T p X Lie (L X T ) p = lim λ 0 (Φ λ T ) p T p λ (1.10) X p 0p x µ X µ = δ µ 0 Φ λx 0 = x 0 + λ, Φ λx i = x i (1.11) (L X T ) µν (x) = 0 T µν (x). (1.12) (Lie ) 1. Lie i) X φ L X φ = dφ(x). (1.13) ii) X, Y L X Y =[X, Y ]. (1.14) iii) a, b X, Y T L ax+by T = al X T + bl Y T. (1.15) iv) a, b X T,S L X (at + bs) =al X T + bl X S. (1.16) v) X T,S L X (T S) =(L X T ) S + T L X S. (1.17)

9 i),ii),iv),v) Lie L X G X F : G X X (1.1) G X i) F g (x) :=F (g, x) F a F b = F ab a, b G. (1.2) ii) e G F e =id. G Lie X F G X Lie 2. F g =id g = e G 3. X 2 x, y g G F g (x) =y G 4. X S g G F g S S G 5. p X O p := {F g (p) g G} (1.3) p G G G G G 6. p X H p := {g G F g (p) =p} (1.4) G p G

10 G 8. G G G M Lie G M L (G) L (G) [X, Y ] Lie G L (G) G Lie M L L = L (G) M Lie G () G X H X G/H G X H dim G =dimx +dimh (1.5) 1.4 Killing () Riemann (M,g) Φ Φ g = g (1.1) Φ (Killing )

11 1 11 Riemann (M,g) ξ µ ξ ν + ν ξ µ =0 (1.2) Killing Φ λ Riemann (M,g) 1 Killing Killing Proof. Φ λ g = g ξ Φ λ 0 = (L ξ g)(x, Y )=L ξ (g(x, Y )) g(l ξ X, Y ) g(x, L ξ Y ) = g( ξ X L ξ X, Y )+g(x, ξ Y L ξ Y )=g( X ξ,y )+g(x, Y ξ) g G g 2. g G M G 3. Riemann (M,g) Lie M n n(n +1)/2 Proof. p Φ p Gauss Φ T p (M) Φ p

12 ξ Killing γ(t) (V γ ) V ξ µ = F µν V ν, (1.3) V F µν = R µν λσv λ ξ σ. (1.4) F µν =( ν ξ µ µ ξ ν )/ () Riemann (M,g) p k p (P ) P dim M > 2 k p (P ) p k p (P ) (M,g) R(X, Y, Z, W) =k[g(x, Z)g(Y,W) g(x, W)g(Y,Z)] (1.5) () 1. n Riemann (M,g) n(n +1)/2 (M,g) 2. i) k = 0: Euclide ii) k>0: iii) k<0: 3. i) k =0: Minkowski ii) k>0: de Sitter iii) k<0: de Sitter

13 (Fubini ) m Riemann k k m(m +1)/2 k <m(m +1)/ k Lie n Riemann (M,g) m (M,g) G k (m) G k (m, S), G k (m, T ), G k (m, N) 1. n G k (n 1,S) n =4Fubini k k =3, 4, 6 n =4,k =6 2. n G (n 1)(n 2)/2 (n 2,S) 3. n G 1 (1,S) S 1 4. n G 1 (1,T) 1.5 Weyl : g µν ĝ µν = e 2Φ g µν (1.1) Weyl

14 1 14 : Weyl ˆR µ νλσ = R µ νλσ +2δ µ [σ λ] ν Φ 2g ν[σ λ] µ Φ 2 ν Φ [ν Φδ µ σ] +2 µ Φ [λ Φg σ]ν 2( Φ) 2 δ µ [λ g σ]ν, (1.2) ˆR µν = R µν g µν 2 Φ (n 2) µ ν Φ +(n 2) µ Φ ν Φ (n 2)( Φ) 2 g µν, (1.3) e 2Φ ˆR = R 2(n 1) 2 Φ (n 1)(n 2)( Φ) 2. (1.4) Weyl : Riemann R µ νλσ = Cµ νλσ + 2 ( ) R µ [λ n 2 g σ]ν R ν[λ δ µ 2 σ] + (n 1)(n 2) δµ [σ g λ]νr (1.5) C µ νλσ Weyl Weyl Weyl Riemann Bianchi C λ µλν =0. (1.6) n =3Weyl n >3 Weyl θ a ( S a := Rb a R 2(n 1) δa b C ab := (1/2)C abcd θ c θ d ) θ b (1.7) C ab = R ab 1 n 2 (Sa θ b S b θ a ) (1.8) 1.6 : Riemann Φ:(M,g) (M,g ) Φ g =Ω 2 g (1.1) Φ (M,g )=(M,g) Φ

15 1 15 Killing : () Φ t Killing dim M = n µ ξ ν + ν ξ µ = 2 n λξ λ g µν. (1.2) n Riemann Killing k n >2 (n +1)(n +2)/2 k =(n +1)(n +2)/2 Proof. Killing ξ µ F µν := 2 [µ ξ ν],φ:= µ ξ m u, φ µ := µ φ V V ξ µ = 1 2 V ν F ν µ + 1 n V µ φ, V F µν = 2R µνλσ V λ ξ σ 4 n V [µφ ν], V φ = V µ φ µ, V µ V φ µ = 2(n 1)(n 2) (2Rφ + n νrξ ν ) 2 n 2 V ν R νµ φ n n 2 V ν σ R νµ ξ σ n n 2 V ν R(µF σ ν)σ. (ξ µ,f µν,φ,φ µ ) ξ µ 1.7 Petrov : A ab + A ab = 1 2 (A ab + i A ab ) (1.1)

16 1 16 A ab = A ab (1.2) + A ab = i + A ab. (1.3) Weyl + C abcd = 1 2 (C abcd + i C abcd ) (1.4) ɛ ab pq C pqcd = ɛ cd pq C abpq (1.5) + C abcd (ab) (cd) Q IJ = + C 0I0J (1.6) Q IJ trace-free : e a null (k, l, m) k = e 0 + e 1,l= e 0 e 1,m= e 2 ie 3 (1.7) l l = k k = m m =0,l k = 1, m m =1. (1.8) (k l) =im m, (k m) = ik m, (l m) =il m (1.9) V := k m, U := l m, W := k l + m m (1.10) U U = V V = U W = V W =0,U V =2,W W = 4. (1.11) + C + C = Ψ 0 U U +Ψ 1 (U W + W U) +Ψ 2 (V U + U V + W W ) +Ψ 3 (V W + W V )+Ψ 4 V V (1.12) Ψ 0 Ψ 4 Ψ 0 = C(k, m, k, m), Ψ 1 = C(k, l, k, m), Ψ 2 = 1 (C(k, l, k, l) C(k, l, m, m)), 2 Ψ 3 = C(l, k, l, m), Ψ 4 = C(l, m, l, m). (1.13)

17 1 17 Petrov : Q Weyl Petrov Petrov type Ψ a I [1 1 1] Ψ 0 =Ψ 4 =(λ 2 λ 1 )/2 Ψ 1 =Ψ 3 =0 Ψ 2 =(λ 1 + λ 2 )/2 D [(1 1) 1] Ψ 0 =Ψ 1 =Ψ 3 =Ψ 4 =0 Ψ 2 = λ 2 II [2 1] Ψ 0 =Ψ 1 =Ψ 3 =0 Ψ 2 = λ 1, Ψ 4 = 2 N [(2 1)] Ψ 0 =Ψ 1 =Ψ 2 =Ψ 3 =0 Ψ 4 = 2 III [3] Ψ 0 =Ψ 1 =Ψ 2 =Ψ 4 =0 Ψ 3 = i O Ψ 0 =Ψ 1 =Ψ 2 =Ψ 3 =Ψ 4 =0

18 : n = m +2 G m(m+1)/2 (m, S) ds 2 = g ab (y)dy a dy b + r 2 (y)dσ 2 m (2.1) dσ 2 m = γ AB(z)dz A dz B (2.2) K m Einstein : G ab = m ( m(m 1) r D K (Dr) 2 ad b r m ) 2 r 2 r r g ab, (2.3a) [ G A B = 1 2 (m 1)(m 2) K (Dr) 2 R + m 1 ] r δ 2 2 r 2 B A (2.3b) r Ḡ aa =0. (2.3c) Weyl : (θ a,θ A ) C ab = m 1 m +1 Xθa θ b, C aa = m 1 m(m +1) Xθa θ A, C AB 2 = m(m +1) XθA θ B. (2.4a) (2.4b) (2.4c) X = 1 2 R + r 2 r + K (Dr)2. (2.5) r 2

19 2 19 : n =4 + C ab = X 6 ɛ ab( θ 0 θ 1 + iθ 2 θ 3 ), (2.6a) + C AB = i 2 ɛab ɛ + ab C ab, (2.6b) + C aa = X (θ a θ A i2 ) 12 ɛ abɛ AB θ b θ B (2.6c) Q 11 = X 6, Q 1A =0,Q AB = X 12 δ AB. (2.7) Petrov D Ψ 2 = X 12. (2.8) 2.2 Birkhoff Einstein-Maxwell : m +2 S EM = d m+2 x g 1 4 F µνf µν (2.1) T µν = 2 g δs EM δg µν = F µλ F ν λ 1 4 g µνf αβ F αβ. (2.2) G m(m+1)/2 (m, S) m +2 F = Edy 0 dy F ABdz A dz B ; (2.3) E = q e r, (2.4) m { qm ɛ F AB = r 2 AB ; m =2 (2.5) 0; m>2 T ab = q2 2r 2m g ab, T AB = q 2 = q 2 e + q 2 m q2 2r 2m g AB (2.6)

20 2 20 Proof. m>2 m F AB =0m =2F AB = Bɛ AB. df = 1 2 a(r m B)(ɛ AB /r m )dy a dz A dz B =0 (2.7) B = q m r m. (2.8) ν F a µ = 1 r m ɛ abd b (r m E) = 0 (2.9) E = q e r m. (2.10) Einstein-Maxwell ( Λ) G m(m+1)/2 (m, S) m +2 (1)Schwarzschild/Reissner-Nordstrom (Dr 0): ds 2 = N 2 (r)dt 2 + N 2 (r) =K 2M r m 1 dr2 N 2 (r) + r2 dσ 2 m ; (2.11) 2Λ m(m +1) r2 + Q2. (2.12) r2m 2 (2) (Dr =0): 2 N 2 m M = N 2 K m. (2.13) N 2 Ricci 2 RK m K/r 2 (K =0, ±1) K r = 2 Q2 Λ+, (2.14) 2 m(m 1) r2m 2 R = 2 m Q2 Λ 2(m 1)2. (2.15) r2m

21 2 21 Proof. Einstein G µν = Λg µν + κ 2 T µν (2.16) ( D a D b r = m 1 K (Dr) 2 2 r (m 1)Q2 + 2r 2m 1 2 R = (m 1)(m 2) K (Dr)2 r r + m +1 λr 2 ) g ab, (2.17) 2(m 1) r r +m(m +1)λ m(m 1) Q2. (2.18) r2m 2Λ λ := m(m +1) (2.19) Q 2 κ 2 q 2 = m(m 1). (2.20) D a D b r = 1 2 rg ab, (2.21) r =(m 1) K (Dr)2 r 2 R = m r (m +1)λr (m 1)Q2 r 2m 1, (2.22) Q2 r +2(m +1)λ 2(m 1)2. (2.23) r2m 2D b D a D b r = D a r D a r = 2 RD a r (2.24) 2 D 2 a R =(m +1) ( 2 R +2λ + 2(2m 1)(m 1)2 m +1 ) Q 2 Da r r 2m r D a [r m+1 ( 2 R 2λ)+2(m 1)(2m 1)Q 2 r 1 m ]=0. (2.25) 2 R =2λ +2m(m 1) M Q2 2(m 1)(2m 1), rm+1 r2m (2.26) (Dr) 2 = N 2 (r) :=K λr 2 2M r + Q2, m 1 r2m 2 (2.27) r =(N 2 ). (2.28)

22 2 22 Dr 0 ds 2 = Vdt 2 + dr2 N 2 + r2 dσ 2 m (2.29) r =(N 2 ) r ln(v/n 2 ) (2.30) V/N 2 t V = N 2 Dr 0 r K r = m +1 2 m 1 λ + Q2, (2.31) r2m 2 R =2(m +1)λ 2(m 1) 2 Q2. (2.32) r2m 2.3 Schwarzschild BH Weyl ds 2 = N 2 (r)dt 2 + dr2 N 2 (r) + r2 dω 2 m ; (2.1) N 2 (r) =1 2M. rm 1 (2.2) Ψ 2 = m(m +1) 12 M. (2.3) rm+1 Future Finkelstein : (v, r) dr r = N 2 (r), (2.4) v = t + r (2.5) ds 2 =2dvdr N 2 (r)dv 2 + r 2 dω 2 m (2.6) r>r H (r m 1 H ξ = t ξ = v =2M) r>0 (2.7)

23 2 23 k = r, l = v N 2 (r) r (2.8) r >r H null ξ k = 1, ξ l = 1 2 N 2 (r), (2.9) k k = l l =0,k l = 1. (2.10) r = r H r>r H future null (future horizon) 2.1: Future Finkelstein Past Finkelstein : (u, r) u = t r (2.11) ds 2 = 2dudr N 2 (r)du 2 + r 2 dω 2 m (2.12) r>r H (r m 1 H =2M) r>0 ξ = u (2.13) k = r, l = u 1 2 N 2 (r) r (2.14) r >r H null ξ k = 1, ξ l = 1 2 N 2 (r), (2.15) k k = l l =0,k l = 1. (2.16) r = r H r>r H past null (past horizon)

24 : Past Finkelstein Szekeres : (U, V )4 (m =2)Null (u, v) ds 2 = N 2 (r)dudv + r 2 dω 2 2. (2.17) r >r H =2M U = 2Me u/4m, V =2Me v/4m (2.18) UV = 2M(r 2M)e r/2m, (2.19) V/U = e t/2m. (2.20) ds 2 = 8M r e r/2m dudv + r 2 dω 2 2, (2.21) d(uv) dr = re r/2m < 0 (2.22) U<0,V > 0 r>0 UV < 4M 2 U =2Mtan T R,V=2Mtan T + R ; 2 2 (2.23) T R <π, T + R <π. (2.24) dudv = 1 UV 4M 2 = M 2 (dt 2 dr 2 ). (2.25) cos 2 T R cos 2 2 T +R 2 cos T cos T R cos T +R 2 2 UV < 4M 2 T <π/2 (2.26)

25 : Szekeres Minkowski 2 Minkowski ds 2 = dt 2 + dx 2 (2.1) null u = t x, v = t + x (2.2) ds 2 = dudv. (2.3) u =tan T X, v =tan T + X 2 2 (2.4) <u,v< D M : T X <π, T + X <π (2.5) ds 2 = 1 ( dt 2 + dx 2 ) (2.6) 4cos 2 T X cos 2 2 T +X 2 : Minkowski : i 0

26 2 26 : I ± : i ± Minkowski : 3 Minkowski ds 2 = dt 2 + dr 2 + r 2 dω 2 m (2.7) dt 2 + dr 2 Minkowski r 0 D M D M : T X <π, T + X <π, X 0 (2.8) ds 2 =Ω 2 [ dt 2 + dx 2 +sin 2 XdΩ 2 m ]; (2.9) 1 Ω=. (2.10) 2cos T X cos T +X 2 2 Minkowski : i 0 : I ± R S m : i ± 2.4: Minkowski ds 2 = N 2 (r)dt 2 + null dr2 N 2 (r) (2.11) u = t r,v= t + r ; (2.12) dr r = (2.13) N 2 (r)

27 2 27 ds 2 = N 2 (r)dudv (2.14) N 2 (r) : N 2 (r) r = r H r 1 <r<r 2 N 2 (r) =(r r H )g(r); g(r) > 0 (2.15) (r 1 <r H <r 2 ) κ κ = 1 2 (N 2 ) (r H ) (2.16) r = 1 2κ ln 2κ r r H + h(r) (2.17) h(r) r H <r<r 2 <v u<2r (r 2 ). (2.18) (U, V ) U = 1 2κ e κu,v= 1 2κ eκv (2.19) UV = 1 2κ (r r H)e h(r), V/U = e 2κt (2.20) (U, V ) I : U<0, V>0, 1 2κ (r 2 r H )e 2κh(r2) <UV <0. (2.21) (U, V ) ds 2 = 2g(r) κ e 2κh(r) dudv, (2.22) I U =0 V =0 d(uv) dr = 1 g(r) eh(r) (2.23) I D 1 : 1 2κ (r 2 r H )e h(r 2) <UV < 1 2κ (r H r 1 )e h(r 1) (2.24) Minkowski D M T = ±X(UV =0) U =0 V =0 r = r H I horizon

28 : f(r) : N 2 (r) r = r H r 1 <r<r 2 N 2 (r) =(r r H ) 2 /g(r); g(r) > 0 (2.25) (r 1 <r H <r 2 ) v u 2 = r = µ r r H + ν ln r r H + h(r r H ). (2.26) µ = g(r H ) > 0,ν = g (r H ) h(r r H ) U 1 2 u = µ U + ν ln U + h 1(U) (2.27) h 1 (U) U =0 h(u) du/du > 0 U u r (U, v) I:U 0, <v<+ D 2 ds 2 = N 1 dudv; (2.28) N 1 =2(h 1 (U) h (U))N 2 (r)+ (r r H) 2 2g(U + r H ) U 2 g(r) (2.29) v U 0 r r H (r r H )/U 1 N 1 (U, v) D 2 v = I r = r H r = r H (U =0) I horizon : κ I t = κ =0 u v u r = r H 2.5

29 Killing (Killing ) (2.11) N 2 (r) = g tt (r) r = r H Weyl (2.4) r>0 X = m(m +1) M Q2 m(2m 1) (2.1) rm+1 r 2m N 2 (r) (2.11) Killing ξ = t () N 2 (r) = ξ ξ. (2.2) r = r H N 2 (r H )=0 Killing (null geodesic generator) Killing Killing Killing ξ Killing ξ ξ µ = ξ ν µ ξ ν = 1 2 µn 2 = 1 2 (N 2 ) (r) µ r (2.3) ( r) 2 = N 2 (r) (2.4) r = r H r//ξ µ r 0, (N 2 ) (r) 0 (v, r) ξ = v, r = v + N 2 (r) r (2.5) r = r H ξ = r (u, r) ξ = r Killing ξ ξ = ±κξ (2.6) κ = 1 2 (N 2 ) (r H ) Killing λ = λ(τ)(ξ µ µ τ =1)λ k ξ = λk k k =0 κ 0 λ/ λ = ±κ λ = ae ±κτ + b (2.7)

30 2 30 ±κ >0(< 0) τ ( ) λ ± τ ( ) λ Killing Killing Killing ξ Szekeres (U, V ) ξ ξ = κ(v V U U ) (2.8) Killing U = V =0 Killing extrem Reissner-Nordstrom κ =0 Killing Cauchy : Σ D D Σ Σ D Cauchy Cauchy M Σ Cauchy D(Σ) Σ Cauchy Cauchy Σ Cauchy : M Σ Cauchy D(Σ) M D(Σ) Σ Cauchy H(Σ) Σ Cauchy I ± Cauchy Schwarzschild Reissner-Nordstrom Cauchy Cauchy Gauss : (M, g) Riemann M Σn

31 2 31 Σ Σ X, Y X Y = X Y K(X, Y )n; X Y //Σ (2.1) [X, Y ] Σ g Σ g Riemann K(X, Y ) Σ K(X, Y )=K(Y,X). (2.2) Weingarten : X Σ n Σ Σ (1, 1) K(X) g(k(x),y)=k(x, Y ) (2.3) X n = ±K(X) //Σ (2.4) : (n +1) ds 2 = N 2 dt 2 + g ij (dx i + β i dt)(dx j + β j dt) (2.5) t = Σ t n n = 1 N ( t β i i ) (2.6) m = Nn K(X, Y ) = ± 1 N g( X m, Y )=± 1 N g([x, m]+ m X, Y ) = ± 1 2N (L mg)(x, Y ) (2.7) K ij = ± 1 2N ( tg ij i β j j β i ) (2.8) Gauss-Codazzi X Y Σ X, Y, Z R(X, Y )Z = X Y Z Y X Z [X,Y ] Z = R(X, Y )Z ± (K(X, Z)K(Y ) K(Y,Z)K(X)) +[ ( X K)(Y,Z)+( Y K)(X, Z)]n. (2.9)

32 2 32 R(X, Y, Z, W) = R(X, Y, Z, W) ± (K(X, W)K(Y,Z) K(X, Z)K(Y,W)), (2.10) R(X, Y, Z, n) = ± (( X K)(Y,Z)) ( Y K)(X, Z)). (2.11) Σ e I Σ e 0 = n M R IJKL = R IJKL ± (K IL K JK K IK K JL ), R 0IJK = n µ Rµ IJK = ±( K K IJ J K IK ). (2.12a) (2.12b) m = Nn = t β i i t = Σ t X, Y L m X =0, L m Y = 0 (2.13) g(n, R(X, n)y )= 1 N 2 g(m, X m Y m X Y ). (2.14) m Y = Y m g(m, X m Y ) = g(m, X Y m)= g(m, X (( Y N)n ± NK(Y ))) = g(m, ( X Y N)n ± N X (K(Y ))) = ±N X Y N N 2 K(X, K(Y )). (2.15) L m X Y //Σ t g(m, m X Y ) = g(m, m ( X Y K(X, Y )n)) = g(m, X Y m m (K(X, Y ))n) = ±N X Y N (L m K)(X, Y ). (2.16) X Y N = X ( Y N)= X (Y i i N)=( X Y ) i i N +( 2 N)(X, Y ). (2.17) g(n, R(X, n)y )=± 1 N (L mk)(x, Y ) K(X, K(Y )) ± 1 N ( 2 N)(X, Y ) (2.18) R 0i0j = 1 N ( K ij (L β K) ij )+K ik Kj k ± 1 N ( 2 N) ij. (2.19)

33 2 33 Note: dimσ=2 2 R IJKL = k(δ IK δ JL δ IL δ JK ) (2.20) 2 R 1212 = k Σ Euclid R abcd =0 Gauss k = K 2 12 K 11 K 22 =detk IJ (2.21) Σ R 1,R 2 K IJ 1/R 1, 1/R 2 Gauss k = 1 R 1 R 2 (2.22) η MN Euclide Σ:η MN X M X N = ±l 2 (2.23) n M = X M /l V n M = V N N n M = 1 l V M (2.24) Σ K(V,W) =±g( V n, W )=± 1 V W, (2.25) l K µν = ± 1 l g µν (2.26) Gauss Σ R µνλσ = ±(K µλ K νσ K µσ K νλ )=± 1 l 2 (g µλg νσ g µσ g νλ ) (2.27) Σ k = ±1/l Einstein Gauss Codazzi 2 G nn =2 R nn R = R + K 2 KjK i j i, (2.28a) G ni = R ni = ±( j K j i ik). (2.28b)

34 2 34 G ij = R ij +( G 0 0 R 0)g 0 ij, (2.29) R ij = R i0j 0 + g kl Rkilj = R ij ± (K ik Kj k KK ij )+ R i0j. 0 (2.30) m = t β i i R 0 i0j = 1 N L mk ij ± K ik Kj k 1 N i j N, (2.31) g ij L m K ij = L m K ± 2NKjK i j i. (2.32) R ij = R ij ± (2K ik K k j KK ij ) 1 N L mk ij 1 N i j N, R 0 0 = 1 N L mk K i jk j i 1 N N, R = R 2 N L mk (K 2 + K i jk j i ) 2 N N. (2.33a) (2.33b) (2.33c) G ij =G ij ± [2K ki K kj KK ij + 12 ] (K2 + K kl K lk)g ij 1 N (L mk ij g ij L m K) 1 N i j N + 1 N Ng ij. (2.34) (L m K) ij = t K ij (L β K) ij (L β K) ij =( β K) ij + K ik j β k + K jk i β k. (2.35) Weyl : dimm = n +1 R 0 i0j = C 0 i0j + 1 n R 0 0g ij (2.36) G ij = R ij ± (K k i K kj KK ij )+ C 0 i0j + G ij = G ij + n 1 ( n [ K k i K kj KK ij G ) 2 R g ij C 0 i0j ( 1 n R ) 2 R g ij (2.37) ] ( ) K 2 Kl k Kl k gij. (2.38)

35 Israel Einstein : (n +1) d s 2 = N 2 dt 2 + g ij dx i dx j. (2.1) (2.33) Einstein G µ ν = Λδµ ν R µ ν = 2Λ n 1 δµ ν (2.2) N(x) g ij (x) n N + 2Λ n 1 N =0, n R ij 1 N i j N = 2Λ n 1 g ij. (2.3a) (2.3b) (N =0) (n 1) N r N =1 2M ( ) 1 +O (2.4) rm 1 r m Λ =0 N 0. Proof. Gauss D dσ N = d n x g n N = 2Λ d n x gn =0. (2.5) n 1 D D N N =0 N D

36 2 36 (m +1) (n = m +1): N 0 N = S m ρ 2 =1/( N) 2 N θ a ds 2 n = ρ 2 dn 2 + γ ab dθ a dθ b (2.6) N = S(N) K ab K ab = 1 2ρ Ng ab, K = 1 ρ N γ γ. (2.7a) (2.7b) ds 2 m = γ abdθ a dθ b S(N) D ( 2 N) 11 = Nρ ρ, 3 (2.8a) ( 2 N) 1a = D aρ ρ, 2 (2.8b) ( 2 N) ab = 1 ρ K ab. (2.8c) (2.33) n R 11 = N K Kb a ρ Kb a ρ ρ, n R 1a = D b Ka b D ak, n R ab = R ab +2K ac Kb c KK ab 1 ρ NK ab 1 ρ D ad b ρ. (2.9a) (2.9b) (2.9c) Einstein (2.3) 1 ρ Nρ = ρk +(m +1)λρ 2 N, (2.10a) 1 ρ N γ = K γ, (2.10b) 1 ρ Nγ ab =2K ab = 2K m γ ab +2ˆK ab, (2.10c) 1 ρ NK = K ρn 1 m K2 ˆK 2 ρ ρ, (2.10d) ( ) 1 ρ ˆK 1 N b a = Nρ + K ˆK b a + Ra b 1 m Rδa b 1 ( D a D b ρ 1 ) ρ m ρδa b, (2.10e) R = 2K Nρ + m 1 m K2 + ˆK 2 + m(m +1)λ, (2.10f) D b ˆKb a m 1 m D ak = D aρ Nρ. 2 (2.10g)

37 2 37 λ = 2Λ m(m +1). (2.11) Israel Λ=0 : k =2 ( t + N ) ρ N, l = 1 ( N 2 t N ) ρ N (2.12) k k = l l =0,k l = 1. (2.13) Killing ξ = t k ξ = 2N 2,l ξ = 1. (2.14) N =0 k ξ k, l null frame Killing ξ ξ = N N = N ρ 2 N. (2.15) N N k = 2N 2 ρ 0(N 0), N N l = 1 ρ (2.16) N N 1 ρ H ξ H (N 0). (2.17) ξ ξ = 1 ρ H ξ. (2.18) κ = 1 ρ H (2.19) κ ρ H ρ H 0.

38 2 38 : R µνλσ Rµνλσ =4 R 0i0j R0i0j + R ijkl Rijkl (2.20) R R0i0j 0i0j R Rijkl ijkl (2.19) R 0i0j = i j N (2.21) N (2.8) R R0i0j 0i0j = 1 ( ) 2 (Dρ)2 + K2 N 2 ρ 4 ρ + Ka b Kb a. (2.22) 2 ρ 2 D a ρ H =0, K ab H =0. (2.23) (2.12) (2.19) R ijkl = n R ijkl N 0 R Rijkl ijkl 4 ρ 2 N Kb a NKa b + R abcdr abcd. (2.24) H ( N K) 2, ( N ˆK) 2 :. (2.25) (2.10) N 0 K N ρ H R, 2 (2.26a) D a ρ 0, N (2.26b) 2 ˆKab N R ab R m δ ab (2.26c) ρ H : N 2 =1 2M ( ) 1 +O, (2.27a) rm 1 r ( m g ij = 1+ 2 ) ( ) M 1 δ m 1 r m 1 ab +O. (2.27b) r m ρ, γ ab,k ab 1 ρ (m 1)M r m, (2.28a) γ ab r 2ˆγ ab, (2.28b) K ab 1 r γ ab, K m r. (2.28c) (2.28d)

39 2 39 : (2.10d) ( γk ) N = m γ (ρ 1/m ) Nρ 1 1/m S(N) ( m 1 γρ 1/m (Dρ) 2 + m ρ ˆK 2 2 ). (2.29) ( ) K 1 N 0. (2.30) N ρ 1 1/m Dρ =0 ˆK ab =0 N (N =0) (N =1) S( ) K N 1 ρ K 1 1/m H N 1 ρ 1 1/m (2.31) 2mΩ m [(m 1)M] 1 1/m ρ 1/m H R (2.32) Ω m m Euclid (2.10d) (2.10f) [ m 1 γ N N ρ = γ H ( KN + 2m )] 1 m 1 ρ [R +(m +1) ln ρ +(m +1) (Dρ)2 + m ρ ˆK 2 2 [ ( 1 N KN + 2m )] 1 N ρ m 1 ρ m 1 S(N) (2.32) S(N) (2.33) ]. (2.34) R. (2.35) 1 dnn R 2mA H 0 S(N) ρ 2 H (2.36) A H

40 2 40 [Killing ]: Killing ξ = t 1 M K := dξ (2.37) 2(m 1)Ω m S( ) Killing Killing Killing (1.2) 2 ξ µ + R ν µ ξ ν = 0 (2.38) d dξ =( 1) m+1 ( 2 ξ µ ν µ ξ ν )dx µ (2.39) d dξ =0 1 M = dξ (2.40) 2(m 1)Ω m H dξ = ( d(n 2 ) dt ) = 2 (dn Ndt)= 2 ρ dσ m (2.41) M = A H (m 1)Ω m ρ H (2.42) (Israel ) Proof. N = Gauss-Bonnet R =8π. (2.43) S(N) (2.32)(2.36)(2.42) m =2 4π 4A H ρ H = 16πM ρ H 4π (2.44) (2.32)(2.36)D a ρ = 0 ˆK ab =0(2.10d) D a K =0 (2.10f) D a R =0S(N) Euclid ρ γ ab Euclid N

41 2 41 Isreal W. Israel(1967, 1968) N = Müller-zum-Hagen (1973, 1974) Robinson(1977) Bach Robinson Bunting Masood-ul-Alam(1987,1993) Reissner-Nordstrom (Majumdar-Papapetrou (1945, 1947) M. Heusler Black Hole Uniqueness Theorems (Cambridge Univ. Press, 1996)

42 Hodge i) ( ω, χ = g ab ω a χ b 1 ω 1,,ω p, χ 1,χ p 2 p ω 1 ω p, χ 1 χ p ω 1 ω p,χ 1 χ p =det ω j,χ k p ω =(1/p!)ω µ1 µ p dx µ 1 dx µp, χ = (1/p!)χ µ1 µ p dx µ 1 dx µp, ω, χ = 1 p! ω µ 1 µ p χ µ 1 µ p. ii) (Hodge ) dx µ 1 dx µp (dx µ 1 dx µp )= 1 (n p)! dxν 1 dx ν n p ɛ ν1 ν n p µ 1 µ p p A p (n p) A n p Hodge p ω Hodge ( ω) µ1 µ n p = 1 p! ɛ ν 1 ν p µ 1 µ n p ω ν1 ν p Ω ω p χ ω χ = ω, χ Ω

43 3 43 (n p) ω, χ A p, η =det(η ab ) Hodge i) 1 =Ω, Ω = η. ii) ω χ = χ ω. iii) ω =( 1) p(n p) η ω. iv) ω, χ = η ω, χ. v) ( d ω) µ1 µ p 1 =( 1) np+p+1 ν ω ν µ 1 µ p X p (p 1) (I X ω)(y 1,,Y p 1 )=ω(x, Y 1,,Y p 1 ) I X i) I X ω =( 1) pn η X ω. ii) I X I Y + I Y I X = η g(x, Y ). iii) L X = di X + I X d. iv) [L X,I Y ]=I [X,Y ].

44 (M,g) G 2 G 2 Killing ξ,η η η (Frobenius) i) n r X 1,,X r r n r X I y p = 0(I =1,,r) n r y p (p = r +1,,n) fjk I (x) [X I,X J ]=f K IJ X K (I,J,K =1,,r) ii) n n r 1 ω P (P = r +1,,n) ω P =Λ P Q df P Λ P Q (x) f P (x) (n r) 2 Ω P Q dω P =Ω P Q ωq Riemann ξ,η ξ η dξ =0,ξ η dη =0 ξ,η Killing ξ d R d[a ξ b η c] =0,η d R d[a ξ b η c] =0

45 3 45 Proof. ξ,η f,g,p,q,r,s ξ = pdf + qdg, η = rdf + sdg Frobenius 1 α, β, γ, δ dξ = α ξ + β η, dη = γ ξ + δ η ξ,η Killing ξ µ = R ν µ ξν L η ξ =0 d( (η ξ dξ) = di η (ξ dξ) =I η d (ξ dξ) L η (ξ dξ) = I η I ξ d( dξ) = I η I ξ ξ = dx µ ɛ νλσ µ η ν ξ λ R σα ξ α. d (ξ η dη = dx µ ɛ µ νλσ ξ ν η λ R σα η α. (η ξ dξ), (ξ η dη) ξ η i) Maxwell F = F + i F df =0 T ab = 1 8π F a c F bc. ii) L ξ Φ=0, L η Φ=0 Φ GF = e 2U [dφ ξ + i (dφ ξ)] e 2U = g(ξ,ξ).

46 3 46 Proof. 0=L ξ F = di ξ F + I ξ df = di ξ F Φ GIξ F = dφ Iξ 2 =0I ξdφ =L ξ Φ=0. 0 =L η dφ= dl η Φ η L η Φ=0. I ξ I ξ F + I ξ I ξ F = g(ξ,ξ)f = e 2U F F = if Ge 2U F = I ξ dφ+i ( I ξ dφ). I ξ dφ =dφ ξ Einstein-Maxwell ds 2 = e 2U [ e 2k (dρ 2 + dz 2 )+W 2 dφ 2] e 2U (dt + Adφ) 2 (3.1) U, k, W, A ρ z F ρ z Φ GF = e 2U [dφ ξ + i (dφ ξ)] ξ = e 2U (dt + Adφ), η= e 2UW 2 dφ + Aξ

47 3 47 ξ µ ξ dξ =0 Proof. ξ µ f = ξ µ = k µ f 1 ξ = ξ µ dx µ ξ dξ =0 (3.1) ξ ξ =0 ξ 1 dξ dξ = χ ξ 1 χ Frobenius ξ = kdf k, f ξ ω := (ξ dξ) =I ξ dξ ω µ := ɛ µνλσ ξ ν λ ξ σ ω µ 1 ξ Killing Weyl 3.1 ξ = e U (dt + Adφ) ξ dξ = e 2U dt dφ da (3.2) A = t+aφ t A =0 ds 2 = e 2U [ e 2k (dρ 2 + dz 2 )+W 2 dφ 2] e 2U dt 2 (3.3)

48 3 48 (3 + 1) R t t = W 1 e 2(U k) (W U), (3.4) R ij = R ij e U i j e U (3.5) (ρ, z) 3 (2 + 1) R φ φ = W 1 e U 2 (We U )= W 1 e 3U 2k 2 (We U ) (3.6) e U 2 φ eu = Γ a φφ au = e 2k W ( W W U) U (3.7) R φφ = We 2k ( 2 W + (W U) ) (3.8) Einstein R µν =0 2 W =0, (3.9) (W U) = 0 (3.10) W = ρ ρ W = ρ a, b = ρ, z R ab = ρ 1 a ρ b k + ρ 1 b ρ a k 2 a U b U [ ] Re 2(k U) ρ 1 ρ (k U) δ ab (3.11) 2 2 R 2 R = 2e 2(U k) 2 (k U) (3.12) Einstein z k =2ρ ρ U z U, (3.13) ρ k = ρ [ ( ρ U) 2 ( z U) 2], (3.14) 2 k +( U) 2 = 0 (3.15) (Weyl)

49 3 49 Einstein 2 (ρ, z) Laplace ρ (ρ ρ U)+ρ 2 z U = 0 (3.16) U(ρ, z) ds 2 = e 2U [ e 2k (dρ 2 + dz 2 )+ρ 2 dφ 2] e 2U dt 2 (3.17) k(ρ, z) z k =2ρ ρ U z U, (3.18) ρ k = ρ [ ( ρ U) 2 ( z U) 2] (3.19) : ρ =Σsinϑ, z =Σcosϑ (3.20) U = k = a n Σ (n+1) P n (cos ϑ), (3.21) n=0 l,m=0 (l +1)(m +1) P l P m P l+1 P m+1 a l a m. (3.22) l + m +2 Σ l+m Schwarzschild Weyl : (ρ, z) Schwarzschild U = 1 2 ln Σ + +Σ 2m Σ + +Σ +2m = 1 2 ln m z +Σ m z +Σ +, (3.23) k = 1 2 ln (Σ + +Σ ) 2 4m 2, 4Σ + Σ (3.24) Σ ± = [ ρ 2 +(z ± m) 2] 1/2 (3.25) Schwarzschild ρ, z ρ = r(r 2m)sinθ, z =(r m)cosθ (3.26) U, k ρ =0, z m r =2m

50 3 50 : Weyl ρ = m (x 2 1)(1 y 2 ),z= mxy (3.27) Σ ± = x ± y (3.28) Schwarzschild Weyl U = 1 2 ln x 1 x +1, (3.29) k = 1 2 ln x2 1 x 2 y 2 (3.30) ( ) dx dρ 2 + dz 2 = m 2 (x 2 y 2 2 ) x dy2. (3.31) 1 y 2 U x ( (x 2 1) x U ) + y ( (1 y 2 ) y U ) =0, (3.32) k k x =(1 y 2 ) x[(x2 1)u 2 x (1 y2 )u 2 y ] 2y(x2 1)u x u y, (3.33a) x 2 y 2 k y =(x 2 1) y[(x2 1)u 2 x (1 y2 )u 2 y ] 2x(1 y2 )u x u y,(3.33b) x 2 y 2 U k x = r m m, sin θ = 1 y 2 (3.34) x =1, y 1 z x 1,y = ± Israel-Kahn Weyl Laplace z Schwarzschild Israel-Kahn (1964) z conic singularity

51 3 51 Schwarzschild BH : z z =0 m z = z 0 m Israel-Kahn m <z<m, m + z 0 < z<m + z 0 z k { 0 ;z< m, z > z0 + m k = 1 ln z2 0 (m+m ) 2 ; m<z<z 2 z0 2 (m m ) 2 0 m (3.35) m <z<z 0 m k <0 z e k φ 2πe k > 2π C-metric C : ] ds 2 1 = [H(y)dt 2 dy2 A 2 (x y) 2 H(y) + dx2 G(x) + G(x)dφ2 ; (3.36) H(y) = λ + ky 2 2mAy 3, (3.37) G(x) =1+kx 2 2mAx 3, (3.38) Λ= 3A 2 (λ + 1) (3.39) Einstein Petrov D λ = 1 ρ = H(y)G(x) xy[ k + ma(x + y)], z = A 2 (x y) 2 A 2 (x + y) (3.40) Weyl e 2U = H(y) A 2 (x y), 2 (3.41) e 2k H(y) = A 4 (x y) 4 (Gρ 2 x Hρ2 y ) (3.42) Weyl : m =0 y2 + λx r = 2 A(x y), ρ = 1+kx 2 (3.43) y 2 + λx 2

52 3 52 C ds 2 = dr 2 r 2 /l 2 λ + r2 ] [ (λρ 2 k)dt 2 + dρ2 λρ 2 k + ρ2 dφ 2 (3.44) l 2 = 1 A 2 (λ +1) (3.45) λ = 1,k = 1 ρ =sinθ (r, θ, φ) (x, y, z) ds 2 = z 2 dt 2 + dz 2 + dx 2 + dy 2 (3.46) Minkowski Rindler T = z sinh t, Z = z cosh t, X = x, Y = y, (3.47) ds 2 = dt 2 + dz 2 + dx 2 + dy 2 (3.48) λ = 1,m 0 r = 1 Ay, t = t A (3.49) A 0 C m Schwarzschild C Horizons: λ = 1,k = 1 (H(x) =G(x)) µ = ma 0 <µ 1 G(x) x 1 <x 2 < 0 <x 3 x 1 1/(2µ), x 2 1 µ, x 3 = 1 µ m =0 D 1 : x 2 <x<x 3, y 1 <y<y 2 (3.50) x = x 2 x = x 3 z Killing y = y 2 Rindler y = 1 y = y 3 A 0 r =2m

53 3 53 Conic singularity: y = x z x = x 2 x = x 2 + G (x 2 )ρ 2 /4 x, φ dx 2 G(x) + G(x)dφ2 dρ 2 + G (x 2 ) 2 ρ 2 dφ 2 (3.51) 4 2π φ 2 φ 2 = G (x 2 )φ/2 (1 2µ)φ (3.52) z φ 3 φ 3 = G (x 3 ) φ/2 (1 + 4µ)φ 2 (3.53) 8µπ Ernst Ernst : Einstein-Maxwell Killing ξ ω ω = I ξ dξ. p I X ω =( 1) p+1 (X ω), I X ω = (X ω) (3.1) Killing ξ ( d dξ) µ = ν ( ν ξ µ µ ξ ν )= 2 ξ µ =2R ν µ ξ ν (3.2) dω = I ξ d dξ + L ξ dξ = I ξ d dξ = (ξ d dξ) = 2 (ξ (Rξ)) (3.3) Einstein R µν =8πGT µν = GF µλ F ν λ (3.4) Rξ = Gdx µ F µλ F ν λ ξ ν = Gdx µ F µλ (d Φ) λ = Gd d ΦF = e 2U I d Φ[dΦ ξ + i (dφ ξ)] (3.5)

54 3 54 dω =2ie 2U (ξ I d Φ (dφ ξ)) = 2ie 2U (ξ (d Φ dφ ξ)) = 2ie 2U I ξ (d Φ dφ ξ) = 2id Φ dφ (3.6) d(ω +2i ΦdΦ) = 0 (3.7) de = d(e 2U )+iω 2 ΦdΦ (3.8) E Ernst Einstein-Maxwell ξ = t t U(y) A i (y) γ ij (y) ds 2 = e 2U (dt + A i dy i ) 2 + e 2U γ ij dy i dy j, (3.9) F t Φ(y) GF = e 2U [dφ ξ + i (dφ ξ )] (3.10) (ξ = ξ µ dx µ = e 2U (dt+a i dy i ))Einstein Maxwell Ernst de =Γ 2 ΦdΦ; Γ = d(e 2U )+iω (3.11) Φ γ ij (y) Σ 3 E = e 2U γ(γ,de ), (3.12) 3 Φ=e 2U γ(γ,dφ), (3.13) 3 R ij = 1 2 e 4U Γ (i Γj) 2e 2U (i Φ j) Φ. (3.14) A i (y) A = A i dy i U ω Σ da 2dU A = e 2U 3 ω. (3.15) 3 ω (Σ,γ) ω Hodge

55 (Neugebauer-Kramer) Einstein-Maxwell (E, Φ,γ ij ) (E, Φ,γ ij ) i) E = α 2 E, Φ = αφ ii) E = E + ib, Φ =Φ iii) E = E, 1+icE Φ = Φ 1+icE iv) E = E 2 βφ β 2, Φ =Φ+β v) E = E 1 2 µφ µ 2 E, Φ = Φ+µE 1 2 µφ µ 2 E α, β, µ b, c Proof. (E, Φ,γ ij ) S = d 3 y γ [3 R 1 2 e 4U γ ij ( i E +2 Φ i Φ)( j Ē +2Φ j Φ) Σ +2e 2U γ ij i Φ j Φ] (3.16) i)-v) Ernst (Ernst ) Einstein-Maxwell ρ, z U, k, A ds 2 = e 2U (dt + Adφ) 2 + e 2U [e 2k (dρ 2 + dz 2 )+ρ 2 dφ 2 ] (3.17) Φ ρ, z Einstein- Maxwell E, Φ (ρ, z) e 2U ρ 1 (ρ E )= E ( E +2 Φ Φ), (3.18) e 2U ρ 1 (ρ Φ) = Φ ( E +2 Φ Φ) (3.19)

56 3 56 U, A, k E, Φ e 2U =ReE + Φ 2, (3.20) ζ = ρe 4U [i ζ (Im E )+ Φ ζ Φ Φ ζ Φ], (3.21) [ ] e 4U ζ k =2ρ 4 ( ζe +2 Φ ζ Φ)( ζ Ē +2Φ ζ Φ) e 2U ζ Φ ζ Φ (3.22) ζ = ρ + iz Kerr-TS class Weyl (ρ, z) ρ = σ(x 2 1) 1/2 (1 y 2 ) 1/2,z= σxy (3.23) Ernst Ξ= 1 E 1+E (3.24) ( Ξ 2 1) [ x {(x 2 1) x Ξ} + y {(1 y 2 ) y Ξ} ] =2 Ξ [ (x 2 1)( x Ξ) 2 +(1 y 2 )( y Ξ) 2]. (3.25) Ξ= β α (3.26) δ =1: α = px iqy, β =1, (3.27) δ =2: α = p 2 (x 4 1) 2ipqxy(x 2 y 2 ) q 2 (1 y 4 ), β =2px(x 2 1) 2iqy(1 y 2 ), (3.28) δ =3: α = p(x 2 1) 3 (x 3 +3x)+iq(1 y 2 ) 3 (y 3 +3y) pq 2 (x 2 y 2 ) 3 (x 3 +3xy 2 ) ip 2 q(x 2 y 2 ) 3 (y 2 +3x 2 y), β = p 2 (x 2 1) 3 (3x 2 +1) q 2 (1 y 2 ) 3 (3y 2 +1) 12ipqxy(x 2 y 2 )(x 2 1)(1 y 2 ). (3.29) p, q p 2 + q 2 =1

57 Kerr-Newman δ =1 Kerr m = σ/p, a = σq/p (3.30) m J/m = a Harrison ν = 1+ µ 2 1 µ = m 2 m2 e, 2 (3.31) p = 2σ 1 1+ν m2 e, 2 (3.32) a q = m2 e 2 (3.33) m a e Kerr Kerr-Newman r m = 2σx, cos θ = y (3.34) 1+ν (r, θ, φ, t) ( ) ( dr ds 2 = Σ dθ2 +sin 2 θ r 2 + a 2 + a2 (2mr e 2 ) Σ 2 2a(2mr e2 ) sin 2 2mr e2 θdφdt (1 Σ 2 Σ 2 ) sin 2 θ dφ 2 ) dt 2 (3.35) Σ 2 = r 2 + a 2 cos 2 θ, =r 2 2mr + a 2 + e 2. (3.36) Kerr-Newmann a 2 + e 2 <m 2 =0 r = r ± = m ± m 2 a 2 e 2 Killing g tt =0g tt > 0 a 2 + e 2 >m (dominant energy condition)

58 3 58 T µν X, Y T (X, Y ) 0 dominant eneryg condition () (M,g,ξ) i) (M,g) Cauchy Σ. ii) ξ I + I Killing M θ t iii) (M,g) Einstein T µν dominant energy condition T µν () (non-rotating) ξ ξ =0(M,g,ξ) (Hawking 1972, HE prop ) Killing Killing k i) ( k Killing ii) ( Killing m

59 (HE prop.9.3.2, prop.9.3.3, Chrusciel & Wald 1994) B(τ) S 2 R τ B(τ) J + (I, M ) J (I +, M ) M [0, 1) S 2 R () Condition 1: DOC := J + (I, M ) J (I +, M ) S 2 R 2 H + := J (I +, M ) S 2 R Condition 2: Non-rotating case DOC ξ ξ< (HE prop.9.3.4) Condition 1 & static Condition (HE prop.9.3.5, Carter 1973) Non-rotating & Condition 2 static [Israel 1967,1968, Muller-zum-Hagen et al 1973,1974, Robinson 1977]

60 3 60 Static & Conditions 1 DOC: (Lindblom 1980) Static, Condition 2 and 3-geometry:conformally flat [Bunting&Masood-ul-Alam 1987, Ruback 1988, Masoodul-Alam 1992] Static and Condition 2 3 Geometry: conformally flat (Uniqueness for Non-rotationg BH) Non-rotating and Condition 2 Schwarzschild or Reissner-Nordstrom (Circular symmetry) [HE prop.9.3.7, Papaetrou 1966, Carter 1969] (M,g): axisymmetric & stationary regular predictable T µν : empty or source-free EM fields Killing ξ,η([ξ,η]=0) 2-surface orthogonal [HE prop.9.3.8, Carter 1971, 1973] & Condition 1

61 3 61 ρ 2 := (ξ η) 2 (ξ ξ)(η η) > 0inDOC(ρ 2 =0on H [Carter 1971, 1973] λ, µ, φ, tdocglobal chart ( ) dλ ds 2 2 =Ξ λ 2 c + dµ2 + Xdφ 2 +2Wdφdt Vdt µ 2 ρ 2 =(λ 2 c 2 )(1 µ 2 ) c = M 2Ω H J Φ H Q. µ = ±1 λ c (Ernst ) [Carter 1970, 1973] Einstein ds 2 2 = dλ2 λ 2 c + dµ2 ( 1 <µ<1,c<λ< ) 2 1 µ 2 X, Y, E, B δs = δ Ldλdµ =0; L = X 2 + Y +2(E B B E) 2 +2 E 2 + B 2 2X 2 X X, Y, E, B. µ ±1 X, λ (E,B,Y ), µ Y +2(E µ B B µ E). λ E = Qµ +O(1/λ),B =O(1/λ), Y =2Jµ(3 µ 2 )+O(1/λ),λ 2 X =(1 µ 2 )(1 + O(1/λ)).

62 (empty case) [Robinson 1975] Prop empty(e = B =0) C, J () [Robinson 1974] Prop C, J, Q (No hair theorem) [Mazur 1982, Bunting 1981, 1983] Condition 1 Einstein rotating, stationary regular predictable Kerr-Newmann