THÈSE Nicolas VASSET
THÈSE Nicolas VASSET THÈSE Nicolas VASSET
h = h rr + h τ . h τ h η h µ {h ri } i=1,2,3 h rθ = ∂hη 1 − ∂θ sin θ h rϕ = 1 ∂h sin θ η ∂hµ + ∂ϕ ∂θ ; ∂h µ , ∂ϕ P = h θθ − h ϕϕ /2 P ≡ h θθ − h ϕϕ 2 = ∂2hW 1 ∂h − ∂θ2 tan θ W ∂θ h θϕ = ∂2hX 1 ∂h − ∂θ2 tan θ X ∂θ ∆θϕ (∆θϕ + 2) h W = ∂2P 3 ∂P + ∂θ2 tan θ ∂θ ∆θϕ (∆θϕ + 2) h X = ∂2 h θϕ 3 + ∂θ2 tan θ ∂h θϕ ∂θ 1 − sin2 θ 1 − sin2 θ − 1 sin 2 θ ∂2hW ∂ − 2 ∂ϕ2 ∂θ ∂2hX ∂ + 2 ∂ϕ2 ∂θ ∂2P 2 − 2P + ∂ϕ2 sin θ 1 ∂h sin θ X ∂ϕ 1 ∂h sin θ W ; ∂ϕ 1 − sin2 ∂ θ 2hθϕ ∂ϕ2 − 2hθϕ − 2 sin θ , θϕ ∂ ∂h hθϕ + ∂ϕ ∂θ tan θ ∂ ∂ϕ , ∂P P + . ∂θ tan θ h η h µ (ℓ = 0) h W h X ℓ = 0 ℓ = 1 h H h H r = ∂hrr 3hrr 1 + + ∂r r H η η ∂h 3hη = ∆θϕ + ∂r r H µ µ ∂h 3hµ = ∆θϕ + ∂r r r (∆θϕh η − h) , + 1 r (∆θϕ + 2) h W + + 1 r (∆θϕ + 2) h X h − hrr 2 , .
T R 3 T ij = ∇ i L j + ∇ j L i + h ij 0 , ∇jh ij 0 = 0 ∇jT ij = 0 ⇐⇒ L i = 0 T ij L h ij 0 A = ∂hX 0 ∂r B = ∂hW 0 ∂r C = ∂h0 ∂r hµ 0 − , r − 1 2r ∆θϕh W 0 − hη0 ∂hrr 0 − ∂r + h0 r r + h0 − hrr 0 4r 3hrr 0 − r − 2∆θϕ A = B = C = 0 ⇐⇒ h0 = 0, , W ∂h0 ∂r + hW 0 r , h (∆h) rr = ∆h rr − 6hrr 4 − r2 r2 ∆θϕh η + 2h r2 (∆h) η = ∆h η + 2 ∂h r η η 2hη 2 ∂h + − ∂r r2 r ∂r (∆h) µ = ∆h µ + 2 ∂h r µ ∂r (∆h) W = ∆h W + 2hW r (∆h) X = ∆h X + 2hX 3hη + r + (∆θϕ + 2) hW r 1 3 + h − 2r 2r hrr , µ 2hµ 2 ∂h 3hµ + − + r2 r ∂r r + (∆θϕ + 2) hX , r 2hη + 2 r 2 , 2hµ + r2 r T r(∆h) = ∆h T r(∆h) ∆h. 2 , H µ H Hη = 0 −hrr /r
- Page 10 and 11: h ij
- Page 13 and 14: e
- Page 15 and 16: 10 5
- Page 17: (−, +, +, +)
- Page 20 and 21: Rµν − 1 2 gµνR = 8πG c 4 Tµ
- Page 22 and 23: ∇µG µν = ∇µ( (4) R µν −
- Page 24 and 25: p ∈ M V ∋ p
- Page 26 and 27: R × S 3
- Page 28 and 29: n µ Σt Σt
- Page 30 and 31: E Jα Sαβ
- Page 32 and 33: N = 1, β i = 0
- Page 34 and 35: M N β i
- Page 36 and 37: ∂ ∂t fij = 0 ˜γij =
- Page 38 and 39: (γij, Kij) Σt0
- Page 40 and 41: Φ = ln(ψ) ∆ S
- Page 42 and 43: h ij ∂2hij 2 N − ∂t2 ψ4 ∆
- Page 44 and 45: (r, θ, ϕ) R 3 (
- Page 46 and 47: f N 0
- Page 48 and 49: H f {fi} H
- Page 50 and 51: R 3
- Page 52 and 53: V R ∀(θ, ϕ) ∀t
- Page 54 and 55: ∇ · D0 = 0 W W µ
- Page 56 and 57: A V r V η
- Page 60 and 61: ∂ 2 A = ∆A, ∂t2 ∂2B C =
- Page 62 and 63: h rr h η h W (ℓ + 2) ∂Eℓm
- Page 64 and 65: A h µ h X ˜
- Page 66 and 67: = R > 0 ∀(θ, ϕ) ∀t
- Page 68 and 69: dt R = 6, Nr = 17, Nθ = 17,
- Page 70 and 71: dt R = 6, Nr = 17, Nθ = 17,
- Page 72 and 73: A, A, ˜ B
- Page 74 and 75: p (M, ηµν) D + (p)
- Page 76 and 77: ∗ = r + 2M r 2M − 1 .
- Page 78 and 79: T = 0
- Page 80 and 81: (x, y, z, t) r(xdx + ydy) −
- Page 82 and 83: ∂ µ ∂t
- Page 84 and 85: = 2M r = 0
- Page 86 and 87: θ (ξ)
- Page 88 and 89: a 2 > M 2
- Page 90 and 91: ∂ µ ∂t ∂ µ µ
- Page 92 and 93: mi Mi N
- Page 94 and 95: X i ψ
- Page 96 and 97: Σ (
- Page 98 and 99: θ (l) µ = 1 16π ζ
- Page 100 and 101: (M, gµν) S 2 × R l µ
- Page 102 and 103: 2π µ ∂
- Page 104 and 105: St θ (l) = 0 θ (k) < 0.
- Page 106 and 107: ¯κ
T R 3 <br />
<br />
<br />
<br />
T ij = ∇ i L j + ∇ j L i + h ij<br />
0 ,<br />
∇jh ij<br />
0 = 0 ∇jT ij = 0 ⇐⇒ L i = 0 <br />
T ij <br />
L h ij<br />
0 <br />
<br />
<br />
A = ∂hX 0<br />
∂r<br />
B = ∂hW 0<br />
∂r<br />
C = ∂h0<br />
∂r<br />
<br />
hµ 0<br />
− , <br />
r<br />
− 1<br />
2r ∆θϕh W 0 − hη0<br />
∂hrr 0<br />
−<br />
∂r<br />
+ h0<br />
r<br />
r + h0 − hrr 0<br />
4r<br />
3hrr 0<br />
−<br />
r<br />
− 2∆θϕ<br />
A = B = C = 0 ⇐⇒ h0 = 0,<br />
, <br />
W ∂h0 ∂r + hW 0<br />
r<br />
<br />
, <br />
<br />
<br />
<br />
<br />
h <br />
(∆h) rr = ∆h rr − 6hrr 4<br />
−<br />
r2 r2 ∆θϕh η + 2h<br />
r2 (∆h) η = ∆h η + 2 ∂h<br />
r<br />
η<br />
η<br />
2hη 2 ∂h<br />
+ −<br />
∂r r2 r ∂r<br />
(∆h) µ = ∆h µ + 2 ∂h<br />
r<br />
µ<br />
∂r<br />
(∆h) W = ∆h W + 2hW<br />
r<br />
(∆h) X = ∆h X + 2hX<br />
3hη<br />
+<br />
r + (∆θϕ + 2) hW<br />
r<br />
1 3<br />
+ h −<br />
2r<br />
<br />
2r hrr<br />
<br />
,<br />
<br />
µ<br />
2hµ 2 ∂h 3hµ<br />
+ − +<br />
r2 r ∂r r + (∆θϕ + 2) hX<br />
<br />
, <br />
r<br />
2hη<br />
+ 2 r<br />
2 , <br />
2hµ<br />
+<br />
r2 r<br />
T r(∆h) = ∆h T r(∆h) ∆h. <br />
2 , <br />
<br />
H µ <br />
H Hη = 0 <br />
−hrr /r
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