THÈSE Nicolas VASSET

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A V r V η µ V ∀t ∈ [0, T ] R ℓm E ℓm A ℓm ∆ rE ℓm = r ∂Aℓm ∂r + 2Aℓm . r = 0 r → ∞ S ∇ · V = 0 S V µ µ S A S S µ A R h h ij (= h ji ) 1(r) 3(ϕ) h 1/r r → ∞ ∀(θ, ϕ) ∀t ≥ 0, ∀r < R, ∂ 2 h ij ∂t 2 = ∆hij , ∀t ≥ 0, ∀r ≤ R, ∇jh ij = 0, ∀r ≤ R, h ij (0, r, θ, ϕ) = α ij 0 (r, θ, ϕ), ∂h ∀r ≤ R, ij = γ ∂t ij 0 (r, θ, ϕ), t=0 ∀t ≥ 0, h ij (t, R, θ, ϕ) = β ij 0 (t, θ, ϕ). α ij 0 , γ ij 0 β ij 0
H hij H i ≡ ∇jh ij ⇐⇒ ⎧ ⎪⎨ ⎪⎩ H r = ∂hrr ∂r H θ = ∂hrθ ∂r H ϕ = ∂hrϕ ∂r + 2hrr r + 3hrθ r + 3hrϕ r rθ 1 ∂h + r ∂θ θθ 1 ∂h + r ∂θ + 1 r 1 ∂h + sin θ rϕ ∂ϕ − hθθ − h ϕϕ + hrθ , tan θ 1 ∂h + sin θ θϕ 1 θθ ϕϕ + h − h ∂ϕ tan θ , θϕ ∂h 1 ∂h + ∂θ sin θ ϕϕ 2hθϕ + . ∂ϕ tan θ h ij h(t, r, θ, ϕ) = ℓ,m ℓm L0 T L0 ℓm ℓm + T0 T T0 ℓm + Eℓm 1 T E1 ℓm + Bℓm 1 T B1 ℓm + Eℓm 2 T E2 ℓm + Bℓm 2 T B2 ℓm , Lℓm 0 , T ℓm 0 , Eℓm 1 , Bℓm 1 , Eℓm 2 , Bℓm 2 (t, r) TE2 TB2 h h rr (t, r, θ, ϕ) = ℓ,m h τ (t, r, θ, ϕ) = ℓ,m h η (t, r, θ, ϕ) = ℓ,m h µ (t, r, θ, ϕ) = ℓ,m h W (t, r, θ, ϕ) = ℓ,m h X (t, r, θ, ϕ) = ℓ,m L ℓm 0 Y m ℓ , T ℓm 0 Y m ℓ , E ℓm 1 Y m ℓ , B ℓm 1 Y m ℓ , E ℓm 2 Y m ℓ , B ℓm 2 Y m ℓ . h τ = h θθ + h ϕϕ
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 58 and 59: h = h rr + h τ . h τ
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
Page 108 and 109:
(Σt, γij, Kij)
Page 110 and 111:
ψ
Page 112 and 113:
ψ
Page 114 and 115:
˜γij = ψ −4 γij; ψ = 1 (γ
Page 116 and 117:
h ij h ij D
Page 118 and 119:
h ij
Page 120 and 121:
A ∆A − ψ4 N 2 LβLβA = A
Page 122 and 123:
∆h µ + 2 ∂h r µ ∂r ∆h η
Page 124 and 125:
h ij h ij
Page 126 and 127:
MHΩ JK a M
Page 128 and 129:
MH JH MADM
Page 130 and 131:
1 − ɛA M2 M 3
Page 132 and 133:
˜γ ij
Page 134 and 135:
h ij A ˜ B
Page 136 and 137:
H IJ A ˜ B
Page 138 and 139:
AHE = 8π(M 2 ADM + M 4 ADM − J
Page 140 and 141:
MHawking(S2) ≥ MHawking(S1) S2
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ɛ P := A 16πM 2 ADM ɛ A := ≤ 1
Page 144 and 145:
R = r + M 2 − a2 + M, 4r R
Page 146 and 147:
H (1−ɛA) r
Page 148 and 149:
ɛA ɛP A JK λ
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(1 − ɛA) NH = 0.5
Page 152 and 153:
ɛP A NH = 0.55
Page 154 and 155:
Σt St
Page 156 and 157:
(b − N) (b − N) = −Cθ
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t0
Page 160 and 161:
S t = t0
Page 162 and 163:
αN = 1000 µN = 1
Page 164 and 165:
ψ 2 α (b−N) = 1000 µ (
Page 166 and 167:
ψ 2 α (b−N) = 1000 µ (
Page 168 and 169:
ψ 2 α (b−N) = 1000 µ (
Page 170 and 171:
ψ 2 α (b−N) = 1000 µ (
Page 172 and 173:
ψ 2 α (b−N) = 1000 µ (
Page 174 and 175:
ψ 2 α (b−N) = 1000 µ (
Page 176:
ψ 2 α (b−N) = 1000 µ (
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THÈSE Nicolas VASSET

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