THÈSE Nicolas VASSET

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∇ · D0 = 0 W W µ W θ ϕ ∂rW η W η W r + − r r W D0 A = ∂W η ∂r + W η r r W − . r D0 = 0 ⇐⇒ W µ = 0 A = 0. V µ A V r V η (∆V) η = ∆V η r V + 2 , r2 (∆V) µ = ∆V µ . µ ∂ 2 V µ ∂t 2 = ∆V µ . A ∂2A = ∆A. ∂t2 V F Φ Ψ F = ∇ × (Ψk) + ∇ × ∇ × (Φk) k eρ
k = er F = − 1 r2 ∆θϕΦer + 1 r 1 sin θ ∂ϕΨ + ∂θ∂rΦ eθ + 1 r −∂θΨ + 1 sin θ ∂ϕ∂rΦ F η F µ F η = 1 r ∂rΦ F µ = − 1 r Ψ A Φ A = 1 r ∂2 r Φ + 1 ∆Φ = ∆ r3 Φ r eϕ. . ∆θϕA = −∆(rF r ). V µ A v0 w0 V µ (t = 0) ∂V µ ∂t A t=0 µ v0 w0 b0 V µ A R [0, T ] V (V r , V η ) V A V r V η A A(t, r, θ, ϕ) = ℓ,m A ℓm (t, r)Y m ℓ (θ, ϕ). ∀ℓ > 0, ∀m − ℓ ≤ m ≤ ℓ, ⎧ ⎪⎨ ⎪⎩ ∂R ℓm ∂r + 2Rℓm r ∂E ℓm ∂r ℓ(ℓ + 1) − E r ℓm = 0 Eℓm Rℓm + − = Aℓm r 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 56 and 57: A V r V η
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
Page 142 and 143:
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
Page 150 and 151:
(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
Page 158 and 159:
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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