THÈSE Nicolas VASSET

h ij

h ij A ˜ B

e
10 40

10 5
2GM/Rc 2

10 5
2GM/Rc 2

10 5
2GM/Rc 2

(−, +, +, +)
0, 1, 2, 3 (i, j, k...)
1, 2, 3
(a, b, c...) 2, 3
(−, +, +, +) gµν
γij
qab
G c 1

(−, +, +, +)
0, 1, 2, 3 (i, j, k...)
1, 2, 3
(a, b, c...) 2, 3
(−, +, +, +) gµν
γij
qab
G c 1

Rµν − 1
2 gµνR = 8πG
c 4 Tµν.
M
gµν g µν g µα gαν = δ µ ν R µν
Rµν = R γ µγν,
R α βµν = Γ α βν,µ − Γ α βµ,ν + Γ α σµΓ σ βν − Γ α σνΓ σ βµ,
Γ α µν = 1
2 gαβ (gβµ,ν + gβν,µ − gµν,β).
∇µT µν = 0 ∇
Γ α µν

∇µG µν = ∇µ( (4) R µν − 1
2 gµν (4) R) = 0.
gµν
gµν
S[gµν] =
Rɛ,
ɛ

∇µG µν = ∇µ( (4) R µν − 1
2 gµν (4) R) = 0.
gµν
gµν
S[gµν] =
Rɛ,
ɛ

M gµν
p ∈ M
p
M gµν
p
t µ
t µ tµ < 0
p t µ p
t µ
I + (p) p ∈ M r ∈ M
λ(t)
t µ λ(0) = p λ(1) = r
I − (p) p λ(t)
p r
J + (p) J − (p)
p λ(t)
S I + (S)∩S = ∅
S

p ∈ M
V ∋ p
M, gµν
ηµν
M = R 4
ds 2 = −dt 2 + dr 2 + r 2 (dθ 2 + 2 θdϕ 2 ).
M, ηµν
( ˜ M, ηµν) ˜
˜ηµν = Ω −2 ηµν,
Ω R 4
d˜s 2 = −dT 2 + dR 2 + 2 R(dθ 2 + 2 θdϕ 2 ).
T R
−π < T + R < π,
−π < T − R < π,
0 ≤ R.
˜ M

R × S 3
S 3 × R
R T
θ ϕ
M
R = 0, T = ±π
i − i +
i 0
I − I + Ω Ω(i 0 ) = Ω(I − ) = Ω(I + ) = 0 ˜ ∇µΩ = 0
I − I + ˜ ∇µ
r = cste t = cste
π
4
R 4

R × S 3
(M, gµν)
S
D + (S) p ∈ M
p S
p S
D − (S) S
D = D + ∪ D − M

t Σt0 t = t0
M = R 4
t Σ × R
Σt0
∇ µ t
Σt1>t0
(M, gµν)
Σt t
(x 1 , x 2 , x 3 )

n µ
Σt
Σt
gµν
n µ
γµν = gµν + nµnν.
D
γµν Σt
Kµν = − 1
2 Lnγµν.
K µν γµν
(Σt) M
Rµν = R α µαν γµν Σt
(4) Rµν = (4) R α µαν gµν
(Σt, γµν)
(M, gµν) Σt
γ µ αγ ν βγ γ ργ σ δ
(4) R ρ σµν = R γ δαβ + K γ αKδβ − K γ βKαδ.
(4) R + 2 (4) Rµνn µ n ν = R + K 2 − KijK ij ,
KijK ij = K µν Kµν
Kij Kµν
g ρ σ = δ ρ σ + n ρ nσ Σt
R
n µ
γ µ αγ ν βγ γ ρn σ (4) R ρ σµν = DβK γ α − DαK γ β.
Σt Σt+δt
(x 1 , x 2 , x 3 )
M

N
nα ∇αt
n α = −N∇ α t,
Σt
t β µ
At(x1 0, x2 0, x3 0)
n µ Σt At+δt(x1 1, x2 1, x3 1)
µ
β µ ∂
∂
∂t
µ
∇µt = 1
µ
∂
∂t
= Nn µ + β µ .
γµν K µν
N β µ
ds 2 = −N 2 dt 2 + γij(dx i + β i dt)(dx j + β j dt).
N β µ
Σt
N
β i
n µ
γαµγ ν βn ρ n σ (4) R µ ρνσ = LnKαβ + 1
N DαDβN + KαµK µ β.
G = c = 1
E = Tµνn µ n ν ,
Jα = −γ µ αTµνn ν ,
Sαβ = γ µ αγ ν βTµν,
∂t

E Jα Sαβ
Σt n µ
n µ
R + K 2 − KijK ij = 16πE.
n µ Σt
DjK j i − DiK = 8πJi.
Σt
∂Kij
∂t −LβKij = −DiDjN +N{Rij−2KikK k j+KKij+4π[(S−E)γij−2Sij]}.
∂γij
∂t − Lβγij = −2NKij.
N β i
γij
(γij, Kij) Σt
γij

DiE i = 0,
∂E i
∂t = −Dj DjAi + D j DiAj
∂A i
∂t = −Ei − DiΦ.
E i A i Φ
t
t0
Σt0
Σt0 (γij, Kij)
Σt1>t0
t = t1
(γij, Kij)
Σt0

N = 1, β i = 0
(γij, Kij) Σt
(M, gµν)
(Σt) (γij, Kij)t
fij
Σt
(x 1 , x 2 , x 3 ) r
(γij, Kij)
γij = fij + O(r −1 ),
∂γij
∂x k = O(r−2 ),
Kij = O(r −2 ),
∂Kij
∂x k = O(r−3 ).

D fij
St,r ∈ Σt
r r
si qab
Σt
MADM = 1
16π
St,r→∞
D j γij − Di(f kl γkl) s i√ qd 2 y.
√ qd 2 y
St,r q qab d 2 y
Σt
(M, gµν)
k µ
MK = − 1
8π
St,r
∇ µ k µ (sµnν − nµsν) √ qd 2 y,
n µ
Σt St
(M, gµν)
φ i
JK = 1
16π
St,r
∇ µ φ µ (sµnν − nµsν) √ qd 2 y = 1
8π
St,r
Kijs i φ i√ qd 2 y.

M
N β i
N = 1 β i = 0.
τ
t
N = 1
N
∇ µ ∇µt = 0.
N
∂
− Lβ
∂t
N = −KN 2 .

−2KN
Σt K = 0
K = cste
DiD i N = N[4π(E + S) + KijK ij ].
K = 0 ∇ µ nµ = 0
Σt
β i
D i Dix k = 0.
ψ γij
fij
fij

∂
∂t fij = 0
˜γij = ψ −4 γij,
ψ =
(γ)
(f)
1
12
.
fij
˜γ ij = ψ 4 γ ij
˜γij
ψ
Kij Kij
A ij = K ij − 1
3 Kγij ,
A ij ζ < 0
à ij = ψ −ζ A ij .
ζ ζ = −4
ζ = −10
Σt (γij, Kij)
ζ = −10
Ãij
X i
à ij = ˜ D i X j + ˜ D j X i − 2
3 ˜ DkX k ˜γ ij + A ij T T ,
= 0.
˜Dj Ãij
T T

˜ D ˜γij
X i
˜Dk ˜ D k ψ = ψ
R − ψ5 2πE −
8 K2
−
12
1
8 Ãkl Ãklψ −7 ,
˜Dk ˜ D k X i + 1
3 ˜ D i DkX ˜ k + R i kX k = 8πψ 10 J i + 2
3 ψ6D˜ i
K.
ψ Xi
˜γ
ÃT T K
t = 0
ζ = −4
à ij = 1
ij ∂˜γ
2N ∂t − Lβ˜γ ij − 2
3 ˜ Dkβ k ˜γ ij
= 1
ij ∂˜γ
2N ∂t + (˜ Lβ) ij
.
ψ β i
˜Di ˜ D i ψ − ˜ R 1
ψ +
8 8 Ãij Ãijψ −7 + 2πψ 5 E − K2
12 ψ5 6 ψ
˜Dj
N
= 0,
(˜ Lβ) ij
+ ˜ 6 ψ ∂˜γ
Dj
N
ij
−
∂t
4
3 ψ6D˜ i
K =
10 i
16πψ J .
∂˜γij ∂t
K ˜γ
˜γ ij = f ij γij
N
∂K
∂t

(γij, Kij) Σt0

Σt
fij
1
γ ij
γ ij = ψ 4 ˜γ ij = ψ 4 [f ij + h ij ],
h ij ˜γ ij
γ ij
ζ = −4
à ij = ψ 4 (K ij − 1
3 Kγij ),
Ãij = ˜γil˜γjkA lk
K = 0
(x 1 , x 2 , x 3 )
∂
∂x j (˜γij ) = 0.
D
Di˜γ ij = Dih ij = 0.
Q = Nψ 2
∆Q = −h kl DkDlQ + ψ 6
+2ψ 2
N
˜R∗
N
8 + ˜ DkΦ ˜ D k Φ
4πS + 3
4
+ ˜ DkΦ ˜ D k N
kl
ÃklA
,

Φ = ln(ψ) ∆
S = S µ µ
˜R∗
˜R∗ = 1
4 ˜γkl Dkh mn Dl˜γmn − 1
2 ˜γkl Dkh mn Dn˜γml.
N
∆N = ψ 4 N
kl
4π(E + S) + ÃklA − h kl DkDlN − 2 ˜ DkΦ ˜ D k N.
β i
∆β i + 1
3 Di Djβ j = 16πNψ 4 J i + 2A ij DjN
−12NA ij DjΦ − 2N∆ i klA kl − h kl DkDlβ i − 1
3 hik DkDlβ l .
∆ i kl
∆ k ij = 1
2 ˜γkl (Di˜γlj + Dj˜γil − Dl˜γij) ,
˜ D i D i
∂Φ
∂t − βk DkΦ = 1
6 Dkβ k .
h ij
A ij
∂hij ∂t − Lβh ij − 2
3 Dkβ k h ij = 2NA ij − (Lβ) ij ,
∂A
ij
∂t − LβA ij − 2
3 Dkβ k A ij = N
2ψ4 ∆hij + S ij
− 1
2ψ6 i jk j ik k ij
D h + D h − D h DkQ.
S ij

h ij
∂2hij 2 N
−
∂t2 ψ4 ∆hij ∂h
− 2Lβ
ij
∂t + LβLβh ij = L ∂β h
∂t
ij
+ 4
k ∂
Dkβ − Lβ h
3 ∂t ij − N
ψ6 DkQ D i h jk + D j h ik − D k h ij
+ 1
∂
∂
− Lβ N − Lβ h
N ∂t ∂t ij − 2
3 Dkβ k h ij + (Lβ) ij
+ 2
∂
− Lβ Dkβ
3 ∂t k − 2
3 (Dkβ k ) 2
h ij + 2NS ij
−
∂
− Lβ
∂t
(Lβ) ij + 2
3 Dkβ k (Lβ) ij ,
(Lβ) ij = D i β j + D j β i − 2
3 Dkβ k f ij .
h ij
h ij A ij
A ij = 1
2N
(Lβ) ij + ∂hij
∂t − Lβh ij − 2
3 Dkβ k h ij
.
h ij
∂h ij
∂t
h ij
h ij

h ij
∂2hij 2 N
−
∂t2 ψ4 ∆hij ∂h
− 2Lβ
ij
∂t + LβLβh ij = L ∂β h
∂t
ij
+ 4
k ∂
Dkβ − Lβ h
3 ∂t ij − N
ψ6 DkQ D i h jk + D j h ik − D k h ij
+ 1
∂
∂
− Lβ N − Lβ h
N ∂t ∂t ij − 2
3 Dkβ k h ij + (Lβ) ij
+ 2
∂
− Lβ Dkβ
3 ∂t k − 2
3 (Dkβ k ) 2
h ij + 2NS ij
−
∂
− Lβ
∂t
(Lβ) ij + 2
3 Dkβ k (Lβ) ij ,
(Lβ) ij = D i β j + D j β i − 2
3 Dkβ k f ij .
h ij
h ij A ij
A ij = 1
2N
(Lβ) ij + ∂hij
∂t − Lβh ij − 2
3 Dkβ k h ij
.
h ij
∂h ij
∂t
h ij
h ij

h ij
∂2hij 2 N
−
∂t2 ψ4 ∆hij ∂h
− 2Lβ
ij
∂t + LβLβh ij = L ∂β h
∂t
ij
+ 4
k ∂
Dkβ − Lβ h
3 ∂t ij − N
ψ6 DkQ D i h jk + D j h ik − D k h ij
+ 1
∂
∂
− Lβ N − Lβ h
N ∂t ∂t ij − 2
3 Dkβ k h ij + (Lβ) ij
+ 2
∂
− Lβ Dkβ
3 ∂t k − 2
3 (Dkβ k ) 2
h ij + 2NS ij
−
∂
− Lβ
∂t
(Lβ) ij + 2
3 Dkβ k (Lβ) ij ,
(Lβ) ij = D i β j + D j β i − 2
3 Dkβ k f ij .
h ij
h ij A ij
A ij = 1
2N
(Lβ) ij + ∂hij
∂t − Lβh ij − 2
3 Dkβ k h ij
.
h ij
∂h ij
∂t
h ij
h ij

(r, θ, ϕ) R 3
(∂/∂r, ∂/∂θ, ∂/∂ϕ)
er = ∂
∂r , eθ = 1 ∂
r ∂θ , eϕ = 1
rθ
∂
.
∂ϕ
Σt
fij = diag(1, 1, 1)
f [a, b]
f
N
{xi}i∈[1,N] ∈ [a, b] x1 = a xN = b
f(xi)

{Ti}i∈N
[−1, 1]
L 2 w = 1
√ 1−x 2
[−1, 1]
f
∀x ∈ [−1, 1], f(x) =
∞
ciTi(x), ci ∈ R(i ∈ N).
i=0
f [a, b]
[−1, 1]
N
∀x ∈ [−1, 1], f(x) ≈
N−1
i=0
ciTi(x), {ci}i∈N ∈ R.
f
N {ci}i∈[0,N−1]
f
(N − 1) {ci}
∀i ∈ N, ci = 2
πδ0i
1
−1
f(x)Ti(x)w(x)dx.
w [−1, 1]
(N + 1) {wi}i∈N (N + 1) {xi} ∈ [−1, 1]
x0 = −1 xN = 1 f
2N − 1
1
N
f(xn)wn.
f(x)w(x)dx =
−1
f
2N − 1
w
xi = cos πi w0 = wN = N π wi = 2N π
N
f f N
INf =
N
˜fiTi(x),
i=0
˜ fi = 1
γi
i=0
N
f(xj)Ti(xj)wj γi =
j=0
N
j=0
T 2
i (xj)wj.

f N
0
f
N
˜ fi
ci
f R 3
(r, θ, ϕ)
∀ℓ ≥ 0, ∀m, 0 ≤ m ≤ ℓ, Y m
ℓ (θ, ϕ) = e imϕ P m ℓ (cos θ),
∀m, −ℓ ≤ m < 0, Y m
ℓ (θ, ϕ) = (−1) m e imϕ P |m|
ℓ (cos θ).
P m ℓ (ℓ, m)
f
f(r, θ, ϕ) =
∞
ℓ
ℓ=0 m=−ℓ
fℓm(r)Y m
ℓ (θ, ϕ).
fℓm r
f
Y m
ℓ
(θ, ϕ)

= 0
f(r, θ, ϕ) =
∞
ℓ
ℓ=0 m=−ℓ
r ℓ
∞
i=0
fiℓmr 2i Y m
ℓ (θ, ϕ).
fℓm(r)
fℓm(r) ℓ
ℓ
f(r, θ, ϕ) =
∞
ℓ
∞
ℓ=0 m=−ℓ i=0
fiℓmTi(r)Y m
ℓ (θ, ϕ),
i ℓ
r ℓ
Hf(r) = s(r).
H
{ ˜si}i∈[0,N−1] N
s { ˜ fi}i∈[0,N−1]
N
xi Hf(xi) = s(xi)
f s N
H
N
H
H
f {si}

H f {fi}
H
H
H
{f(xi)} {s(xi)}
H f f(xi)
∀r ∈ [1, 2], θ ∈ [0, π], ϕ ∈ [0, 2π[,
∆f(r, θ, ϕ) = S(r, θ, ϕ),
∀θ ∈ [0, π], ϕ ∈ [0, 2π[, f(1, θ, ϕ) = α f(2, θ, ϕ) = β.
f(r, θ, ϕ)
∆f = ∂2f 2 ∂f 1
+ +
∂r2 r ∂r r2 ∆θϕf,
∆θϕf = ∂2f cos θ ∂f 1
+ +
∂θ2 sin θ ∂θ sin2 ∂
θ
2f .
∂ϕ2
f S
∀(ℓ, m), ∆θϕY m
ℓ = −ℓ(ℓ + 1)Y m
ℓ .

f
s
r {ℓ, m}
Ofℓm = ∂2fℓm(r) ∂r2 2 ∂fℓm(r) ℓ(ℓ + 1)
+ −
r ∂r r2 fℓm(r) = Sℓm(r),
[1, 2] [−1, 1]
f S
O f Oℓm
N
{ ˜ fi}i∈[0,N−1] { ˜ Si}i∈[0,N−1]
f S
fℓm(r) =
N−1
i=0
˜fiℓmTi(r) Sℓm(r) =
N−1
i=0
˜SiℓmTi(r).
Oℓm N − 1
N −2
N−1
(−1) i fiℓm
˜ = α
i=0
N−1
i=0
˜fiℓm = β
Oℓm
{ ˜ fi}i∈[0,N−1]
{ ˜ Si}i∈[0,N−1]
f

R 3
(ξ, θ ′
, ϕ ′
)
θ = θ ′
ϕ = ϕ ′
ξ
0 ≤ r ≤ r0
r = α0ξ, ξ ∈ [0, 1], , α0 = r0.
(1 ≤ j ≤ n) r ∈ [rj−1, rj]
r = αjξ + βj, ξ ∈ [−1, 1], αj = rj − rj−1
2
βj = rj + rj−1
.
2
rn ≤ r ≤ ∞
u = 1/r
u = 1
r = αn+1(1 − ξ), ξ ∈ [−1, 1], αn+1 = 1
2rn
.

[0, 1] [−1, 1]
u
(t, r, θ, ϕ)

V
R ∀(θ, ϕ)
∀t ≥ 0, ∀r < R,
∂ 2 V
= ∆V,
∂t2 ∀t ≥ 0, ∀r ≤ R, ∇ · V = 0,
∀r ≤ R,
∀r ≤ R,
∀t ≥ 0,
V(0, r, θ, ϕ) = v0(r, θ, ϕ),
∂V
∂t = w0(r, θ, ϕ),
t=0
V(t, R, θ, ϕ) = b0(t, θ, ϕ).
v0 (w0, b0)
∆
(∆V) r = ∂2V r 4 ∂V
+
∂r2 r
r r 2V 1
+ +
∂r r2 r2 ∆θϕV r − 2
r
(∆V) θ = ∂2V θ 2 ∂V
+
∂r2 r
θ 1
+
∂r r2
∆θϕV θ r θ ∂V V
+ 2 −
∂θ sin2 cos θ
− 2
θ sin2 ∂V
θ
ϕ
,
∂ϕ
(∆V) ϕ = ∂2V ϕ 2 ∂V
+
∂r2 r
ϕ 1
+
∂r r2
∆θϕV ϕ + 2 ∂V
sin θ
r cos θ
+ 2
∂ϕ sin2 ∂V
θ
θ ϕ V
−
∂ϕ sin2
,
θ
Θ,
Θ
Θ ≡ ∇ · V =
∂V r
∂r
+ 2V r
r
+ 1
r
∂V θ
∂θ
θ V 1
+ +
tan θ sin θ
∂V ϕ
∂ϕ
.
∇ · v0 = ∇ · w0 = 0.
V
V(t, r, θ, ϕ) = ℓm E
E (t, r)Yℓm + B ℓm (t, r)Y B ℓm + R ℓm (t, r)Y R
ℓm ,
ℓ,m
∀ℓ > 0, ∀ − ℓ ≤ m ≤ ℓ, Y E ℓm = r ∇Y m
ℓ ,
∀ℓ > 0, ∀ − ℓ ≤ m ≤ ℓ, Y
B ℓm = er × Y E ℓm,
∀ℓ ≥ 0, ∀ − ℓ ≤ m ≤ ℓ, Y
R ℓm = Y m
ℓ er;

∇ Y E ℓm Y B ℓm
Y R ℓm
V
V η (t, r, θ, ϕ) =
ℓ,m
V µ (t, r, θ, ϕ) =
ℓ,m
ℓ,m
E ℓm Y m
ℓ ,
B ℓm Y m
ℓ ,
R ℓm Y m
ℓ = V r .
(V η , V µ )
∆θϕV η =
∆θϕV µ =
V θ η ∂V 1
= −
∂θ sin θ
V ϕ = 1 ∂V
sin θ
η µ ∂V
+
∂ϕ ∂θ ;
θ ∂V
∂θ
ϕ ∂V
∂θ
θ V 1
+ +
tan θ sin θ
∂V µ
,
∂ϕ
∂V ϕ
∂ϕ
ϕ V 1
+ −
tan θ sin θ
,
∂V θ
.
∂ϕ
θ ∈ [0, π], ϕ ∈ [0, 2π[
V θ , V ϕ
(V η , V µ ) V η V µ
ℓ = 0
(V r , V η , V µ )
W
Θ =
∂W r
∂r
+ 2W r
r
+ 1
r ∆θϕW η ;
W η
R 3
W = ∇φ + D0,

∇ · 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
.

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

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

∂ 2 A
= ∆A,
∂t2
∂2B C
= ∆B − , 2 2
∂t 2r
∂2C 2C
= ∆C +
∂t2 r
r2 .
B C
A B C
A(t, r, θ, ϕ) =
ℓ,m
B(t, r, θ, ϕ) =
ℓ,m
C(t, r, θ, ϕ) =
ℓ,m
2 + 8∆θϕB
A ℓm (t, r)Y m
ℓ (θ, ϕ),
B ℓm (t, r)Y m
ℓ (θ, ϕ),
C ℓm (t, r)Y m
ℓ (θ, ϕ).
˜ B Ĉ
˜B(t, r, θ, ϕ) =
2B
ℓ,m
ℓm (t, r) + Cℓm
(t, r)
Y
2(ℓ + 1)
m
ℓ (θ, ϕ),
Ĉ(t, r, θ, ϕ) = ℓm ℓm m
C (t, r) − 4ℓB (t, r) Yℓ (θ, ϕ).
ℓ,m
∂2B˜ ∂t2 = ˜ ∆ ˜ B,
∂2Ĉ ∂t2 = ˆ ∆Ĉ;
f(r, θ, ϕ) =
(ℓ,m) f ℓm (r)Y m
ℓ (θ, ϕ)
˜∆f = ∂2f 2 ∂f
+
∂r2 r ∂r
ˆ∆f = ∂2f 2 ∂f
+
∂r2 r ∂r
1
+
r2
+ 1
r 2
ℓm
ℓm
−ℓ(ℓ − 1)f ℓm Y m
ℓ
−(ℓ + 1)(ℓ + 2)f ℓm Y m
ℓ
,
.
1 ℓ
∆θϕ ˜ ∆ ˆ ∆

h ij
˜γ ij = f ij + h ij 1
h
h
B C
C B
∂C
∂r
+ 2C
r
+ 2∆θϕ
∂B
∂r
+ 3B
r
C
− = ∆h.
4r
h B
r C 0
˜ B Ĉ
h
A ˜ B
h = 0
A(t, r, θ, ϕ) ˜ B(t, r, θ, ϕ)
h t
A ˜ B
A H µ = 0
h µ h X h
∂B ℓm
2
∂r
∂B ℓm
1
∂r
Bℓm 1
−
r = Aℓm ,
3Bℓm 1
+
r
2 − ℓ(ℓ + 1)Bℓm 2
+
r
= 0.
˜ B
Hr = Hη = 0

h rr h η h W
(ℓ + 2) ∂Eℓm 2
∂r
∂L ℓm
0
∂r
∂E ℓm
1
∂r
3Lℓm 0
+
r
3Eℓm 1
+
r
2
+ ℓ(ℓ + 2)Eℓm
r
ℓ(ℓ + 1)Eℓm 1
−
r
Lℓm 0
−
2r
2Eℓm 1
−
r −
1
2(ℓ + 1)
∂L ℓm
0
∂r
− ℓ + 4
ℓ + 1
L ℓm
0
2r = ˜ B ℓm ,
= 0,
2 − ℓ(ℓ + 1)Eℓm 2
+
r
= 0.
h
rh X A
h µ h X
ℓ r
r ℓ−2 , 1
rℓ+3 1
r ℓ+1
h rr , h η
h W h τ h rr
h
A V µ
V r V η
V µ

A
V r V η
A
r = R
∂V η
∂r
A
1
r ∆θϕA = − ∂2V r
∂t
∂A A
+
∂r r = ∂2V η
∂t
2 ,
, .
2
A
A
V µ V r V µ V η
V µ V η
∀t ≥ 0, V µ (t, R, θ, ϕ) = b µ
0(t, θ, ϕ),
∂V r (t, R, θ, ϕ)
∂r
V η (t, R, θ, ϕ) = b η
0(t, θ, ϕ),
+ 2
r V r (t, R, θ, ϕ) = − 1
r
∆θϕb η
0(t, θ, ϕ).
V i

A
h µ h X
˜ B h rr h W h η
h τ
A ˜ B
h µ h X
ℓ ≥ 2
r ℓ−2
h rr h η h W
A ˜ B
(∆θϕ + 2)A = − ∂2 h µ
∂A
∂r
+ 2A
r = ∂2hX ∂t
,
∂t2
, . 2
h µ h X

∂ 2 L ℓm
0
∂ 2 E ℓm
1
∂t 2
∂ 2 E ℓm
2
∂t 2
∂ 2 (L ℓm
0 + T ℓm
0 )
∂t 2
∂t2
1 (ℓ + 1)(ℓ + 2)
= −
Ĉ
(2ℓ + 1)r 2
ℓm − ℓ(ℓ + 1)(ℓ − 1) ˜ B ℓm
1
=
(ℓ + 1)(ℓ − 1)
(2ℓ + 1)r
˜ B ℓm ℓ + 2
+ Ĉ
2
ℓm
1 (ℓ + 1) ∂
=
2ℓ + 1 2
˜ Bℓm 1 ∂
−
∂r 4
Ĉℓm (ℓ + 1)(ℓ + 2) ˜B
−
∂r 2
ℓm ℓ − 3 Ĉ
−
r 4
ℓm
r
1 (ℓ + 1)(ℓ + 2) ∂
=
2ℓ + 1 2
Ĉℓm
∂r − ℓ(ℓ + 1)(ℓ + 2)∂ ˜ Bℓm ∂r + ℓ(ℓ + 1)(ℓ − 1)2 ˜ Bℓm r
+ 1
Ĉℓm
(ℓ + 1) [ℓ(ℓ − 3) + ℓ + 4] .
2 r
∂ 2 E ℓm
2
∂t 2
=
1
(ℓ + 1)
2ℓ(ℓ + 1)(2ℓ + 1)
∂Ĉℓm
∂r + 2ℓ(ℓ + 1)∂ ˜ Bℓm ∂r
− ℓ(ℓ + 1)(ℓ − 3) ˜ Bℓm
r
(ℓ + 1)(ℓ + 4) Ĉ
+
2
ℓm
r
.
˜ B
˜ B
Ĉ
ℓ
˜B ℓm
r
=
(ℓ + 1)(ℓ − 1)
∂2Lℓm 0
∂t2 + (ℓ + 1)∂2 Eℓm 1
∂t2
,
˜B
h µ h X
A
˜ B
h rr h η h W
β ij
0
H

= R > 0
∀(θ, ϕ)
∀t ≥ 0, h ij (t, R, θ, ϕ) = ζ ij
0 (t, θ, ϕ).
A ˜ B
h µ h X
r = 0
h rr h η h W
R 3
h µ
h X h rr h η h W
ϕ θ
P m ℓ (cosθ)
∆θϕ

Nr R = 6, dt =
0, 00032, Nθ = 17, Nϕ = 4
∂2φ = ∆φ ⇐⇒
∂t2 ⎧
⎪⎨
∂φ
= ψ,
∂t
⎪⎩ ∂ψ
= ∆φ.
∂t
V i
0 (r, θ, ϕ) z =
r cos(θ)
V x
0 = −V y
0 = cos(z),
V i
0
V x (t, r, θ, ϕ) = −V y (t, r, θ, ϕ) = cos(t) cos(z),

dt
R = 6, Nr = 17, Nθ = 17, Nϕ = 4
A V µ
bi 0(t, θ, ϕ)
(br 0, b η
0, b µ
A µ
V r = b r 0
(Nr, Nθ, Nϕ)
t ∈ [0, 2π]
V i
ϕ
ϕ
Nr
Nθ Nr
dt
0)

Nr
R = 6, dt = 0.00032, Nθ = 17, Nϕ = 4
O(dt 3 )
α ij
0 (r, θ, ϕ)
α xx
0 = −α yy
0 = cos(z),
α ij
0
γ ij
0 = 0
h xx (t, r, θ, ϕ) = −h yy (t, r, θ, ϕ) = cos(t) cos(z),
A ˜ B
β ij
0 (t, θ, ϕ)
(β rr
0 , β η
0, β µ
0 )
A ˜ B
β ij
0

dt R = 6, Nr =
17, Nθ = 17, Nϕ = 4
t ∈ [0, 2π]
Nr
Nθ Nr
O(dt 3 )

A, A, ˜ B

A, A, ˜ B

e
Rg M
Rg = 2GM
c 2 .
M R < Rg

p (M, ηµν) D + (p)
p
( ˜ M, ˜gµν) (M, gµν)
(M, gµν)
˜T = D − (I + ) ⊂ ˜ M
T ⊂ M
Σt
T = M
M
B = M\T.

B
B
M
gµν
ds 2 = gµνdx µ dx ν
= − 1 − 2M
dt
r
2
+ 1 − 2M
−1 dr
r
2 +r 2 (dθ 2 +sin 2 θdϕ 2 ).
gµν
∂ µ
∂t
M
M
r = 0 r = 2M g00
g11
r > 2M r < 2M

∗ = r + 2M r
2M
− 1 .
u = t + r ∗ v = t − r ∗ .
U = e −u/4M V = e v/4M .
T =
U + V
2
R =
U − V
2
.
r
1/2 T = ± − 1 e
2M r/4M
t
,
4M
r
1/2 R = ± − 1 e
2M r/4M
t
(r ≥ 2M),
4M
T = ± 1 − r
1/2 e
2M
r/4M
t
,
4M
R = ± 1 − r
1/2 e
2M
r/4M
t
(r ≤ 2M),
4M

(R, T )
(r, t)
ds 2 = 32M 3 e r/2M
r
(−dT 2 + dR 2 ) + r 2 (dθ 2 + 2 θdϕ 2 ).
r = 2M
(r, t, θ, ϕ) r = 0
R µνρσRµνρσ
M 2
r6
(t, r) ↔ (T, R)
(t, r)
(T, R)
(M∗ , g∗ µν)
r = 2M
M
M
(R, T )

T = 0
r > 2M
(M, gµν)
r = 2M
r < 2M
M
r = 0
r
r = 2M
dτ 2 = −ds 2
(M, gµν)
r = 0 r = 2M
R = T = 0
T = 0
r = 2M

ds 2 = −
+
1 − 2M
1
+ O
r r2
dt 2 2 4J θ 1
− + O
r r2
dϕdt
1 + 2M
1
+ O
r r2
[dr 2 + r 2 (dθ 2 + 2 θdϕ 2 )],
J
ds 2 = ρ 2
2 dr
+ dθ2
∆
+ (r 2 + a 2 ) 2 θdϕ 2 − dt 2 + 2Mr
ρ 2 (a sin2 θdϕ − dt) 2 .
ρ 2 = r 2 + a 2 2 θ ∆ = r 2 − 2Mr + a 2 .
ϕ t
∂ µ
µ
∂
∂t
∂ϕ
M a
a 0
a
a
a = J
M
a 2 > M 2
ρ 2 = r 2 + a 2 2 θ = 0,
r = 0 θ = π/2
r = 0

(x, y, z, t)
r(xdx + ydy) − a(xdy − ydx)
ds 2 = dx 2 +dy 2 +dz 2 −dt 2 + 2Mr3
r 4 + a 2 z 2
r 2 + a 2
+ zdz
r
2 + dt .
r
(x, y, z)
r 4 − (x 2 + y 2 + z 2 − a 2 )r 2 − a 2 z 2 = 0.
r
r = cste
r
x 2 + y 2 = a 2 z = 0.
RµνρσR µνρσ
r < 0
a 2 < M 2
∆
r± = M ± √ M 2 − a 2 .
r = 2M
(r+ = cste
r− = cste)
a 2 > M 2 a 2 < M 2
a 2 > M 2

+ r−
∂ µ ∂t
I +
a 2 ≤ M 2
r
r = r+
r < 2M r ≤ r+
r r = r−
r ≤ r−

∂ µ
∂t
r = M + √ M 2 − a 2 2 θ.
r = r+
∂ µ
∂t
r < M + √ M 2 − a 2 2 θ
u µ ∇ µ t
Ω = dϕ
dt
= − gθϕ
gϕϕ
=
a(r2 + a2 − ∆)
(r2 + a2 ) 2 − ∆a22 .
θ
r → r+
ΩH =
a
r2 ,
+ + a2
ΩH a M
ΩH
r = r+
χ µ =
∂
∂t
µ
+ ΩH
∂
∂ϕ
µ
χ µ
χ µ
r = r+
r = 2M
∂
∂t
µ

M a
r > 2M

= 2M
r = 0
ρ = c5
G 2 = 5 × 1096 . −3 ,
r > 2M

ξ µ
θ (ξ) = ∇ µ ξ ν hµν.
hµν
ξ µ
hµν = gµν + ξµξν.
θ (ξ) ξ µ
ξ µ p ∈ M
ξ µ ξ µ
hµν

θ (ξ)
(M, ηµν)
rs = 1 ℓ µ
k µ
θ (ℓ) = 2
r > 0 θ(k) = − 2
< 0
r
(M, gµν)
ℓ µ k µ
θ (ℓ) < 0 θ (k) < 0.
r < 2M
r < 2M
r− < r < r+
θ (ℓ) = 0
r = 2M r = r+
Tµν
u µ
[Tµν − 1
2 T gµν]u µ u ν ≥ 0.
u µ
(M, gµν)

dA
(Σ, hab, Kab) Σ
D + (Σ)
r < 2M
(Σ ′ , h ′ ab , K′ ab ) (Σ, hab, Kab)
(Σ ′ , h ′ ab , K′ ab )

a 2 > M 2
(Σ, hab, Kab)
(M, gµν)

a2 > M 2
(M, gµν)
T ⊂ M T ⊂ B
a2 < M 2
Σ1 Σ2 Σ2 ∈ D + (Σ1)
B˙ ∩ Σ2 ˙ B ∩ Σ1

∂ µ
∂t
∂ µ
µ
∂ + Ω
∂t
∂ϕ
t

Kij = 0
γij ˜ = 0
∆ψ = 0,
ψ
ds 2
= 1 + α
r
ψ = 1 + α
r
α ∈ R
4 [dr 2 + r 2 (dθ 2 + 2 θdϕ 2 )].
{ri}i∈[1,N]
ψ = 1 +
N
i=1
mi
.
2|r − ri|
{ri}

mi Mi
N
Mi = mi 1 +
i=j
mj
2|ri − rj| .
K = 0
Kij
X i = − 1
4r [7P i n i njP j ] + 1
r 2 ɛijk njSk.
n i
ɛ ijk P i

S i
Ãij = 3
2r 2 [niPj + njPi + nkP k (ninj − δij)] − 3
r 3 (ɛilknj + ɛjlkni)n l S k ,
P i S i
Kij
{ri}
ψ = ψBL + u =
N
i=1
mi
+ u,
2|r − ri|
u Σ R 3 ψBL
u
∆u + 1
8ψ7 Ã
BL
ij Ãij
1 + u
ψBL
−7 = 0.
R 3 C 2 1
u

X i ψ

X i ψ

Σ
( ∂
∂ϕ )µ
H Σ S
l µ = ( ∂
∂t )µ + Ω( ∂
∂ϕ )µ
k µ S
kµl µ = −1
κ = −∇µlνk ν l µ .
δM = κ
8π δA + ΩδJ,
A t J
Ω

St H l µ k µ
St
qµν gµν
St v µ St
Θ (v)
µν = ∇αvβq α µq β ν = 1
2 qµ αq ν βLvqµν.
qµν
v
θ (v) σ (l)
µν
v
θ (v) = q µ Θ (v)
µν σ (l)
µν = Θ (v)
µν − 1
2 θ(v) qµν.
θ (v) ɛ S µν qµν St
Lvɛ S µν = θ (v) ɛ S µν,
σ (l)
µν
l H
Ω (l) µ = −kα∇βl α q β µ.
q α Ω
µLl
(l) α
8π + θ(l) Ω(l) µ
8π = Tαβl α q β µ + 1 (2)
Dµ
8π
κ + θ(l)
−
2
1 (2)
Dασ
8π
(l)α µ,
κ
(2) Dµ qµν St
q α µLlπ (l)
α + θ (l) π (l)
µ = − (2) DµP + 2µ (2) Dασ (l)α µ + ζ (2) Dµθ (l) + f.
Πµ = − 1
8π Ω(l) µ
κ
8π
= P
Tαβlαq β µ = fµ

θ (l) µ = 1
16π
ζ = − 1
16π
I +
ζ
Σt

θ (l) µ = 1
16π
ζ = − 1
16π
I +
ζ
Σt

(M, gµν)
S 2 × R l µ
θ (l) = 0

l µ ∇µθ (l) − κθ (l) − 1
2 θ(l) + σ (l)
µνσ (l)µν + Rµνl µ l ν .
σ (l)
µνσ lµν + Rµνl µ l ν = 0.
σ (l) µν = Θ (l)
µν = 0
Llqµν = 0,
qµν H (0, +, +)
qµν St H
Σt (M, gµν)
St
ˆ ∇ H
[l]
H
∀V µ ∈ H (Ll ˆ ∇ − ˆ ∇Ll)V µ = 0.
H
κ
(H, qµν, ˆ ∇)
l µ
SO(2)
St
SO(3)
St
qab St
SO(2) φ µ

2π
µ
∂
∂ϕ
St
JH =
St
Ωaφ a ɛ S ,
ɛ S = √ qd 2 y
H φ µ
St
t µ t µ
EH
m µ = t µ + Ωφ µ ∈ [l],
Ω t µ [l]
l
t µ EH AH
JH
δEH = κH(AH, JH)
δAH + ΩH(AH, JH)δJH,
8π
κH ΩH
EH
κH ΩH [l]
m µ
RH
m µ
MH(RH, JH) = MKerr(RH, JH) =
κH(RH, JH) = κKerr(RH, JH) =
ΩH(RH, JH) = ΩKerr(RH, JH) =
R 2 H = AH
,
4π
R 4 H + 4J 2 H
2RH
R4 H − 4J 2 H
2R3
4
H RH + 4J 2 H
2JH
4
RH RH + 4J 2 H
,
,
.

κ
Ω
q α µLlΩ (l) α = (2) Dµκ,
St
l k
θ (l) = 0 θ (k) < 0 Lkθ (l) < 0.
Lkθ (l) < 0
θ (l)
θ (k) < 0
h µ

St
θ (l) = 0 θ (k) < 0.
φ µ
St H
J (φ)
S
= 1
8π
Kabφ
S
a h b ɛ S ,
Kab Σt St h b
St H
φ µ

φ µ
St Σt
φ µ S1(t1) S2(t1)
J (φ)
S2
− J (φ)
S1
= J (φ)
M
(φ)
+ J G ,
J (φ)
M J (φ)
G
St
κS(t) = κKerr(RS(t), J (φ)
S (t)) ΩS(t) = ΩKerr(RS(t), J (φ)
S (t)).
t
M(t) = MKerr(RS(t), J (φ)
S (t)) =
R 4 S (t) + 4J 2 S (t)
2RS(t)
AS(t)
∂2AS + ¯κ∂AS
∂t2 ∂t
= B(t),
B(t) ¯κ
St κ(t)
∂ 2 AS
− ¯κ∂AS
∂t2 ∂t
= C(t).

¯κ
h µ = l µ − Ck µ
H Lht = 1 C m µ = l µ + Ck µ
H
q α µLhΠ (l)
α +θ (h) Π (l)
µ = −Tαβm α q β µ+ θ(k) (2)
DαC+
8π
1 (2)
Dµ
8π
−κ + θ(h)
+
2
1 (2)
Dασ
8π
(m)α
µ.
θ (l)
l
f = −Tαβmαq β µ +θ (k) /8π (2) DαC
1 (2)
Dµθ 16π
(h) = ζ (2) Dµθ (h)
Σt

φ µ
φ µ
φ µ

(Σt, γij, Kij)
Σt
St
s i
St
q i j = γ i j − s i sj.
l
Θij = N(Dmsn − Kmn)q m iq n j,
θ (l) = N(Dis i + Kijs i s j − K).
St ˜s i = ψ 2 s i θ (l) = 0
4˜s i ˜ Diψ + ˜ Di˜s i + ψ −2 Kij˜s i ˜s j − ψ 2 K = 0.

ψ
∂
∂t
µ = Nn µ + β µ
St
s i
˜s i
β i = bs i − V i = ˜ b˜s i − V i ,
St
˜
N
b = .
ψ2
σ (l)
ab
St V i
∂
∂t ˜γij = 0,
St
qbc (2) DaV c + qac (2) DbV c − qab (2) DcV c = 0,
(2) D St
V i
κ
κ = s i DiN − NKijs i s j + L(l)(N),
St

ψ
∂
∂t
µ = Nn µ + β µ
St
s i
˜s i
β i = bs i − V i = ˜ b˜s i − V i ,
St
˜
N
b = .
ψ2
σ (l)
ab
St V i
∂
∂t ˜γij = 0,
St
qbc (2) DaV c + qac (2) DbV c − qab (2) DcV c = 0,
(2) D St
V i
κ
κ = s i DiN − NKijs i s j + L(l)(N),
St

ψ
∂
∂t
µ = Nn µ + β µ
St
s i
˜s i
β i = bs i − V i = ˜ b˜s i − V i ,
St
˜
N
b = .
ψ2
σ (l)
ab
St V i
∂
∂t ˜γij = 0,
St
qbc (2) DaV c + qac (2) DbV c − qab (2) DcV c = 0,
(2) D St
V i
κ
κ = s i DiN − NKijs i s j + L(l)(N),
St

ψ
∂
∂t
µ = Nn µ + β µ
St
s i
˜s i
β i = bs i − V i = ˜ b˜s i − V i ,
St
˜
N
b = .
ψ2
σ (l)
ab
St V i
∂
∂t ˜γij = 0,
St
qbc (2) DaV c + qac (2) DbV c − qab (2) DcV c = 0,
(2) D St
V i
κ
κ = s i DiN − NKijs i s j + L(l)(N),
St

H
(0, +, +)
ℓ µ
S
H t
qab S ℓ µ
k µ
S
ℓ µ
k µ S
θ (ℓ) = 0 θ (k) ≤ 0
σ (ℓ)
ab S
H

˜γij = ψ −4 γij; ψ =
1
(γ) 12
,
(f)
fij
h ij
˜γ ij = f ij + h ij .
 ij = ψ 10 (K ij − 1
3 Kγij );
Dk˜γ ki = Dkh ki = 0,
ψ Nψ β i h ij
∆ψ = − 1
∆(Nψ) = Nψ
8 ÂijÂijψ −7 + 1
7
8 ÂijÂijψ −8 + ˜ R
8 − hijDiDj(Nψ) 8 ψ ˜ R∗ − h ij DiDjψ,
∆β i + 1
3 Di Djβ j = 2ψ 6 A ij DjN − 12Nψ 6 A ij DjΦ − 2N∆ i klψ 6 A kl
,
− h kl DkDlβ i − 1
3 hikDkDlβ l ,
∂2hij 2 N
−
∂t2 ψ4 ∆hij ∂h
− 2Lβ
ij
∂t + LβLβh ij = S ij
hij(N, ψ, β i , Âij , h ij ).
S ij
hij
Âij =
ψ−10Kij
Âij
 ij = ψ6
2N
(Lβ) ij + ∂hij
∂t − Lβh ij − 2
3 Dkβ k h ij
,

L fij
h ij γij Kij
( ∂
∂t )i
h ij
∆h ij − ψ4
N 2 LβLβh ij = S ij
2 (h ij , N, ψ, β, A ij ).
γij ˜

h ij
h ij
Dih ij = 0,
˜γ ij
(˜γ ij ) = (h ij + f ij ) = 1.
Dih ij
T
h ij = D i W j + D j W i + h ij
T ,
= 0
h ij
T
h ij
T
A ˜ B
A ˜ B
∆h ij = S ij
∆A = AS,
˜∆ ˜ B = ˜ BS,

AS BS S ij
˜ ∆
∆A − ψ4
N 2 LβLβA = AS(h ij , N, ψ, β, A ij ),
˜∆ ˜ B − ψ4
N 2 LβLβ ˜ B = ˜ BS(h ij , N, ψ, β, A ij ),
LβA = βiDiA
hij
A ˜ B
A ˜ B
hij
˜ C
˜ C
˜ B

h ij
θ (ℓ) = 0 σab = 0
rH
s i
Σt
4˜s i ˜ Di ln(ψ) + ˜ Di˜s i + ψ −2 Kij˜s i ˜s j = 0,
˜s i = ψ 2 s i ˜ Di
ψ
ψ
β i = ˜ b˜s i − V i = bs i − V i V i
S
˜ b = N
ψ 2
σ ℓ ab V i

(θ, ϕ)
V i
V i = Ω
∂
∂ϕ
i
,
Ω ϕ
a
M Ω
Ω
a
a
= JK
M 2 ,
ADM
MADM
MADM JK
i ∂
∂ϕ
NH
hij h ij

A
∆A − ψ4
N 2 LβLβA = AS(h ij , N, ψ, β, Âij ).
A
A
ψ 4
N 2 LβLβA = ψ4
N 2 (βr ) 2 ∂ 2 r A + ψ4
N 2 (LβLβA) ∗ ;
A =
AlmYlm(θ, ϕ),
(l,m)
Ylm
(l, m)
∆θϕYlm = −l(l + 1)Ylm
β r = ˜ b
√˜γ rr =
N
Ψ2√ .
˜γ rr
l = 0 (β r )(l=0) =
( N
ψ2 )(l=0)
l = 0
rH
ψ 4
N 2 (βr ) 2
∂
(l=0)
2 r A = [1 + α(r − rH) +
δ(r − rH) 2 + O(r − rH) 3 ]∂ 2 r A,
α δ
N ψ β r

A l
−α(r − rH) − δ(r − rH) 2 ∂ 2
−
l(l + 1)
r 2
∂r 2 Alm + 2
r
∂
∂r Alm
Alm = AS + ψ4
∗∗
(LβLβA)
N 2 lm,
∗∗
ψ4
N 2 (β r ) 2
Qαδ α δ
Qαδ = −α(r − rH) − δ(r − rH) 2 ∂ 2
+ 2
r
∂
∂r
∂r 2
− l(l + 1)
r 2 I,
R 3
Qαδ
Qαδf = Sf
4
˜Q
ψ
=
N 2 (βr ) 2
(l=0)
∂2 2
+
∂r2 r
∂
∂r
− l(l + 1)
r 2 I
˜ B
˜ ∆
˜ B
h ij
h rr h η h µ h ij

∆h µ + 2 ∂h
r
µ
∂r
∆h η + 2 ∂h
r
η 2hη
+
∂r r
∆h rr − 6hrr 4
−
r2 2hµ ψ4
+ −
r2 N 2
r
N 2
2hrr ψ4
+ − 2 2
r2 ∆θϕh η + 2h
r
LβLβh ij µ µ ij
= S2
LβLβh ijη η ij
= S2 N 2
2 − ψ4
LβLβh ij rr
= S rr
2 ,
(µ, η, rr)
A
Qαδ
h µ
Qαδ(h µ ) + 2 ∂h
r
µ 2hµ
+
∂r r2 = S ijµ
ψ
2 + 4
N 2
LβLβh ij µ(∗∗)
,
Qαδ r = rH
h µ
4 ∂h
r
µ
∂r + (∆θϕ + 2)
r2 h µ = S ijµ
ψ
2 + 4
N 2
LβLβh ij µ(∗∗)
.
h rr h η
h ij
h ij
S ij
2
A
˜ B
˜ B

h rr h η h µ
h ij
h ij
h ij
A ˜ B
Qαδ
h ij
Nr × Nθ × Nϕ = 33 × 17 × 1
Nϕ = 4
rH
r = rH = 1

h ij
h ij
rH
ψ V i
˜ b
Ω
0 ≤ NH ≤ 1
Ω
0 0.22
h ij = 0
Ω
Ω
NH

MHΩ
Nr = 33
Nθ = 17 Nϕ = 1 NH = 0.55
a
M
NH Ω a
M
NH = 0.8
NH
NH Ω
a 0.85
M
a
M

MHΩ
JK a
M
NH = 0.55
∂ i
∂t

MH =
R 4 H + 4J 2 H
2RH
.
RH H MH
JH JK
MADM
MH
10 −7
a
M
a
M
ɛA =
A
8π(M 2 ADM + M 4 ADM − J 2 ≤ 1.
K )
A JK
MADM
a
M
ɛA
JK−JH
JK

MH
JH MADM

ϕ a
i ∂ ∂ϕ
(ζ, ϕ)
q H ab = R 2 −1
H f(ζ) DaζDbζ + f(ζ)DaϕDbϕ ,
RH f(ζ)
ϕa
ϕ
( ∂
∂ϕ )i ζ
Daζ = 1
R2 ɛbaϕ
H
b .
H ζd2V = 0
ζ = cos θ
n
Mn = Rn H MH
8π
Jn = Rn−1
H
8π
S
S
{RPn(ζ)}d 2 V,
P ′
n(ζ)Kabs a ϕ b d 2 V.

1 − ɛA
M2
M 3
J3
M 4

M0 = MH J1 = JH JH
M0 J1
β i ψ Nψ
h ij
h ij
h ij

˜γ ij
qab = ω 2 fab,
fab
Di˜γ ij = V i V i
V i = 0

˜γ ij
qab = ω 2 fab,
fab
Di˜γ ij = V i V i
V i = 0

h ij
A ˜ B
h ij A ˜ B
h ij
H i = Dih ij ,
H i = 0
H i
h ij
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
,
.
A ˜ B
h ij
⎧
⎪⎨
∂X hµ
− = A,
∂r r
⎪⎩ ∂h µ 3hµ 1
+ +
∂r r r (∆θϕ + 2) h X = 0;
⎧
˜B
⎪⎨
⎪⎩
lm = B lm + Clm
2(l + 1) ,
∂hrr 3hrr 1
+ +
∂r r r (∆θϕh η − h) = 0,
∂hη
3hη 1
+ + (∆θϕ + 2) h
∂r r r
W
h − hrr
+ = 0.
2
B C
hij h
R3
HIJ A ˜ B

h ij
A ˜ B h h ij
H IJ A ˜ B

H IJ A ˜ B

H IJ A ˜ B

AHE = 8π(M 2
ADM + M 4 ADM − J 2 K ),
MADM JK
AHE
AHA
AHA ≤ 16πM 2 ADM, ,
AHA

MH
Kij = 0
S0
H
S
MHawking(S) =
AS
1 −
16π
1
H
16π S
2

MHawking(S2) ≥ MHawking(S1) S2 S1
MHawking(S0) =
AS0
,
16π
S0
Kij = 0
JK
AHA ≤ 8π(M 2
ADM + M 4 ADM − J 2 K ).

J 2 K ≤ M 4 ADM.
J 2 K = M 4 ADM
J 2 K /M 4 K
J 2 K /M 4 K 1
|JK| ≤ 1
8π AHA.
JK

ɛ P := A
16πM 2
ADM
ɛ A :=
≤ 1 , ɛ D := |J|
A
M 2
ADM
8π|J|
:= A
8π(M 2
ADM +√M 4
ADM −J2 ) ≤ 1 , ɛP A
Σ0
φi
˜γij
≤ 1
≤ 1 .
S i = 1
φ ɛijk φj ˜ Dkω Lφω = 0,
φ φ iɛ ijk γij ˜ Di
˜γij = ψ −4 γij ψ
ω
ω ˜γij
S i φ i
A ij = 2
φ [Si φ j + S j φ i ]
˜DiA ij = 0.
K ij Σ0
K ij = ψ −10 A ij .

ψ
˜Dk ˜ D k ψ = ψ 1
R −
8 8 Ãkl Ãklψ −7 ,
˜γij
(γij, Kij)
(r, θ, ϕ) φi i ∂ = ∂ϕ
JK
S ni
JK = 1
8π S→∞
Sin i ɛ S = 1
(ω(r, θ = π) − ω(r, θ = 0)).
4
r = rH
JK
ω = ωBY (JK) = JK( 3 θ − 3θ) ˜γij = fij,
fij
ψ
r = rH
ω
ω = ωKerr(JK, M) = ωBY (JK) − ωcorr(JK, M)
ωcorr(JK, M) = Ma3 4 θθ
ρ 2 ,
ρ2
M
a = JK/M
˜γijKerr(JK)

R = r + M 2 − a2 + M,
4r
R r
ψ
Σ0(γij, Kij) M
JK
˜γijKerr(JK)
λ
ω(JK, MKerr, λ) = ωBY (JK) − λωcorr(JK, MKerr),
λ
MKerr
JK r = rH
[γij(λ, JK), Kij(λ, JK)]
JK λ λ =
1 λ = 0
˜γij
JK λ
˜γij
r = rH
r = rH

H r = rH
Ω NH
JK MADM
˜γ ij = f ij + h ij
Dih ij = 0 (˜γij) = 1.
AKerr(NH, Ω) ˜ BKerr(NH, Ω)
h ij
χ
Aχ(NH, Ω) = χA(NH, Ω) ˜ Bχ(NH, Ω) = χ ˜ B(NH, Ω).
χ
h ij
χ (Aχ, ˜ Bχ)
hχ
h ij
χ Aχ ˜ Bχ
(µχ, ηχ, h rr
χ ) h ij
χ Aχ ˜ Bχ
h ij χ
(NH, Ω, χ) h ij
χ
˜γ ij
(NH, Ω)
χ = 1
χ = 0
NH Ω

H
(1−ɛA) r = 1
JK
(1 − ɛA)
r = 1 JK
λ ɛA = 1
λ = 1 λ
λJ > 1 JK (1 − ɛA)
JK λ > λJ ɛA
1 JK
r = 1
r = 1
r = 1 λ JK = 5

o = 4300rH λ = 10 14
ɛA ≤ 1
λ (1 − ɛA)
r = 1
ɛA ɛP A JK λ
ɛA
λ = 1
ɛA = 1
ɛA = 1
ɛP A ≤ 1
ɛP A
λ = 1 JK JK
λ = 1
a/M → 1 ɛP A
ɛD
ɛP A JK
λ = 1 JK
λ
(ɛA − 1)
λ → ∞ ɛA → 1
λ JK

ɛA ɛP A
JK λ
λ = 1
σ ab
ℓ
S
σ 2
ℓ =
S
|σ ab
ℓ σℓab|ɛ S .
λ
(Mn, Jn)
M2 M4 M6 M8 J3
M 3 M 5 M 7 M 9 M 4
ɛA ɛP A
8π|JK| ≤ AHA ≤ 8π
M 2 ADM +
M 4 ADM − J 2 K
.
ɛD
ɛP A ≤ 1
8π |JK| + M 4 ADM − J 2
K ≤ AHA

log(X)
1
0
-1
-2
-3
-4
-5
J=5
-6
4 6 8 10 12 14
log(lambda)
1 - epsA
Max shear
delta_M_2
delta_M_4
delta_M_6
delta_M_8
delta_J_3
λ → ∞
(1 − ɛA) σ2
M2 M4 M6 M8 J3
M 3 M 5 M 7 M 9 M 4
ɛA
ɛA
rH
(1−ɛA) (NH, Ω)
Ω
ɛA = 1

(1 − ɛA)
NH = 0.55
χ = 1
(1 − ɛA)
r = rH
ɛP A
(NH, Ω)
Ω ɛP A
Ω
ɛP A
ɛD
χ

ɛP A
NH = 0.55

ɛP A
NH = 0.55

ɛP A
NH = 0.55

Σt
St
H
h i H
h i = Nn i + bs i ,
n i s i
n i
s i
Σt b β i
H
θ (l) = 0

ψ
h
Lhθ (l) = 0
qab St (2) Da
(2) R
(2) Da (2) D a + 2L a (2) Da −
(2)
R
2 + (2) DaL a + LaL a + 8πTab ˆl aˆ
b
k (b − N) =
1
2 σ(ˆ l)
ij σ(ˆ l)ij
+ 4πTab ˆl aˆ
b
l (b + N),
La = Kbcs b q c a, ˆ l a = n a + s a , ˆ k a = (n a − s a )/2.
Tab St ˆ l a ˆ k a
St
b N b = N
(b − N) (b + N)
h i b ≥ N
ˆ l i
σ (ˆ l)
ij = Tab ˆ l aˆ l b = 0.
b = N
V a
qbc (2) DaV c + qac (2) DbV c − qab (2) DcV c = 2σ h ab.
Lhσ (h)
ab = Sij Sij
V a
N
(b − N)
(b + N)
H

(b − N)
(b − N) = −Cθ ˆ (k) C
h ij
h ij
(b − N)

= rH
V i

t0
t = t0
t ≤ t0
t ≥ t0
V i
t ≥ t0

t = t0
F
S r = rS
1 ∂F
N ∂t = µ(∆θϕF − ∆θϕF∞) − α(F − F∞).
µ α
F∞
F
S N = N0 ∈ R +
∞ ℓ
F0 =
F0ℓmY m
ℓ (θ, ϕ). t = t0
ℓ=0 m=−ℓ
∀(t ≥ t0), ∀(ℓ, m), Fℓm(t) = F∞ℓm + (F0ℓm − F∞ℓm)e −N(α+µℓ(ℓ+1))(t−t0) .
α µ

S t = t0
t = t0
S θ (l)
0 (θ, ϕ)
θ (l)
∞ = 0
1 ∂θ
N
(l) (t)
∂t = µθ∆θϕθ (l) (t) − αθθ (l) (t).
ψ
4˜s i ˜ Diψ + ˜ Di˜s i + ψ −2 Kij˜s i ˜s j − ψ 2 K = θ (l) (t)ψ 2 ,
ψ
ψ ψ
∂ψ
∂t = βi ˜ Diψ + ψ
6 ( ˜ Diβ i − NK),
β i

V i
(b − N)
1 ∂(b − N)
= µ(b−N)∆θϕ(b − N) − α(b−N)(b − N),
N ∂t
b = N
V a
1 ∂V
N
i
∂t = αV ( (2) ∆V i + (2) R i jV j ),
V i
b = 1
s i ˜ Diψ
∂ψ
∂t − V i ˜ Diψ ψ
6 ( ˜ Diβ i − NK)
.
N0(θ, ϕ) N∞
1
N
∂N
∂t = µN∆θϕN − αN(N − N∞).
N0(θ, ϕ)
S t = t0

αN = 1000 µN = 1
S r = rH S
θ (l)
0
(b − N) N V i
Ω
r = rH
rH

ψ θ (l) (b−N) N
S
r = rH
N θ (l)
N (b − N) ψ 2
S
t = 0
S θ (l) = −0, 05
N = 0, 5 (b − N) = 0, 05 V i
Ω = 0.15
θ (l) = 0 (b − N) = 0 N = 0, 45
V i
V i

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1

ψ 2
α (b−N) = 1000 µ (b−N) = 1