Sur la forme de Painlevé d`une métrique à symétrie sphérique

ds2=c2dt2−dr2−r2dω2,
dω2=dθ2+sin2(θ)dφ2
v(r) = −q2M
r
ds2=c2dt2−(dr +s2M
rdt)2−r2dω2.

t
dt dr
r > 2M
r > RoRo
toro
vo
ds2=c2dt2−(dr −qv2
o−2M
ro+2M rdt)2
1 + v2
o−2M
ro−r2dω2.
v2
o−2M
ro
Φ(r) = M/r v(r) = qv2
o−2M
ro+2M rK(r) =
v(r)2−2Φ(r)
ds2=c2dt2−(dr −v(r)dt)2
1 + K(r)−r2dω2.
vo>0
vo≤0v(r) = −qv2
o−2M
ro+2M r
vo
ro−qv2
o−2M
ro+2M rv(r) = −qv2
o+2M
r

p=1
(1+v2
o)dT =√pdt
ro=vo≤0
(ro, vo)
vo≤0
voro
ds2=c2dt2−R2(t)(dx2+f2
k(x)dω2),
fk(x) = x, sin(x) sinh(x)k= 0,1−1
ds2=c2dt2−R2(t)
R2(to) dy2+R2(to)f2
k(y
R(to))dω2!.
r=R(t)fk(y
R(to))
ds2=c2dt2−(dr −r H(t)dt)2
1 + (1 −Ω(t))H2(t)r2−r2dω2.
H(t)>0
r < c
H(t)√Ω(t)
(1 −Ω(t)) H2(t)r2=H2(t)r2−Ω(t)H2(t)r2

1/2mH2r2−1/2mH2Ωr2
v(t, r) = H(t)rΦ(t, r) = Ω(t)H2(t)r2/2K(t, r) = v(t, r)2−2Φ(t, r)
ds2=c2dt2−(dr −v(t, r)dt)2
1 + K(t, r)−r2dω2.
γ(t, r) = dv
dt =∂v ∂t +v∂v ∂r
γ=dv
dt =∂v ∂t +v∂v ∂r =∂Φ ∂r +∂Φ v ∂t .
v(t, r) Φ(t, r)
ds2=c2dt2−(dr −v(t, r)dt)2
1 + v(t, r)2−2Φ(t, r)−r2dω2.
v(t, r) Φ(t, r)
∂v
∂t +v∂v ∂r =∂Φ ∂r +∂Φ v ∂t .
K(t, r) = v(t, r)2−2Φ(t, r)
Rµ ν R Gµ ν
Gµ
νGµ ν;µ
G00=−2Φ (t, r) + r∂
∂r Φ (t, r)
r2

G01= 2 −∂
∂r Φ (t, r)v(t, r)−∂ ∂t Φ (t, r) + ∂ ∂r v(t, r)(v(t, r))2+v(t, r)∂ ∂t v(t, r)
r1+(v(t, r))2−2 Φ (t, r)
G10= 2
∂
∂t Φ (t, r)
r
G11G22=G33
G01= 0
G00=−2Φ+r∂
∂r Φ
r2
G10= 2
∂
∂t Φ
r
G11=−2vr ∂
∂r Φ+vΦ+r∂ ∂t Φ
r2v
G22=G33=−
2v2∂
∂r Φ+r∂2
∂r2Φv2+v∂
∂t Φ+rv ∂2 ∂r∂t Φ−r(∂ ∂r v)∂ ∂t Φ
rv2
Tµ
ν
Gµ
ν;µ= 0
t−> r(t)
K(t, r(t))
d
dt K(t, r(t)) = 0
v(t, r) Φ(t, r)K(t, r) =
v2−2Φ
Φ
K(t, r(t)) t−> r(t)
G01= 0
ΦGµ
ν