linet b. cours de relativite generale

IR
4
10
−18
10
−4
IR
O
t
t
t t
t
O e
i
(i = 1, 2, 3)
t t
y
i
(i = 1, 2, 3) e
i
OP =
3

i=1
y
i
e

i
soit OP = y
i
e
i
.
y
i
e
i
O
C
O
P
e
e
1
2
espace au temps t
t
O
origine du
temps
O
i
(t)y
y
i

|y

i
|  

e
i
t e
i
t y
i
t
t y
i
(t, y
i
) O C
y
i
(t)
v
i
=
dy
i
dt
et a
i
=
d
2
y

i
dt
2
.
(t, y
i
) (t

, y
i

)
T
IR
t
T
T
t
T = t
O O

O
O’ O
t
t = 0
t
O t e
i
t
(y

i
) e
i

O O

e
i
e
i

(t, y
i
) (t

, y
i

)
t

= t + d
0
T

= aT + b a > 0
d
0
e
i

e
i

R
j

i
(t)
e
i
=
3

j

=1
R
j

i
(t)e
j

soit e
i
= R
j

i
(t)e

j

.
d
i

(t) O

y
i

= R
i

j
(t)y
j
+ d
i

(t).
e
i
e
i
(y
i
) O
O


x
i
(t) = v
i
t + x
i
0
x
i
0
v
i
O

O v
i
(t, y
i
) (t

, y
i

) O
O

t

= t et y
i


= R
i

j

y
j
− v
j
t − y
j
0

R
j

i
e
i
= R
j

i
e
j

m f
i
(x

j
)
m
d
2
x
i
dt
2
= f
i
(x
j
(t)).
f
i

(x
k

) = R
i

j
f
j
(x
l
(x
k


, t))
x
l
(x
k

, t)
a f
i
(ab)
b
f
i
(ba)
b a f
i
(ba)
= −f
i
(ab)
η
e
a
(a = 0, 1, 2, 3)
η(e
a
, e
b
) = η

ab
avec η
ab
= diag(−1, 1, 1, 1).
X
X =
3

a=0
X
a
e
a
soit X = X
a
e
a
.
X
a
X X
η(X, X) =
3

a=0
3

b=0
η
ab

X
a
X
b
soit η(X, X) = η
ab
X
a
X
b
.
X
η(X, X) < 0 X
η(X, X) = 0 X
η(X, X) > 0 X
x
µ
(τ) τ dx
µ
/dτ
dx
µ
/dτ x
µ
(τ)
dx
µ
/dτ x
µ
(τ)

dx
µ
dτ x
µ
(τ)
e
a
x
µ
(µ = 0, 1, 2, 3)
OM = x
0
e
0
+ x
i
e
i
.
(x
µ
)
M
1
M
2
s
2
(M
1

, M
2
) = η(M
1
M
2
, M
1
M
2
).
M
1
M
2
x
µ
1
x
µ
2
s
2
(M
1
, M
2
) = η
µν
(x

µ
1
− x
µ
2
)(x
ν
1
− x
ν
2
).
η
µν
ds
2
x
µ
x
µ
+ dx
µ
ds
2
= η
µν
dx
µ
dx
ν

.
s
2
(M
1
, M
2
)
C
M
(O, e
a
) e
0
C
+
M
C
−
M
X
X
0
> 0 η(X, e
0
) < 0
s dx
µ
/ds −1
η

µν
dx
µ
ds
dx
ν
ds
= −1 et
dx
0
ds
> 0,
s
u
µ
(O, e
a
) e
0
O
x
i
= 0 (i = 1, 2, 3)
e
0
s x
0
t
t =
1

c
x
0
t =
s
c
c
t x
0
= ct
e
i
e
i
x
i
= 0
e
i
e
1
e
2
e
0
O
x
0
= 0
x

i
= 0 espace au temps t = x
0
/ c
(x
µ
) x
0
c t
x
i
y
i
O

e
a

e
0

(x
µ

)
(x
µ
) (x
µ


)
x
µ

= L
µ

ν
x
ν
+ D
µ
avec L
0

0
> 0
D
µ
L
b

a
e
a
= L
b

a
e

b

ds
2
= η
µν
dx
µ
dx
ν
c = 2, 997 924 58 × 10
8 −1
η
ρσ
= L
µ

ρ
L
ν

σ
η
µ

ν

η
ρ


σ

= η
ρσ
e
a
e
a

det L
µ

ν
=
1
L
0
(4) L
↑
+
(4)
L(4)
x
0

=
x
0
− wx
1

√
1 − w
2
x
1

=
x
1
− wx
0
√
1 − w
2
x
2

= x
2
x
3

= x
3
w
O

x
1


= 0 x
1
(x
0
) = wx
0
O v
1
= cw
O O

c v
1
/c  1 v
1
x
1
/c  x
0
x
1
= 0 x
1

= 0
O
e
0
e
0’

e
1’
e
1
cone de lumière
x
1
x
1’
x
0
x
0’
1
2 1 2
e
µ
0

= u
µ
u
µ
u
0
> 0
O
e
0’
e

0
e
1
e
1’
cone de lumière
x
1
x
0 O’ O
t
s
t =
s
c
.
s
s < x
0
∆
O
s
x
1
x
0
∆ x
0
s x
µ

(τ)
s τ
i
τ
f
s =

τ
f
τ
i

−η
µν
dx
µ
dτ
dx
ν
dτ
dτ,
τ x
0
c
s =

x
0
f
x

0
i

1 −
v
2
c
2
dx
0
car
dx
i
dx
0
=
v
i
c
et donc s < x
0
.
e
i
(y
i
)
(y
i
)

|y
i
|   avec  =
c
2
| a
i
|
 a
i
(a, 0, 0) x
2
= x
3
= 0
x
0
(s) =
c
2
a
sinh
as
c
2
, x
1
(s) =
c
2

a
cosh
as
c
2
u
µ
u
0
= cosh
as
c
2
, u
1
= sinh
as
c
2
.
e
1
e
0
1
= sinh
as
c
2
, e

1
1
= cosh
as
c
2
.
a
µ
= du
µ
/ds u
µ
(a/c
2
, 0, 0)
e
i
a
µ
= (a/c
2
)e
µ
1
x
2
= x
3
= 0

 ≈ 10
16
10
−2
O
u
e
1
cone de lumière
observateur
espace orthogonal
x
1
x
0
m
(x
µ
)
F
µ
(x
λ
)
m
du
µ
ds
= F
µ

(x
λ
(s)) avec η
µν
u
µ
F
ν
= 0
x
µ
(s) s u
µ
d
ds
(η
µν
u
µ
u
ν
) = 0 soit mη
µν
u
µ
du
ν
ds
= 0.
p

µ
p
µ
= mcu
µ
.
F
µ

(x
ρ

) = L
µ

ν
F
ν
(x
σ
(x
ρ

)).
η
µ

ν

u

µ

F
ν

= 0
L
0
(4)
c
k
µ
c
x
1
x
0
(λ) = k
0
λ (k
0
> 0) x
1
(λ) = k
1
λ mais k
0
= k
1
donc x

1
(x
0
) = x
0
.
(x
µ
)
A(x
ν
) = a
0
exp(−iη
µν
k
µ
x
ν
)
k
µ
η
µν
k
µ
k
ν
= 0 k
0

> 0
ν[u]
u
µ
ν[u] = −
c
2π
η
µν
u
µ
k
ν
,
η
µν
u
µ
k
ν
< 0
n
µ
[u]
u
µ
n
µ
[u] =
1

(−η
ρσ
k
ρ
u
σ
)
(k
µ
+ η
ρσ
k
ρ
u
σ
u
µ
)
η
αβ
n
α
[u]n
β
[u] = 1 η
αβ
n
α
[u]u
β

= 0
e
i
n
µ
= n
i
e
µ
i
k
µ
u
µ
u
µ
u
µ
ν
e
ν
o
=
η
µν
k
µ
u
ν
e

η
µν
k
µ
u
ν
o
.
[u]
k
n
µ
µ
u
espace orthogonal
µ
observateur
A k
µ
ν
e
/ν
o
u
µ
ν[u] n
µ
[u]
ν[u]  |a
i

| /c
a m
a
x
i
a
(t)
m
a
d
2
x
i
a
dt
2
= f
i
(x
k
a
(t))
f
i
f
i
a
f
i
= m

(p)
ga
g
i
m
(p)
ga
a g
i
b
m
(a)
gb
g
i
g
i
(x
k
) = −Gm
(a)
gb
(x
k
− x
k
b
)
|x
i

− x
i
b
|
3
G
b
U(x
k
) = Gm
(a)
gb
1
|x
i
− x
i
b
|
.
a V
a
= −m
(p)
ga
U
m
(p)
ga
m

(a)
gb
= m
(p)
gb
m
(a)
ga
m
ga
m
gb
Gm m
m
d
2
x
i
dt
2
= m
g
g
i
.
g
i
m = m
g
.

m m
g
g
i
d
2
x
i
dt
2
= g
i
.
a
i
x
i
(t) = a
i
t
2
/2
(t, y
i
) (t

, y
i

)

t

= t et y
i

= y
i
−
1
2
a
i
t
2
.
y
i
(t) = v
i
t + y
i
0
y
i

(t)
d
2
y
i


dt
2
= −a
i
.
g
i
G  6, 673 ×10
−11
3 −1 −2
g
i
a
i
= −g
i
d
2
y
i

dt
2
= −a
i
+ g
i
g
i

−a
i
+ g
i
= 0
∂
ij
U
L T
T = 2π

m
m
g

1/2

L
g

1/2
g
m/m
g
10
−3
10
−12
a b
a

b
c
i
|c
i
| = Rω
2
cos φ
g
i
g  9, 8
−2
φ
c
g
ω
i
i
i
F
i
a
= m
ga
g
i
+ m
a
c
i

et F
i
b
= m
gb
g
i
+ m
b
c
i
m
ga
m
gb
m
a
m
b
n
i
T
i
= T n
i
et C
i
= Cn
i
T

i
C
i
bati
fil de torsion
A
B
n
F
F
a
b
i
i
i
F
i
a
+ F
i
b
= T
i
et 
ijk
(OA)
j
F
k
a

+ 
ijk
(OB)
j
F
k
b
= C
i
C C
i
T
i
C
i
T
i
= 
ijk
(AB)
i
F
j
a
F
k
b
C =
1
| F

i
a
+ F
i
b
|

ijk
(AB)
i
F
j
a
F
k
b
=
m
ga
m
b
− m
a
m
gb
| F
i
a
+ F
i

b
|

ijk
(AB)
i
g
j
c
k
.

ijk
= 1 i, j, k 
ijk
= −1 i, j, k

ijk
= 0
C F
i
a
F
i
b
m
ga
/m
a
= m

gb
/m
b
180
o
a b C  −C C = 0
a b
10
−15
10
−17
Ψ m
V
i
∂
∂t
Ψ = −

2
2m
Ψ + V Ψ

v
x
i

∂
∂t

− v

∂
∂x


Ψ = −

2
2m
∂
2
∂x
2
Ψ.
Ψ

= Ψ exp −
i


mvx

+
1
2
mv
2
t


.

V = −
G
∞
m
a
m
b
r

1 + α
ab
exp −
r
λ

G
∞
λ α
ab
a b |α
ab
|< 10
−4
B/µ
  1, 054573 ×10
−34
a
1
x
i


∂
∂t

− a
1
t

∂
∂x


Ψ = −

2
2m
∂
2
∂x
2
Ψ.
Ψ

= Ψ exp −
i

ma
1

x


t

+
1
6
a
1
t
3

i
∂
∂t

Ψ

= −

2
2m
∂
2
∂x
2
Ψ

+ ma
1
x


Ψ

.
m g
1
x V = −m
g
g
1
x m
g
i
∂
∂t
Ψ = −

2
2m
∂
2
∂x
2
Ψ − m
g
g
1
xΨ.
m = m
g

a
1
=
−g
1
S
M
A B
D
C
h
l
v
h
Ψ ∝ exp

−i
Et

+
i


x
p(x)dx

avec p(x) =

2m(E − V (x))
m

δ
SAB
φ =
l

√
2mE et δ
DCM
φ =
l


2m(E − mgh).
δφ = δ
SAB
φ − δ
DCM
φ g
δφ ≈
mglh
√
2mE
2E
soit δφ ≈ mglh
1
v
.
δφ h
g
ρ

v
i
p U
U = −4πGρ
∂
∂t
ρ +
∂
∂x
i

ρv
i

= 0
∂
∂t
(ρv
i
) +
∂
∂x
k
(ρv
k
v
i
) = ρ
∂
∂x

i
U −
∂
∂x
i
p
m  1, 7 ×10
−27
λ
SAB
 1, 4