relativité restreinte et principes variationnels
DESCRIPTION
Optique relativisteTransformation de Lorentz, effet Doppler, etc.TRANSCRIPT
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ct = γ(ct + βx)x = γ(βct + x)y = y
z = z
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β 1⇒ γ = 1 +O(β2) et
ct = ct + β x
x = x + β ct
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Λ(α) =
coshα sinh α 0 0sinh α coshα 0 0
0 0 1 00 0 0 1
tanhα = βR/R
X = Λ(α1)X
X = Λ(α2)X X = Λ(α1) Λ(α2) X = Λ(α1 + α2)X
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βR/R = tanh(α1 + α2) =βR/R + βR/R
1 + βR/R βR/R
X =
ctxyz
X = Λ(β)X Λ(β) =
γ γβ 0 0γβ γ 0 00 0 1 00 0 0 1
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ct = γ(ct + β · r)r = γ(r + βct)r⊥ = r⊥
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∆x = xB − x
A = 0 ∆t = tB − tA = 0E1-'/+)(('%N'%12345#467462#`M'0K/.%)K%M^M'%(,'Ka%
>)10%&% c∆t = γ(c∆t + β∆x) = γc∆t
∆t = γ∆t
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t =d
vt = δ +
d
v
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γ(v) v
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γ(v) v
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TA(aller) =D
v
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TA(aller)
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28/04/11 11:25http://upload.wikimedia.org/wikipedia/commons/7/78/Bondi_diagramme_3.svg
Page 1 sur 1
A B
O
P
k(kT)
T
kT
t
x
T'
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k =
1 + β
1− β≥ 1
28/04/11 11:25http://upload.wikimedia.org/wikipedia/commons/0/0d/Bondi_diagramme_1.svg
Page 1 sur 1
A B
O
T
kT
T
kT
t
x
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N'%_5V% β =v
c=
d
c t=
k2 − 1k2 + 1
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d = ck2T − T
2t =
k2T + T
2
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∆tr =
1 + β
1− β∆te = 3∆te
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∆tr =
1− β
1 + β∆te =
13
∆te
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γ =20061.5
≈ 1337 =⇒ 1− β ≈ 2.8 10−7
γ =25× 365
15≈ 608 =⇒ 1− β ≈ 1.3 10−6
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βAC =k2
AC − 1k2
AC + 1=
βAB + βBC
1 + βAB βBC
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βAB + βBC + βCA = −βAB βBC βCA
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tr = te +D(te)
c
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r c= (1− β cos θ) dte
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νobs =νp
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νobs =ν0
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V⊙/FDC = 371± 1 km/s
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ct = γ(ct + βx)x = γ(βct + x)y = y
z = z
wx =dx
dt=
γ (β cdt + dx)γ (cdt + β dx)
=w
x + v
1 + wx v/c2
wy =dy
dt=
dy
γ (cdt + β dx)=
wy
γ (1 + wx v/c2)
w⊥ =w⊥
γ (1 + w · v/c2)w =
w + v
1 + w · v/c2
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wx = c cos θwz = c sin θ
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wx =w
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1 + wx v/c2
=⇒ cos θ =cos θ + β
1 + β cos θ
wz =w
z
γ (1 + wx v/c2)
=⇒ sin θ =sin θ
γ (1 + β cos θ)
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w
x = c cos θ
wz = c sin θ
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cos θ = βsin θ = 1/γ ∆θ = θ − θ ≈ β
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v⊕c≈ 10−4
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dΩ
dΩ=
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(1− β cos θ)2
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tr = te +D(te)
c
vapp =v sin θ dte
dtr= c
β sin θ
1− β cos θ
dtr = dte −r · v dte
r c= (1− β cos θ) dte
vapp maximum quand cos θ = β et alors vapp = γβc
vapp > c si β >1√2
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