Éléments de théorie de la relativité

E
Bqv
F
F=qE+qv∧B.
E=ρ
ε0
B= 0
E+∂B
∂t = 0
B−ε0µ0
∂E
∂t =µ0j
ε0
µ0
ρ C.m−3
jA.m−2
∂ρ
∂t +j= 0.
c= 299792458m.s−1
µ0= 4π10−7kg.m.A−2.s−2
ε0= 1/µ0c2

E=−∂
∂t B=−ε0µ0
∂2
∂t2E.
E=∇E−∆E
E=−∆E,
∆−ε0µ0
∂2
∂t2E= 0,
1+1
E(t, x) = E−x−1
√ε0µ0
t+E+x+1
√ε0µ0
t,
E+E−tE
c= 1/√ε0µ0
.
c
E=ρ
B= 0
E+1
c
∂B
∂t = 0
B−1
c
∂E
∂t =1 cj.
B
A
B=A.
E−c−1∂tA= 0 E−c−1∂tA
φ
E+1
c
∂A
∂t =−∇φ.
(φ, A)χ
φ
A7−→ φ+c−1∂tχ A− ∇xχ

E B χ
A+1
c
∂φ
∂t = 0.
Aφ
=1
c2
∂2
∂t2− ∇2,
φ=ρ, A=j.
O(1,3)
R4x0=ct
x= (x1, x2, x3) = (x, y, z)
gµν =gµν = (−1,1,1,1).
g
xµ=gµν xν, pµ=gµν pν.
∂µ=∂/∂xµ
(∂0, . . . , ∂3) = (c−1∂t,∇x),(∂0, . . . , ∂3)=(c−1∂t,−∇x),
∂2=∂2
0−∂2 1−∂2 2−∂2 3=c−2∂2 t− ∇2 x.
A F
j
Aµ= (φ, A), i.e. Aµ= (φ, −A),
Fµν =∂µAν−∂νAµ, i.e. F µν =∂µAν−∂νAµ,
jµ= (ρ, j)i.e. jµ= (ρ, −j).
∂µAµ= 0, ∂µjµ= 0,
∂κFµν +∂µFνκ +∂νFκµ = 0,
∂2Aµ=jµ.

A1×A3
A= (tA,xA)M= (tM,xM)
tA=tM
A M

−
−
−
V
4
i) (t, x, y, z) (t0, x0, y0, z0)
R R0
R R0
R R0
ii)R R0
c=V=dr
dt =dr0 dt0,dr2=dx2+dy2+dz2
dr02=dx02+dy02+dz02.