θ
t
t
′
=
∂
t
′
∂
t
=
1
θ
t
x
′
=
∂
x
′
∂
t
=
−
ω
x
sin
ω
t
−
ω
y
cos
ω
t
θ
x
x
′
=
∂
x
′
∂
x
=
cos
ω
t
θ
y
x
′
=
∂
x
′
∂
y
=
−
sin
ω
t
θ
t
y
′
=
∂
y
′
∂
t
=
ω
x
cos
ω
t
−
ω
y
sin
ω
t
θ
x
y
′
=
∂
y
′
∂
x
=
sin
ω
t
θ
y
y
′
=
∂
y
′
∂
y
=
cos
ω
t
{\displaystyle {\begin{matrix}\theta _{t}^{t'}&=&{\frac {\partial t'}{\partial t}}&=&1\\\theta _{t}^{x'}&=&{\frac {\partial x'}{\partial t}}&=&-\omega x\sin \omega t-\omega y\cos \omega t\\\theta _{x}^{x'}&=&{\frac {\partial x'}{\partial x}}&=&\cos \omega t\\\theta _{y}^{x'}&=&{\frac {\partial x'}{\partial y}}&=&-\sin \omega t\\\theta _{t}^{y'}&=&{\frac {\partial y'}{\partial t}}&=&\omega x\cos \omega t-\omega y\sin \omega t\\\theta _{x}^{y'}&=&{\frac {\partial y'}{\partial x}}&=&\sin \omega t\\\theta _{y}^{y'}&=&{\frac {\partial y'}{\partial y}}&=&\cos \omega t\\\end{matrix}}}
modifier
det
g
=
−
c
2
{\displaystyle \det g=-c^{2}}
modifier
g
t
t
=
−
c
−
2
g
t
x
=
g
x
t
=
−
c
−
2
ω
y
g
t
y
=
g
y
t
=
c
−
2
ω
x
g
x
y
=
g
y
x
=
c
−
2
ω
2
x
y
g
x
x
=
1
−
c
−
2
ω
2
y
2
g
y
y
=
1
−
c
−
2
ω
2
x
2
{\displaystyle {\begin{matrix}g^{tt}&=&-c^{-2}\\g^{tx}=g^{xt}&=&-c^{-2}\omega y\\g^{ty}=g^{yt}&=&c^{-2}\omega x\\g^{xy}=g^{yx}&=&c^{-2}\omega ^{2}xy\\g^{xx}&=&1-c^{-2}\omega ^{2}y^{2}\\g^{yy}&=&1-c^{-2}\omega ^{2}x^{2}\end{matrix}}}
modifier
g
t
t
,
x
=
2
ω
2
x
g
t
t
,
y
=
2
ω
2
y
g
t
x
,
y
=
g
x
t
,
y
=
−
ω
g
t
y
,
x
=
g
y
t
,
x
=
ω
{\displaystyle {\begin{matrix}g_{tt,x}&=&2\omega ^{2}x\\g_{tt,y}&=&2\omega ^{2}y\\g_{tx,y}=g_{xt,y}&=&-\omega \\g_{ty,x}=g_{yt,x}&=&\omega \end{matrix}}}
Γ
t
|
t
x
=
Γ
t
|
x
t
=
ω
2
x
Γ
x
|
t
t
=
=
−
ω
2
x
Γ
t
|
t
y
=
Γ
t
|
y
t
=
ω
2
y
Γ
y
|
t
t
=
−
ω
2
y
Γ
x
|
t
y
=
Γ
x
|
y
t
=
−
ω
Γ
y
|
t
x
=
Γ
y
|
x
t
=
ω
{\displaystyle {\begin{matrix}\Gamma _{t|tx}=\Gamma _{t|xt}&=&\omega ^{2}x\\\Gamma _{x|tt}=&=&-\omega ^{2}x\\\Gamma _{t|ty}=\Gamma _{t|yt}&=&\omega ^{2}y\\\Gamma _{y|tt}&=&-\omega ^{2}y\\\Gamma _{x|ty}=\Gamma _{x|yt}&=&-\omega \\\Gamma _{y|tx}=\Gamma _{y|xt}&=&\omega \\\end{matrix}}}
Γ
t
t
x
=
−
ω
2
x
Γ
t
t
y
=
−
ω
2
y
Γ
t
y
x
=
Γ
y
t
x
=
−
ω
Γ
t
x
y
=
Γ
x
t
y
=
ω
{\displaystyle {\begin{matrix}\Gamma _{tt}^{x}&=&-\omega ^{2}x\\\Gamma _{tt}^{y}&=&-\omega ^{2}y\\\Gamma _{ty}^{x}=\Gamma _{yt}^{x}&=&-\omega \\\Gamma _{tx}^{y}=\Gamma _{xt}^{y}&=&\omega \end{matrix}}}
x
¨
=
ω
2
x
t
˙
2
+
2
ω
y
˙
t
˙
y
¨
=
ω
2
y
t
˙
2
−
2
ω
x
˙
t
˙
{\displaystyle {\begin{matrix}{\ddot {x}}&=&\omega ^{2}x{\dot {t}}^{2}+2\omega {\dot {y}}{\dot {t}}\\{\ddot {y}}&=&\omega ^{2}y{\dot {t}}^{2}-2\omega {\dot {x}}{\dot {t}}\end{matrix}}}
Pour les vitesses petites devant la vitesse de la lumière, on a
d
s
2
≈
−
c
2
d
t
2
{\displaystyle ds^{2}\approx -c^{2}dt^{2}}
d
2
x
d
t
2
≈
ω
2
x
+
2
ω
y
˙
d
2
y
d
t
2
≈
ω
2
y
−
2
ω
x
˙
{\displaystyle {\begin{matrix}{\frac {d^{2}x}{dt^{2}}}&\approx &\omega ^{2}x+2\omega {\dot {y}}\\{\frac {d^{2}y}{dt^{2}}}&\approx &\omega ^{2}y-2\omega {\dot {x}}\end{matrix}}}
On retrouve les termes classiques d'accélération centrifuge et d'accélération de Coriolis .
Nul. Le calcul peut se faire à partir de la formule ou plus simplement en remarquant que le tenseur de courbure de l'espace pseudo-euclidien de métrique constante
D
i
a
g
(
−
c
2
,
1
,
1
)
{\displaystyle \mathrm {Diag} \left(-c^{2},1,1\right)}
0
{\displaystyle 0}
0
{\displaystyle 0}
