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M R R
ρ » 10´26 ¨ ´3
RΞ“ 4πGρR2{p3c 2q1{2
R “
ˆ
3c 2
8πGρ
˙1{2
» 6 ,
13,8
M@ ¨ ´3
3ˆ 10´6 6ˆ 103 5ˆ 103 10´9
1 7ˆ 105 103 10´6
0,1´ 0,4 „ 104 „ 1010 „ 10´3
1´ 3 „ 10 „ 1018 „ 0,2
Á 3´ 30 9´
M3M@
¯
0 0,5
106 ´ 1010 2´
M108M@
¯
0 0,5
1024 14 10´26 0,5
front d'onde
événement
émission d'un flash
lignes d'universévénement
temps
espace
accident de parachutisme
surface
pp p
q
q p q p
p q q p
p q
p
p pp
pp
p
p
p
ailleurs
futur
cône de lumière
passé
futur
passé
simultané
physique pré-relativiste relativité restreinte
p p
particule
physique pré-relativiste relativité restreinte
photon
tem
ps
abso
lu
espace absolu particule
x , y, z
t , x , y, z
Ot , x , y, z O 1
v x O pt “ x “ y “ z “ 0 t 1, x 1, y 1, z 1
t 1“ t ,
x 1“ x ´ v t .
O 1
Ot 1
“ γ pt ´ v x{c 2q ,x 1
“ γ px ´ v t q ,
γ ” p1´v2{c 2q´1{2
O t “
O 1 t 1 “
O O 1 c
relativité restreintephysique pré-relativiste
x
𝒪′𝒪
t = t' = const
𝒪′𝒪′v = c
v > c
v <
c
𝒪′𝒪t
p p
t = const
t' = const
v <
c
∆t∆t
}∆x}
I
I “ ´c 2p∆t q2
` p∆x q2
` p∆yq2
` p∆z q2 .
I
I 1“ ´c 2p∆t 1
q2
` p∆x 1q2
“ ´c 2p∆t q2
` p∆x q2
“ I .
I
Δt
physique pré-relativiste relativité restreinte
I > 0
I < 0
I =
0
∆𝑥
I
I
physique pré-relativiste relativité restreinte
invariance de c
?
Électromagnétisme(Maxwell, 1862)
Mécanique relativiste(Einstein, 1905)
Mécanique classique(Galilée, Newton)
Gravitation universelle(Newton, 1687)
miF “ mi a
m gF g “ m g gg mi “ m g a “ g
10´8
10´13
10´15
19001920
19401960
19701980
19902000
10-8
10-9
10-10
10-11
10-12
10-13
10-14
TLL
2010
Matter waves
Free-fall
Princeton
Boulder
Renner
Eöt-Wash
Eötvös
Eöt-WashMoscow
η
η “ 2pa1 ´ a2q{pa1 ` a2q
Géométrieeuclidienne
Géométrieminkowskienne
Géométrieriemannienne
Géométrielorentzienne
+ temps
+ temps
+ courbure
+ courbure
Physiquepré-relativiste
Relativitérestreinte
Relativitégénérale
principed'équivalenc e
principe de relativité
espace courbe espace-tempscourbe
espace plat espace-temps plat
Rn n
V n PN˚
RnV
p nU V p U
ϕ : U Ă V ÝÑ ϕpU q ĂRnp ÞÝÑ px 1, ¨ ¨ ¨ , x nq .
𝒱
𝒰
𝜙 (𝒰)
𝜙
ℝn
(x α)p
RnRn
pxαq ” px 1, ¨ ¨ ¨ , x nq
U F α :Rn ÑR 1 ď α ď npyαq ” py1, ¨ ¨ ¨ , y nq yα “ F αpx 1, ¨ ¨ ¨ , x nq
URn
n “ 2R2 S2 T2
n “ 4 R4
R4
ℝ2 22
U Ă V ϕpU q ĂRn
Vn “ 2 R3
V pUk ,ϕkq1ďkďKK P N˚ Uk V ϕk UkUk V
Kď
k“1
Uk “ V .
Rn VUi XU j ‰ ∅
ϕi ϕ j
ϕi ˝ϕ´1j :ϕ j pUi XU j q ĂRn ÝÑϕi pUi XU j q ĂRn
C 8
Rn RnRn
ϕk
𝜙i ∘ 𝜙j–1
𝒱
𝜙i
𝜙j
𝒰i𝒰j
ℝn
ℝn
𝜙j (𝒰j)
𝜙i (𝒰i)
R3Rn n P N˚
C VP :RÝÑ V
λ ÞÝÑ p “P pλq PC .
ℝ
λ𝒫 𝒞
p
C “ tP pλq|λ PRu P λpxαq
n X α :RÑRC xα “ X αpλq α P t1, ¨ ¨ ¨ , nu
C R2 tx “ X pλq, y “ Y pλqu CR2 v C
p “P pλq pdX {dλ, dY {dλq
f : R2 Ñ Rf C
d f |C “ pB f {Bx qdX ` pB f {B yqdY
d fdλ
ˇ
ˇ
ˇ
ˇC“ v ¨ ∇ f .
vf
𝒞
x
y
p
v→
C V P pλq Cv P pλq p P C
f : V Ñ R pd f {dλ
v p f q ”d fdλ
ˇ
ˇ
ˇ
ˇC“ lim
ϵÑ0
1ϵ
!
f pP pλ` ϵqq ´ f pP pλqq
)
.
pxαq p P V np xα
Cα α P t1, ¨ ¨ ¨ , nu Cα λ “ xαp β ‰ α xβ BBBαCα xα
f
BBBαp f q “d fdxα
ˇ
ˇ
ˇ
ˇCα“
B fBxα
.
fpxαq f V
v
𝒱
𝒞1
pp
𝝏2𝝏1𝒞2
𝒞
Cpxαq V
v p f q “
nÿ
α“1
B fBxα
dX α
dλ“ BBBαp f q
dX α
dλ,
Σf v
v “ vα BBBα , vα “dX α
dλ.
p P Vp
n R pBBBαq TppV q
V p pBBBαq
pxαq vα “ dX α{dλv
pTppV q
p V R3R3
𝒱
Tp (𝒱 )
p q
Tq (𝒱 )
px 1αq Vv “ v 1αBBB
1α
vα v 1α pBBBαq pBBB1αq
BBBβp f q “ pBx 1α{BxβqBBB1αp f q
f
v 1α“
Bx 1α
Bxβvβ .
pvαq
pxαq ÞÑ px 1αq
p q
C p q P pλq Cp “P pλq q “P pλ`dλq dλ
λ p q
dl ” v dλ ,
v C p P pλq
v P TppV q dλ PR dl p
f
dl p f q ” v p f qdλ“d fdλ
ˇ
ˇ
ˇ
ˇCdλ“ f pqq ´ f ppq .
dl p qCp q
pxαq p pxα0 q
p pxα0 ` dxαq q dl p f q “
d f “ pB f {Bxαqdxα “ BBBαp f qdxα
dl “ dxα BBBα .
dlpBBBαq pxαq dxα
𝒞
dl
p =𝒫(λ)
=𝒫(λ+dλ)q
p P Vω : TppV q ÝÑR
v ÞÝÑωpv q
ωpλu ` v q “ λωpuq `ωpv q u , v P TppV q
λ P R TppV q
n R TppV q T ˚p pV q
pBBBαq TppV q
T ˚p pV q p xαq
xα`
BBBβ˘
“ δαβ ,
δαβ δαβ “ 1 α “β 0 α ‰β p xαq
pBBBαq p xαq
xα dxα
xα`
dl˘
“ xα`
dxβBBBβ˘
“ dxβ xα`
BBBβ˘
looomooon
δαβ
“ dxα .
xα P T ˚p pV q
dxα PR xα p xαq
ω P T ˚p pV q ωα
pxαq
ω “ωα xα .
ω BBBαωα “ωpBBBαq
ω v
ωpv q “ωα xα`
vβBBBβ˘
“ωαvβ xα
`
BBBβ˘
looomooon
δαβ
“ωαvα .
px 1αq Vω “ ω1
α x 1α
ωα “ωpBBBαq ω1α “ωpBBB
1αq p xαq p x 1αq
pxαq ÞÑ px 1αq
pBBBαq pBBB1αq BBB
1α “ pBxβ{Bx 1αqBBBβ
ω1
α “Bxβ
Bx 1αωβ .
pxαq ÞÑ px 1αq
TppV q peαq
Vpe αq
T ˚p pV q
e αpeβq “ δαβ .
pe αq peαq
eα “ BBBα e α “ xα
f : V Ñ R f ∇ fv P TppV q
p∇ f qpv q ” v p f q “d fdλ
.
pdxαq
pxαq
p∇ f qpBBBαq “ BBBαp f q “ B f {Bxα
∇ f “B fBxα
dxα .
fn pxαq
d f “B fBxα
dxα .
pk , l q pk , l q P N2k l
T :
khkkkkkkkkkkkkikkkkkkkkkkkkj
T ˚
p pV q ˆ ¨ ¨ ¨ ˆT ˚
p pV qˆ
lhkkkkkkkkkkkikkkkkkkkkkkj
TppV q ˆ ¨ ¨ ¨ ˆTppV q ÝÑRpω1, ¨ ¨ ¨ ,ωk , v1, ¨ ¨ ¨ , vl q ÞÝÑ T pω1, ¨ ¨ ¨ ,ωk , v1, ¨ ¨ ¨ , vl q
k` lp0,1q
T ˚˚p pV q » TppV q v
T ˚p pV q
v : T ˚
p pV q ÝÑRω ÞÝÑ v pωq ”ωpv q .
p1,0q
pR
b T pk , l q Spm,nq T b S pk ` m, l ` nq
k ` m ω1, ¨ ¨ ¨ ,ωk`m l ` n v1, ¨ ¨ ¨ , vl `n
T b S pω1, ¨ ¨ ¨ ,ωk`m , v1, ¨ ¨ ¨ , vl `nq “ T pω1, ¨ ¨ ¨ ,ωk , v1, ¨ ¨ ¨ , vl qˆ S pωk`1, ¨ ¨ ¨ ,ωk`m , vl `1, ¨ ¨ ¨ , vl `nq .
T bS ‰ SbT
peαq TppV q pe αq T ˚p pV q
nk`l`
eα1b¨ ¨ ¨b eαkb eβ1b¨ ¨ ¨b eβl˘
pk , l q p T pk , l q
T “ T α1¨¨¨αkβ1¨¨¨βl
eα1b ¨ ¨ ¨ b eαk b eβ1 b ¨ ¨ ¨ b eβl .
nk`l T α1¨¨¨αkβ1¨¨¨βl
T peαq
T peαq
pe αq
T α1¨¨¨αkβ1¨¨¨βl
“ T`
e α1 , ¨ ¨ ¨ , e αk , eβ1, ¨ ¨ ¨ , eβl
˘
.
pxαq eα “ BBBα e α “ xα
pxαq ÞÑ px 1αq
pk , l q
T 1α1¨¨¨αkβ1¨¨¨βl
“Bx 1α1
Bxρ1¨ ¨ ¨
Bx 1αk
BxρkBx σ1Bx 1β1
¨ ¨ ¨Bx σlBx 1βl
T ρ1¨¨¨ρkσ1¨¨¨σl
.
pT α1¨¨¨αkβ1¨¨¨βl
q
pxαq ÞÑ px 1αq
T pk , l q C Tpk ´ 1, l ´ 1q T α1¨¨¨αk
β1¨¨¨βlT
peαq C T
pCT qα1¨¨¨αk´1
β1¨¨¨βl ´1“ T α1¨¨¨γ ¨¨¨αk
β1¨¨¨γ ¨¨¨βl,
γpeαq
C T
TppV q pV p
p P V TppV q
p0,0q
T pk , l q
T α1¨¨¨αkβ1¨¨¨βl
T pk , l q
T a1¨¨¨akb1¨¨¨bl
ai b j
b
p1,0q v pvαq v a
p0,1q ω pωαq ωa
pk , l q T`
T α1¨¨¨αkβ1¨¨¨βl
˘
T a1¨¨¨akb1¨¨¨bl
p0,0q ωpv q ωαvα ωav a
p2,1q u b v bω`
uαvβωγ
˘
uav bωc
p0,2q R pv , ¨, ¨,ωq`
R δαβγ
vαωδ
˘
R dab c v aωd
p2,1q C 21 S p¨, ¨, v , ¨q
`
Sαβγβδε
vδ˘
S ab cbd e v
d
SabSab “ Sba Aab
Aab “ ´Aba Tab p0,2q
Tab “ Tpabq`Trabs
Tpabq ”12
pTab `Tbaq ,
Trabs ”12
pTab ´Tbaq .
Srabs “ 0 Apabq “ 0 Ta1¨¨¨alp0, l q
Tpa1¨¨¨al q”
1l !
ÿ
π
Taπp1q¨¨¨aπpl q,
Tra1¨¨¨al s”
1l !
ÿ
π
επTaπp1q¨¨¨aπpl q,
π a1, ¨ ¨ ¨ ,anεπ
Tpabqrc d s p0,4q
n “ 3 RR3
V
p VTppV q gab
gab TppV qˆTppV q ÑRp0,2q
gab uav b “ gab v au b ua v agab “ gba
gab uav b “ 0 v a ua “ 0
gabTppV q p P V
ua v a gab
gab uav b
“ 0 .
pBαqa TppV q pdxαqa
gabgαβ
gab “ gαβ pdxαqapdxβ
qb , gαβ “ gabpBαqapBβq
b .
gab gαβua v a
uα vα pBαqa
gab uav b
“ gαβuαvβ .
gab VTppV q T ˚
p pV q v a
va ” gab vb .
gab g ab p2,0q
g ab gb c “ δac δa
c TppV q TppV q
g ab “ g ba
ωa
ωa” g abωb .
ua v agab uav b “ ub v b “ g ab ub va “ uava
R3fab f ab
p fαβq “ p f αβq “ p`1,`1,`1q ,
gab
pk , l q
pk´1, l `1q gab pk`1, l ´1q
g ab
T ab cd e p3,2q
T a c d eb gb f g d h g e jT a f c
h j
gab p gαβq
p P V peαqa TppV q
gab
p gαβq “ p˘1, ¨ ¨ ¨ ,˘1q .
s n´ s
gab
gab “ p´, ¨ ¨ ¨ ,´looomooon
s
,`, ¨ ¨ ¨ ,`looomooon
n´s
q .
s “ 0 gabgab uau b ě 0 ua gab uau b “ 0
ua “ 0p`,`q
s “ 1 gab uau b ua
ua
gab uau b ă 0 gab uau b “ 0 gab uau b ą 0TppV q I
gab p v a P I λ PR λv a P I
p´,`,`,`q
métrique riemannienne métrique lorentzienne
espace
temps
lumière
I
p q Vdl a P TppV q
p q gab
ds 2 ” gab dla dl b .
V gabp
qds 2
dl a
pxαq p dxαp q
dl a pBαqa
ds 2 “ gαβ dxαdxβ .
p qp˘ds 2q1{2
pE , gabq E n “ 4 gabp0,2q
p´,`,`,`q
E “R4 gab “ ηab pR4,ηabq
gab
relativité restreinte relativité générale
E, gab)(ℝ4, ηab)
p qTppV q TqpV q
gab
∇pk , l q pk , l `1q
T a1¨¨¨akb1¨¨¨bl
pk , l q ∇cTa1¨¨¨ak
b1¨¨¨blpk , l `1q
∇ T
pT , S q pk , l q λ PR
∇c
`
λT a1¨¨¨akb1¨¨¨bl
` S a1¨¨¨akb1¨¨¨bl
˘
“ λ∇cTa1¨¨¨ak
b1¨¨¨bl`∇c S
a1¨¨¨akb1¨¨¨bl
;
T S pk , l q pm,nq
∇c
`
T a1¨¨¨akb1¨¨¨bl
S a1¨¨¨amb1¨¨¨bn
˘
“`∇cT
a1¨¨¨akb1¨¨¨bl
˘
S a1¨¨¨amb1¨¨¨bn
`T a1¨¨¨akb1¨¨¨bl
`∇c Sa1¨¨¨am
b1¨¨¨bn
˘
;
T pk , l q
∇d
`
T a1¨¨¨c ¨¨¨akb1¨¨¨c ¨¨¨bl
˘
“∇dTa1¨¨¨c ¨¨¨ak
b1¨¨¨c ¨¨¨bl;
f v a
v p f q “ v a∇a f ;
f
∇a∇b f “∇b∇a f .
∇a
Rn
T cab
a b ∇a∇b f “∇b∇a f `T cab∇c f
T cab ‰ 0
pxαq
BaT a1¨¨¨ak
b1¨¨¨blT α1¨¨¨αk
β1¨¨¨βl
pxαq BcTa1¨¨¨ak
b1¨¨¨blBpT α1¨¨¨αk
β1¨¨¨βlq{Bx γ
Bapx 1αq
BcTa1¨¨¨ak
b1¨¨¨bl
BpT 1α1¨¨¨αkβ1¨¨¨βl
q{Bx 1γ T 1α1¨¨¨αkβ1¨¨¨βl
T a1¨¨¨akb1¨¨¨bl
px 1αq
v a
Bv 1α
Bx 1β‰
Bx 1α
BxρBx σ
Bx 1β
Bvρ
Bx σ.
Ba
∇a ∇a∇a ∇a
`∇a ´ ∇a
˘
f “ 0
f ∇a´∇aωa ω1
a
`∇a ´ ∇a
˘`
f ωb `ω1
b
˘
“ f`∇a ´ ∇a
˘
ωb ``∇a ´ ∇a
˘
ω1
b ,
∇a´∇aC c
ab p1,2q
∇aωb “ ∇aωb ´C cabωc .
∇a ∇aC c
abωb “∇b f “ ∇b f
C cab∇c f “ ∇a∇b f ´∇a∇b f .
∇a∇b f ∇a∇b f
C cab “ C c
ba .
n C cab n2pn ` 1q{2
n “ 4∇a´∇a v a
f “ωb v b
`∇a ´ ∇a
˘
pωb vbq “ ´C c
abωc vb
`ωb
`∇a ´ ∇a
˘
v b .
b cωb
∇avb
“ ∇avb
`C bac v
c .
∇a pk , l q ∇a C cab
∇aTb1¨¨¨bk
c1¨¨¨cl“ ∇aT
b1¨¨¨bkc1¨¨¨cl
`
kÿ
i“1
C biadT
b1¨¨¨
iÓ
d ¨¨¨bkc1¨¨¨cl
´
lÿ
j“1
C dac jT b1¨¨¨bk
c1¨¨¨dÒ
j
¨¨¨cl.
∇a “ Ba C cab Γ cab
v a ωa
∇avb
“ Bavb
` Γ bac vc ,
∇aωb “ Baωb ´ Γ cabωc .
Bapxαq ∇a “ Ba C c
ab “ Γ cab∇a Γ cabΓ cab∇a Ba
pxαq ÞÑ px 1αq
Ba Ñ B1a Γ cab Ñ Γ 1c
abΓ cab pxαq
Γ 1cab px 1αq
∇aC t a v aC
t a∇avb
“ 0
C v a
t a T a1¨¨¨akb1¨¨¨bl
pk , l q
C t c∇cTa1¨¨¨ak
b1¨¨¨bl“ 0
𝒞v
v
t
t
t aBav b ` t aΓ bac vc “ 0 λ
t a Cdvα
dλ` t βΓ αβγ v
γ“ 0 .
v a
p PC∇aC p q
TppV q TqpV q
gab
ua v at a
gab uav b
t a∇ap gb c ub v c q “ 0 .
ua v aua v a
∇a gb c “ 0 .
∇a gabg ab ∇a g b c “ 0
𝒞u
v v
u
Bafab
Baηab
Ba fb c “ 0 Baηb c “ 0
fab ηab gab
Ba Ba ∇a
Ba fb c “ 0 Baηb c “ 0 ∇a gb c “ 0
∇agab ∇a∇a gb c “ 0
pxαq Bagab ∇a “ Ba C c
ab “ Γ cab
0 “∇a gb c “ Ba gb c ´ Γ dab gd c ´ Γ dac gbd .
Ba gb c ` Bb gac ´ Bc gab “ pΓcab ` Γbac q ` pΓc ba ` Γab c q ´ pΓbac ` Γac bq .
2Γcab
Γ cab “12g c d
pBa gbd ` Bb gad ´ Bd gabq .
gabΓ cab
pxαq
∇a gab
M δ | M | “ pM´1 ˆδM q
Γβαβ
“12g βγ
B gβγBxα
“12 g
B gBxα
“B
Bxαln
b
| g | ,
g ” p gαβq gαβgab g
∇ava
“ Bαvα
` Γβαβ
vα “1
a
| g |
B
Bxα`
b
| g |vα˘
.
R3
gab∇a t a
c b ” t a∇a tb
“ 0 .
p
q
t
c
t
pxαq
v at a t α
t α “ dX α{dλ
d2X α
dλ2` Γ αβγ
dX β
dλdX γ
dλ“ 0 .
nn X αpλq
X αpλ0q pdX α{dλqpλ0q
gabp q
L qp dl a p qL ds 2 “ gabdl adl b
p q p q L
dτ ”`
´ds 2˘1{2 .
P pλq
L dl a v a
dτ “`
´ gab vav b˘1{2 dλ .
dτ
p
q
dl { tics
horloge
L
p “P pλq q “P pλ1q
LL
τpp, qq ”
ż q
pdτ “
ż λ1
λ
`
´ gab vav b˘1{2 dλ .
pq
p q
L ua P TppE q p PL
ua”
dl a
dτ,
dl a p L dτ
ua Lτ
λ “ τ v a “ ua
gab
gab uau b
“ ´1 .
v
}v}
ua
ua
u
u
τ
L
papa ua
m
pa “ mua .
OL p L q p
O p qO
u1 qu2 t L p
p q
t “12
pt1 ` t2q ,
t1 t2 u1 u2 Lp q
dt1 u1 pdt2 p u2
p
{{dt1
q
u2
u1
dt2
L
dt ´dt ua
p u1 dt ua p u2 dt na pq pu1, qq pq , u2q
dt ua `dt na dt ua ´dt na
pua` na
qpua ` naq “ 0 ,pua
´ naqpua ´ naq “ 0 .
uana “ 0 .
p E pO p
ua OTppE q ua gab
TppE q
O pO
p
u
n
L
p q
u1
u2
dt u
dt n
− dt u
O O 1 L L 1 p
O O 1 L L 1
p τ τ1 O O 1 pdτ1 O 1 p q 1 τ`dτ
O q 1
dτ dτ1
γ O 1 Odτ “ γ dτ1 .
ua u 1a
O O 1 q L q 1 Odτ ua p q dτv a
q q 1 ua
O v a O 1 OO 1 O dτ1u 1a “ dτua ` dτv a
dτ dτ1
u 1a“ γ pua
` v aq .
ua ua
ua v a uaua “ ´1 v aua “ 0
γ “ ´uau 1
a .
u 1au 1a “ ´1 uaua “ ´1 v aua “ 0
γ “ p1´ v avaq´1{2 .
v a v ava ą 0 Oγ ą 1 dτ ą dτ1
u
dτ
p
q q'
dτ'
dτv
u'{ {LL'
O ua L v aua ua
v a “ ´ pv b ubq ualooooomooooon
ua
`pv a ` pv b ubq uaq
loooooooomoooooooon
ua
.
v a ua Oua
k ab ” ´uaub ,
hab ” δa
b ` uaub ,
v a “ k ab v
b ` hab v
b
"
k ab u
b “ ua
hab u
b “ 0
"
k ac k
cb “ k a
b ,ha
c hcb “ ha
b .
q pL dl a p q
O ua
dl a “ k abdl
b` ha
bdlb .
p qdt ” }k a
bdlb} g dr ” }ha
bdlb} gO
dt “ ´dl aua ,
dr “`
dlahabdl
b˘1{2 .
pk abdl
bqphacdl c q “ 0 Op q
ds 2 “ ´dt 2 ` dr 2 .
O 1 L 1 L pp q dt 1 ‰ dt dr 1 ‰ dr
´dt 12 `dr 12 “ ds 2
p
uv
h(v)
k(v)p
u
dl
dr
dt
q
L L
O L ua
pa L 1 L p
O p paua
pa “ ´ pp b ubqlooomooon
E
ua` ha
b pb
loomoon
P a
.
OE ” ´paua pa ua
OP a ” ha
b pb
pa P aua “ 0pa
E 2“ P aPa ` m2 .
m “ 0 E “ pP aPaq1{2
m ą 0 u 1a pa “ mu 1a
E “ ´pauaP a “ pa ´E ua v a
OE “ mγ ,P a
“ mγ v a .
L L 1 ua u 1a
v a “ 0 γ “ 1P a “ 0 E “ m
u
p
p
P
E u
LL'
p
ua pL
ab ua
accélération nulle accélération non nulle
p L c b ” ua∇au b
uaua “ ´1 ua c b ub “ 12 u
a∇apu b ubq “ 0ab
c b
ab “ ua∇aub .
p q
δτpp, qq “ 0 .
p qτpp, qq ą τ1pp, qq ą τ2pp, qq
p
q
ττ'τ''
a “ ∇U
Fab
me ua
f a “ eF ab ub
mua∇aub
“ eF b c uc .
f aua “ 0F ab uaub “ F rabsupaubq “ 0
Fab
∇raFb c s “ 0 ,
∇bFab
“ j a .
j a
∇a ja
“ 0 ,
ua µ” ´ j aua
J a ” hab j
b
j a “ ´ p j b ubqlooomooon
µ
ua` ha
b jb
loomoon
J a
.
v a n ej a “ e nv a
ua v aµ“ e n J a
FabEa Ba
Ea ” Fab ub ,
Ba ” ϵab c d ubF c d ,
ϵab c d ϵab c d ϵab c d “ ´4!
E aua “ 0 B aua “ 0
Fab “ 2uraEbs ´ ϵab c d ucB d .
µ0 “ 1
ua
Fab “ Aab eiαψ ,
Aab ψ α
Aab
αÑ `8 ψpiαψq
j a “ 0
∇raAb c s ` iα∇raψAb c s “ 0 ,
∇aAab
` iα∇aψAab“ 0 .
ॠ`8
ka ”∇aψ
kraAb c s “ 0 ,
kaAab
“ 0 .
k a 13pk akaqAb c “ 0
Ab c k aka “ 0 k a
k a
k a
k a∇akb “ k a∇a∇bψ“ k a∇b∇aψ“ k a∇b ka “12∇bpk akaq “ 0 .
ψ“ k a “∇aψ
pa “ ħhk a ħh
O ua
k a“ ´ pk b ubq
looomooon
ω
ua` ha
b kb
loomoon
K a
.
ω ” ua∇aψOO
E “ ´paua “ ħhω k a
K a ” hab k
b OpaP a “ ħhK a
Tab
ua uaua “ ´1ρ nρ“ mn m
ua ρ
ua∇aub
“ 0 .
mV ua
S1 S2 uaş
S1pρuaqdSa dSa
S1 V´
ş
S2pρuaqdSa
ż
BVpρua
qdSa “ 0 ,
ua dSa “ 0V
ş
V∇apρuaqdV “ 0 V
∇apρuaq “ 0 .
∇aua
ua∇aρ“ ´ρ∇aua
u
u
u
dS
dS
dS
𝒱
ρ élevé
ρ faible
S2
S1
ρ u b
ρua∇aub
` u b∇apρuaq “ 0 .
ub ub u b “ ´1ub∇au b “ 0
∇aTab
“ 0 ,
T ab ” ρuau b
´T ab u
b “ ρua “ n pa
ua εp
ε p
p “ ppεq
ua ε p
Tab “ pε` pquaub ` p gab .
ppεq “ 0 ε“ ρ
Tab “ εuaub ` phab hab “ gab `uaubua
∇aT ab “ 0pε` pqua∇au b ` hab∇a p ` pu b∇aua ` u b∇apεuaq “ 0
ub
∇apεuaq “ ´p∇au
a .
ε“ ρ
∇aua ă 0 ∇aua ą 0
εhb c
pε` pqua∇aub
“ ´h b c∇c p .
ua
p “ 0ε` p
Fab
Tab “ FacFc
b ´14gabFc d F
c d .
g abTab “ 0O ua
´T ab u
b“
12
`
E bEb `B bBb
˘
ua` ϵab cEb Bc ,
ϵab c ” ϵab c d ud
O ua
´T ab u
b
ua
∇aTab
“ F b c∇aFac ` F a
c∇aFb c
´12Fc d∇bF c d
“ F b c∇aFac ´32Fc d∇rbF c d s
“ ´F b c jc .
j a “ 0Fab j a
j aFabT ab “ ρuau b
∇aTab
“ u b∇apρuaq `ρua∇au
b“ F b c jc .
Tab Oua ´T a
b ub O
´T ab u
b“ pTb c u
b u cq
loooomoooon
ua´Tb c u
b h calooooomooooon
,
paTab uau b
O ´Tb c u b h ca ua
∇aTab “ 0 .
pE , gabq
pxαq gαβx α
B gαβBx α
“ 0 .
x α Ñ x α`
x αpBαq
a
pxαq
pxαq
k a ” pBαqa kα “ δαα
pxαq k cBc gab “ 0Ba pxαq
k cBc gab “ k c`∇c gab ` Γ dac gbd ` Γ db c gad
˘
“ gbd`
Bakd
` Γ dac kc˘
` gad`
Bb kd
` Γ db c kc˘
“ gbd∇akd
` gad∇b kd ,
Bak b “ 0
∇pakbq “ 0 .
k a
k
k
k
paL
k a paka Lpa∇app
b kbq “`
pa∇a pb˘
kb ` pa p b∇akb “ pa p b∇pakbq “ 0 .
pap b
k a pa Lpxαq k a “ pBαq
a k a pa “ pαpBαq
a
x α
paBa pc “12
`
Bc gab˘
pa p b .
α“ α dpα{dτ “ 0
k a
TabT ab kb
∇apTab kbq “ p∇aT
abqkb `T ab∇akb “ T ab∇pakbq “ 0 .
T ab
T ab kb
Tab “ 0 S1 S2V S1 S2
ż
S2pT ab kbqdSa ´
ż
S1pT ab kbqdSa “
ż
BVpT ab kbqdSa “
ż
V∇apT
ab kbqdV “ 0 .
Q ”ş
S pT ab kbqdSa S
k a Qk a
Q
Tab ≠ 0
Tab = 0
dS
dS
𝒮1
𝒮2
𝒱
pR4,ηabq
ηαβ “ p´1,`1,`1,`1q .
pE , gabq
gabpxαq
p P E
gαβppq “ ηαβB gαβBx γ
ˇ
ˇ
ˇ
ˇ
p“ 0 .
p
pΓ γαβ
ˇ
ˇ
p “ 0 ,
ppd2X α{dλ2qppq “ 0
E, gab)gαβ ≈ ηαβ
p pxαq
peαqa
gabpeαqapeβq
b“ ηαβ .
pe0qa
pei qa i P t1,2,3u
p pe0qa ua
U Ă Ep C q PU p
na pq
xα ” λ nα ,
nα na peαqa λ“ s
λ“ τ p q C
p
n λ
x 1
x 0
x 2
p
q
e2e1
e0
𝒞
U
pxαq pBαqa
peαqa p
gαβppq “ ηαβ .
C xα “ X αpλq
dX α{dλ “ nα d2X α{dλ2 “ 0p
Γ γαβ
ˇ
ˇ
p nαnβ “ 0 .
na
Γ γαβ
ˇ
ˇ
p “ 0B gαβ{Bx γ |p “ 0
Oua L L
pxαq
x 0 “ τ Lx i O p PL
O peαqa
pe0qa ua
p pei qa
L peαqa
ua∇apeαqb
“ 0 ,
Lpe0q
a“ ua
Lpxαq “ pτ, x i q L
L x i “ 0
u
L
e1
e1
e2
e2
ux i = 0
τ
n ě 24
ě 1
f
ωaf ωc
∇a∇bp f ωc q “∇apωc∇b f ` f ∇bωc q
“ p∇a∇b f qωc `∇b f ∇aωc `∇a f ∇bωc ` f ∇a∇bωc ,
a b`∇a∇b ´∇b∇a
˘
p f ωc q “ f`∇a∇b ´∇b∇a
˘
ωc .
R dab c p1,3q
∇a∇bωc “∇b∇aωc ` R dab c ωd .
v a f “ωav a ωa
`∇a∇b ´∇b∇a
˘
pωc vcq “ωc
`∇a∇b ´∇b∇a
˘
v c ` R dab c v cωd .
c dωc
∇a∇b vc
“∇b∇avc
´ R cabd v d .
T c1¨¨¨ckd1¨¨¨dl
pk , l q
∇a∇bTc1¨¨¨ck
d1¨¨¨dl“∇b∇aT
c1¨¨¨ckd1¨¨¨dl
´
kÿ
i“1
R ciab e T c1¨¨¨
iÓe ¨¨¨ck
d1¨¨¨dl
`
lÿ
j“1
Re
abd jT c1¨¨¨ck
d1¨¨¨ eÒ
j¨¨¨dl
.
R δαβγ
R dab c
pxαq
∇a BaΓ cab
∇a∇bωc “ Ba`
Bbωc ´ Γ db cωd
˘
´ Γ eab`
Beωc ´ Γ de cωd
˘
´ Γ eac`
Bbωe ´ Γ db eωd
˘
.
a, b Γ cab
R dab c “ ´2BraΓ
dbsc ` 2Γ ec raΓ
dbse .
a b
pxαq
R δαβγ
“ ´BΓ δβγBxα
`BΓ δαγBxβ
´ Γ δϵαΓϵβγ ` Γ δϵβΓ
ϵαγ .
gαβpxαq g αβ
pxαq
Γ γαβ
“12g γδ
ˆ
B gδβBxα
`B gαδBxβ
´B gαβBxδ
˙
.
p Γ γαβ
ˇ
ˇ
p “0BΓ γ
αβ{Bxδ
ˇ
ˇ
p
p Rab c d “ gd eR eab c
Rαβγδˇ
ˇ
p “12
ˆ
B2 gαδBxβBx γ
´B2 gαγ
BxβBxδ´
B2 gβδBxαBx γ
`B2 gβγ
BxαBxδ
˙
.
S pB pλ,σq S p p 1 q
q 1 S p0,0q pδλ,0q pδλ,δσq p0,δσq
δλ ua δσ s a p p 1 q 1
v a p v a v aB δp2qv a ” v a ´ v a
limδλÑ0δσÑ0
δp2qv a
δλδσ“ u b s c v d R a
b c d .
Bq p
q'
q
v vS
p
p'
δδσ
R dab c “ 0
espace plat espace courbe
p qp
q
Rpabqc d “ 0 ðñ Rab c d “ ´Rbac d ,
Rabpc d q “ 0 ðñ Rab c d “ ´Rabd c ,
Rrab c sd “ 0 ðñ Rab c d ` Rcabd ` Rb cad “ 0 ,
Rab c d “ Rc d ab ,Rarb c d s “ 0 .
nn4
n2pn2´1q{1220
Rbdca
Rcabd
Radcd
Rcdab
Radbc
Rcbda
ωc
Rab c d ωcωd
“ωc`∇a∇b ´∇b∇a
˘
ωc “`∇a∇b ´∇b∇a
˘
pωcωc q “ 0 .
ωcωd ωc
`∇a∇b ´∇b∇a
˘
gc d “ R eab c ge d ` R e
abd gc e “ Rab c d ` Rabd c .
a b c
R drab c s
ωd “ 2∇ra∇bωc s “ 0 ,
∇a∇bωc
ωd
∇raRb c sd e “ 0 .
Rac ” R bab c
Rab “ Rba
˘Rab
R ” g abRab “ R abab
g c e
∇aRbd ´∇bRad `∇cRc
abd “ 0 .
g ad 2∇aRab “∇bR
∇a´
Rab ´12R gab
¯
“ 0 .
Gab ” Rab ´ 12 R gab
pLσqσPI σI Ă R Lσ λ P R Σ Ă E
n “ 2 pxαq “ pλ,σq
Σ ua ” pBλqa
ua∇aub
“ 0 .
s a ” pBσqa Σ
dσ dσ s a p PLσpλ,σq q PLσ`dσ pλ,σ ` dσq
Ba Γ cabpxαq pBαq
a
ua∇a sb
“ s a∇aub .
dσ s aLσ Lσ`dσ
s aua Lσuaua
ua∇apsb ubq “
`
ua∇a sb˘
ub “`
s a∇aub˘
ub “12s a∇apu
b ubq “ 0 .
Lσ s aua “ 0s a ua
LσLσ ua∇a s b
u
s
Lσ
p q
σ
λΣu
s
Lσ+dσ
Lσ ua∇a s b
Lσ u c∇c pu b∇b s aq
ua∇a s b
u c∇c pub∇b s
aq “ u c∇c ps
b∇b uaq
“ pu c∇c sbq∇b u
a` u c s b∇c∇b u
a
“ ps c∇c ubq∇b u
a` u c s b∇b∇c u
a´ R a
c bd s b u c ud
“ s c∇c pub∇b u
aq ´ R a
c bd s b u c ud
“ R ab c d s b u c ud .
b c
R dab c “ 0
u b∇b s a “ 0
espace plat espace courbe
Rab c d ‰ 0
ua
9” ua∇a
:s a “ K ab s
b ,
Kab ” Rb c d au c ud ua Kab “ Kba Kab u b “ 0
p pe ai ,κi q
K ab
K ab e
bi “ κi e
ai .
e ai κi e ai ua “
Kab ua e bi “ 0
:e ai “ κi eai .
κi e aiLσ |κi |
u b∇b s a “ 0L T L
L
e ai
u
e1 e2
e3
e1 e2
e3
espace-temps espace local de repos
e1 e2
e1 e2
u
LT
GabTab
x pt q:x “ ∇U U ą 0 s ” x1 ´ x2
} s }
:s “ ps ¨ ∇q∇U .
R ab c d u c ud
ÐÑ BaBbU .
∆U ” BaBaUρ
∆U “ ´4πρ .
G “ c “ 1
Tab
Tab uau b
ÐÑ ρ .
Rab uau b “ 4πTab uau b
ua
Rab?
“ 4πTab .
∇aTab “ 0 ,
Rab Tab∇aR “ 0 R“ g abRab T ” g abTab
Rab ´12R gab “ 8πTab .
R “ ´8πT
Rab “ 8π´
Tab ´12T gab
¯
.
Tab uau b » ´T » ρRab uau b “ 4πTab uau b
pxαq
gαβpx q gab
gαβ
Fab j agab
Tab
T ab “
ρuau b ∇aG ab “ 0 ∇aT ab “ ρua∇au b `u b∇apρuaq “ 0∇apρuaq “ 0 ua∇au b “ 0
Rab c d “ 0 ER4
gab “ ηab
Tab “ 0 Rab “ 0Rab c d “ 0
Gc 8πG{c 4
8πGc 4
» 2ˆ 10´43´2
¨ ´3.
pua L p
θ ”∇aua
“1V
dVdτ
Vθ
θτ ua
dθdτ
“ ´13θ2
´ Rab uau b .
RabTab
dθdτ
“ ´13θ2
´ 4π pε` 3pq ,
ε p ε`3p ą 0θ
p
σab ”∇raubs
ωab ”∇paubq ´ 13 θ hab
Eabp g q “ Tab ,
Eabp g q
∇aEab “ 0 ,
n “ 4 Eabp g q
Eab “ αGab `β gab ,
α,β PRGab `Λ gab “ 8πTab
Λ
gab ηab
gab “ ηab ` hab ,
hab
pxαq
ηab ηαβ “ p´1,`1,`1,`1q
|hαβ| ! 1 .
ˇ
ˇBhαβ{B tˇ
ˇ !ˇ
ˇBhαβ{Bx i ˇˇ ,
Γ γαβ
“12ηγδ
ˆ
BhδβBxα
`BhαδBxβ
´BhαβBxδ
˙
.
Gab
G00 “ ´∆h00 .
px i q
Tab
T00 “ ρ .
U “12h00 .
pxαq
d2X α
dτ2` Γ αβγ
dX β
dτdX γ
dτ“ 0 ,
xα “ X αpτq
dX α{dτ p1,0,0,0q
τ t
d2X i
dt 2“ ´Γ i00 “
12
Bh00Bx i
“BUBx i
,
Gc
G » 6,67ˆ 10´11 3¨
´1¨
´2 ,
c “ 299 792 458 ¨´1 .
G “ 1 c “ 1
M
M p q “Gc 2
ˆ M p q “ 7,42ˆ 10´28¨
´1ˆ M p q .
M@ “ 1,47MC “ 4,43 MK “ 54,5
M 2M3
G c
G
c
qpt q 9q ” dq{dtLpq , 9qq
t1 t2
S rqpt qs ”
ż t2
t1
Lpqpt q, 9qpt qqdt .
δq qpt q δqpt1q “ δqpt2q “ 0δL “ pBL{Bqqδq `pBL{B 9qqδ 9q Lpq , 9qq
δ 9q “ dpδqq{dt
δS “
ż t2
t1
dt δL “
ż t2
t1
dt"
BLBq
´ddt
ˆ
BLB 9q
˙*
δq ,
δS “ 0δqpt q
ddt
ˆ
BLB 9q
˙
“BLBq
.
Lpqn , 9qnq nqnpt q 9qn “ dqn{dt n
pqn , 9qnq
L P pλq L v apxαq L
xα “ X αpλq v apBαq
a vα “ dX α{dλ” 9X α
p “P pλq q “P pλ1q L
τpp, qq “
ż λ1
λ
`
´ gαβ 9X α 9X β˘1{2 dλ .
LpX α, 9X αq ”
`
´ gαβpX γq 9X α 9X β˘1{2 ,
BLBX γ
“ ´12L
B gαβBx γ
9X α 9X β ,
BLB 9X γ
“ ´1Lgγβ 9X β .
Lλ“ τ L “ 1
pt , q , 9qq Ñ pτ,X γ , 9X γ q
ddτ
´
gγβ 9X β¯
“12
B gαβBx γ
9X α 9X β .
gαβx α gαβ 9X β L
gγβ :X β`
ˆ
B gγβBxα
´12
B gαβBx γ
˙
9X α 9X β“ 0 .
α βg σγ g σγ gγβ “ δσ
β
:X σ`
12g σγ
ˆ
B gαγBxβ
`B gγβBxα
´B gαβBx γ
˙
9X α 9X β“ 0 .
λ“ τΓ σαβ
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