http://letiec.yolasite.com

10´43

M R

Ξ”GMc 2R

G c

ΞC „ 10´9

Ξ@ „ 10´6

Ξ „ 10´3

Ξ „ 0,21,4Md

γ3Md

mM R v

mv2{2 “ GmM {Rv “ c

2GMc 2R

“ 1 ðñ Ξ“12.

MR

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

M RGM {pc 2Rq

M R “ 2GM {c 2

„ 30Md

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

R4

R2

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

t agab t a t b

p P Ep

p ppE , gabq

pa

gab pa p b

“ 0 .

L

pa

pa

pa

gab pa p b

“ ´m2 ,

m ą 0m Ñ 0

p

p

p

p

p

p

ligne d'univers

L

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

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dt ua `dt na dt ua ´dt na

pua` na

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uana “ 0 .

p E pO p

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p

u

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p τ τ1 O O 1 pdτ1 O 1 p q 1 τ`dτ

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ua u 1a

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q q 1 ua

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dτ dτ1

u 1a“ γ pua

` v aq .

ua ua

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γ “ ´uau 1

a .

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γ “ p1´ v avaq´1{2 .

v a v ava ą 0 Oγ ą 1 dτ ą dτ1

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v a “ ´ pv b ubq ualooooomooooon

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loooooooomoooooooon

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.

v a ua Oua

k ab ” ´uaub ,

hab ” δa

b ` uaub ,

v a “ k ab v

b ` hab v

b

"

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k ac k

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b .

q pL dl a p q

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dl a “ k abdl

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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

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pa L 1 L p

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pa “ ´ pp b ubqlooomooon

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ua` ha

b pb

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.

OE ” ´paua pa ua

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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

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L L 1 ua u 1a

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u

p

p

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p

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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 .

m ua

f a

f b“ mab .

ua

f aua “ 0 .

ff

f a

ua f a

ab “ 0

ua∇aub

“ 0 .

L

a = 0

a

u

a

L

p

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

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∇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

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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

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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

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∇aua

ua∇aρ“ ´ρ∇aua

u

u

u

dS

dS

dS

𝒱

ρ élevé

ρ faible

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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

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b ´14gabFc d F

c d .

g abTab “ 0O ua

´T ab u

b“

12

`

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˘

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ϵab c ” ϵab c d ud

O ua

´T ab u

b

ua

∇aTab

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c∇aFb c

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“ 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

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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 ,

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a“ ua

Lpxαq “ pτ, x i q L

L x i “ 0

u

L

e1

e1

e2

e2

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τ

pE , gabq pE , gabq

pκ1 κ2 e a1 e a2na

R3

n e2

1/κ1

1/κ 2

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n

e1

e1

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n ě 24

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f

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“ 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

,

m F “ m ∇UU

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

m

l

t

s

τ

E p { q2

P {

L 2{ 2

L 2{ 3 0

ε {p 2q ´2

p {p 2q ´2

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

Egab

ds 2

∇aΓ cab

Lpa

m

τ

ua

E

P a

L

Tab

ρ

ε

p

k a

Rab c d

Rab

R

Gab

Λ