Сборник лекций по общей теории относительности(С. Н. Вергелес)
.pdf
(a, b, c, d)
(k) = Rbc da + Rbd ac + Rba cd = 0,
(h) = Rcd ab + Rca bd + Rcb da = 0, (l) = Rda bc + Rdb ca + Rdc ab = 0.
1
2 {(i) + (k) − (h) − (l)} = Rad bc − Rbc ad = 0,
|
|
n = 2, 3, 4 |
n = 2 |
|
R12 12 |
n = 3 |
|
Rab ab |
Rab ac |
c |
a b |
a |
|
a |
bc
n = 3
Rab = Rac cb.
{ea} |
Rab |
|
|
n = 4 |
|
|
Rab ab |
|
Rab ac |
c |
a b |
a |
|
|
|
a |
b |
c |
|
|
|
|
Rab cd |
R01 23 R02 31 R03 12
{ea}
gνρ ρµλ = νµλ
gµν ,λ = νµλ + µνλ.
(µ, ν, λ)
gνλ ,µ = λνµ + νλµ, gλµ ,ν = µλν + λµν .
νλ = |
2gµσ |
∂xλ |
+ ∂xν |
− ∂xσ |
≡ |
||
µ |
1 |
|
∂gνσ |
∂gλσ |
|
∂gνλ |
|
νλ = |
µ |
− 2Tνλ − |
2gµσ (Tλνσ |
|
µ |
ν λ |
1 µ |
1 |
|
ωa
Cbca = −Ccba
ωba = γbca ωc
1
Cabc = 2(γabc − γacb).
Cabc = ηadCbcd γabc = ηadγbcd , γabc = −γbac
ν λ
.
µ
Tλµν ≡ gλρTµνρ
+ Tνλσ) .
dωa = Cbca ωb ωc
ωa
(a, b, c)
ωab = (Cabc − Cbac − Ccab) ωc.
(Tbac −
Tabc + Tcab)ωc/2 |
Tbac = ηbdT d |
|
ac |
R3
d s2 = a2(d θ2 + sin2 θ d φ2) = ωθ2 + ωφ2.
(eθ, eφ)
ωθ = a d θ, ωφ = a sin θ d φ.
d ωφ = a cos θ d θ d φ d ωφ−a−1 ctg θωθ ωφ = 0 d ωφ + ωφθ ωθ = 0
ωθφ = −ωφθ = −a−1 ctg θωφ + γωθ.
d ωθ = 0
γ = 0
ωθφ = −a−1 ctg θωφ.
d ωθφ = a−2ωθ ωφ.
Rθφ θφ = a−2.
|
|
|
|
|
|
|
|
|
|
|
|
|
µJµ |
||||
|
|
|
|
|
∂Jµ |
|
|
|
|
|
|
|
|||||
|
|
µJµ |
= |
|
|
|
|
+ νµµ Jν , |
|||||||||
|
|
∂xµ |
|||||||||||||||
|
|
µ |
|
|
1 |
|
|
µσ ∂gµσ |
|
|
|
|
|||||
|
|
ν µ |
= |
|
|
|
g |
|
|
|
|
|
. |
|
|
|
|
|
|
2 |
|
|
∂xν |
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
g = det gµν |
|
|
|
|
|
|
|
|
d g = ggµν d gµν |
||||||||
gµν |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
d g |
|
|
|
|
|
|
|
|
|
|
|
|
|
d gµν |
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
∂x |
|
|
|
|
|
|
|
|
|
|
|
|
|
p |
|
|
gµσ |
∂gµσ |
= g−1 |
∂g |
|
, µ |
= |
∂ ln |g| |
. |
|||||||||
|
∂xν |
|
|
||||||||||||||
|
ν |
|
|
|
|
|
|
νµ |
|
|
|
∂xν |
|||||
|
|
|
1 |
|
|
|
∂ |
|
|
|
|
|
|
|
|||
|
µJµ = |
|
|g| |
∂xµ |
|
|
|g| Jµ . |
||||||||||
|
|
|
|
|
|
|
|
|
|
|
p |
|
|
|
|
||
p
a
ωθφ ωφ = 0 →
ggµν
φ
gµν ∂φ/∂xν
φ ≡ φ;;µµ = |
|
|
g |
∂xµ |
|g|gµν ∂xν . |
|||||
|
1 |
|
|
∂ |
|
p |
|
|
∂φ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| |
| |
|
|
|
|
|||
|
p |
|
|
|
ν Tµν |
|
T µν |
|||
Jµ(x)
εµ1µ2...µn
gµ′ν′
ν Tµν = ∂x∂ν Tµν + νλν Tµλ − λµν Tλν =
= ∂x∂ν Tµν + Tµν ∂x∂ν ln p|g| − λµν T λν ,
T λν = T νλ
λµν T λν = |
1 |
( λµν + νµλ)T λν = |
1 |
T λν |
∂ |
gλν . |
||||||||||||||
2 |
|
µ |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
∂x |
|||||
|
|
|
|
|
1 ∂ |
|
|
|
1 |
|
|
∂ |
||||||||
|
|
|
|
|
(p|g|Tµν ) − |
|
||||||||||||||
ν Tµν = |
|
|
|
|
|
|
|
T λν |
|
gλν . |
||||||||||
|
p |
|
|
|
∂xν |
2 |
∂xµ |
|||||||||||||
|
|g| |
|||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(n − 1) |
||||
ωJ = |
|
|
1 |
|
|
Eµ1µ2...µn Jµ1 d xµ2 |
. . . d xµn . |
|||||||||||||
|
|
|
|
|
||||||||||||||||
(n |
− |
1)! |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
Eµ1µ2... |
µn (x) |
|||||||
p
Eµ1µ2...µn (x) = |g(x)|εµ1µ2...µn ,
0, ±1 |
ε012...(n−1) = 1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(n, 0) |
|
|||||
|
|
|
|
|
|
∂xµ1 |
|
|
∂xµn |
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
Eµ1′ ...µn′ (x(x′)) = |
|
|
|
|
|
|
|
p |
|
|
|
|
|
|||||||
∂x |
|
|
|
|
|
|
|
|||||||||||||
|
∂xµ1′ |
. . . ∂xµn′ |
|
|g(x)|εµ1...µn |
|
|||||||||||||||
|
|
|
= det ∂x′ |
p |
|
|
|
|
|
|
||||||||||
|
|
|g(x)| |
εµ1′ ...µn′ . |
|
|
|
|
|
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
det |
∂x′ |
2 |
|||||
(x′(x)) = ∂xµ′ ∂xν′ gµν (x) → g′(x′(x)) = |
||||||||||||||||||||
g(x). |
||||||||||||||||||||
|
∂xµ |
|
∂xν |
|
|
|
|
|
|
|
|
|
|
|
|
|
∂x |
|
|
|
|
|
|
|
Eµ1′ ...µn′ (x(x′)) = ±p |
|g′(x′)| |
εµ1′ ...µn′ . |
′ |
||||||
|
|
|
|
|
|
|
|
|
|
|
|
det (∂x/∂x ) |
|
|
|
|
|
|
(n − 1) |
|
|
|
|
ωJ |
|
||
d ωJ = " |
1g |
|
|
|g|Jµ1 |
# |
|
|
|
|
|
d xµ2 . . . d xµn . |
||
| |
∂xµ |
(n |
1 |
1)! |
|
|g|εµ1...µn d xµ |
|||||||
|
| |
∂ |
p |
|
|
− |
|
p |
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
p
|
|
|
|
|
|
µ |
µ1 |
|
|
|
|
|
|
|
|
|
µ2, . . . , µn |
= " |
|
|
|
|
µ = µ1 |
|
# |
|
|
|
|
|
|||
d ωJ |
|
|
g |
|
∂xµ |
|
|g|Jµ |
|
n!Eµ1...µn d xµ1 |
d xµ2 . . . d xµn . |
||||||
|
| |
|
| |
|
|
p |
|
|
|
|
|
|
|
|
||
|
1 |
|
|
|
∂ |
|
|
|
|
|
1 |
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
p |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
X |
Y |
|
|
|
|
|
|
|
|
|
|
|
n |
Y |
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
X |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Z |
d ωJ = Z |
ωJ . |
|
||
Y∂Y
|
ZY ( µJµ) d V = Z∂Y Jµ d Sµ. |
||||
(n − 1) |
|
|
|
|
|
d Sµ = |
|
1 |
|
Eµµ1...µn−1 d xµ1 |
. . . d xµn−1 . |
|
|
|
|||
(n |
− |
1)! |
|||
|
|
|
|
|
|
Y |
0 |
≤ t |
0 |
|∂Y = 0 |
|
|
|
|
|
0 ≤ x |
|
d x |
|
|
|
µ |
|||
|
|
|
d S0 |
|
|
|
|
x |
|
x0 = t |
|
|
|
|
|
|
|
|
|
|
|
|
|
d S0 |
= |
|
t dx1 |
. . . dxn−1 |
= dS(t). |
|
|
|
|
|g| |
|||||
|
|
|
|
|
p |
|
|
|
|
Z
Q = J0p|g| d x1 . . . d x(n−1) = Const.
t
Q |
Jµ |
Y
∂+Y ∂−Y ∂+Y
Z
Q = Jµ d Sµ
∂+Y
µJµ = 0
|
|
xa |
p X |
|
|
|||
|
|
p |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
K0 |
|
|
|
p |
|
|
|
|
{ea} |
p |
|
|
|
|
s |
p |
||
|
|
|
|
|
|
|||
|
p |
|
|
|
|
|
|
|
|
|
|
p s = 0 |
|
|
|
|
|
|
|
|
va |
a d xµ(s) |
p. |
|
||
|
|
|
= eµ |
|
|
|
||
|
|
|
d s |
|
||||
µ |
|
p |
|
|
|
|
|
p |
d x (s)/ds |
{va} |
|
|
|
|
|||
|
| |
|
|
|
|
|
|
|
|
|
|
xa(s) = vas, |
|
|
|||
|
|
p |
|
|
va |
p |
||
s |
|
|
|
|
|
|
|
|
p
xa |
xµ |
xµ(s) = vµs
d xµ(s) = vµ = const, d s
d2 xµ(s) |
µ |
d xν (s) d xλ(s) |
µ |
||
|
+ νλ x(s) |
|
|
|
= 0 → νλ vs vν vλ = 0. |
d s2 |
d s |
|
d s |
||
{ea} |
xµ = 0 |
xµ(s) |
ωµab(vs) d(vµs) = 0 → ωµab(vs)vµs = 0.
(∂µeaν ) (vs) + ωbµa (vs)ebν (vs) − λνµ(vs)eaλ(vs) = 0.
v |
µ |
|
|
|
|
|
|
|
|
|
µ ν |
(8.26) |
vµ vν |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
vν vµ (∂µeνa) (vs) = 0. |
|
|||||
|
|
|
|
vµ∂µ |
|
|
vν vµ∂µ = vµ∂µvν |
||||||
|
|
|
|
|
|
|
vµ∂µ (eνavν ) (vs) = 0. |
|
|||||
|
|
|
|
|
|
eνavν(vs) = eνavν (s = 0) = va. |
|
||||||
0 ≤ t ≤ 1 |
|
|
|
X × I |
I |
X × I |
p |
||||||
|
|
|
{xµ, t} |
xµ |
|||||||||
|
|
|
|
|
|
|
|
|
X × I → X , |
|
|||
|
|
|
|
|
|
|
|
|
|
|
µ = txµ. |
|
|
|
|
|
|
|
|
|
|
|
|
x |
|
||
|
|
|
µ |
|
|
|
|
|
|
|
|
|
p X |
|
|
x |
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
Ω X → Ω (X × I) , |
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Ω X |
|
|
|
|
|
|
|
d |
|
µ = xµ d t + t d xµ. |
|
|||
|
|
|
|
x |
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
ωab |
ωa |
|
|
xµ → |
|
µ, d xµ → d |
|
µ |
|
|
|
||||
|
|
x |
x |
|
|
|
|||||||
ωµab(x) (xµ d t + t d xµ) = tωµab(x) d xµ = ωab, ωab|t=0 = 0.
ωa = eaµ(x) (xµ d t + t d xµ) = eaµ(vst) (vµs d t + t d xµ) = xa d t + ωa, ωa|t=0 = 0.
xa |
ωab ωa |
dxa |
d t |
|
|
|
|
|
|
|
|
|
ab |
|
|
|
|
∂ω |
|||||
d ωab = d t |
|
|
|
+ δω |
ab, d ωa = d t |
|
∂t |
||||||
δ |
|
|
|
|
|
|
xa
|
|
a |
|
|
|
|
|
|
|
|
ba |
|
|
|
|
|
|||
∂ω |
|
a |
|
a b |
|
|
∂ω |
a |
c |
|
d |
|
|||||||
|
|
|
|
|
|
|
|
|
|||||||||||
|
|
|
= dx + ωb x , |
|
|
|
|
|
|
= Rb cdx ω |
, |
||||||||
|
|
|
|
|
∂t |
||||||||||||||
∂t |
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
O(t3) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a |
|
|
a |
1 |
|
3 a |
b c |
d |
|
|||||
|
|
|
|
|
|
|
|
||||||||||||
|
|
|
ω |
|
= t d x + |
|
t Rb cdx x |
d x , |
|
||||||||||
|
|
|
|
6 |
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
t = 1 ωa = ωa, ωab = ωba
− d xa + |
|
|
a |
|
|
|
∂ω |
|
|
a. |
|||
|
|
|
+ δω |
|||
∂t |
||||||
d t
ωab|t=0 = ωa|t=0 = 0.
t
ωab = 12t2 Rba cdxc d xd.
|
2 |
a |
|
b |
a |
b |
1 |
c d a |
|
1 |
2 a c |
d |
|
d s |
|
= ηabω |
ω |
|
= gab d x |
d x , gab = ηab + |
|
Rac dbx x , ωb |
= |
|
t Rb cdx |
d x . |
|
|
|
3 |
|
||||||||||
|
|
|
|
|
|
|
|
|
2 |
|
|
||
p
Rab cd = 0 |
p |
p |
|
|
Rab cd = 0 |
µνλ
K0
(t′, x′, y′, z′)
z
ω
t = t′, |
z = z′ |
x = x′ cos ωt′ + y′ sin ωt′, |
y = −x′ sin ωt′ + y′ cos ωt′. |