Сборник лекций по общей теории относительности(С. Н. Вергелес)

√

gµν

−ggµν

√

1√

µν

λρ

λµ

ρν

δ −g =

−gg

δgµν ,

δg

=

−g

g

δgµν −→

2

−→ δ √−ggλρ =

√−g

2gλρgµν − gλµgρν δgµν .

1

gµν

c3

1

δgSg =

Z d(4) x√−g Rµν

−

gµν R δgµν .

16πG

2

Λ =

1

εabcd Rab ωc ωd

(13.17)

4

εabcd

ε0123 = 1

Rab = Rab dxµ

dxν

ωab dxµ

ω

a

µν a

µ

µ

= eµ dx

εabcd eλc eρd = (det eσc ) εµνλρ eaµebν

gµν = ηab eµa eνb

g = (det ηab)(det eµa )2 = −(det eµa )2

det ea

> 0

det ea

= √

g

µ

µ

−

εabcd eλc eρd = Eµνλρ eaµebν

(13.18)

Eµνλρ

Λ =

1

dxµ dxν dxλ dxρ Eστλρ eaσebτ Rµνab

4

eaσebτ Rµνab = Rµνστ ,

dxµ dxν dxλ dxρ = dx0 dx1 dx2 dx3 · εµνλρ

1

√

0

1

2

3

στ

Λ =

−g dx dx dx

dx

· εµνλρεστλρ Rµν

4

εµνλρ εστλρ = 2 (δσµδτν − δτµδσν ) ,

Λ = dV · R ,

R = Rµνµν ,

dV = √

d4x

−g

(13.19)

c

Sg = −

Z

d4x √−g R

(13.20)

2κ2

gµν

c

c

Sg = −

Z

Λ = −

Z

d4x √−g eaµebν Rµνab

(13.21)

2κ2

2κ2

δ ( Sg + S ) = 0

(13.22)

S

ωab

, ea

q

q

µ

µ

δΛ = d

2 εabcdωc ωd δωab − 2

εabcd T c ωd δωab + 2 εabcd Rab ωc δωd

(13.23)

1

1

1

T c

ωab = 0

d δωab =

1

δRab ,

dωc =

1

T c

2

2

1

1

εabcd δRab ωc

ωd =

εabcd ωc ωd dδωab =

4

2

= d

2 εabcd ωc ωd

δωab − 2 εabcd T c ωd δωab

1

1

ωab = 0

S

ωab

δωab

εµνλρεabcd eνc Tλρd

= 0 ,

eλ

a

Tµνa

= (eµa Tνλb − eνa Tµλb ) ebλ

(13.24)

eν

T a

eν

= 0

a

µν

a

∂µeνa − ∂ν eµa + ωµabebν − ωνabebµ = 0

(13.25)

∂′∂xµ

Rab

µν

1

√

µν

√

a

a

ν

λ ν

c

−g g

δ gµν = δ

−g ,

δ gµν = eaµ δeν + eaν δeµ , δeb

= −eb ec

δeλ

(13.26)

2

δSg

δeµa

c

4

√

µ

ν

1 µ

d

x

−g (Rν

ea

−

ea R )

(13.27)

κ2

2

δgµν

δeµa

δS

1

4

√

µν

−

d

x

−g T

eaν

(13.28)

c

κ2

Rµν − 2 gµν R − c T µν eaν = 0 ,

(13.29)

c

1

1

c

δSg = −

Z

d (εabcd ωc ωd δωab )+

4κ2

c

1

+

Z

d4x √−g (Rµν −

gµν R ) δgµν

(13.30)

2κ2

2

Sg = −

c3

Z

d4x √−g ( R + 2Λ )

(13.32)

16π G