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

τ

xa = (x0, xα) , α = 1, 2, 3

x0

xα(x0) = 0, d xα = 0

x0

xi = const

d τ =

1

√

d x0.

g00

c

x0

g00 > 0

x0

g00 > 0

g00 > 0

p

xµ

q

xµ + d xµ

p

q

d s2 = gµν d xµ d xµ = 0

d x0

d x(1,2) = −

g00

s

−g00

+

g002

d xi d xj .

0

g0i d xi

gij

g0ig0j

q(1)

x0 + d x(1)0 , xi + d xi

p

0

0

i

(x0, xi i)

q(2)

x

+ d x(2)

, x + d x

q(1)

q(2)

d x0 = d x(2)0 − d x(1)0

q(1)

p

q(2)

c/2

d l

p

q

q(1)

q(2)

x0

(xi + d xi)

g0ig0j

d l = pγij d xi d xj , γij = −gij +

.

g00

γij

(x0′ = x0, xi′ = fi(xj ))

γij

d2xµ

µ

dxν dxλ

+ νλ

= 0

ds2

ds

ds

λ

f = ae−iψ

a = const, ψ = kµxµ, kµkµ = 0

kµ =

∂ψ

, gµν kµkν = 0.

µ

∂x

ψ

kµ

dkµ = 0

( kk)µ = 0

d kµ

µ

ν

λ

µ

µ

+ νλk

k

= 0,

d x = k

d λ.

d λ

δS = −mcδ Z1

2 d s = 0,

d s =

gµν (x) d xµ d xν

.

2

q

2gµν d xν d

δxµ

gλν,µ d xλ d xν δxµ

δS = −mc Z1

+

d s .

2 d s

2 d s · d s

δS = −mc Z1

2

gµν uν d δxµ + 2

+ gλν,µuλuν δxµ d s .

1

δS = −mc gµν uν δxµ 1 + mc Z

ds δxµ

ds(gµν uν ) − 2 ∂xµ

· uλuν .

2

d

1 ∂gλν

d

ν

d uν

d xλ

λ

d uν

1

λ

ν

(gµν u

) = gµν

+ gµν,λ

u

= gµν

+

(gµν,λ

+ gµλ,ν ) u

u

,

d s

d s

d s

2

d s

2

d uν

1

δS = −mc gµν uν δxµ 1 + mc Z d sδxµ gµν

+

(gµν,λ + gµλ,ν − gλν,µ) uλuν =

d s

2

µ 2

d u

ρ

ν

µ

ρ

λ

u

ν

= −mc gµν u δx

1 + mc Z

d sδx gµρ d s

+ λν u

.

1

1

d xµ d xν

δS = −

δ Z

d τ

gµν (x)

+ e(τ)(mc)2

.

2

e(τ)

d τ

d τ

τ

e(τ)

hµν

0

hµν

u

0

00i =i − i 00

x = ct

d s = c d t

= 1 |u | 1

d2 xi

2

i

2

= −c

00

= c

i 00.

d t2

i 00 = −

1 ∂g00

∂g0i

+

.

2 ∂xi

∂x0

d2xi

= −

c2

∂g00

.

dt2

2

∂xi

q

x0 − Zl g00 d xi .

g0i

x0

p

x0

q

p

x0

p q

τ = 1 +

φ

x0

.

c2

c

x0

ω0 = c∂ψ/∂x0

ψ

x0

ω0

ω =

∂ψ

=

∂ψ ∂x0

= (g00)−1/2 ω0.

∂τ

∂x0

∂τ

ω =

1 − c2

ω0

ω0

= −c2 .

φ

ω

φ

ω2 = (1 + φ0/c2)ω1

φ0

ω(1+φ/c2) = const

ω

= −

φ

.

ω

c2