3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.2. . . . . . . . . . . . . . . . . . . . 17

6.3. - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.5. - . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.6. - ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.7. . . . . . . . . . . . . . . . . . . . . . . . 44

6.8. , , . . . . . . . . 50

6.9. . . . . . . . . 65

7.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2. ,

. . . . . . . . . . . . . . . . . . . . . . . . 87

7.3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.1. . . . . . . . . 98

8.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.3. C

. . . . . . . . . . . . . . . . . . . . . 118

8.4.

, . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9. . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.1. ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.2.

- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.3. - -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9.4. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Borodin

Borodin.. ( )

Borodin .

Borodin

Borodin8.2. ()

Borodin.

Borodin

ChirokSticky Note

4

9.5.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

9.6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

10.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

10.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

10.2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

10.3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

a). . . . . . . . . . . . . . . . . . . . . . . . . . . 217

b). . . . . . . . . . . . . . . . . . . . . . . . . 234

11.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

11.1. . . . . . . . . . . . . . . . . . 239

11.2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

11.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

12.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

12.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

12.2. . . . . . . . . . . . . . 264

12.3. . . . . . . . . . . . . . . . . . . 275

12.4. ,

, . . . . . . . . . . . . . . . . . . . . . . 280

12.5.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

13.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

13.1. . . . . . . . . . . . . . . . . . . . . . 292

13.2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

13.3. - . . . . . . . . . . . . . . . 302

13.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

13.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

13.6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

13.7.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Shliakhovskaya .

Borodin

Borodin

Borodin : " "

5

13.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

13.9.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

13.10.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

13.11. . . . . . 351

13.12. . . . . . . . 354

13.13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

13.14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

13.15.

. . . . . . . . . . . . . . . . . . . . . . . . . 360

13.16.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

13.17.

. . . . . . . . . . . . 367

14. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

14.1. ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

14.2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

14.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

14.4. , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

14.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

14.6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

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, 2014.

Borodin

6.1. 15

6

C

B -

(). -

, , -

[1].

() -

, -

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.

()

,

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6.1.

-

4- ( ),

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.

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, , F .

.

- -

(+) V = + + VF , (6.1) V - 4 - ,

gVV = 1, (6.2)

g - ,

= (V) V(F), (6.3) = [V] V[F], (6.4)

F = VV. (6.5)

, , -

, - .

,

.

6.2. 17

(6.1) :

1. , 4 - V , , - -

.

, , F (6.3-6.5), (6.1) .

2. , , F. - (6.1)

V g . (6.1) (6.2) ,

.

4 -

2Vxx

=2Vxx

. (6.6)

2[]V = 2[]V +(x

x+

)V,

c (6.1-6.6),

R,V = 2[] + 2[] + 2[(V]F). (6.7)

(10.1), (10.7), R, - , , -

- , -

. (6.7)

() [7], [8].

6.2.

-

.

, .

. va

vaxb

+vbxa

= 0

vaxb vbxa

= 0

18 6.

dvadt

=vat

+ vbvaxb

= aa = const. (6.8)

(6.8)

, .. .

, .. ,

. (6.8)

va = aat+ v0a, (6.9)

v0a - . -, , (6.8)

va =

t0

aa()d + v0a. (6.10)

,

-

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20 6.

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[11].

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, ,

dS2 = (dx0)2 (dx1)2 (dx2)2 (dx3)2, (6.11) x0 = ct , x1 , x2 , x3 - , :

x1(y1, t) = y1 + (c2/a0)[

1 + a02t2/c2 1],x2 = y2, x3 = y3, x0 = y0 (6.12)

x1(y1, ) = y1 + c2/a0[cosh(a0/c) 1],x2 = y2, x3 = y3, t = (c/a0) sinh(a0/c), (6.13)

(6.12)

, (6.13) - . (6.12) (6.13) (6.11) [11]

dS2 =c2dt2

1 + a02t2/c2 2 a0tdtdy

1

(1 + a02t2/c2)1/2

(dy1)2 (dy2)2 (dy3)2, (6.14)

dS2 = c2(d)2 2 sinh(a0/c)cddy1 (dy1)2 (dy2)2 (dy3)2, (6.15)

(6.14), (6.15) [133] -

kl = gkl + g0kg0l/g00, " "

dl2 = (1 + a02t2/c2)(dy1)2 + (dy2)2 + (dy3)2, (6.16)

dl2 = cosh2(a0/c)(dy1)2 + (dy2)2 + (dy3)2. (6.17)

, , -

, .

6.2. 21

x1(y1, T ) = y1 cosh(a0T/c) + c2/a0[cosh(a0T/c) 1],

x2 = y2, x3 = y3, t = c/a0(1 + a0y1/c2) sinh(a0T/c), y

0 = cT,(6.18)

dS2 = (1 + a0y1/c2)2c2(dT )2 (dy1)2 (dy2)2 (dy3)2. (6.19)

(6.12), (6.13), (6.18) -

,

(6.11).

-

-

.

( ) -

( ),

- .

(6.1) ,

= = 0, gVV = 1, gF

F = a02/c4 (6.20) (6.7) (6.3) (6.4) , 0

c . C (6.1) -

V = VF (6.21)Ee -

,

Vk = Vk = 0, V 0 = g00

1/2, V0 = g001/2. (6.22)

-

x1.

dS2 = D(X1)(dX0)2 A(X1)(dX1)2 (dX2)2 (dX3)2. (6.23) A(X1) - ,

g0k - [134]. (6.21)

A(X1) =c4

4a20D2

(dD

dX1

)2. (6.24)

22 6.

(6.24) (6.7) -

. Xi yi

dy1 = A1/2dX1/2, X0 = y0, X2 = y2, X3 = y3,

dS2 = exp

(2a0y

1

c2

)(dy0)2 (dy1)2 (dy2)2 (dy3)2. (6.25)

a0 y1[7].

(6.25)

F 1 =DV 1

dS=dV 1

dS+ 100

(V 0)2

=1

g00100 =

g11

2g00

g00y1

=a0c2. (6.26)

4- .

(6.25)

y2 = y3 = 0, y1 = const. ,

s = exp

(a0y

1

c2

), (6.25a)

s - , - .

, . -

[6 c.109] (6.25) -

. -

, -

(6.19).

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.. [15 . 281],

.

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[133]

R, =1

2

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+2gyy

2g

yy

2gyy

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+g

(

)= gR

., , (6.27)

, =1

2

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gy

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. =1

2g(gy

+gy

gy

), (6.29)

,

(6.25),

R10,10 = 12

[2g00

y12 1

2g00

(g00y1

)2]= a0

2

c4exp

(2a0y

1

c2

). (6.30)

R = gR,

R00 = R10,10, R11 = a02

c4, R10 = 0. (6.31)

R = 2a0

2

c4. (6.32)

,

.

Shliakhovskaya

24 6.

E (6.25) (6.30) g00 =(1+a0y

1/c2)2, , R10,10 = 0, , , -

(6.19), (6.11) -

( )

(6.18). -

(6.7),

-

. (6.25), (6.30)

[7] [13, 14].

(6.25 - 6.32) -

,

"". -

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6.3. - 25

6.3. -

(6.25) -

-

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x, -

x = x(y), (6.33)

(6.25)? (6.25)

-

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(6.11)

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(6.25) . , -

.

4 -

, .. -

. (6.25)

d2y

dS2+

dy

dS

dy

dS= 0, (6.34)

(6.25) - . ,

y1,

ddS

(dy1

dS

)=

1

2

g00y1

(dy0

dS

)2. (6.35)

(6.25)

1 +

(dy1

dS

)2= g00

(dy0

dS

)2. (6.36)

(6.35) , (6.36)

dy1

dS= tan(a0S/c2), y1 = x1 + c

2

a0ln | cos(a0S/c2) | . (6.37)

dy0

dS=

exp(a0x1/c2)cos2(a0S/c2)

, y0 =c2

a0tan(a0S/c

2) exp(a0x1/c2). (6.38)

6.3. - 27

, S = 0, dy/dS = 0, - y1 - x1, . S/c = x0/c = t ,

- . ,

, .

(6.37) (6.38) (6.25) , y2 = x2, y3 =x3

dS2 = c2dt2 cos2(a0t/c)(dx1)2 (dx2)2 (dx3)2. (6.39)

, ,

-

. [1-4],

, , -

, -

. (6.37),

(6.38) x1, x0, -

,

x1 = y1 c2

2a0ln

1 a02y02c4 exp(2a0y1/c2),

x0 =c2

a0arcsin

(a0y

0

c2exp(a0y

1/c2)

). (6.40)

(6.40) (6.39) (6.25).

(6.40) , -

x1 = y1 c2

a0ln | cos(a0t/c) |, t = t. (6.41)

(10.41) t v1 -

v1 = c tan(a0t/c), v =

(g11v1v1) = c sin(a0t/c). (6.42) v t1 = c/(2a0). (6.39)

011 =1

2c

tcos2(a0t/c),

101 =

1

2c

tln cos2(a0t/c). (6.43)

28 6.

4 - V (10.40)

V = x

y0, (6.44)

4-. 4 -

V1 = sin(a0t/c), V 0 = V0 = 1cos(a0t/c)

, = exp(a0y1/c2).(6.45)C (6.45) (6.39) ,

(6.3) (6.4) , 4 - gF

F = a20/c4 (10.5) ., (6.39) - -

- , 4 - (6.45)

- . (6.39) (6.42) ,

dS2 = c2dt2 (1 v2/c2)(dx1)2 (dx2)2 (dx3)2. (6.46) -

. -

(6.36), (6.37), (6.19),

, -

(6.11), .

: " -

, -

, (6.46)

?"

.

6.4.

, ,

, -

.

- ,

-

. -

,

. ,

(10.1) = 0,

6.4. 29

= 0 , F = 0. V = (const),

x = V S + y, (6.47)

y - , , , S = , - . y, S = 0 V, Vy

= 0 (6.47)

xk = V kS + yk, x0 = V 0S + Vkyk/V0. (6.48)

(6.48) (6.11) 3 - vk

dS2 = (dy0)2 (kl vkvl/c2)dykdyl. (6.49) x1 - , , (6.49) , (6.46), -

.

yk yk y1 = V 0y1 , y2 = y2, y3 = y3 , (6.48) ,

(6.49) (6.11).

,

, -

V0 = 1/(1 v2/2)1/2. (6.46) , ..

(6.46) - . (6.48) , -

~v ,

, ~v .

yk = yk + yk, y

k = y

pvkvp/v2, yk = yp(kp vkvp/v2), (6.50) yk - ~v, y

k - - ~v. , - yk : yk = V

0yk , yk = y

k.

yk = yk yp vkvp

v2+ yp

vkvp

v2V 0. (6.51)

(6.51) (6.48)

~r = ~R+~v

1 v2/c2 +(

11 v2/c2 1

)~v(~v~R)v2

,

30 6.

t = + (~v~R)/c2

1 v2/c2 , (6.52)

[1], [15],

(6.49) . ,

(6.52) (6.48) -

, .. -

. (6.46) , .. v .

, -

. :

(

) .

-

-

, -

. [13], "

- -

"

.

6.5. -

, -

,

. (6.25)

, -

-

. -

- , -

,

.

- -

, - (6.35) (6.36), , -

, S = 0 - . vk

- , -

[133], ,

vk =cdyk

(g00)1/2(dy0 + (g0k/g00)dyk), (6.53)

6.5. - 31

,

,

- .

(6.25)

v1 = c exp(a0y1

c2)dy1

dy0, v =

11v1v1. (6.54)

(6.35), (6.36)

dy1

dS= tan(a0S/c2 + 1), y1 = c

2

a0

[ln | cos(a0S/c2 + 1) | +c2

]. (6.55)

dy0

dS=

exp(c2)cos2(a0S/c2 + c1)

,

y0 =c2

a0

[tan(a0S/c

2 + c1) + c3

]exp(c2),

y2 = z2, y3 = z3, s = z0, (6.56)

c1, c2, c3 .

(6.54) - (6.56) ,

v1 = c sin(a0S/c2 + c1). (6.57) S = 0, v1/c = = const.

sin c1 = . (6.58)

S = 0 y1 - z1 -

y1(0) = f(z1, ), c2 =af

c2 ln | cos c1|. (6.59)

y1 =c2

a0ln

cos(a0S/c2 + 1)cos c1+ f(z1, ). (6.60) f(z1, ) ( (6.35) (6.36) ) c3 , -

, :

g00(z) =

y

z0y

z0g = g00

(y0

z0

)2(y1

z0

)2= 1,

g01(z) =

y0

z0y0

z1g00 y

1

z0y1

z1= 0,

32 6.

g11(z) = g00

(y0

z1

)2(y1

z1

)2=

(f

z1

)2cos2(a0z

0/c2 + c1). (6.61)

g01 = 0

2

a0

c3z1

= c3f

z1, (6.62)

c3 = exp

(a0f

c2

)(), (6.63)

() . f(z1, ) () , .. ,

a0 (6.39) .

() = tan c1, f(z1, ) = z1

cos c1. (6.64)

-

y1 =c2

a0ln

cos(a0z0/c2 + c1)cos c1+ z1cos c1 ,

y0 =c2

a0cos c1 exp

( a0z

1

c2 cos c1

)[tan (a0z

0/c2 + c1) tan c1 exp(

a0z1

c2 cos c1

)],

sin c1 = v0/c = . (6.65)

(6.65) a0 0, , .

y1 =z1 z0

1 2 , y0 =

z0 z11 2 . (6.66)

c1 = 0, z = x (6.37), (6.38). (6.65) (6.25),

dS2 = (dz0)2 cos2(a0z

0/c2 + c1)

cos2 c1(dz1)2 (dz2)2 (dz3)2. (6.67)

6.5. - 33

-

, (6.65), (6.37) (6.38) y1 y0.

x0 =c2

a0arcsin[sin z0 tan c1 cos z0 exp(z1)],

x1 =c2

a0

[z1 ln | cos c1U |

]U =

1 + 2 tan z0 tan c1 exp(z1) tan2 c1 exp(2z1),

z0 = a0z0/c2 + c1, z

1 =a0z

1

c2 cos c1, x2 = z2, x3 = z3. (6.68)

(6.68) - ,

,

. -

a0 0 .

x1 =z1 z0

1 2 , x0 =

z0 z11 2 . (6.69)

(6.68)

z0 + c1c2/a0 = u

0, z1/ cos c1 = u1, z2 = u2, z3 = u3, (6.70)

(6.67)

dS2 = (du0)2 cos2(a0u0/c2)(du1)2 (du2)2 (du3)2. (6.71)

(6.68) z0 z1 u0 u1 z0 = a0u

0/c2, z1 = a0u1/c2, , (6.39) ,

, (6.39) (6.71).

-

- ,

, [11]:

-

- , . -

-

,

- .

- .

34 6.

- , -

,

x0 =c2

a0arcsin

[sin

(u0a0c2

) tan c1 cos

(u0a0c2

)exp

(a0u

1

c2

)],

x1 = u1 c2

a0ln | cos c1P |,

P =

1 + 2 tan

(u0a0c2

)tan c1 exp

(a0u1

c2

) tan2 c1 exp

(2a0u1

c2

).

(6.72)

(6.72) - - -

.

6.6. -

- -

,

- , -

. -

- , -

,

. , -

, -

- . -

- ,

-

. -

.

. -

"

- ,

.

- (6.39), ,

a0 = 0 , - . , xk

t - . - , "-

" - t = f(zk, T ) ,xK = (zr) ( .. [12] - - ),

0 = 0.

6.6. - 35

, "-

" (6.39). 3 -

: ( )

( ).

-

. , , -

,

. , -

.

-

, , -

. (6.37),

(6.38) - , (6.40)

- .

, , ,

- (6.39).

, , -

- (6.37),

(6.38).

(6.39) -

(6.11) - (6.39).

, -

[135]. ,

, , -

, , . -

, -

.

( -

), ( -

), , .

, -

, . ,

: -

, ( )

(6.11) ,

( ), -

, -

(6.39).

, -

. -

( )

. -

36 6.

-

. ,

""

" ". .., -

n m, - m . , m n . -

(6.12) (6.41), (6.12) t T , T- , -

, T t -.

t =c

a0arccos

[exp

(1

1 +

a02T 2

c2

)]. (6.73)

- (6.39) -

dS2 = g00c2dT 2 g11(dx1)2 (dx2)2 (dx3)2,

g00 =

2 exp

(2(1 (1 + 2)1/2)

)(1 + 2)

[1 exp

(2(1 (1 + 2)1/2)

)] ,

g11 = exp

(2(1 (1 + 2)1/2)

), =

a0T

c. (6.74)

(6.25),

[133]

=1

c(g00)

1/2y0

(6.40), (6.41), (6.73).

=c

a0

[1 exp

(2(1 (1 + 2)1/2)

)]=

c

a0sin(a0t/c). (6.75)

, ""

- ,

- ,

. a0 0 (6.74) ( (6.39)) - , (6.39)

(6.74) .

Borodin , -

6.6. - 37

-

"" 0 = 10 /2.

- (6.75),

m = (c/a0) ln (+(1+2)1/2) [133] - (0/). (6.75)

, . (6.75)

" , .

, , = c/a0 = 347.22 . , , ,

, -

. (6.73), (6.75) , - t = pic/(2a0).

. 1 -

t a0t/c T a0T/c, - (6.73). , -

, "" pi/2, tmax = 347.22 .

.2 -

T1(T ) =a0c

T a0T/c -

T2(T ) =ma0c

= arsinha0T

c.

, -

""

T2(T ) = 1, max = 347.22 . -

[133]

m =c

a0ln

2a0T

c.

, -

t = pic/(2a0) ,

38 6.

. 1:

. 2: :

T2(T ) , - T1(T )

6.6. - 39

- l

l =

t0

(g11 x

1

t

x1

t

)dt =

c2

a0(1 cos(a0t/c)) =

=c2

a0(1 exp

(1

1 +

a02T 2

c2

). (6.76)

(6.39) (6.41).

t = pi/20 l = c2/a0. - l - (6.39) (6.46), -

.

-

(6.76) (6.11) (6.12)

t T ,

l1 =

T0

(g11 x

1

T

x1

T

)dT =

c2

a0

(1 +

a20T2

c2 1). (6.77)

l = a0t2/2, l1 = a0T

2/2, T .

. 3

F (T ) =l(T )

l1(T )=

1 exp(

1

1 + a02T 2

c2

)

1 +a20T

2

c2 1, (6.77a)

(6.76) (6.77).

(6.42) ,

(6.75), c -, -

=v

a0. (6.78)

c, , -

. > c/a0

40 6.

. 3: -

, .. ""-

- .

[12]. -

u1 [16]

u1 =cg

1/200 dx

1

g0kdxk=

cdx1

g1/200 dT

= c tan(a0t/c), (6.79)

u =

(gi0gj0g00

gij)uiuj = c sin(a0t/c), (6.80)

(6.42).

. -

-, -

. ,

( )

6.6. - 41

10 /c2, , -. , -

-, ,

. :

1.

a0 ( ) [133].

2. , .

:

1. , -

-

x . x(T ) T [133], (6.77)

x(T ) = l1 =

T0

(g11 x

1

T

x1

T

)dT =

c2

a0

(1 +

a20T2

c2 1). (6.81)

T1 - - ( -

). cT0 = x(T1), T0 = 4.

(6.81),

T1 = T0

1 +

2c

a0T0= 1.215T0 = 4.86 (6.82)

.

1, , [133]

1 =c

a0arsinh

a0T1c

= 0.555T0 = 2.22 (6.83)

.

v(T1) T1

v(T1) =a0T1

1 +a20T

21

c2

= 0.981c. (6.84)

2. , , -

- -

. ,

42 6.

, " "(, -

, ), .

, -

, -

. -

.

(6.73). ,

(6.73), (6.82) ,

t1 T1,

t1 =c

a0arccos

[exp

(1

1 +

a02T 21c2

)]=

=c

a0arccos

[exp

(a0T0

c

)]= 0.37T0 = 1.48. (6.85)

.

2 (6.75)

2 =c

a0sin(a0t1/c) = 0.238T0 = 0.952 = 347.17cym (6.86)

.

, -

.

, 347

.

4 347.17 ,

347.22

. , 72 , -

10 /2, . -

,

- , 10 /2 , - 347 1 12

, c2/a0 = 9 1012 , , - 9.46 1012 . - (6.46). -

, , -

( ) ,

-

. , ,

, -

6.6. - 43

. x, - . c2/a0 = 9 1012 - x, .

, - / ,

.

(6.77) T = T1. (6.82),

F (T1) =l(T1)

l1(T1)=

1 exp(a0T0c

)a0T0c

, (6.77b)

,

F (T1) =l(T1)

l1(T1)=

1 exp( T0c2/a0

)cT0c2/a0

, (6.77c)

cT0 = 4 9.46 1012 =37.84 1012 , c2/a0 = 9 1012 ,

F (T1) =1 exp(4.2)

4.2= 0.2345. (6.77d)

, , -

, 4.26

, , -

" . ,

, -

, -

,

.

, -

, .

,

( ) -

. , -

, , -

, , (6.16) (6.17).

, L0 -

L,

L = L0

1 +

a20T21

c2= 5.2L0. (6.87)

44 6.

5.2 . -

,

, .

.

6.7.

,

,

-

. .. (6.74) ,

~e :

e() =|g| , e() =

|g|, (6.88)

. "". ,

u(k), , - ,

u(k) = cdx(k)

dx(0)= c

e(k) dx

e(0) dx

. (6.89)

u(1) = c

|g11|dx1|g00|dx0 = c sin(a0t

c

). (6.90)

,

- :

1. , , ,

, , (6.25),

F(y).

2. (6.37), (6.38) -

- , (6.39).

3. (6.73) -

- .

4. (6.88), -

F()()(x) (6.74).

6.7. 45

4 -

.

- . -

.

, -

(6.39). -

F , "- " . ,

~E, x1,

F(0)(1) = F(1)(0) = E = const. (6.91)

(10.87)

F01 = F10 = E cos(a0x0/c2), (6.92)

, ,

, (6.39).

- -

[133]

Fx

+Fx

+Fx

= 0,1g

x

(gF) = 4picj. (6.93)

(6.93),

g = cos(a0x0/c2), F 01 = g00g11F01 = Ecos(a0x0/c2)

,

1g(gF 10)x0

0, (6.94)

.

mcDV

ds=e

cFV

. (6.95)

, -

, 4- (6.45)

(6.43) (6.1) - (6.5)

46 6.

V V = F = V (Vx

V), F0 =

a0c2

tan(a0t/c),

F1 = a0c2, = = 0, gF

F = a02

c4.

(6.92) -

(6.95),

a0 =eE

m. (6.96)

, ,

-

, -

- . -

,

(6.25).

(6.40),

F(y) =

x

yx

yF .

F01 = E exp(a0y1/c2). (6.97)

"" -

F()() = e()e

()F , (6.98)

F(0)(1) = E = const. (6.99)

- ,

, -

. , -

,

~E.

, -

:

1. -

, . -

-

(6.12), (6.13)

(6.14), (6.15).

6.7. 47

2. , -

- (6.39) (6.74).

(6.40) ( (6.12),

). -

- (6.25).

, " , -

- " -

" ".

,

- -

, -

. -

, -

, (6.11),

, . -

[1], ,

. ,

- - ,

. ,

- (

- ), -

- -

(6.74) (6.11), -

( ) .

-

[1]. , -

- ,

,

. , .. -

.

dx - - - , , -

(6.74) (6.11)

. ,

- . -

(6.11) (6.74)

, , -

, - ,

~A , (6.11)

(6.74) [1]. (6.11) -

. , ,

- , " ".

48 6.

5. , -

, - (6.74), -

(6.88),

(6.11).

- .

,

,

- (6.98) -

(6.11).

, ""

- ,

(

[17], [18]), , -

- -

,

.

.

-

2E, D1 = E, a0 = Ee/m. - "

. -

T = 2Ee. , -

(6.95)

mcDV

ds=e

cFV

.

(6.13) (6.15) ( a0 2a0).

(6.25) ( -

) (6.93) -

(6.25) ( ).

F01 = E1 = D1 exp

(a0y

1

c2

)= D1

g00, D1 = E = const. (6.100)

6.7. 49

4- (6.2)

dV 1

dS a0c2

(1 + V 1

2)

=eE

mc2

1 + V 12. (6.101)

eE/m = a0, a0/c2 = ,

V 1 = tanx(S) (6.102)

dV 1

dS=

1

cos2 x

dx

dS=

(1

cos2 x+

1

cosx

). (6.103)

(6.103) , x = 0, S = 0,

tan

(x

2

)= S, (6.104)

V 1

V 1 =dy1

dS=

2S

1 2S2 . (6.105)

V 0 4- -, (6.105) .

V 0 =dy0

dS=

1 + 2S2

1 2S2 exp(a0y

1

c2

). (6.106)

(6.105) (6.106) , S = 0 - y1 = x1,

y1 = x1 c2

a0ln

1 a02S2c4, y0 = S exp(a0x1/c2)1 a02S2/c4 , (6.107) x1 - , S/c - - .

(6.25) ( -

),

dS2 = dS2 (1 S22)dx12 2SdSdx1 dx22 dx32. (6.108)

50 6.

(6.108) ,

S = exp(a0x1/c2),, [363], -

. y0 ,yk xk (6.25) . , .. -

-

,

, ,

. -

,

y1 = x1 + a0t2,

,

a0.

, "" -

. , -

, (6.7)

: ,

- .

, - .

6.8. , ,

-

, -

r0, 0, z0, t0 r, , z,t :

r0 = r, 0 = + t, z0 = z, t0 = t,

z -.

dS2 =

(1

2r2

c2

)c2dt2 2r2ddt dz2 r2d2 dr2. (6.109)

, r/c < 1. [19] ( [87]) , -

, .. , ,

c r > c/.

6.8. , , 51

, -

.

[20], [88], , -

,

, . -

. , -

.

e [21] , -

r c, r/c 1 v = r. - , (

) v = r, = const, (6.109). - r , v(r) < c .

-

, (6.1) (6.7)

= 0 -,

.

=

22

c2= const. (6.110)

, -

,

V 1 = V 2 = V 3 = 0, V 0 = D1/2, V1 = V3 = 0, V2 = PV 0dS2 = D(r)c2dt2 2P (r)cdtd dz2 r2d2 dr2,

g00 = D, g02 = P, g11 = 1, g22 = r2, g33 = 1,det g = g = P 2 r2D K, g00 = r

2

K, g02 = P

K

g11 = 1, g22 = DK, 12 = K

1/2

cD1/2,

F 1 =1

2D

dD

dr, F 2 = F 3 = F 0 = 0. (6.111)

(6.1)

P

D

dD

dr dPdr

= 2c

(Dr2 + P 2

)1/2. (6.112)

52 6.

(6.7)

(6.112), -

,

dD

dr = 2

cDP

(Dr2 + P 2

)1/2. (6.113)

(6.110) -

, -

[12],

i = c1/2

eijkjk, e123 = 1, (6.114)

= det kl , kl (6.3) [133]

kl = gkl + g0kg0lg00

. (6.115)

(6.110) -

[21].

=

(1

2r2

c2

)1.

(6.112) (6.113) , , (6.112) (6.113) (6.109), -

-, -

. = + (r)t (r), (r)r < c, , -

.

(6.111) -

, 0 r , -.

(6.112), (6.113) -

. , r/c 1 (6.111) (6.109). r/c 1 (6.112),(6.113)

D = D0 exp

(2r

c

), P =

c

. (6.116)

6.8. , , 53

r/c > 10 D0 = 5, = 1.7. r - (6.111) ,

, -

[133].

g00 > 0, g, , , , -

.

-

a = c2F 1 = cPDr2 + P 2

, (6.117)

r , r a =c. (6.111), (6.112), (6.113)

R10,10 = 2D

Kc2

[2P 2 Dr2 + DP

2r2

K

] D

2Pr

cK3/2,

R20,20 = DrcK1/2

[Dr3

cK1/2 P

],

R12,10 =PDr

cK

[P

K1/2 r

c

(2 +

Dr2

K

)],

R12,12 =2D2r6

K2c2 2Dr

3P

cK3/2+P 2

K. (6.118)

C R = ggR,

R =2DP 2

K2

{1

2r2

c2

[2 +

Dr2

K

(1 r

2D

P 2

)]

+rP

K1/2c

(1 2Dr

2

P 2

)}. (6.119)

-

, (6.115),

dl2 = dr2 +

(1 +

P 2

r2D

)r2d2 + dz2. (6.120)

z = const -

12,12 = P2

K+2r2

c2

[P 2

Dr2 2Kr2D

+P 2 Dr2

K

(2 P

2

K

)]

BorodinR 12,12 - ??

Borodin+ (. . .)

54 6.

+rP

cK3/2

(2Dr2 P 2

). (6.121)

=P12,12

1122 212. (6.122)

= DP2

K2 2

2

c2+2P 2

c2K+DPr

K5/2c

(2Dr2 P 2

)

+

(2 P

2

K

)(P 2 Dr2

)D2r2

c2K. (6.123)

r = 2/c2. ,

dS2 = D(r)c2dt2 dz2 dr2, (6.124)

(6.116) (6.117) (10.25) a0 = c.

dS2 = D0 exp

(2r

c

)c2dt2 dz2 dr2, (6.125)

, -

-

. ,

. -

,

(6.25) -

,

.

(6.109).

(6.112) (6.113).

:

x rc, D Z, Y Px

r. (6.126)

6.8. , , 55

(6.112) (6.113)

dY

dx=

2Zx2Zx2 + Y 2

, (6.127)

dZ

dx= 2ZY

Zx2 + Y 2. (6.128)

U ,

U YxZ, (6.129)

dU

dx+U

x=

2 + U21 + U2

, (6.130)

1

Z

dZ

dx= 2U

1 + U2. (6.131)

, ,

v U1 + U2

(6.132)

dv

dx+v

x(1 v2) = (2 v2)(1 v2), (6.133)

Z = D(x) = exp

(2 x

0

v(x)dx

). (6.134)

v(x). , - (6.120)

dl2 = dr2 +1

1 v2 r2d2 + dz2. (6.135)

dl2 = dr2 +1

1 2r2c2r2d2 + dz2. (6.136)

(6.135) (6.136) ,

v(x) -, -

- .

56 6.

(6.120) (6.136), -

P(x).

P (x) =xv(x)

D(x)

1 v(x)2 c

. (6.136a)

(6.133). v, , v = x, -

. D(x)

D = exp

(2vdx

)= exp(x2) = 1 x2, (6.137)

(6.109).

(6.133) , x - v = 1. , .

(6.133) . . 100 -

v(x). , x, v(x) x, x, v(x) 1 x v(x)

v(x) = 1 (x), (x) 1. (6.138) 2 , (x)

d

dx= 2

1 xx

. (6.139)

= 0x2e2x, v(x) = 1 0x2e2x, (6.140) 0 - .

D(x)

D(x) = D0 exp

(2x e2x(x2 + x+ 1/2)

), (6.141)

D0 = 5 (6.116) ( (6.116) (6.141) ).

P (x)

P =

D00

c

, 0 = 1.73, (6.142)

6.8. , , 57

. 4: -

0 (6.116). - , -

D(x) P0(x) = P (x) /c . 5. (6.109) -

(6.111), , D(x) - Dk(x) = 1 x2, P0(x) Pk = x

2.

Dk(x) 0 < 1, D(x) 0 x . , ,

. -

4 -

.

. "-

", ( ) -

.

1

1 =r

c=

2pir

cT=

2picT0cT

= 9173.45.

58 6.

. 5: -

-

4 . -

2

2 =2r

c=

2pir

cT2=

2picT0cT2

= 8pi = 25.13.

,

(6.109), .

(6.111), -

.

-

. -

,

. [133],

. ,

, , -

t = 1c

g02g00

d =2pi

c

vr1 v2 exp

(vdx

). (6.143)

6.8. , , 59

, c, - , [22] (-

).

, , -

.

,

. -

,

, .

t0 t.

t0 =4pi

c

vr1 v2 exp

(vdx

). (6.144)

v = x -

[22].

t0 =4S

c2, (6.145)

S - .

t0 =4pi

D0exp (2x). (6.146)

.

- -

-

, . -

-

c. . [21] .

[21] .

{x}, = 1, 2, 3, 4, x4 = ict. , , - .

, x3. -

h() , ~h(4) 4 - V, ..

ih(4) = V. (6.147)

(6.147) ,

ih(4)h(a) = i(4a) = V (a) = 0, (6.148)

.. , (6.147),

.

60 6.

4 -

Va = V (r)aknk, V3 = 0, V4 =i

1 v2/c2 , V (r) =1

c

1 v2/c2 ,

a, k = 1, 2, nk =xkr, r2 = x21 + x

22, ak = ka, 12 = 1. (6.148) v(r) . , -

(6.4) - .

= [V] V[F], (6.149) ,

(ab) = [V]h(a)h(b). (6.150)

,

~h(3) x3, ha(2) na.

~h(3) ~h = (3), ~h2 ~ha = ha(2) = na. (6.151)

~h(3) (ab) (a, b = 1, 2) , -

~h(3) .

h()h() = (), (6.152)

h() =

iV4n2 iV4n1 0 iVn1 n2 0 00 0 1 0iV4n2 iV n1 0 iV4

, (6.153) (6.153) - , - . - (6.150)

(ab) =0c(ab), (ab) = (ba), (12) = 1, (6.154)

, -

. 0

. , -

(ab) =0c

[h(1)h(2) h(2)h(1)]. (6.155)

,

~h(3) ~h(4), -

,

~h(1) ~h(2) .

6.8. , , 61

, (6.155) -

~h(1) ~h(2). , ~h(3) ~h(4) ~h(1) vech(2) - x1x2. (

) x1x2

() =

cos sin 0 0 sin cos 0 00 0 1 00 0 0 1

, (6.156)h(

) = ()h() = iV4n2 cos+ n1 sin iV4n1 cos+ n2 sin 0 iV cosiV4n2 sin+ n1 cos iV4n1 sin+ n2 cos 0 iV sin0 0 1 0iV n2 iV n1 0 iV4

.(6.157) , h(

) (6.155). -

. , -

:

~h(3),

,

~h(4), 4-. - .

. -

, -

.

(6.154) 0 = const, -,

V

r+V

r+V 3

r= 2i0V4

c. (6.158)

v

v

r+v

r= 20(1 v2/c2). (6.159)

[19], [20]. (6.159) , v(0) = 0

v(r) =c2

20

d

drln

I0(20rc), (6.160) I0(x) - - [51]. , -

I0(x) = 1 +x2

4+

1

2!2x4

24+ ...

62 6.

,

v(r) =0r

1 + 20r2/c2

0r. (6.161)

0r/c 1, -

I(x) =

1

2pixex,

v c(

1 c40r

)< c. (6.162)

, (6.160)

, -

.

, -

.

-

. ,

(r) v(r)

(r) =v(r)

r=

c2

20r

d

drln

I0(20rc) (6.163) -

r0, 0, z0, t0 r, , z,t :

r0 = r, 0 = + (r)t, z0 = z, t0 = t,

(r) z - r,

dS2 =

(1

2r2

c2

)c2dt2 2r2ddt dz2 r2d2

2r2tddr

(d+ dt) dr2(

1 + r2t2(d

dr

)2)(6.164)

dl2 (6.115),

dl2 = dz2 +

r2d2 + dr2[1 2r2c2 + r2t2

(ddr

)2]+ 2r2drdtddr

1 2r2c2. (6.165)

6.8. , , 63

, -

, .. -

, . , -

- -

.

"-

..

(6.148), r = r0, - x3. (6.154)

(ab) = 0, r > r0. (6.166)

-

~ =1

2 ~v = 0, (6.167) . , -

, ,

.

(6.158)

V

r+V

r+V 3

r= 0, (6.168)

r > r0

V =r0

r

1 r20/r2, (6.169)

v = cr0r, va = vabnb. (6.170)

, [1]

.

-

. 4-

(6.148), , v = v(r, t). 4- -

dVdS

= a, dS = cdt

1 v2/c2 (6.171)

, (6.153),

h(b)dVdS

=a0(b)

c2, (6.172)

64 6.

a0(b) - , - .

(10.172)

V2

r=a0(2)

c2, i

V

x4=a0(1)

c2. (6.173)

(6.173) a0(2) = 0(r, t)2r,

V =0(r, t)r

c,

0t

=a0(1)

r= 0(r, t). (6.174)

v(r, t) =0r

1 + o20r2/c2

, (6.175)

0 (6.174), - 0(r, t). , 0 = 0, (6.175) -

r 0. 0 = 0(t) t = 0 0 = 0,

v(r, t) =r t

00(x)dx

1 + r2

c2

[ t00(x)dx

]2 . (6.176)

0 = const,

v(r, t) =0rt

1 +20r

2t2

c2

(6.177)

, -

. -

-

, -

.

, ,

.

6.9. 65

6.9.

, -

-. -

(6.1), (6.7) ,

. -

, , -

, , "" -

.

-

[23],

g = g g g = 0, (6.178) ,

.

(6.178) ,

= + T

.. , (6.179)

=1

2g(g + g g

), (6.180)

T .. = S.. S... + S..., (6.181)

S.. []. (6.182)

, (6.179) -

S... (6.178) - (6.181)

g = g g g = 0. (6.183)

, - (6.180).

(6.6) (6.7)

R,.V 2S.a = 2[a] , a V . (6.184)R,. = 2[] + 2[||] . (6.185) [24], (6.185)

R,. = R,. + 2[T]

66 6.

+2T[||.T]. + 2S

.T

.. (6.186)

(6.1), (6.184), (6.186)

R,.V = 2[a] 2T[||.a]

+2V

([T]. + T[||.T ].S.T.

), (6.187)

R,. - , (6.180).

(6.1)

V = T .V + + + VF . (6.188)

, -

,

, -

-

g , V, S.

7.1. ... 67

7

, , -

,

-,

( ) -

. -

-, -

-

. -

.

7.1

,

,

, -

,

- . , -

, -

, , -

- , -

. -

.

,

-

,

,

[25], [26].

-

-

() .. [12] [18].

-

, -

( ), -

. "-

" ,

68 7.

, ( -

), 4 - V , - .

.

x = (yk, 0), (7.1)

x - , yk - , -

, (1/c)0 - , , . , -

, - -

. (7.1) yk 0, -

. , x/0

x/yk - . (7.1)

, "" "" -

,

4- yk 0. ,

hk

=

( V V

)

yk, h

0=

0= V ,

hk =yk

x, h0 = V. (7.2)

-

[23]

= hh

y= h

h

y, (7.3)

/y - ,

y0=

0= V

x,

yk= h

k

x(7.4)

(7.3) , -

. -

= h

h

y+ h

h

h, (7.4a)

7.1. ... 69

- .

C .

C .

= []

=1

2

(h

h

y h

h

y

)=

1

2hh

(hx

h

x

). (7.5)

[23],

=

{

}+ T

, T

= C

+ gg

C .

+ ggC .. (7.6)

- -

, . , -

(11.2)

,

[23]

R...

= 2[

]+ 2[||

]

+ 2C

0. (7.7) (7.6) (7.7)

R...

= 2[T ] 2T[||T

] 2C

T . (7.8)

(7.8) -

{

},

g = ghh

, g00 = 1, g0k = 0, (7.9)

g - . -

,

. -

, -

(7.6).

,

R...

.

-

[23]

2

y y

2

yy= 2C

y. (7.10)

70 7.

-

,

. ,

,

(7.2), -

C 0kl

= kl, 2C00k

= Fk, Ck

= 0, (7.11)

kl = h

khl, Fk = Fh

k. (7.12)

(7.12) (6.4)

(6.5), 4 - -

-

. (7.11) -

(7.10)

2

ykyl

2

ylyk= 2lk

y0,

2

yky0

2

y0yk= Fk

y0. (7.13)

(7.13)

[27]. (7.9), -

(7.6) (7.2) {0

00

}=

{k

00

}=

{0

0k

}= 0,

{0

kl

}= kl,{

k

nl

}= k

nl,

{k

0n

}= kn, T

00k.

= Fk, T k00. = F k,

T kml.

= Tml,k = T0k0.

= 0, T k0l.

= T kl0

= kl., T 0

kl= kl. (7.14)

kl + kl = hkhlV , (7.15) , y0,

y0

(kl+kl

) gmn

(ln+ln

)(km+km

)+kFlFkFl, (7.16)

, ,

y0kl [kFl]. (7.17)

7.1. ... 71

(7.16)

y0kl gmn

(ln + ln

)(km + km

)+ (kFl) FkFl. (7.18)

,

, -

,

. -

,

- . ,

R...,

+ R...,

+ R...,

= 2

[C .

{

}+ C

.

{

}+ C .

{

}](7.18a)

R..., + R...

,+ R...,

= 2C .R...

,+ 2C

.R..., + 2C

.R..., , (7.19)

-

.

(7.8)

R...,

= 2[

{

]

}+ 2

{

[||

}{

]

}+ 2C

.

{

}

K ...,

+ 2C .

{

}. (7.20)

K ..., -

, -

. -

-

.

,

72 7.

(7.6) (7.3), -

. ,

-

g = g = 0. (7.21) . -

0 = []g = C .gy

K,(). (7.22)

(7.20),

R,() = 0. (7.23)

(7.18) ,

R[] = C.

{

}+ C

.

{

}+ C .

{

}. (7.24)

(7.24) , -

.

,

R, 6= R, . (7.25) , ,

-.

R, =1

2

(R, + R,

), (7.26)

, -

-. , -

,

R, + R, + R, = 0, (7.27)

R . -

R, R, = C 0.g

y0+

+C 0.

g

y0+ C 0.

g

y0+ C 0.

g

y0. (7.28)

7.1. ... 73

R,

Rab,cq = qbac qabc 2abcq,

Rab,c0 = 2[ab]c + 2abFc 1

2Fbac +

1

2Facb,

R0b,c0 = FbFc (bFc) 2n(bc)n + ncnb.. (7.29) -

Rab,cq = Rab,cq 2q[ab]c, (7.30)

Rab,cq - - . -

. (7.29) (7.30)

Rab,cq = 2q[ab]c + qbac qabc 2abcq. (7.31)

(7.31)

[28].

V = hV =

0, []V = 0 =

= []V T [].V T [||.]V,

kVl = kl + kl, 0Vl = Fl, kV0 = 0V0 = 0. (7.32) (7.32)

[ab]c + [ab]c = abFc. (7.33)

a, b, c (7.33), - , (7.33),

abc + bca + cab + Fabc + Fbca + Fcab 0. (7.34) [12].

(7.18) (7.33), (7.29)

Rab,c0 = [ab]c 1

2Fbac +

1

2Facb,

R0b,c0 = bcy0

+ ncbn. (7.35)

74 7.

(7.29) (7.35) , -

.

1

Rbc = bcy0

+ nnbc + 2ncbn + Rbc

FbFc (bFc) 2n(bc)n + nbnc.,

Rb0 = aab. 2abF a 1

2Faab +

1

2Fb

cc

aab baa +1

2Fb

aa

1

2F aab,

R00 = b

b

y0 ncnc FnF n nF n + nbbn. (7.36)

R = 2FnFn 2nF n nbbn. (7.37) (7.19) .

(R 1

2gR

)= R[]

+2C .

(R + R[]

)+ C,R

,. (7.38)

(7.38) , ,

,

,

.

..

[12], , [12].

gab = hab, ab = 1

cDab, c = Aca, aa =

1

cD,

nc =1

cDnc ,

c. =

1

cAac., Fb =

1

c2Fb, F

a = 1c2F a,

1

, -

. , [133] [135]

,

[23]. [23], [135]

, [133]

.

[133], [23] , [135] .

7.1. ... 75

yk=xk

,

y0=

1

c

t. (7.38a)

(7.36),

Dikt

(Dij +Aij

)(Djk +A

jk.

)+DDik DijDjk

+3AijAjk. + (iFk)

1

c2FiFk + c

2Rik 0,

j(hijD Dij Aij

)+

2

c2FiA

ij 0,

Dt

+DjkDkj +AjkA

kj + jF j 1c2FjF

j 0. (7.39) (7.39) -

, . -

.

. ,

, . -

-

- .

,

, (7.8) -

,

. -

. ,

(7.33)

[ab]Vc = 1

2R0ab,c.

.

a c, b,

Rb0 -.

-

-, -

,

, -

. 4- V , - -

- , (6.1) -

. -

(7.1) ,

76 7.

"" , -

, -

4- x yk, ,

, -

. -

"" -

,

.

-

. (6.11) (7.2)

dS2 = dy02

+ g

yn

ykdyndyk, g = g VV . (7.40)

g - , -

.

dy0 = d0 + V

yndyn = Vdx

. (7.41)

(7.41) , dy0 , .. y0

- .

(7.40) -

,

dy0 = Vdx ,

, - -

.

[12] [29].

V k = hkV =

dyk

d0= 0,

Vk = V

yk= gkV

= gk0V0 =

gk0g00

, (7.42)

-

dl2 =

(gn0gk0g00

gnk)dyndyk. (7.43)

(7.43) -

[133]. , (7.42) (7.43)

(7.2).

7.1. ... 77

-

. -

, -

-

.

(7.6) -

.

(7.9)

{

}=

1

2g(g + g g),

y

. (7.44)

, (7.44)

, (7.44)

. -

, c (7.2)

= h

x=

y+ L

s, L V V,

V = hV =

0, V = V

x

y, (7.45)

y0

0 s .

(7.45) , L0 = 0.

(7.6) (7.45) {,

}= , + T , T = L + L L ,

=1

2

gs

, 00 = 0k = 0, (7.46)

, - , -

(7.9). ,

(7.6),

=

+ ,

= T

+ T

. (7.47)

78 7.

(7.7) -

(7.47), (7.45) (7.6)

R...

= 2[

]+ 2[

]

+2L[

]

s+ 2[||

]

+ 2C

0. (7.48) (7.48)

K ...

= 2[

]+ 2[||

], (7.49)

-

-. (7.48)

(7.49)

K ...

= 2[] + 2[||

]

+ 2C

+ 2L[

]

s. (7.50)

(7.50) :

1. (7.50)

( )

,

.

2. , ()

(7.50), -

, -

.

, -

: -

, (7.3), -

{

}, -

(7.6) , ,

(7.46).

, ,

(7.9)

. , . (7.3)

(7.9),

g = g g g , (7.51)

, g 0.

7.1. ... 79

(7.6) (7.51)

g = g + T g + T g . (7.52)

2

, g = g = 0.

g = g + T , + T , 2L ,

T ,

= L + L L (7.53),

T ,

+ T ,

2L = 0. (7.54)

g = g = g = 0. (7.55)

, -

(7.1), , (7.1) -

.

, , . -

,

. , (7.1), 0 - . 4 - V

V =

0, (7.56)

4 - .

2 =1

g

0

0

, (7.57)

C

hk

=

( V V

)

yk, h

0=

0=V

,

2

, T

. -

{

} -

.

80 7.

hk =yk

x, h0 = V. (7.58)

-

(7.3)

(7.6) C

(7.5).

, -

.

(7.58)

C 0kl

= kl, 2C00k

= Fk ln

yk, C k

= 0, (7.59)

kl = h

khl, Fk = Fh

k. (7.60)

(7.58),

g = ghh

, g00 =

1

2, g0k = 0, (7.61)

g - . -

,

. -

, (7.10),

2

ykyl

2

ylyk= 2lk

y0= 2lk

s,

2

yky0

2

y0yk=

(Fk

ln

yk

)

y0=

1

(Fk

ln

yk

)

s. (7.62)

T0k,0 = (Fk

ln

yk

)1

2, T00,k =

(Fk

ln

yk

)1

2,

Tml,k = Tk0,0 = 0, T0l,k = Tl0,k =1

lk, Tkl,0 =

1

kl. (7.63)

-

, .

= h

x=

y+ L

0,

7.1. ... 81

L (0 V

), V = V

x

y. (7.64)

(7.6) -

.

-

(7.9) (7.44) , (7.44)

, (7.44)

.

, c (7.2)

(7.64) {,

}= , + T ,

T =1

[L + L L

],

=1

2g

0, 00 = 0k = 0, (7.65)

, - , -

(7.61),

0,kl = 1

2

gkl0

, 0,0l =1

2

g00yl

, 0,00 =1

2

g00y0

,

n,0l =1

2

gnly0

, n,00 = 1

2

g00yn

, nkl

= nkl, (7.65a)

nkl , -

kl = gkl. -

(7.49) (7.50). (7.49)

K0k,lm =g00(mkl lkm),

Kik,lm = Pik,lm (klim kmil),

K0k,0m = g00(kms

gqrkqmr)

12

[2g00ykym

12g00

g00yl

g00ym

nkm

g00yn

]. (7.65b)

(7.65), (

.. = 0)

82 7.

, -

(. (7.20)).

xa =

t0

v() d + ya, x0 = ct = 0. (7.66)

0 . -

(7.61)

dS2 =1

2d0

2+ g

yn

ykdyndyk, g = g VV . (7.67)

g - , -

.

dS2 (7.40) -

, , -

dy0, (7.41),

d0 = cdt. , - . ,

,

,

, , -

.

1/2 (7.57). (7.66) -

(7.66)

dS2 =1

V 20dx0

2 (nk + VnVk)dyndyk. (7.68)

C (7.68) .

. [1], yk -

k

.

-

, -

, , (7.68),

. [136]

7.1. ... 83

. , -

,

.

(7.68)

(7.67)

xa = a(yk, x0), x0 = ct = 0, (7.69)

[25]

k

yndyn = dXk, (7.70)

, , -

, -

t. -

.

(7.69) -

dS2 =1

V 20d0

2 (mn + VmVn)m

yln

ykdyldyk. (7.71)

,

(7.69) (7.71)

1. .

, -

, -, -

[133].

, -

r0, 0, z0, t0 r, , z, t :

r0 = r, 0 = + t, z0 = z, t0 = t,

z -.

(7.71) , -

.

dS2 =

(1

2r2

c2

)c2dt2 dr2 r

2d2

1 2r2c2 dz2. (7.72)

84 7.

-

dS2 =

(1

2r2

c2

)c2dt2 2r2ddt dz2 r2d2 dr2. (7.73)

, r/c < 1 - , ( -

). :

(7.72) , -

(7.73) - . t = const - (7.72) ""

(7.43). (7.72) (7.73) g0k - ,

[133].

- t .

d2 =

(1

2r2

c2

)dt2 (7.74)

, ,

.

(7.72) .

, -

, -

:

( ) -

"".

2. () .

6, -

,

: -

.

,

(6.14), (6.15).

(6.12) (7.71) (

(7.71) yk yk)

dS2 =c2dt2

1 + a02t2/c2 (1 + a02t2/c2)(dy1)2 + (dy2)2 + (dy3)2. (7.75)

(6.13) (7.71) (c = 1, t )

7.1. ... 85

dS2 = c2d2 cosh2(a0

c2

)(dy1)2 + (dy2)2 + (dy3)2. (7.76)

(7.75) . . -

[1], , , ,

. , (7.76) (7.75) -

, (6.13) -

t = (c/a0) sinh(a0/c).

(7.75) (7.76) -

(6.14) (6.15) , -

( ) -

,

(

) "",

(6.16) (6.17).

(6.14) (6.15),

(6.11) (6.12) (6.13), -

g01 . , ,

, t = const (6.14) = const (6.15) .

,

, , "" "-

", ..

. g0k = 0 g00 = 1 [135] [133]. , -

-

,

. "" -

-

!

.

, -

,

.

-

.

, - , "-

" , -

86 7.

, .

. ,

, "" .

" " ""

(.. )

, ""

. . , -

, .

.

(6.25) -

(7.76) ?

, -

-. -

(6.25) , (7.76) -

-.

(6.25) -

,

(7.76)

, -.

, -

(7.76) ( -

) .

R001,1.

= a20

c4cosh

(a0

0

c2

)(7.77)

R = 2a20

c4(7.77a)

-

, ,

, .

[133],

, , .. -

. -

"" -

.

-

, (7.40)

7.2. , ... 87

dy0

yk (7.41).

dS2 = d02

+ g

yn

ykdyndyk, g = g VV , (7.78)

(7.40). -

,

(.. (7.78))

, (. (7.40).

,

, , ,

-, .

7.2. ,

-

-

.

-

. 4 -

- V . -, 4 - U V . .

(7.2),

hdU

dS=

1

m0chf

. (7.79)

(7.79) m0 - , f- 4 - .

(7.2) - (7.6),

h = hT ., h = 0 (7.80)

dU

dS+

{

}U U =

1

m0cf T

.U U . (7.81)

, U = V (7.14)

88 7.

(7.81). , -

, - -

. (7.81) ,

dS2

= dy02 dl2 =

(1 u

2

c2

)dy0

2, (7.81a)

u - . - (7.14) (7.38), -

[27]

dE

d+mDiku

iuk mFiui = c2Vf

1 u2

c2. (7.82)

dpk

d+ k

nlpnul + 2m(Dk

i+A.k

i)ui mF k = cf k

1 u

2

c2. (7.83).

(7.82) (7.83) :

E =m0c

21 u2c2

, pi =m0u

i1 u2c2

, m =E

c2, d =

dy0

c, (7.84)

E - ( (..) [27]) -

, m - (..) , pi - (..). (7.82) (7.83) -

[27].

. ,

, (7.1) -

Vf = f 0 = V0f

0 +Vkfk =g00f

0 +g0kg00

f k = g0f =

f0g00

, (7.85).

-

(7.82) [27]. (7.83) .

,

, -

.

(7.81) h - ,

. (7.14),

hDU

dS K = f

m0c 2g(UV)U[V]. (7.86).

7.2. , ... 89

(7.86) K - 4- - , 4- U,f/(m0c) - 4- , (7.8) - . ,

(7.86) -

, -

.

, 4-

4--

(7.6).

(7.8) R... -

. -

T . = T

. T .T . + T .T

.+ T

.T ., (7.87)

(7.8)

R...

= 2[T ]. 2T[||.T

].. (7.88)

(7.80), (7.88) -

,

.

hhh

h

R

...

,= R..., = 2[T]. 2T [||.T].. (7.89)

T .

T . = FVV V ( + FV) + V. + V.. (7.90)

T, =1

2[FV + FV + FV ], F = 2[V]. (7.91)

, -

, - -

,

-

, . ,

(7.89)

.

(7.89) -

-, -

( ) -

, (7.6) -

(7.91).

90 7.

,

.

, 1/c2. (7.29) (7.36)

Rab,cq = 0, Rab,c0 0, R0b,c0 (bFc),

Rbc (bFc), Rb0 0, R00 nF n. (7.92)

F b - 4-, -

F b = ab/c2, ab - .

nan = 4pik, (7.93)

k - , - .

R00 =4pik

c2, Rb0 = 0, Rbc =

1

c2(bFc). (7.94)

(7.94) -

,

- .

-

. (7.91)

T . ,

T . = FVV V FV = gV

(Vx

Vx

). (7.95)

, -

. ,

-.

R = T

x F

x+

V V FV . (7.96)

R

R = 2F

x+

. (7.97)

7.3. ... 91

G

G = T

x F

x+

V V

FV + g F

x 1

2g

. (7.98)

-

R =(F V )

x F

x. (7.99)

G =(F V )

x F

x+ g

F

x= 2

x

(g[F]

), (7.100)

g - , (11.40). (7.100) -, ,

G

x 0, (7.101)

.

7.3

-

, ,

v2/c2 . (7.40)

dS2 = 2dt2 mn m

ykn

yldykdyl. (7.102)

(7.102) -

, gmn = mn, t - - . , (7.102)

.

, (7.102)

.

, -

.

Borodin

92 7.

, -

. ,

-

. ,

t1, t2. , - , (t2 t1). , , ,

. ,

,

(7.102). (7.102) -

(6.11) xn = n(yk, t), - xn t, .. . , -

-

, ,

-

.

-

. -

(6.11) -

xn = n(yk, t), xn . ( (7.102)) ,

, g00 - g0k . - . , -

.

(7.102)

kl = mnm

ykn

yl. (7.103)

[30]

dkldt

= 2kl, (7.104)

kl .

yk -

, dyk/dt = 0,

dkldt

=klt

+klym

dym

dt=klt

= 2kl. (7.105)

7.3. ... 93

-

,

.

vat

+ vkvaxk

= ga,

t+

xa(va) = 0. (7.106)

(7.106) xb,

t(ab+ab)+(kb+kb)(ak+ak)+v

k

xk(ab+ab) =

gaxb

, (7.107)

d

dt(ab + ab) + (kb + kb)(ak + ak) =

gaxb

, (7.108)

ab =1

2

(vaxb

+vbxa

), ab =

1

2

(vaxb vbxa

)(7.109)

(7.109) ab, ab - .

(7.107) a, b,

taa +

kbbk

=gaxa

. (7.110)

, ab = 0.

ab =vaxb

=vbxa

,baxb

=bbxa

.

bba abb = 0, (7.111)

(7.103).

(7.102) -

[133] -

.

R00 = 1

c2

(

taa +

kbbk

), (7.112)

R0a =1

c

(bba abb

), (7.113)

Rab =1

c2

(

t(ab) + ab

kk 2ka bk

). (7.114)

94 7.

,

tab =

(dkldt

+ 2mlmk

)k

yal

yb, (7.115)

, (7.114)

Rab =1

c2

(dkldt

+ klmm

)k

yal

yb. (11.116)

(7.108),

Rab =1

c2

(gkxl

+ klmm kmml

)k

yal

yb. (7.117)

, , ga = /xa,

- , -. (7.110), (7.112)

R00 =4pik

c2. (7.118)

(7.113) (7.111)

R0a = 0. (7.119)

(7.117)

Rab =4pik

c2ab + Fab, Fab =

(gm

xmkl +

gkxl

+klmm kmml

)k

yal

yb. (7.120)

(7.118) - (7.120) , -

Fab = 0, , [133]. -

, Fab 6= 0,

.

, -

, -

,

, . (7.103) ,

-

.

7.3. ... 95

,

Fab = 0. (7.121)

- , -

va = v(r, t)na, na =xar, nana = 1. (7.122)

(7.106), (7.122), -

(7.121)

1

r

v

t= ,

1

r

v

t+v2

r2+

2

r

r= . (7.123)

:

1.

- , -

, ,

= 0, = kM0r

, v2 = 2kM0

(1

r 1r0

)+ v20 , (7.124)

M0 - , , v0 - r = r0. (7.123) (7.124)

v

t= 0, v2 = 2kM0

1

r. (7.125)

(7.125) (7.124) ,

. (7.125),

r = (

3c

2

)2/3F 1/3(t0 t)2/3, F 2kM0

c2= rg, (7.126)

rg - . , - c -

,

.

. ""

"" . t0 , t = 0 r = r0, r0 - - . ,

r(r0, t) t < t0.

96 7.

(7.102)

dS2 = c2dt2 (r

r0

)2dr20 r2(d2 + sin2 d2), (7.127)

(7.126),

R 23

r03/2

rg1/2, (7.128)

dS2 = c2dt2 dR2[

32rg

(R ct)]2/3

[

3

2(R ct)

]4/3rg

2/3(d2 + sin2 d2), (7.129)

[133]. -

,

-

- . ,

, , -

. , ,

.

(7.129)

, ,

, ,

. -

, .

, T - r1 r(r1, T ) -

T =2

3

[r1c

(r1rg

)1/2 rc

(r

rg

)1/2], (7.130)

, -

[31]. -

t.

.

7.3. ... 97

2. [31], [133], -

,

v(r, t) = H(t)r. (7.132)

(11.123), ,

1

r

v

t= 4pik, 3

r

v

t+v2

r2+

2v

r

v

r= 4pik. (7.133)

(7.132), (7.133)

H

t= 4pik, H

t+H2 = 4

3pik. (7.134)

H2 =8

3pik, (7.135)

-

. -

[31] ,

[31] ,

r = r0

(t tt0 t

)2/3, (7.136)

(t0 t) - "" . (7.136) (7.127)

dS2 = c2dt2 (t tt0 t

)4/3[dr20 r02(d2 + sin2 d2)

], (7.137)

()

.

, , -

,

,

, .

98 8.

8

, . -

"" -

6

. -

, " -

. -

, -

. ,

.

8.1.

6, ,

(6.1),

.

[133]

F

x= 4pi

cj, F + F + F = 0, F = 2[A]. (8.1)

(8.1) F - , j - - , A - 4-.

, (7.1), (7.2)

F = 4picj, F + F + F = 0,

F = F + 2C0A0, F = 2[A], A = h

A, (8.2)

F = h

hF , j

= h j . (8.2a)

8.1. 99

(8.2) , -

F -

"". F

F = 2[A] = 2[A], (8.3)

(7.6).""

A0 c , . - (7.11). , -

, ..

"".

.

(8.2)

F = F + T F + T F = 4pi

cj,

F = 1g(gF )y

.

F k0

0+ l(F kl + 2klA0)

y0(F kA0) Fl(F kl + 2klA0)

+ll(F k0 F kA0) =

4pi

cjk,

k(F 0k + F kA0) + kl(F kl + 2klA0) = 4pi(UV ), (8.4) k - , - , - ,j = cU. , A -. , , -

A

x= 0, (8.5)

A0

0+ kAk + kkA0 FkAk = 0. (8.6)

~E,

~D,

100 8.

~H ~B , -

[133] -

.

Ek = F0k, Bkl = Fkl, Dk = g00F 0k, H kl = g00F kl (8.7) [133] -

(7.9). :

(~ ~E)a = 12eabc

(Ec

y Eyc

), H a = 1

2eabcHbc,

12ecab(FaEb FbEa) = ~F ~E, a = 2 e

abcbc,

(~ ~H)a = 12eabc

(Hc

y H

yc

), ~ ~E = 1

ya(Ea). (8.8)

(8.8)

abc =eabc,

abc =1eabc, e123 = e123 = 1,

abc - -.

, (8.2) -

, (7.1),

( -

), .

~ ~E = 1

( ~H)

0 ~F ~E, ~ ~E = 2

c~ ~H + 4pi,

~ ~H = 1

( ~E)

0 ~F ~H, ~ ~H = 2

c~ ~E. (8.9)

, -

.

1

()

0= 0. (8.10)

, (8.2), -

, -

, (8.9) ( )

8.1. 101

4 - ,

.

, -

, 3-

, ..

[16].

-

. -

,

.

~a(y) (y) - :

(~ (~ ~a))k = (~(~ ~a))k ak + Rk.lal + 2kn

an

0,

~ ~ = 2~

c

0, ~ (~ ~a) = 2

k

c

ak0

,

~ ~a0

=

0(~ ~a) + D

c~ ~a ~F ~a

0,

~ ~a0

=

0(~ ~a) D

c(~F ~a) ~F ~a

0 ~ac ~D. (8.11)

, (8.11) (7.38).

(8.7),

-

( ~E)k = kl Al0 (~A0)k (~FA0)k. (8.12)

~H = ~ ~A+ 2~

cA0. (8.13)

(8.12) (8.13) -

(8.9). , ,

(8.1), (8.1) ,

F = 2[A].

(8.9) -

. (8.11),

102 8.

(7.17), (7.34), -

2

c

(~)

0 ~ ~F = 0, ~ ~ ~ ~F = 0. (8.14)

(8.6),

2A0 +

0(~F ~A+ D

cA0) +

1

c~A ~D + D

c(~F ~A) + ~F

~A

0

2ca(AkDak) ~(A0 ~F ) =

42

c2A0 +

2~

c [~ ~A] + 4pi. (8.15)

2Ak k(A00

+ ~F ~A+ DcA0)

+ Rk.lAl + 2kn

An

0

+

[~

(2A0

~

c

)]k= D

c

(2

cDklAl +

Ak

0+ kA0 + F kA0

)

0

[2

cDklAl+kA0 +F kA0

][~F[~ ~A

]]k 2A0

c

[~F~

]k. (8.16)

(8.15) (8.16)

2 =2

02 klkl (8.17)

- -

, Rbc = aqRab,cq, Rab,cq -

, (7.31). -

,

, . ,

,

, .

8.2.

-

,

kl = 1

cDkl = 0. (8.18)

8.2. ... 103

2A0 +

0(~F ~A) + ~F

~A

0 ~(A0 ~F )

=42

c2A0 +

2~

c [~ ~A] + 4pi. (8.19)

2Ak k(A00

+ ~F ~A)

+ Rk.lAl

+2knAn

0+

[~

(2A0

~

c

)]k=

0

[kA0 + F kA0

][~F

[~ ~A

]]k 2A0

c

[~F ~

]k. (8.19a)

,

"" ,

? ( , -

"").

, -

, ,

, 0 . .. (8.18) :

kl0

= 0,F

0= 0. (8.20)

(7.17), (8.20) F0 = 0, 0 = 0

y0 [F] = [F] = [F] = 0. (8.21)

hh [F] = 0, (8.22)

Fx

Fx

= 0. (8.23)

(8.23) - -

.

(8.23) V , , = 0

dFdS

+ FFV F = 0. (8.24)

104 8.

4- g,

g 2e2

3c

(dF

dS+ FF

V F.)

(8.25)

. -

e - , "" , ( - ).

, , = 0 - g [133]. , g = 0.

, -

.

). , , -

( ) -

. , 2,

, -

.

(6.18)

(6.19),

T , - [9]. -

,

, "-

", , -

, -

( ). ,

(7.2) ,

V

yk= 0, 4 - V =

0, 0 = cT , -

:

hk

=

yk, h

0=

0= V , hk =

yk

x,

h0 = V, =1

1 + a0y1

c2

, gkl = kl, g00 = 1,

F 1 =a0c2

, F 2 = F 3 = 0, Fk = ln

yk. (8.26)

, (8.26), ( -

(6.19))

ds2 =(d0)2

2, (8.27)

8.2. ... 105

-

.

[9], 4 - -

-

V 1 =a0t

c

(1 + a0x

1

c2

)2 a02t2c2

. (8.28)

, (8.28) -

(8.23). , -

.

C -

(8.19), (8.19) (8.11).

A0 +~(A0 ~F ) = 4pi. (8.29)

~A+ ~

(~F ~A

)=

[~F

[~ ~A

]]. (8.29a)

(8.6)

+kAk FkAk = 0. (8.30)

(7.16), -

kFl FkFl. (8.31)

.. k = k, , (8.30) (8.31), Ak

Ak = F k, = const. (8.32)

(8.13), (8.14) (8.32) , ,

"" , -

(8.23), -

, ..

~H = 0. (8.33)

(8.19) -

, .

106 8.

A0 = h

0V (12.26), -

A0

= A0/, (8.29)

2A0

ykyk+ Fk

A0

yk= 4piQ(y1)(y2)(y3). (8.33a)

(8.33a)

.. (. [32] ).

A0

=Qa0c2

2 + (y1 + c2/a0)2 + c4/a0

2{[2 + (y1 + c2/a0)2 c4/a02

]2+ 4c4/a022

}1/2 , (8.33b) 2 = (y2)2 + (y3)2.

-

A = hA = V A0 +

y1A1.

(8.32) , - Q. , (6.18), -

, 4- V 1 (8.28), (

1 +a0y

1

c2

)2=

(1 +

a0x1

c2

)2 a

20t

2

c2, (8.34)

A0 = Q

{ (x1 + c2/a0)[2 + (x1 + c2/a0)2 c2t2 + c4/a02][(x1 + c2/a0)2 c2t2

]R

ct(x1 + c2/a0)2 c2t2

},

A1 = Q

{ct[2 + (x1 + c2/a0)2 c2t2 + c4/a02][(x1 + c2/a0)2 c2t2

]R

x1 + c2/a0

(x1 + c2/a0)2 c2t2}, 2 = (x2)2 + (x3)2,

8.2. ... 107

R =

[2 + (x1 + c2/a0)2 c2t2 c4/a02

]2+ 42c4/a02. (8.35)

(8.35) [33],

[34]. C

, (8.35) .

). ,

(8.23). -

, "" -

, .. ""

, (8.25) g = 0.

.

. -

, -

(8.9)

~E ~H.

~H

~H = ~ ~A+ 2~

cA0 = ~ ~F, (8.36)

~E

~E = ~ ~F, A0. (8.37) (8.14), (8.36) (8.37),

(8.9)

+ ~ ( ~F +

2~

c

)= 0, (8.38)

+ ~ (~F 2

~

c

)= 4pi. (8.39)

= + i, i2 = 1. (8.40)C, i (8.38), (8.37) -

+ ~ (

~F +2i~

c

)= 4pi. (8.41)

108 8.

(8.41) , ""

.

, (7.10)

(7.11) , -

/yk

/yk.

. -

. (8.4)

l(F kl + 2klA0) Fl(F kl + 2klA0) = 0,

k(F 0k + F kA0) + kl(F kl + 2klA0) = 4pi, (8.42) (8.42)

lF kl = FlF kl, F kl = (F kl + 2klA0) (8.43) (8.43) (7.33),

1

2(abc bac) = abFc, (8.44)

lkl = 2Flkl, (8.45) (8.43) (8.45) (8.43)

F kl = kl. (8.46)

(8.46) (8.43) ( ln

yl+ Fl

)kl = 0, (8.47)

ln

yl= Fl. (8.48)

(8.48) -

. (7.17),

. ,

Shliakhovskaya

8.2. ... 109

,

0 =

y0kl [kFl]. (8.49)

-

. -

dS2 = c2dt2 dr2 r2d2 dz2, (8.50) g -

g00 = 1, g11 = 11 = 1, g22 = 22 = r2, g33 = 33 = 1,y1 = r, y2 = , y3 = z, y0 = c,

x1 = r, x2 = , x3 = z, x0 = ct. (8.50)

x2 = y2 +

cx0, y2 = x2

cx0, x1 = y1, x3 = y3. (8.51)

4- V

V0 =1

1 2 = V0, V 2 =

cV 0, V2 = r

2

cV 0,

V 3 = V3 = 0, V1 = V1 = 0, r

c. (8.51a)

.

h0 = V, hk =

yk

x, h1 =

1, h

3 =

3,

h2 = 2

c0, h

0= V , h2

k=

1

1 2 2k,

h1k

= 1k, h3

k= 3

k, h0

k=

r2

c(1 2)2k. (8.52)

,

.

g = ghh

, g11 = 11 = 1, g22 = 22 =

r2

1 2 ,

110 8.

g11 = 11 = 1, g00 = 1, g11 = 11 = 1, g00 = 1.

g22 = 22 = r2(

1 2). (8.53)

, -

(7.72) g00, . ,

, (7.72) -

. ,

, -

(7.73). ,

, -

(7.13). -

.

, -

(8.42). . -

(r). 4- F . -

(7.44) {0

00

}=

{k

00

}=

{0

0k

}=

{0

kl

}= 0,

{1

22

}= 1

2

22r

= r(1 2

)2 ,{

2

12

}=

1

222

22r

=1

r

(1 2

) .(8.54) -

(8.54) = 0. , - 4- F . - -

,

4- F 1, - . ,

F 1 =DV 1

ds=dV 1

ds+

{1

22

}V 2V 2 =

2r

c2(1 2) , F1 = F1. (8.55)

-

F1 = h

1F = F1 =

2r

c2(1 2) . (8.56)

8.2. ... 111

(8.48),

ln =

2r

c2(1 2) dr, = c3

1 2, (8.57)

- 3 - , .

, (8.43)

F kl = c3

1 2kl. (12.58) (8.42)

k(F 0k + F kA0) + c3

1 2klkl = 4pi, (8.59)

klkl =

22

r2(1 2)2 , (8.60) -

, ,

,

2

r2+

1 2r2

2

2+2

z2+

1

r

1 + 2

1 2

r+

2c32

r2(1 2)2 = 4pi

V0,

A0 = V0, V0 =1

1 2 . (8.61)

(8.61) -

,

, -

, ,

-

. , -

,

. -

, , -

.

, .

(8.61)

dP

dr+

1

r

1 + 2

1 2P = 2c3

2

r2(1 2)2 , P =

r. (8.62)

,

(8.62) -

P =c1(1 2)

c3

r, (8.63)

112 8.

c1 - , . ,

V0 r

= A0r

+ A0F1 = F01. (8.64)

F .

F = hh

F = h

kh

l Fkl + h

0h

l F0l + h

0h

lFl0 =

= +A0r

(V1 V1) + A0(VF VF). (8.65) F01 = F10 F12 = F21 , - :

F01 = 01 (A0r

V0 A0V0F1)

=1

(c3

c c1

), (8.66)

F12 = 12 + V2V0

r= c1r. (8.67), -