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Ужгородский национальный университет

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[НЕСОРТИРОВАННОЕ]

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x2 – x1 = D = l0 = g×(x'2 – x'1)

x¢2 - x1¢ = l¢ =

= l0 × 1 -

.pdf

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§2.

( ,

, ).

(Vy=Vz=0 ; . 2).

t=t¢=0,

, = ¢=0

r= t; – ¢

. 2 r'= t¢ ( = ¢!).

”

x2 + y2 + z2 = c2t2		(3)
x¢2 + y¢2 + z¢2 = c2t¢2	'.	(4)
( “ ”)

. ,

( , , ).

, ( )

x¢ = g×(x–Vxt), y¢ = y, z¢ = z; (5) t¢ = a×t + b×x, (6)

g, a b – , V (

, ,

: b=0, a=g=1,

). (5) (6) (4), :

g2×(x Vxt)2 + y2 + z2 = 2×(a×t + b×x)2,			(7)
(g2 2) – g22xV	t	+ g2V 2×t2 + y2 + z2 = c2a2×t2 + c2×2a×t×bx + ( 2b2× 2)
(g2 – 2b2)× 2 + y2 + z2 – 2t×(g2V + c2ab)×x = (c2a2 – g2V 2)×t2.			(7 )

g, a b , (7 )

(3) ( 3 4 ).

(g2 – 2b2) = 1; (g2V + c2ab) = 0; (c2a2 – g2V 2) = 2.

V 2

− 1

= c2 -V 2

= 1 - (V / c 2);

a = g =

ç1

- c 2 ÷

;

b = -g c2 .

(8)

(9)

(8)

( )

(g2 – 2b2) = 1;

(2)

(g2V + c2ab) = 0;

(1)

(c2a2 – g2V 2) = 2. (3)

(8)

(1- )

b = -

g2V

(2)

g2 - c2

g4V 2

® g2 (1 -

g2V 2

) =1.

a2c4

ac2

a2c2

(3- ) ,

c2a2 = 2 + g2 V2

2- :

g2V 2

+ g2V 2 - g2V 2

2 2

(1 -

) = g

(

) =1 Þ g

= c

+ g V

2 2

+ g

2 2

+ g V

c2 -V

-V 2

1 -

V 2

g (3)

= c

® a

V 2

=1 Þ

c2 -V 2

-V 2

=1 +

V 2

-V 2 +V 2

= g

c2 -V 2

-V 2

a = g.

a (1)

2 2			V
g V + c	gb = 0	® b = -g
			c2

-----------------------------------------------------------------------------------------------------

g.
	x=ct, '	x'=g(x-Vt)
:	x'=ct'	x=g(x'+Vt')

(1)ct=g(x'+Vt')=g(ct'+Vt')=g(c+V)t'

(2)ct'=g(x-Vt)=g(ct-Vt)=g(c-V)t

(1) (2) 2tt' = g2(c2-V2)t't		c2 = g2(c2-V2)
g2 = c2/(c2-V2) g =	1

	1 -		V 2
			V 2
			c2
			c2

----------------------------------------------------------------------------------------------------

x' = g(x–Vt), x = g(x'+Vt') x' = g[g(x'+Vt')–Vt] = g2x'+g2Vt'–gVt;

-1)

-1

t =

+ gt'= g(t'+

) = g[t'+

gVt = x'(g

–1)+g

Vt'

×(1 -

)] =

= g[t'+

×(1 -1

V 2

)] = g(t'+

x'V

) .

(

t' = ….)

(9) (5) (6),

¢,

(

x¢

= g(x -Vxt) =

x -Vxt

x = g(x¢ +Vxt¢) =

x¢ +Vxt¢

1 - (V / c)2

1 - (V

/ c)2

= y,

z = z

y = y ,

z = z

t¢ = g × (t -

Vx × x

) =

- x ×Vx / c2

;

t =

t¢

+ x¢×Vx / c2

(10)

c 2

1 - (Vx / c)2

¢ ( ) ( )

(10) “ ” “ ”.

, Vx<< c .

( , ).

, ( V),

, .

g =

³1 – , –

1 - v2

, v<c –

§3. 4- .

. ,

, – .

4- ( –

): , , z, –


	t, t = t ,
	,
	. ( , , z, t)
	.
		. 3
	(
				).
	–		–
. 3	( ,t), ,

, . , 0τ – ,

= 0. 0 ,

, , ,

t=0, ( 3- –

0 0z). = t, =τ,

45 . ,

0 – .

′ τ′ ,

x′=γ(x Vt) ′=0. =Vt=βτ, β=V/ ,

, ϕ τ (tgϕ=β).

′ ( ′ τ′).

′ – , ′ ,

′=0 t′=0. t′=γ(t xV/c2)

t′=0, t=xV/c τ=β . , ′ ϕ

, , ′ .

, ′

. .3, ′ ′ ). , D,

, ,

′.

( )

– ( ) 4-

( ):

S = [(τ1-τ2)2 – l2]1/2 = [(τ′1-τ′2)2 – l′2]1/2 →

→S2 = c2(t1-t2)2 – [(x1-x2)2 + (y1-y2)2 + (z1-z2)2] =

′	2		′2	′2	′2
= (ct12 )		− (x12		+ y12	+ z12 ) . ( t'1–t'2= t' = t'12) (11)
,			,
, :

1) (l > ct12) –

, : (11) t′12 = 0,

( t12)2 – l2 = –l′2 < 0; .3 ′

( ′, ), '
( ′ –
	'=0),

.
2) (ct12 12 > l) –
′, ,	. 3

, : l′ = 0, ( t12)2 – l2=(ct′12)2 > 0; D 1 2 ( .3 ), '

′, t'.

3) (ct12 = l) –

===========================================================

x′2+y'2+z'2=c2t'2.

( x'=x Vt, y'=y, z'=z ) :

(x–Vt)2+y2+z2=c2t2 x2–2xVt+V2t2+y2+z2=c2t2,

(1)

, ' .

, =const.

t'=t+fx, (1) :

x2–2xVt+V2t2+y2+z2 = c2t2+2c2ftx+ 2f2x2

, f = –V/c2, , xt ,

t'=t–Vx/c2,

x2(1–V2/c2)+y2+z2 = c2t2(1–V2/c2)

(1–V2/c2),

x2 i t2. ,

x' = (x–Vt)/(1–V2/c2)1/2, y' = y, z' = z;

t' = (t–Vx/c2)/(1–V2/c2)1/2.

. .

. . .

. .

1. . ¢,

V ,

x1¢ 2¢ , t'0,

2 ). :

x1¢

×V

x¢2 ×V

)

Dt = t1 - t2

= g

(1)

t1 = g(t0

t2 = g(t0

Dx .

x1¢ ¹ x2¢ , t1 ¹ t2,

Dt. , , ,

2. :

. ¢ , V

, ¢ , Dt¢=t2¢-t1¢=t0 (t0 –

– , '

t1 t2 ( )

×V

t1 = g(t1 +

) i t2

= g(t

) ,

Dt = t2 – t1 = g×(t2¢ – t1¢) = gDt¢ =

Dt > 0.

(2)

1 -V 2 / c2

. t0 ,

, , . , ,

(

). Dt 0 V (

), . 1.

v=100000 , 0=50 .

	t =		t0			=	50	=	50	=	150	=	150	= 53,2 .
													2,82
				v2			(105 )2		9 -1		8
		1 -		v2	1 -		(105 )2		9 -1		8
									9
				c2			(3 ×105 )2		9
L=v×t=1,6777152×1017 = 17,73 . .
(1 = 31536000 ;						1 . = 9,4608×1015 ).
	v=290000 50 195,3 ( =3,9057),
188,8 . ( L = 1,7862×1018 ).
	, .

(L¢=L/ »1,79/3,9 » 0,46) , , .

3. : (

). ¢ ,

l0 = D ¢ = x2¢–x1¢ (

, ' ).

V .

, ( t) ( )

1 2 .

x1¢ = g(x1–Vt); x2¢ = g(x2–Vt).

x2¢ – x1¢ = D ¢ = l0 = g×(x2 – x1)

- x = l =

= l

V 2

(3)

, , g

( , ).

, 1 2

, ' (

)? , , “ ”

t' '1 '2 l, -

. 1 2

: x1 = g(x'1+Vt'); x2 = g(x'2+Vt'). :

. 2.

t/τ0 m/m0

3,0

2,5

2,0

1,5

1,0

0,0 0,2 0,4 0,6 0,8 V/c

. 1

l/l0
1,0
0,8
0,6
0,4
0,0	0,2 0,4 0,6	0,8 V/c
	. 2

4..

¢ ( V=V

) vx¢ ¢ ( ). vx

v¢x	=	dx¢	;	vx	=	dx	.
		dt¢
						dt

( x=g(x¢+Vt¢) )

( t=g(t¢+Vx¢/c2) ), dx i dt :

dx=g×(dx¢+Vdt¢),

dt = g ×(dt¢ +

V × dx¢

) ;

+V)

dt (

vx =

g ×(dx

+Vdt )

= (v¢x

+ V)

; (4)

Vdx¢

g ×(dt¢ +

)

dt¢(1 +

dx¢

)

1 +

v¢x

c 2

+ Vv¢x

dt¢

vx < v'x + V – .

, v ¢ , v ¢ vz¢,

( , Vy=0 i Vz=0, a dy=dy¢ dz=dz¢):

vy =

dy¢

v¢y

;

vz =

v¢z

(4 )

g(dt¢ +Vdx¢/ c2 )

1 +Vv¢x / c2

v ,

vx -V

c ×

c 2 -V 2

v¢x =

;

v¢y =

;

v¢z =

vz .

(4 )

1 -

c 2 -Vvx

1 -

c 2

1 -

, ,

, v'x×V/ 2 << 1

. , ,

! v'x = ,

vx =		c +V			=	c +V	×c = c .
vx =			V		=		×c = c .
			V			c +V
	1 +			c
	1 +		c2	c

, , ,

( !).

5. .

2 ö 2

;

ç1

;

y,z

, z

ç1

ay, z

. ( )

, .

0=10-8 . (

, ) l ~ × 0 = 3 .

~ 0,995 (

!),

t = gt0 =		10−8				=	10−8	=10−7 c ,

		æ 0,995c ö2
		æ 0,995c ö2			1 - 0,99
	1 - ç			÷
	1 - ç		c	÷
	è		c	ø

10 »30 ,

l12 t1 t2:

S12 = c2 (t1 - t2 )2 - l122 .

, , -

. 4-

, 4-

( , ).

, .

. 2- 3- . .

. .

. . .

. .

§1.

: , ,

; , 320

( 1687 .) “ ” (

, ,

) .

, , –

: ,

, .

: , ;

: ,

, .

, , .

, .

. F – ,

. , ,

( ) , ,

, .

. ,

1): F = mg, – (

), g – .

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