Геометрия Лобачевского и ее применение в специальной теории относительности. Часть 1. Сосов Е.Н. - 22 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

S(< 0; 0 >, r)
x
1
= 0
< x
1
; x >∈ S(< 0; 0 >, r)
x
2
1
+ x
2
= r
2
, θ(< x
1
; x >) =
r
r + x
1
< 0; x > .
Π
+
= R
∗
+
× E
ρ
Π
+
(< x
1
; x >, < y
1
; y >) = ρ
P
(θ
−1
(< x
1
; x >), θ
−1
(< y
1
; y >)),
ρ
Π
+
Π
+
ρ
Π
+
(< x
1
; x >, < y
1
; y >) = kArch
x
2
1
+ y
2
1
+ (x − y)
2
2x
1
y
1
.
(θ(< x
1
; x >))
2
=
r
2
((r + x
1
)
2
+ x
2
)
2
(r
4
+ (x
2
)
2
−2x
2
1
(r
2
−x
2
) + 2r
2
x
2
+ x
4
1
) =
r
2
((r − x
1
)
2
+ x
2
)
(r + x
1
)
2
+ x
2
, r
2
− (θ(< x
1
; x >))
2
=
4r
3
x
1
(r + x
1
)
2
+ x
2
,
r
2
+ (θ(< x
1
; x >))
2
=
2r
2
(r
2
+ x
2
1
+ x
2
)
(r + x
1
)
2
+ x
2
.
θ
−1
= θ
ρ
Π
+
(< x
1
; x >, < y
1
; y >) = ρ
P
(θ
−1
(< x
1
; x >), θ
−1
(< y
1
; y >)) =
kArch
(r
2
+ (θ(< x
1
; x >))
2
)(r
2
+ (θ(< y
1
; y >))
2
) − 4r
2
(θ(< x
1
; x >), θ(< y
1
; y >))
(r
2
− (θ(< x
1
; x >))
2
)(r
2
− (θ(< y
1
; y >))
2
)
=
kArch
(r
2
+ x
2
1
+ x
2
)(r
2
+ y
2
1
+ y
2
) − (r
2
− x
2
1
− x
2
)(r
2
− y
2
1
− y
2
) − 4r
2
(x, y)
4r
2
x
1
y
1
=
kArch
x
2
1
+ y
2
1
+ (x − y)
2
2x
1
y
1
. 
Îòìåòèì, ÷òî ïðè ýòîé èíâåðñèè ñôåðà S(< 0; 0 >, r) îòîáðàçèòñÿ íà ãè-
ïåðïëîñêîñòü ñ óðàâíåíèåì x1 = 0.
      Äåéñòâèòåëüíî, åñëè < x1 ; x >∈ S(< 0; 0 >, r), òî
                                                       r
             x21 + x2 = r2 ,      θ(< x1 ; x >) =          < 0; x > .
                                                    r + x1
Åñëè ìû ââåäåì íà âåðõíåì îòêðûòîì ïîëóïðîñòðàíñòâå Π+ = R∗+ × E
ìåòðèêó

        ρΠ+ (< x1 ; x >, < y1 ; y >) = ρP (θ−1 (< x1 ; x >), θ−1 (< y1 ; y >)),

òî ïîëó÷èì ìîäåëü Ïóàíêàðå ïðîñòðàíñòâà Ëîáà÷åâñêîãî â îòêðû-
òîì ïîëóïðîñòðàíñòâå åâêëèäîâà ïðîñòðàíñòâà.
   Ëåììà 1. Ìåòðèêà ρΠ+ íà Π+ èìååò âèä
                                                 x21 + y12 + (x − y)2
            ρΠ+ (< x1 ; x >, < y1 ; y >) = kArch                      .
                                                         2x1 y1
  Âû÷èñëèì ñëåäóþùèå âåëè÷èíû

                2        r2
(θ(< x1 ; x >)) =           2   2 2
                                    (r4 + (x2 )2 − 2x21 (r2 − x2 ) + 2r2 x2 + x41 ) =
                  ((r + x1 ) + x )
        r2 ((r − x1 )2 + x2 )                                     4r3 x1
                              ,    r2 − (θ(< x1 ; x >))2 =                    ,
          (r + x1 )2 + x2                                     (r + x1 )2 + x2
                     2                    2r2 (r2 + x21 + x2 )
                                          2
                    r + (θ(< x1 ; x >)) =                      .
                                           (r + x1 )2 + x2
Òîãäà, ó÷èòûâàÿ, ÷òî θ−1 = θ, ïîëó÷èì

      ρΠ+ (< x1 ; x >, < y1 ; y >) = ρP (θ−1 (< x1 ; x >), θ−1 (< y1 ; y >)) =
      (r2 + (θ(< x1 ; x >))2 )(r2 + (θ(< y1 ; y >))2 ) − 4r2 (θ(< x1 ; x >), θ(< y1 ; y >))
kArch                                                                                       =
                        (r2 − (θ(< x1 ; x >))2 )(r2 − (θ(< y1 ; y >))2 )
        (r2 + x21 + x2 )(r2 + y12 + y 2 ) − (r2 − x21 − x2 )(r2 − y12 − y 2 ) − 4r2 (x, y)
kArch                                                                                      =
                                             4r2 x1 y1
                                  x21 + y12 + (x − y)2
                            kArch                      .
                                          2x1 y1
Çàäà÷è.
                                           21

Страницы