Minkowski Uzay-Zamanda Timelike Eğriler Arasındaki Bäcklund Dönüşümü Üzerine

Year 2018, Volume: 22 Issue: 3, 1143 - 1150, 20.09.2018

Melek Erdoğdu Mustafa Özdemir

https://doi.org/10.19113/sdufenbed.469297

Abstract

Bu çalışmanın amacı, Minkowski
uzay-zamanda timelike eğriler arasında Bäcklund dönüşümünü tanımlamaktır. Bu
amaç doğrultusunda, timelike Bäcklund eğrilerin Frenet çatıları arasında
ilişkiyi ortaya koyan dönme matrisinin seçimine bağlı olarak dönüşümü
inceledik. İkisi spacelike hiperdüzlemde küresel dönme ve biri ise timelike
hiperdüzlemde hiperbolik dönme olmak üzere üç farklı dönme matrisi durumu söz
konusudur. Her durum için, timelike Bäcklund eğrilerinin eğrilik fonksiyonları
arasındaki ilişki ortaya konmuştur. Bu arada, işaret farkı gözeterek timelike Bäcklund
eğrilerin eşit ikinci burulma fonksiyonuna sahip olması gerektiği
ispatlanmıştır. Bu aynı zamanda; Bäcklund dönüşümün bir sabit ikinci burulmaya sahip
timelike eğriyi bir başka sabit ikinci burulmaya sahip timelike eğriye taşıyan
dönüşüm olduğu anlamıma gelir.

Keywords

Minkowski uzay-zaman, Timelike eğriler, Bäcklund dönüşümü

On Bäcklund Transformation Between Timelike Curves in Minkowski Space-Time

Abstract

The aim of this paper is to define
Bäcklund transformation between two timelike curves in four dimensional
Minkowski space. For this purpose, we examine the transformation depending on
the choice of rotation matrix which determines the relations between Frenet
frames of timelike Bäcklund curves. There are three different cases for
rotation matrix; two of them are spherical rotations on the spacelike hyperplane
and one of them is hyperbolical rotation on the timelike hyperplane. For each
case, we get the relations between curvature functions of timelike Bäcklund
curves. By the way, we prove that timelike Bäcklund curves must have equal constant
second torsion functions up to sign. This also means that Bäcklund
transformation is a transformation which maps a timelike curve with constant
second torsion to another timelike curve with constant second torsion.

Keywords

Minkowski space-time, Timelike curves, Bäcklund transformation

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