Minkowski Uzay-Zamanda Timelike Eğriler Arasındaki Bäcklund Dönüşümü Üzerine

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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