Determination of time-like helices from intrinsic equations in Minkowski 3-Space

TL;DR

This paper investigates the intrinsic equations of time-like helices in Minkowski 3-space, deriving their position vectors from curvature and torsion functions, and provides examples illustrating this process.

Contribution

It establishes a differential equation framework for time-like curves in Minkowski space and derives parametric forms of general helices from intrinsic equations.

Findings

Position vectors satisfy a fourth-order differential equation.

Explicit parametric forms of general helices are obtained.

Examples demonstrate the method of deriving position vectors.

Abstract

In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E$^3_1$ satisfies a vector differential equation of fourth order. The general solution of mentioned vector differential equation has not yet been found. By special cases, we determine the parametric representation of the general helices from the intrinsic equations (i.e. curvature and torsion are functions of arc-length) of the time-like curve. Moreover, we give some examples to illustrate how to find the position vector from the intrinsic equations of general helices.