The Study mapping for directed space-like and time-like lines in Minkowski 3-space \(R_1^3\) (Q1275777)

Let \(L^3\) be the Minkowski 3-space with the Lorentzian inner product \(\langle \;,\;\rangle \) of signature \((+,+,-)\) and the Lorentzian cross product \(\wedge\). A vector \(\vec a\in L^3\) is spacelike (timelike) if \(\langle \vec a,\vec a\rangle =1\) (\(\langle \vec a,\vec a\rangle =-1\)). A directed spacelike (timelike) line \(l\) in \(L^3\) is given by \(\vec y=\vec x+\lambda \vec a\), where \(\vec x\) is a position vector and \(\vec a\) is the direction spacelike (timelike) vector of \(l\). The moment vector of \(l\) is \(\vec a_0=\vec x\wedge \vec a\). In this paper, the authors show that there is a bijection between the set of directed spacelike (timelike) lines in \(L^3\) and the set of ordered pairs of vectors \((\vec a,\vec a_0)\) such that \(\langle \vec a,\vec a\rangle =1\) (\(\langle \vec a,\vec a\rangle =-1\)) and \(\langle \vec a,\vec a_0\rangle =0\).