We have the following informations:
The components of a metric tensor are given by $g_{11}=-2z, g_{22}=4x^{2}+2z, g_{33}=1,g_{23}=g_{32}=2x$, where $(x,y,z)$ are standard cartesian coordinates where $z$ is positive.
The equation of a parametric curve is
$\gamma \left(s\right)=\left(\surd s,- \surd s,s\right),s>0$
By calculation we obtained
$\kappa = \tau = \frac{1}{s}$, this is a generalized helix in a $3$-dimensional lorentzian space.
Using Mathematica the curve is ploted in $3$-dimensional Euclidean space as $ParametricPlot3D \left[ \left\lbrace \surd s, -\surd s, s \right\rbrace , \left\lbrace s, 0, 30 \right\rbrace \right]$
$\gamma \left( s\right)$ in a $3$-dimensional Euclidean space (figure)
then I want to plot a parametric curve $\gamma \left(s\right)$ in a Lorentzian space. I am new to mathematica and I have no idea how to do this, so could someone please help me to solve this. \end{document}
The description of a parametric curve $\gamma: I \to \mathbb{R}^3$ is entirely independent of the metric you put on $\mathbb{R}^3$. That is to say, once you have chosen a coordinate system on your Minkowski $\mathbb{R}^3$, as a (smooth, topological) manifold it is entirely indistinguishable from the Euclidean $\mathbb{R}^3$. So the same plot should do exactly the job.