I am working on space curves in Minkowski space and de-Sitter space. To comprehend it better I am trying to come up with some examples of timelike and spacelike curves.
A curve $x$ is locally spacelike if $\left\langle x^{\prime },x^{\prime }\right\rangle >0$ and timelike if $\left\langle x^{\prime },x^{\prime }\right\rangle <0.\ $de-Sitter 3-space can be defined as $\mathbb{S}_{1}^{3}=\left\{ x\in \mathbb{R}_{1}^{4}:\left\langle x,x\right\rangle =1\right\} ,~$where $\mathbb{R}_{1}^{4}$ is a Minkowski space defined by a usual metric as $\left\langle x,y\right\rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.$
For example I find $x=\left( \cos s,\sin s,1,1\right) $ as an example for a unit speed spacelike curve in de-Sitter 3-space. It is in $\mathbb{S}_{1}^{3}$,~since $\left\langle x,x\right\rangle =\cos ^{2}x+\sin ^{2}x+1-1=1~$and also it is spacelike since $x^{\prime }=\left( -\sin s,\cos s,0,0\right) ~$and $\left\langle x^{\prime },x^{\prime }\right\rangle =\sin ^{2}s+\cos ^{2}s=1>0.$ However I could not find any example in de-sitter 3-space that is a unit speed timelike.
Couple of my attemp failed as the following. I firstly thought that $x=\left( \cosh s,0,0,\sinh s\right) ~$satisfy that it is in $\mathbb{S}% _{1}^{3}$ and also $\left\langle x^{\prime },x^{\prime }\right\rangle <0\ $ however the curvature of this curve equals to zero which I do not want this either. Then I tried $x=\left( \cos s,\sin s,0,\sqrt{2}s\right) :~$this satisfies $\left\langle x^{\prime },x^{\prime }\right\rangle <0$ however it is not in $\mathbb{S}_{1}^{3},~$that is $\left\langle x,x\right\rangle \neq 1.$
Can you help me figure out with a good example?
I realized that my spacelike curve example has also zero curvature. So I failed on this one too. Curvature for spacelike and timelike curve can be found respectively by $\kappa =\left\Vert T^{\prime }+\alpha \right\Vert ,\ $
$\kappa =\left\Vert T^{\prime }-\alpha \right\Vert , $where $T$ is tangent of
the curve.
Thanks for your help.