# Can we actually have null curves in Minkowski space?

I know that this sounds really stupid but, when I think of the Minkowski space I cannot imagine a null curve, only null lines. For me, the only possible way to have one is to change the basis of the space for one that is not orthogonal, and that don't make any practical sense for me. And almost the same goes to null surfaces... I just can't think of any other than null planes and null cones. And because of this, I also have doubts of what I think is a null curve and null surface in general relativity.

• A line is a curve. – Slereah Oct 20 '15 at 11:20
• Is it just the terminology that is confusing you, i.e. the terms line and curve? If so, the term curve includes straight lines. – John Rennie Oct 20 '15 at 11:21
• No, for instance. A lot of books consider a congruence of null curves that can have some shear... If are lines, how is it possible to they to have shear? and then what is the purpose of the Newman-Penrose formalism in special relativity? – raul Oct 20 '15 at 11:26
• I'm reading about Twistors by the way... – raul Oct 20 '15 at 11:27

Regarding null curves in flat space, how about $$X(t) = (t,x,y) = (\tau, \cos(\tau), \sin(\tau)) .$$ Then $$V(t) = (\dot t, \dot x, \dot y) = (1,-\sin(\tau), \cos(\tau))$$ in which case $V^2 = 0$.
Any particle moving in $\mathbb{R}^3$ along any curve with constant speed $|v|=c$ will trace a null curve in Minkowski space.