Table 1 All types of null curves in the lightlike cone \(\mathbb {Q}^{3}_{2}\)
Cases | \(x_{1}-x_{3}\) | \(x_{1}+x_{3}\) | \(x_{4}-x_{2}\) | \(x_{4}+x_{2}\) | The type of the curve |
|---|---|---|---|---|---|
1 | 0 | 0 | 0 | 0 | Impossible |
2 | 0 | 0 | 0 | ≠0 | Straight line |
3 | 0 | 0 | ≠0 | 0 | Straight line |
4 | 0 | 0 | ≠0 | ≠0 | Impossible |
5 | 0 | ≠0 | 0 | 0 | Straight line |
6 | 0 | ≠0 | 0 | ≠0 | Type 2 |
7 | 0 | ≠0 | ≠0 | 0 | Type 1.3 |
8 | 0 | ≠0 | ≠0 | ≠0 | Impossible |
9 | ≠0 | 0 | 0 | 0 | Straight line |
10 | ≠0 | 0 | 0 | ≠0 | Type 1.2 |
11 | ≠0 | 0 | ≠0 | 0 | Type 1.1 |
12 | ≠0 | 0 | ≠0 | ≠0 | Impossible |
13 | ≠0 | ≠0 | 0 | 0 | Impossible |
14 | ≠0 | ≠0 | 0 | ≠0 | Impossible |
15 | ≠0 | ≠0 | ≠0 | 0 | Impossible |
16 | ≠0 | ≠0 | ≠0 | ≠0 | Type 2 |