Consider $\mathbb{R}^4$ equipped with the Lorentz inner product: $$\eta(X,Y)=x^0y^0-x^1y^1-x^2y^2-x^3y^3$$
Let $X,Y\in\mathbb{R}^4$, $X\not=0$ and $Y\not=0$, two future-causal (this means: $\eta(X,X)\le0$ and $x^0>0$) vectors. I want to show that the following points are equivalent:
- $\eta(X,Y)=0$
- $\eta(Z,Z)=0$ for $Z:=X+Y$ (this means $Z$ is lightlike)
- $\eta(X,X)=\eta(Y,Y)=0$ and the vectors $X$ and $Y$ are parallel
I cannot find a "starting point" for the proof. I always think that a special condition is missing. For example: If I assume that 1. and 2. holds, than it is easy to prove the second point:
$$0=\eta(X,X)+2\eta(X,Y)+\eta(Y,Y)=\eta(X,X+Y)+\eta(X+Y,Y)=\eta(X+Y,X+Y)$$
In addition I'm confused: The first point means, that $X$ and $Y$ are orthogonal. So how is it possible, that $X$ and $Y$ are parallel (which is claimed in the third point)?
Your help is greatly appreciated.