Demonstration of non orthogonality of time-like vectors on Minkowski space

Question

I want to demonstrate that if two 4-vectors, $A^k$ and $B^k$ are orthogonal, then they aren't time-like.

So, if we suppose that $A^k$ and $B^k$ are time-like, we have that:

$$\sum_k A^k A_k <0, \hspace{15pt} \sum_k B^k B_k <0$$

But this seems not enough to prove that:

$$(\vec{A}.\vec{B})=A^1B_1+ A^2B_2 + A^3B_3 + A^4B_4\neq 0$$

Inner product on Minkowski space is a "pseudo-inner product", and so, I'm not sure if it is right to use Cauchy-Schwarz inequality.

What would you do to complete this proof?

Answer 1

Suppose $\vec{A}$ is timelike. There is a frame where it has only a time component: $A^\mu = (A^0, \mathbf{0})$, with $A^0 \neq 0$. Now also suppose that $\vec{A}\cdot\vec{B} = 0$. In components, $\vec{A}\cdot\vec{B} = -A^0 B^0 = 0$, which implies $B^0 = 0$. But then $B_\mu B^\mu = -(B^0)^2 + \mathbf{B}^2 = \mathbf{B}^2 \geq 0$, showing $\vec{B}$ is not timelike. Furthermore, it can't be null either, except the case where $\vec{B} = 0$.