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

So, if we suppose that $A^k$ and $B^k$ are time-genre, 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 you would do to complete this demonstration?

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?