My problem is the following. I am working in general relativity (1+3 dimensional spacetime) with an unknown metric. I have one fixed timelike vector $u^a$ and another timelike vector $\xi^a$ so that $\xi^a\xi_a=-1$, but $\xi^a$ can vary. The question is, what can I say about the possible values of $\xi^a u_a$?
My intuition is that the product of these vectors will always be timelike and will minimal when both of them are aligned but I don't know if this is right or how to prove it. Any help is appreciated.
First of all, you should assume that $u$ and $v$ are tangent vectors at the same point of your manifold, otherwise the question makes no sense. Then, assuming $u^a u_a=-p^2$, where $p>0$, you get the inequality $$ \xi^a u_a \in (-\infty, -p] $$ and all these values can be realized. To prove this, do the calculation in the $(1,1)$-Lorentz space.