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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.

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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.

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  • $\begingroup$ Indeed, I didn't put that they were defined in the same point because it was obvious. I guess because my vectors are defined at a point and I can always choose specific coordinates so that my spacetime looks like Minkowski in that spacific point, I am safe? $\endgroup$
    – user404720
    Mar 7, 2018 at 22:40
  • $\begingroup$ @user404720: Yes, this is a basic theorem of linear algebra that after some linear change of coordinates all (nondegenerate) quadratic forms of the given signature on $R^n$ are the same. Hence, it is safe to assume that you are in $R^n$ with the "standard" Lorentzian form. $\endgroup$ Mar 7, 2018 at 22:58

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