I'm going over an exam I had a couple months back, over the exercises I didn't manage to get right and I'm kinda stuck with the following subtask:
Let $\xi$ be a 4-vector with the Minkowski scalar product $(\xi_1, \xi_2) = \sum_{\mu, \nu = 0}^{3} \xi_1^{\mu}g_{\mu \nu} \xi_2^{\nu}$ where g is $diag(-1,1,1,1)$ . I'm now supposed the following equality:
A 4-vector $\xi \neq 0,$ is a timelike vector (i.e. $(\xi,\xi) > 0$) if and only if there is a frame of reference in which $\hat{\xi}^i = 0 $ for $i \in \{ 1,2,3\}$
(Note that, $\hat{\xi} = A \xi$, where A is a Lorentztransformation which satisfies $A^T g A = g)$
I've tried calculating all sorts of bits but haven't found anything coherent, and nothing that shows the implication from both sides. I'd appreciate any help.
Cheers!
So you've done the "if" direction.
In the "only if" direction your strategy should probably be first to apply a spatial rotation such that only one of the space coordinates is nonzero (such rotations clearly always preserve the Minkowski product), and then to find an appropriate Lorentz boost that will get you to the rest frame of the rotated vector.
The exact shape of a Lorentz boost doesn't drop directly out of the algebra (or at least not without a certain amount of inspired fiddling), but for an SR exam you're probably supposed just to know what it looks like already. So all you need to show is that it works -- i.e. that it preserves the Minkowski product (depending on context you may be able to assume thid as known), and that there's always an appropriate boost to find under the "timelike" condition.