currently I am reading the following paper by Halvorson and Clifton (https://arxiv.org/abs/quant-ph/0103041) where they try to argue that localizable particles are inconsistent with relativistic constraints. In the process, they define the following axiom, which they call No Absolute Velocity:
[No Absolute Velocity] Let $M$ be an affine space and $\textbf{a}$ a spacelike translation in $M$. Then, there is a pair $(\textbf{b}, \textbf{c})$ of timelike translations in $M$ such that $\textbf{a}=\textbf{b}-\textbf{c}$
I interpret this in the following way: This condition ensures that, even if two spatial sets are spacelike seperated, one can "connect" them through two timelike translations. A quick sketch reveals that this condition is satisfied in Minkowski spacetime. So far, so good. My problem is: What does this condition have to do with it's name? Why would this condition be violated if there was an absolute velocity?
