It is a common argument in the theory of kinematic groups (the groups of motions for a spacetime) that the subgroups generated by boosts must be non-compact[1][2][3]. This is true of all commonly used kinematic groups such as the Poincaré group, Galilean group, De Sitter group, Anti De Sitter group, and also the less commonly used examples.
Why is this a requirement of kinematic group? The only idea that I have been able to come up with so far is that compactness may spoil the causal structure of a spacetime otherwise.
Consider this argument : Take some spacetime vector space $V$, with one time dimension and the rest spatial. Now take the oriented projective space $OV \cong V\setminus \{0\} / \mathbb{R}+$, where we contract every type of vector onto the $n-1$ sphere. Vectors of various types, future timelike, past timelike, null, and spacelike if they exist, all form separate regions on that sphere. In the Poincaré case, it's two disks on that sphere for past and future vectors, in the Galilean case the sphere is split in two regions by a circle of null vectors, etc.
The stabilizer subgroup of the kinematic group $\mathrm{Stab}(K)$ spans the entire region for each : the orbit space of any vector of any type by $\mathrm{Stab}(K)$ is the entire region. However, if the stabilizer subgroup is entirely compact (rotations are usually assumed compact as well), then the orbit generated will be compact as well, by continuity of the action map $g \to g \cdot x$. Since the projective space $OV$ is compact and connected, there exists only two possible compact open sets on it, which are the whole space itself or the empty set. Therefore if our stabilizer subgroup is compact, there is no clear differentiation of vector types as in the other cases.
Does this sound like a proper argument for it, and also, what would be a good argument for the case where only one of the subgroups generated by boosts was compact?
