Partial order on Grothendieck group of an abelian category

It's not true that this always gives a translation invariant partial order on $\mathscr{G}(\mathcal{C})$. If it did, then unless $\mathscr{G}(\mathcal{C})=0$ the partial order could not be trivial, since $[a]\geq[0]$ for every object $a$ of $\mathcal{C}$.

But there are abelian categories whose Grothendieck groups are nontrivial torsion groups (see this question, for example), and so do not admit any nontrivial translation invariant partial order.