Proper definition of world lines in Galilean and Minkowskian spacetime

 P: 71 I posted several questions on Galilean and Minkowskian spacetime on this forum lately, but I just don't seem to be able to get a real grip on things. I noticed that the core of my problems mostly arise from the definition of world lines. Therefore I tried formulating a definition of them in both spacetime's and my question is whether these definitions are correct/complete. 1. In Galilean space, world lines are defined as curves (continuous maps) $$\bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (t(\tau),\bar{x}(\tau))$$ for which a curve in Euclidean space $$\bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon \tau \mapsto \bar{x}$$ and an injective map (because a world line shouldn't contain simultaneous events) $$t\colon \mathbb{R}\to \mathbb{R}\colon \tau \mapsto t$$ We used the fact that $\mathbb{R}^{3}$ has the Euclidean structure and that a basis was choosen in $\mathbb{R}^{4}$ so that all vectors $(0,\bar{x})$ form a subspace of Galinean space $\mathbb{R}^{4}$ which is isomorphic with $\mathbb{R}^{3}$ (i.e. Euclidean inner product defined on this subspace). 2. In Minkowskian spacetime with signature (-+++), world lines are defined as differentiable curves $$\bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (x^{0}(\tau),x^{1}(\tau),x^{2}(\tau),x^{3}(\tau))$$ which are timelike (because a world line shouldn't contain simultaneous events) meaning that the velocity of the world line is a timelike vectors ($\eta(\bar{w}',\bar{w}')<0$) or in other words $$(\frac{dx^{1}}{d\tau})^{2}+(\frac{dx^{2}}{d\tau})^{2}+(\frac{dx^{3}}{d\ tau})^{2} <(\frac{dx^{0}}{d\tau})^{2}$$ We used the fact that $\mathbb{R}^{4}$ has an inner product $\eta$ which is non-degenerate instead of the usual positive-definite. 3. It seems that we always choose $t(\tau)=\tau$ (Galilean) and $x^{0}(\tau)=c\tau$ (Minkowskian) but I'm not sure how these choices are justified.