Proper definition of world lines in Galilean and Minkowskian spacetime

Wox

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)
[tex]
\bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (t(\tau),\bar{x}(\tau))
[/tex]
for which a curve in Euclidean space
[tex]
\bar{x}\colon \mathbb{R}\to \mathbb{R}^{3}\colon \tau \mapsto \bar{x}
[/tex]
and an injective map (because a world line shouldn't contain simultaneous events)
[tex]
t\colon \mathbb{R}\to \mathbb{R}\colon \tau \mapsto t
[/tex]

We used the fact that [itex]\mathbb{R}^{3}[/itex] has the Euclidean structure and that a basis was choosen in [itex]\mathbb{R}^{4}[/itex] so that all vectors [itex](0,\bar{x})[/itex] form a subspace of Galinean space [itex]\mathbb{R}^{4}[/itex] which is isomorphic with [itex]\mathbb{R}^{3}[/itex] (i.e. Euclidean inner product defined on this subspace).

2. In Minkowskian spacetime with signature (-+++), world lines are defined as differentiable curves
[tex]
\bar{w}\colon \mathbb{R}\to \mathbb{R}^{4}\colon \tau \mapsto (x^{0}(\tau),x^{1}(\tau),x^{2}(\tau),x^{3}(\tau))
[/tex]
which are timelike (because a world line shouldn't contain simultaneous events) meaning that the velocity of the world line is a timelike vectors ([itex]\eta(\bar{w}',\bar{w}')<0[/itex]) or in other words
[tex]
(\frac{dx^{1}}{d\tau})^{2}+(\frac{dx^{2}}{d\tau})^{2}+(\frac{dx^{3}}{d\ tau})^{2} <(\frac{dx^{0}}{d\tau})^{2}
[/tex]

We used the fact that [itex]\mathbb{R}^{4}[/itex] has an inner product [itex]\eta[/itex] which is non-degenerate instead of the usual positive-definite.

3. It seems that we always choose [itex]t(\tau)=\tau[/itex] (Galilean) and [itex]x^{0}(\tau)=c\tau[/itex] (Minkowskian) but I'm not sure how these choices are justified.

Matterwave

You have to be a little careful with the notation and logic. The tau you wrote in the Galilean case is some universal time function that every observer can measure with a good clock (up to affine re-parametrizations for origin and units). The tau you wrote in the Minkowski case is the proper time as measured by a clock carried by that specific observer. There is a major difference in logic there.

Otherwise, I don't see any problem with what you've written down.